Anne-Sophie Auguères
email: anne-sophie.augueres@ecoex-moulis.cnrs.fr

Michel Loreau

Can organisms regulate global biogeochemical cycles?

Keywords: biogeochemical cycles, nutrient cycling, regulation, Earth system, resource access limitation, anthropogenic impacts

Préface Remerciements

Pour commencer, je voudrais dire un très grand merci à Michel Loreau, mon directeur de thèse. Merci tout d'abord pour ton encadrement, l'autonomie que tu m'as laissée dans le choix du sujet, merci pour toutes les discussions et bien sûr les (très nombreuses) relectures de papiers. Merci de m'avoir appris à prendre du recul sur mes équations. Merci d'avoir été présent et de m'avoir encouragée et remotivée durant ces trois années, bien que j'aie vraiment un problème avec la notion de planning et de deadline. Merci de m'avoir secouée quand c'était nécessaire, mais toujours avec le sourire et la manière ! Je voudrais tout particulièrement remercier les membres de mon comité de thèse, Claire de Mazancourt, Sergio Vallina et Jérôme Chave, qui m'ont guidée lors de la réunion du comité de thèse. Claire, merci particulièrement pour tes conseils avisés sur les différentes parties de ma thèse ainsi que pour la relecture des papiers.

Merci à toutes les personnes avec qui j'ai partagé à un moment le bureau : Tomas Revilla, Jarad Mellard, Shaopeng Wang, Romain Bertrand, Audrey. Un grand merci tout particulier à Anne-Sophie Lafuite, copine de "qu'est-ce qu'on crève de chaud dans ce bureau !", pour les pauses thé du matin (et de l'après-midi) et toutes les conversations sérieuses ou pas qui vont avec, et pour l'écoute aussi. Merci pour ces moments de partage sur nos thèses respectives.

Merci à Camille, nouvel adepte de la pause thé, de m'avoir écoutée parler quand j'avais besoin de m'éclaircir les idées. Merci pour la compagnie lors des nocturnes au labo, de m'avoir engraissée avec des mikados, et m'avoir protégée des envahisseurs à coup de raquette tennis.

Merci aussi à Morgane pour l'animation pendant les nocturnes au labo, le ravitaillement en chocolat et les petits nounours lancés par la fenêtre de mon bureau. D'ailleurs merci au chocolat, sans qui je ne serais très probablement pas arrivée au bout de cette thèse ! Merci à toute l'équipe des thésards, avec qui j'ai été pendant les premiers mois avant de m'exiler bien loin de la bonne vieille salle thésards. Merci à Elvire Bestion et Elise Mazé.

Merci à Lisa Fourtune, pour ta bonne humeur et pour m'avoir fait connaître la magnifique chanson de la poubelle que j'ai dans la tête depuis deux semaines. Merci à Alexis Rutschman, mon voisin préféré, notamment pour avoir fait du baby-sitting de chat pendant mes escapades.

Un grand grand merci à Louis Sallé pour le soutien et les nombreuses soirées, tout simplement pour toutes nos discussions, ta bonne humeur... et ton short rose ! Un très très grand merci à mes visiteurs préférés pendant la période de rédaction. D'abord Alice : ça a commencé par les pauses café pendant notre stage de master, puis les pauses thé, on a fini en coloc et un joli voyage en Malaisie en perspective. Merci pour le café du matin, pour toutes les petites attentions que tu as eues pendant la période de la rédaction de cette thèse (pas forcément évidente), merci pour ton soutien moral et toutes nos discussions.

Merci de m'avoir fait découvrir des séries intellectuelles (pièce dans le pot !!!). Merci tout simplement pour cette belle amitié. Ensuite, merci à Nico, mangeur de lulu, pour ses nombreuses visites au labo pendant la rédaction (même le dimanche !), pour avoir animé un peu les fins de journée en tuant des vieux en appuyant sur un bouton. Et merci pour tes blagues nulles, quoique je ne voudrais pas t'encourager à continuer... Merci à toutes les autres personnes que j'ai pu rencontrer pendant ces trois années à Moulis, à toutes les personnes qui m'ont écoutée râler sur ma thèse. Elsa, mon poteau préféré, pour les consultations psy par téléphone, pour toutes les soirées qu'on a passées ensemble et toutes nos discussions. Merci d'avoir été là dès que j'en avais besoin ces deux dernières années. Merci pour ces magnifiques chansons apprises en ta compagnie, et merci pour toutes les danses qui m'ont valu de me casser un bout du pied !! Merci aussi à Yolan, fournisseur officiel de chocolat et de jurançon et surtout très bon ami. Merci à Fred pour les blagues pourries, pour m'avoir fait découvrir les euproctes, tritons et autres bestioles, pour toutes les soirées qu'on a pu faire ensemble, et le cocktail au chou-fleur que je n'oublierai pas. Et merci à Benoit... pour avoir été un ami puis avoir partagé ma vie, et pour m'avoir supportée pendant les périodes de stress.

Merci à ma famille. A mon père, Patrick, parce que quoi que je cherche tu as toujours un livre dessus dans un rayon de ta bibliothèque. Merci de m'avoir encouragée et remotivée quand j'étais en baisse de régime. Merci à ma mère, Marie-Jo, pour le soutien tout au long de cette thèse, et pour toutes les relectures de dernière minute des parties en français. Merci aussi à Marie, ma soeurette, pour ton soutien et les moments de partage qu'on a pu avoir les dernières années (et avant aussi d'ailleurs !).

Merci à tous les autres qui ont croisé mon chemin pendant ces trois années dans la montagne ariégeoise...

MERCI !

Avant-propos

Cette thèse de l'école doctorale SEVAB de l'Université Toulouse III a été réalisée au Centre de Théorie et Modélisation de la Biodiversité appartenant à la Station d'Écologie expérimentale du CNRS de Moulis (USR 2936), sous la direction de Michel Loreau. Elle a été financée par une bourse de la région Midi-Pyrénées. Les recherches ont été financées par le Laboratoire d'Excellence TULIP (ANR-10-LABX-41). La thèse a commencé le 1er septembre 2012.

Cette thèse, rédigée en anglais, est constituée d'une introduction générale, de 4 chapitres sous forme d'articles scientifiques et d'une discussion générale. Pour assurer la cohérence de l'ensemble du manuscrit, j'ai développé entre les différents chapitres le lien qui relie le chapitre concerné avec le suivant. Sur les 4 articles composant la thèse, un article est paru dans Global Biogeochemical Cycles (Chapitre 3) et un a été accepté dans Ecosystems (Chapitre 1). Les deux autres sont encore en préparation (Chapitres 2 et 4).

This thesis of the SEVAB doctoral school of Toulouse III University was realised at the Centre for Biodiversity Theory and Modelling, in the Station d'Écologie Expérimentale du CNRS in Moulis (France), under the supervision of Michel Loreau. The PhD was funded by the Midi-Pyrénées region. This work was supported by the TULIP Laboratory of Excellence (ANR-10-LABX-41). The PhD began the 1st of September 2012. This thesis, written in english, is composed of a general introduction, 4 chapters presented as scientific articles and a general discussion. Connecting statements were used between chapters to provide a coherent and logical whole. Among the 4 articles in this thesis, one is published in Global Biogeochemical Cycles (Chapter 3) and one other is in press in Ecosystems (Chapter 1). The remaining 2 articles are still in preparation (Chapters 2 and 4).

Résumé général

Les activités anthropiques affectent les cycles biogéochimiques globaux, principalement par la modification des apports de nutriments dans les écosystèmes, que ce soit par l'utilisation d'engrais dans l'agriculture ou l'émission de composés volatiles dans l'atmosphère. Il est donc important de comprendre dans quelle mesure les cycles biogéochimiques globaux peuvent être régulés. Les processus abiotiques, physiques ou chimiques, sont responsable d'une partie de la régulation des cycles biogéochimiques, mais les organismes jouent aussi un rôle important dans la régulation de ces cycles. A l'échelle locale, un des processus majeurs conduisant à la régulation de la concentration de nutriments est la consommation des ressources inorganiques par les autotrophes. Cependant, une grande partie des ressources dans un écosystème sont inaccessibles aux organismes à l'échelle globale, à barrières physiques ou chimiques.

Durant ma thèse, j'ai étudié la manière dont les organismes autotrophes répondent à l'augmentation des apports de nutriments dans les écosystèmes à l'échelle globale, et comment ils régulent la concentration de ces nutriments lorsque leur accessibilité est limitée. La méthodologie pour aborder ces sujets consiste à développer des modèles théoriques basés sur la théorie de l'accès limité aux ressources. L'efficacité de la régulation des cycles biogéochimiques est étudiée en réponse à une variation des apports des nutriments dans le système concerné.

Le premier objectif est de déterminer le potentiel de régulation du cycle d'un nutriment limitant par les autotrophes, dans un contexte d'accès limité aux ressources (Chapitre 1).

Pour cela, je présente un modèle générique dans lequel un nutriment limitant la croissance des autotrophes n'est accessible qu'en partie. Ce modèle est applicable à la fois à l'inaccessibilité chimique et à l'inaccessibilité physique des nutriments. Ce modèle prédit que les variations des apports du nutriment limitant sont parfaitement régulées dans le réservoir accessible, mais seulement partiellement régulées voire amplifiées dans le réservoir inaccessible. De plus, les organismes ne peuvent pas exercer un contrôle fort sur le réservoir accessible et réguler efficacement le réservoir inaccessible simultanément. Ces résultats suggèrent que les organismes ne peuvent pas réguler efficacement les réservoirs de nutriments du système dans son ensemble à cause de l'inaccessibilité d'une partie des ressources.

Le second objectif est de comprendre la manière dont le couplage entre les cycles des nutriments affecte leur régulation par les autotrophes (Chapitre 2). Pour cela, je développe une extension stoechiométrique du modèle précédent, qui décrit la dynamique d'un nutriment limitant et d'un nutriment non-limitant consommé par une population d'autotrophes. Ce modèle montre que la régulation des réservoirs du nutriment non-limitant est plus faible que pour le nutriment limitant. Les variations des apports du nutriment non-limitant n'ayant pas d'impact sur la croissance des autotrophes, elles n'affectent pas les concentrations du nutriment non-limitant. Au contraire, les variations des apports du nutriment non-limitant affectent la croissance des autotrophes et donc la quantité de nutriments -limitant et non-limitant -consommés, et ont donc un impact sur les concentrations du nutriment non-limitant dans les réservoirs accessible et inaccessible. Ces résultats suggèrent que les interactions entre les cycles d'un nutriment limitant et d'un nutriment non-limitant peuvent soit augmenter, soit diminuer la régulation des réservoirs de nutriments par les organismes, selon que les apports des deux nutriments varient dans le même sens ou non.

Le troisième objectif est d'évaluer l'impact de la compétition entre groupes fonctionnels sur la régulation des cycles biogéochimiques, et plus principalement sur la régulation des rapports de Redfield dans l'océan global (Chapitre 3). Je présente un modèle des cycles océaniques de l'azote et du phosphore, ainsi que deux groupes fonctionnels de phytoplancton (i.e. les fixateurs et les non-fixateurs). Ce modèle permet d'étudier précisément les mécanismes qui contrôlent la valeur du rapport N:P des eaux profondes et qui contribuent à sa régulation en réponse à l'augmentation d'origine anthropique des apports de N et de P dans l'océan de surface. En accord avec les résultats des deux chapitres précédents, les simulations numériques montrent que la régulation des concentrations de N et de P dans les eaux profondes est partielle. Malgré cela, le rapport N:P reste quasiment constant. Le modèle montre que la valeur du rapport N:P des eaux profondes est contrôlé par le rapport N:P des non-fixateurs, et dans un second plan par le recyclage et la dénitrification. Ces résultats suggèrent que la similarité entre le rapport N:P moyen du phytoplancton et celui des eaux profondes serait due à un équilibre entre la dénitrification et la fixation de N 2 . Le rapport N:P du phytoplancton contrôle la valeur du rapport N:P des eaux profondes, mais étonnament pas l'efficacité de sa régulation.

Le quatrième objectif est de déterminer l'impact des variations des apports de nutriments sur la production primaire à l'échelle globale (Chapitre 4). Dans ce chapitre, j'explore par un modèle des cycles océaniques du N, du P et du Fe lequel de ces trois nutriments contrôle la production primaire océanique totale à de grandes échelles de temps. Pour s'assurer de la robustesse des résultats obtenus, la limitation proximale des fixateurs et des non-fixateurs est décrite par la loi du minimum de Liebig dans un premier temps, et par des formulations de colimitation par plusieurs nutriments dans un second temps. Lorsque la croissance des non-fixateurs est limitée par le N et celle des fixateurs est limitée par le P ou le Fe, le nutriment qui contrôle la croissance des fixateurs est celui qui limite la production primaire océanique totale à de grandes échelles de temps. Une colimitation de la production primaire océanique par le P et le Fe peut cependant émerger, que la croissance des deux types de phytoplancton soit limitée par un ou plusieurs nutriments localement. Ces résultats suggèrent que les apports anthropiques de Fe et de P dans les écosystèmes marins peuvent avoir des effets importants à court comme à long terme sur la production primaire océanique et donc sur les cycles biogéochimiques.

De manière générale, les organismes ne régulent donc pas efficacement les réservoirs de nutriments. Le couplage des cycles biogéochimiques et la compétition entre groupes fonctionnels peuvent altérer, négativement ou positivement, la régulation des cycles biogéochimiques globaux par les organismes. Une régulation inefficace des concentrations de nutriments n'exclut pas, par contre, une forte régulation des rapports entre ces nutriments, comme dans le cas des rapports de Redfield. L'ajout de nutriments dans les écosystèmes dû aux activités anthropiques risque donc de fortement impacter la production primaire et les cycles biogéochimiques globaux, à court comme à long terme.

General introduction

Impact of global change on biogeochemical cycles

Biogeochemical cycles are under pressure of anthropogenic forcing both at local and global scales [START_REF] Denman | Couplings between changes in the climate system and biogeochemistry[END_REF][START_REF] Canadell | Interactions of the carbon cycle, human activity, and the climate system: a research portfolio[END_REF][START_REF] Ciais | Carbon and other biogeochemical cycles[END_REF], resulting in the alteration of chemical, physical and biological ecosystem processes [START_REF] Gruber | Warming up, turning sour, losing breath: ocean biogeochemistry under global change[END_REF][START_REF] Ciais | Carbon and other biogeochemical cycles[END_REF].

The coupling of biogeochemical cycles of key elements such as carbon (C), nitrogen (N) and phosphorus (P) to each other and to climate involve complex interactions between the geosphere, atmosphere and biosphere [START_REF] Falkowski | The global carbon cycle: a test of our knowledge of Earth as a system[END_REF][START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF].

Terrestrial and marine biogeochemical cycles of key nutrients such as iron (Fe), nitrogen and phosphorus are heavily affected by industrialization and increased food production, mainly through increased nutrient supply to ecosystems [START_REF] Seitzinger | Sources and delivery of carbon, nitrogen, and phosphorus to the coastal zone: An overview of Global Nutrient Export from Watersheds (NEWS) models and their application[END_REF][START_REF] Duce | Impacts of atmospheric anthropogenic nitrogen on the open ocean[END_REF][START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF][START_REF] Raiswell | The iron biogeochemical cycle past and present[END_REF]. In this section we will focus on how the increase in nutrient supply to ecosystems affects biogeochemical cycles both directly (i.e. eutrophication) and indirectly (e.g. through an increase in temperature and acidification).

Direct effects of increasing nutrient supply on global biogeochemical cycles

Increasing nutrient supply to ecosystems occurs mainly through fossil fuel burning and land-use changes, which contribute to the release of radiatively active gases to the atmosphere (e.g. [START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF][START_REF] Rees | Pressures on the marine environment and the changing climate of ocean biogeochemistry[END_REF], as well as the use of fertilizers in agriculture (e.g. [START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF][START_REF] Bouwman | Human alteration of the global nitrogen and phosphorus soil balances for the period 1970-2050[END_REF].

Agricultural activities are a major factor of ecosystem eutrophication [START_REF] Bouwman | Human alteration of the global nitrogen and phosphorus soil balances for the period 1970-2050[END_REF][START_REF] Vitousek | Nutrient imbalances in agricultural development[END_REF][START_REF] Van Vuuren | Phosphorus demand for the 1970-2100 period: A scenario analysis of resource depletion[END_REF]. N-rich and P-rich fertilizers are used to compensate for losses due to soil erosion and harvesting, but less than 50 % are assimilated by crops [START_REF] Peoples | Minimizing gaseous losses of nitrogen[END_REF][START_REF] Smil | Phosphorus in the environment: natural flows and human interferences[END_REF][START_REF] Van Drecht | Global modeling of the fate of nitrogen from point and nonpoint sources in soils, groundwater, and surface water[END_REF]. The remaining part is released in aquatic and coastal ecosystems through river run-off [START_REF] Benitez-Nelson | The biogeochemical cycling of phosphorus in marine systems[END_REF][START_REF] Carpenter | Eutrophication of aquatic ecosystems: Bistability and soil phosphorus[END_REF][START_REF] Bergström | Atmospheric nitrogen deposition has caused nitrogen enrichment and eutrophication of lakes in the northern hemisphere[END_REF][START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF]. For example, the global N cycle has been strongly altered by human activities over the last century [START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF][START_REF] Canfield | The evolution and future of Earthâs nitrogen cycle[END_REF]; Figure A1). Anthropogenic activities (e.g. fertilizer use and fossil fuel burning) currently transform 203 Tg/a of atmospheric nitrogen into reactive N that can be assimilated by autotrophs [START_REF] Fowler | The global nitrogen cycle in the twenty-first century[END_REF]Figure A1). The human-induced release of nitrogen from land to the ocean corresponds to 50-70 TgN/a, in addition to the natural flow estimated to 30 TgN/a [START_REF] Galloway | Transformation of the nitrogen cycle: Recent trends, questions, and potential solutions[END_REF][START_REF] Fowler | The global nitrogen cycle in the twenty-first century[END_REF]; Figure A1). In the ocean, coastal areas are heavily affected by anthropogenic activities and eutrophication as they are closest to human centres [START_REF] Benitez-Nelson | The biogeochemical cycling of phosphorus in marine systems[END_REF][START_REF] Seitzinger | Sources and delivery of carbon, nitrogen, and phosphorus to the coastal zone: An overview of Global Nutrient Export from Watersheds (NEWS) models and their application[END_REF]. However, pelagic areas are also impacted due to the large-scale atmospheric transport and deposition of atmospheric dusts enhanced by land-use changes [START_REF] Benitez-Nelson | The biogeochemical cycling of phosphorus in marine systems[END_REF][START_REF] Jickells | Global iron connections between desert dust, ocean biogeochemistry, and climate[END_REF][START_REF] Krishnamurthy | Impacts of atmospheric nutrient inputs on marine biogeochemistry[END_REF].

The increase in the supply of a nutrient to terrestrial and marine ecosystems can affect the growth of primary producers, and thus alter the biogeochemical cycles of other elements.

An example is the excessive loading of N and P in lakes, which is widespread in the Northern hemisphere [START_REF] Bergström | Atmospheric nitrogen deposition has caused nitrogen enrichment and eutrophication of lakes in the northern hemisphere[END_REF] and can result in a shift from N to P limitation depending on the ratio in which both nutrients are supplied [START_REF] Elser | Shifts in Lake N:P Stoichiometry and Nutrient Limitation Driven by Atmospheric Nitrogen Deposition[END_REF].

Box A: Anthropogenic alteration of the global nitrogen cycle

Figure A1: Impact of anthropogenic activities on the global nitrogen cycle on land and in the ocean and coupling with phosphorus and carbon cycles. Blue fluxes denote ânaturalâ (unperturbed) fluxes; orange fluxes denote anthropogenic perturbation. The numbers (in Tg N per year) are values for the 1990s. Few of these flux estimates are known to better than ±20 %, and many have uncertainties of ±50 % and larger. Extracted from [START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF].

In marine ecosystems, a major impact of eutrophication is induced by the coupling between the biogeochemical cycles of nutrients and that of oxygen. Microorganisms indeed use oxygen for remineralization of organic matter. When primary production is enhanced by the supply of nutrients in coastal or pelagic waters, the increase in the intensity of the recycling flow can lead to an excess consumption of oxygen and a deoxygenation of seawater [START_REF] Diaz | Spreading dead zones and consequences for marine ecosystems[END_REF][START_REF] Stramma | Expanding oxygen-minimum zones in the tropical oceans[END_REF][START_REF] Rabalais | Dynamics and distribution of natural and human-caused hypoxia[END_REF][START_REF] Gruber | Warming up, turning sour, losing breath: ocean biogeochemistry under global change[END_REF].

Indirect effects of increasing nutrient supply on global biogeochemical cycles

Anthropogenic activities thus have direct impacts on biogeochemical cycles through the intensification of the supply of nutrient supply to terrestrial, aquatic and marine systems.

However, they also have indirect impacts as they can alter the physical properties of the environment. The most common example is the release of carbon dioxide (CO 2 ) in the atmosphere, whose atmospheric partial pressure (pCO 2 ) rose from 280 ppm before the industrial revolution to 390.5 ppm in 2011 [START_REF] Ciais | Carbon and other biogeochemical cycles[END_REF] corresponding to an increase in mean global temperature of 0.78 • C [START_REF] Hartmann | Observations: atmosphere and surface[END_REF]. The increase in temperature has two main effects on element cycling: (1) it affects the strength of most of the biogeochemical and biological processes in terrestrial, aquatic and marine ecosystems, and (2) it results in an increase in water column stratification and thus a decrease in the vertical mixing in marine ecosystems [START_REF] Ruardij | The impact of thermal stratification on phytoplankton and nutrient dynamics in shelf seas: a model study[END_REF]Gruber 2011, Figure B1b). Enhanced rainfall, melting of sea ice and river run-off at high latitudes further enhance water column stratification through a decrease in salinity (Gruber 2011, Figure B1c). As stratification lowers the mixing between the surface layer and the deep ocean and thus the supply of nutrients to the surface ocean, one could think that the warming of the surface ocean will decrease the growth of autotrophs.

However, the response of marine productivity to sea-surface warming, and thus its effect on nutrient cycles, is likely to depend on latitude. At high latitudes, the growth of phytoplankton is often light-limited, thus the increase in light availability in the surface mixed layer is likely to enhance primary production and export of organic matter to the deep ocean. In contrast, at low latitudes, the availability of nutrients often limits the growth of phytoplankton, thus the additional decrease in nutrient availability due to a lowered vertical mixing will probably reduce oceanic primary production [START_REF] Riebesell | Sensitivities of marine carbon fluxes to ocean change[END_REF]. The increase in water column stratification is also likely to affect the oceanic oxygen cycle as it decreases the supply of oxygen to the ocean's interior, an effect that is further reinforced by the decrease in oxygen solubility induced by increasing temperature [START_REF] Gruber | Warming up, turning sour, losing breath: ocean biogeochemistry under global change[END_REF].

Another consequence of the increase in atmospheric pCO 2 is the alteration of oceanic pH.

Ocean-atmosphere interactions are a key determinant of the buffering of atmospheric CO 2 concentrations as oceanic pCO 2 follows variations in atmospheric pCO 2 [START_REF] Denman | Couplings between changes in the climate system and biogeochemistry[END_REF]Gruber 2011, Figure B1a). A fraction of the CO 2 dissolved in seawater dissociates and releases protons, thereby resulting in ocean acidification [START_REF] Caldeira | Oceanography: Anthropogenic carbon and ocean pH[END_REF][START_REF] Orr | Anthropogenic ocean acidification over the twenty-first century and its impact on calcifying organisms[END_REF][START_REF] Gruber | Warming up, turning sour, losing breath: ocean biogeochemistry under global change[END_REF][START_REF] Rees | Pressures on the marine environment and the changing climate of ocean biogeochemistry[END_REF]. Atmospheric inputs of N can also alter seawater (2009). Extracted from the Chapter 3 of the fifth IPCC assessment report [START_REF] Rhein | Observations: Ocean[END_REF], page 301. alkalinity and pH, thereby affecting the physical and carbonate pumps in the ocean [START_REF] Doney | Impact of anthropogenic atmospheric nitrogen and sulfur deposition on ocean acidification and the inorganic carbon system[END_REF][START_REF] Krishnamurthy | Impacts of atmospheric nutrient inputs on marine biogeochemistry[END_REF]. Ocean acidification modifies the calcium carbonate (CaCO3) equilibrium in oceans by affecting the saturation state of seawater with respect of CaCO3 [START_REF] Feely | Impact of anthropogenic CO2 on the CaCO3 system in the oceans[END_REF]). Ocean acidification is thus likely to constrain the growth and spatial distribution of calcifying phytoplankton [START_REF] Feely | Impact of anthropogenic CO2 on the CaCO3 system in the oceans[END_REF][START_REF] Orr | Anthropogenic ocean acidification over the twenty-first century and its impact on calcifying organisms[END_REF][START_REF] Hall-Spencer | Volcanic carbon dioxide vents show ecosystem effects of ocean acidification[END_REF]) and hence their impact on nutrient cycles, although the variability in the calcification degree of different morphotypes could lead to more complex responses to seawater acidification [START_REF] Iglesias-Rodriguez | Phytoplankton calcification in a high-CO2 world[END_REF][START_REF] Beaufort | Sensitivity of coccolithophores to carbonate chemistry and ocean acidification[END_REF].

Box B: IPCC record for changes in oceanic properties

Impact of organisms on biogeochemical cycles at local scales

As biogeochemical cycles are heavily altered by anthropogenic activities, it is crucial to understand the extent to which biogeochemical cycles are regulated at both local and global scales. Regulation of biogeochemical cycles occurs through physical and chemical processes, e.g. the dissolution of CO 2 in the ocean that buffers part of the variations of atmospheric pCO 2 [START_REF] Falkowski | The global carbon cycle: a test of our knowledge of Earth as a system[END_REF][START_REF] Denman | Couplings between changes in the climate system and biogeochemistry[END_REF]), but biotic processes also play a major role.

Determining feedbacks between nutrient flows and biotic processes (e.g. primary production) is crucial to understand how ecosystems will respond to climate change and eutrophication at both small and long timescales [START_REF] Arrigo | Marine microorganisms and global nutrient cycles[END_REF]Boyd et al. 2010;[START_REF] Vitousek | Terrestrial phosphorus limitation: mechanisms, implications, and nitrogen-phosphorus interactions[END_REF][START_REF] Moore | Processes and patterns of oceanic nutrient limitation[END_REF]. In this section, we will focus on how organisms can alter and thus possibly regulate their local environment, with a special interest in the control of nutrient availability by autotrophs at local scales and the processes that could influence the strength of this control.

Biotic alteration of the local environment

Ecosystem engineers are organisms that affect the availability of resources in the environment directly or indirectly, either through their physical structure or through their metabolic activity [START_REF] Jones | Organisms as ecosystem engineers[END_REF][START_REF] Jones | Positive and negative effects of organisms as physical ecosystem engineers[END_REF]. An example is bioturbators that modify the physical structure of sediments and allow an increase in oxygen supply to the upper layer of the sediments. By modifying the remineralization rate of organic matter and thus the availability of dissolved inorganic nutrient in seawater, bioturbators can facilitate the development of other organisms [START_REF] Lohrer | Bioturbators enhance ecosystem function through complex biogeochemical interactions[END_REF][START_REF] Erwin | Macroevolution of ecosystem engineering, niche construction and diversity[END_REF]. Another example is nitrogen fixers, whose metabolic activities release fixed nitrogen in the environment, thereby increasing nitrogen availability for other organisms (e.g. [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Menge | Evolutionary tradeoffs can select against nitrogen fixation and thereby maintain nitrogen limitation[END_REF].

Niche construction theory integrates the concept of ecosystem engineers in ecological and evolutionary theory. Niche-constructing processes create feedbacks that affect both biotic and abiotic selecting pressure on niche constructors and/or other organisms that live in the same environment [START_REF] Odling-Smee | Niche construction[END_REF][START_REF] Laland | Evolutionary consequences of niche construction and their implications for ecology[END_REF][START_REF] Laland | Perspective: Seven reasons (not) to neglect niche construction[END_REF]. By modifying their environment through a wide range of processes such as resource consumption, metabolism and habitat modification, organisms create strong feedbacks with, and possibly regulation of, their local environment (Kylafis andLoreau 2008, 2011).

Feedbacks between autotroph growth and nutrient availability

Autotrophs are known to modify nutrient cycles locally, mainly through resource consumption. However, resource consumption by autotrophs can be restricted by the scarcity of nutrients in ecosystems (e.g. [START_REF] Cullen | Hypotheses to explain high-nutrient conditions in the open sea[END_REF][START_REF] Sañudo-Wilhelmy | Phosphorus limitation of nitrogen fixation by Trichodesmium in the central Atlantic Ocean[END_REF][START_REF] Sperfeld | Multiple resource limitation theory applied to herbivorous consumers: Liebigâs minimum rule vs. interactive co-limitation[END_REF][START_REF] Moore | Processes and patterns of oceanic nutrient limitation[END_REF]. We can distinguish two types of limitation of primary production by nutrients, depending on the temporal and spatial scales considered [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF]Menge et al. 2009;[START_REF] Vitousek | Terrestrial phosphorus limitation: mechanisms, implications, and nitrogen-phosphorus interactions[END_REF]. The proximate limiting nutrient constrains primary production at small spatial and temporal scales, and its addition stimulates biological processes directly.

In contrast, the ultimate limiting nutrient controls primary production at large spatial and temporal scales, i.e. over hundreds or thousands of years [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Vitousek | Terrestrial phosphorus limitation: mechanisms, implications, and nitrogen-phosphorus interactions[END_REF].

Here, we focus on the proximate limiting nutrient as we examine the control of autotrophs on nutrient cycles at local scales.

Proximate limitation of primary production has been studied for more than a century, first in the field of agricultural sciences [START_REF] Liebig | Chemistry and its application to agriculture and physiology[END_REF]. The proximate limiting nutrient can be studied indirectly by measuring the availability of nutrients in the soil or in the water [START_REF] Beardall | Approaches for determining phytoplankton nutrient limitation[END_REF][START_REF] Vitousek | Terrestrial phosphorus limitation: mechanisms, implications, and nitrogen-phosphorus interactions[END_REF], the investment of autotrophic organisms to assimilate nutrients (e.g. development of roots and production of extracellular phosphatases; Tredeser and Vitousek 2001;[START_REF] López-Bucio | The role of nutrient availability in regulating root architecture[END_REF], nutrient uptake kinetics [START_REF] Beardall | Approaches for determining phytoplankton nutrient limitation[END_REF]) and/or the concentration of the nutrient or the ratio of elements in the tissues of autotrophs [START_REF] Beardall | Approaches for determining phytoplankton nutrient limitation[END_REF][START_REF] Wardle | Ecosystem Properties and Forest Decline in Contrasting Long-Term Chronosequences[END_REF][START_REF] Wassen | Endangered plants persist under phosphorus limitation[END_REF]). As it is not necessarily the nutrient in highest demand for autotrophs or the scarcest one in the environment that controls primary production proximately, but rather a balance between nutrient supply and demand, these indirect methods are often not sufficient to infer proximate nutrient limitation [START_REF] Karl | Nutrient dynamics in the deep blue sea[END_REF][START_REF] Vitousek | Terrestrial phosphorus limitation: mechanisms, implications, and nitrogen-phosphorus interactions[END_REF]. Another way of determining the proximate limitation of autotrophs is to look at the response of biological processes, such as primary production, to nutrient enrichment (e.g. [START_REF] Blain | Effect of natural iron fertilization on carbon sequestration in the Southern Ocean[END_REF][START_REF] Elser | Global analysis of nitrogen and phosphorus limitation of primary producers in freshwater, marine and terrestrial ecosystems[END_REF][START_REF] Vitousek | Terrestrial phosphorus limitation: mechanisms, implications, and nitrogen-phosphorus interactions[END_REF]. Liebig's law of the minimum states that the growth rate of organisms is not controlled by the total amount of available nutrients but by the scarcest nutrient with respect to organism requirement (Von Liebig, 1842). Liebig's law implies that the growth of organisms is limited by the most limiting nutrient, and the switch between limiting nutrients occurs abruptly [START_REF] Sperfeld | Multiple resource limitation theory applied to herbivorous consumers: Liebigâs minimum rule vs. interactive co-limitation[END_REF]Figure C1a). However, the concept of proximate nutrient limitation is often extended to limitation by more than one nutrient, as limitation of the growth of organisms by several nutrients simultaneously is common in ecological systems (O'Neill et al. 1989;[START_REF] Gleeson | Plant allocation and the multiple limitation hypothesis[END_REF][START_REF] Harpole | Nutrient co-limitation of primary producer communities[END_REF]. For example in the ocean, primary production is locally often either limited by N, P or Fe (e.g. [START_REF] Wu | Phosphate depletion in the Western North 197 Atlantic Ocean[END_REF][START_REF] Moore | Iron cycling and nutrient-limitation patterns in surface waters of the World Ocean[END_REF][START_REF] Aumont | An ecosystem model of the global ocean including Fe, Si, P colimitations[END_REF][START_REF] Moore | Relative influence of nitrogen and phosphorus availability on phytoplankton physiology and productivity in the oligotrophic sub-tropical North Atlantic Ocean[END_REF], or colimited by N and P or P and Fe [START_REF] Mills | Iron and phosphorus co-limit nitrogen fixation in the eastern tropical North Atlantic[END_REF][START_REF] Thingstad | Nature of phosphorus limitation in the ultraoligotrophic Eastern Mediterranean[END_REF][START_REF] Moore | Relative influence of nitrogen and phosphorus availability on phytoplankton physiology and productivity in the oligotrophic sub-tropical North Atlantic Ocean[END_REF]. Colimitationalso called multi-nutrient limitation -considers that there are interactions between limiting nutrients and thus the shift from limitation by one nutrient to a limitation by another is progressive [START_REF] Sperfeld | Multiple resource limitation theory applied to herbivorous consumers: Liebigâs minimum rule vs. interactive co-limitation[END_REF]Figure C1b). When one nutrient is much more abundant than the other, colimitation and Liebig's law of the minimum give similar predictions, but the difference is important around colimiting conditions [START_REF] Tilman | Resources: a graphical mechanistic approach to competition and predation[END_REF][START_REF] Sperfeld | Multiple resource limitation theory applied to herbivorous consumers: Liebigâs minimum rule vs. interactive co-limitation[END_REF]Figure C1). Colimitation of the growth of organisms can arise from different mechanisms: (1) Elements that are biogeochemically mutually independent can be scarce in the environment due to extremely low abiotic supply (independent nutrient colimitation; [START_REF] Arrigo | Marine microorganisms and global nutrient cycles[END_REF][START_REF] Saito | Some thoughts on the concept of colimitation: Three definitions and the importance of bioavailability[END_REF]). An example is the colimitation of phytoplankton growth by N and P in marine and aquatic systems (e.g. Benitez-Nelson 2000 for marine systems; Sterner 2008 for aquatic systems). ( 2) Elements can substitute for each other because they play the same biochemical role, either directly within the same macromolecule or indirectly by substituting one macromolecule for another, thereby resulting in colimitation of the growth of the organism (biochemical substitution colimitation; [START_REF] Saito | Some thoughts on the concept of colimitation: Three definitions and the importance of bioavailability[END_REF]). An example of biochemical substitution colimitation is the case of Fe and manganese in a marine microorganism [START_REF] Tabares | The single superoxide dismutase of Rhodobacter capsulatus is a cambialistic, manganese-containing enzyme[END_REF]. (3) The assimilation of one nutrient can depend on the availability of another nutrient (biochemically dependent colimitation; [START_REF] Saito | Some thoughts on the concept of colimitation: Three definitions and the importance of bioavailability[END_REF]). An example is marine phytoplankton that needs to assimilate Fe before fixing atmospheric N2 [START_REF] Falkowski | Evolution of the nitrogen cycle and its influence on the biological sequestration of CO2 in the ocean[END_REF].

Box C: Liebig's law of the minimum vs. colimitation

(4) Autotrophs can adapt their stoichiometry to nutrient availability to use resources in an optimal ratio (e.g. Geider and LaRoche 2002;[START_REF] Sterner | Ecological stoichiometry: The biology of elements from molecules to the biosphere[END_REF][START_REF] Klausmeier | Phytoplankton stoichiometry[END_REF]). ( 5) Different nutrients limit the growth of different species or functional groups within an ecosystem (community colimitation; [START_REF] Arrigo | Marine microorganisms and global nutrient cycles[END_REF][START_REF] Vitousek | Terrestrial phosphorus limitation: mechanisms, implications, and nitrogen-phosphorus interactions[END_REF].

Community colimitation often occurs in terrestrial and marine systems, for example through the interactions between N and P biogeochemical cycles when P limits the growth of N-fixers.

The enrichment of the system with P increases the intensity of N-fixation, thereby increasing the amount of available N for non-fixers. The addition of N in the environment also increases the growth of non-fixers. Thus in this case, primary production increases with both P and N addition as the result of the limitation of the two functional groups by different nutrients [START_REF] Elser | Global analysis of nitrogen and phosphorus limitation of primary producers in freshwater, marine and terrestrial ecosystems[END_REF][START_REF] Vitousek | Terrestrial phosphorus limitation: mechanisms, implications, and nitrogen-phosphorus interactions[END_REF][START_REF] Harpole | Nutrient co-limitation of primary producer communities[END_REF].

Adaptable stoichiometry of autotrophs and its impact on local nutrient pools

Constraints on mass balance occur in inorganic chemical reactions, biochemical transformations as well as ecological interactions [START_REF] Elser | Organism size, life history, and N:P stoichiometry[END_REF][START_REF] Sterner | Ecological stoichiometry: The biology of elements from molecules to the biosphere[END_REF]. By consuming resources in a particular ratio, autotrophs allow a coupling between biogeochemical cycles and thus alter the availability of a nutrient depending on the availability of the others. At the base of the food chain, autotrophs have the particularity to adapt their elemental content to nutrient availability (Geider and LaRoche 2002;[START_REF] Sterner | Ecological stoichiometry: The biology of elements from molecules to the biosphere[END_REF][START_REF] Han | Biogeography and variability of eleven mineral elements in plant leaves across gradients of climate, soil and plant functional type in China[END_REF]. This highly adaptive stoichiometry is characterized by luxury consumption (i.e. the absorption of a nutrient in excess of the immediate growth requirement of the organism), which occurs when a different limiting factor (light or another nutrient) limits the growth of the autotroph [START_REF] Sterner | Ecological stoichiometry: The biology of elements from molecules to the biosphere[END_REF]. For example, after a long period of P starvation, the N-fixing bacteria Trichodesmium consumes P in excess compared to its requirement for immediate growth. When P is scarce again, variations in the internal N:P ratio of Trichodesmium shows that P stocks are used by the cell to enhance its growth [START_REF] White | Flexible elemental stoichiometry in Trichodesmium spp. and its ecological implications[END_REF]. However, luxury consumption can in some cases have a negative impact on autotroph growth. For example, elevated N concentrations in the soil lead to luxury consumption in seedlings and saplings of some tree species in Northeastern US forests. Luxury consumption of N seems to be advantageous for species that are under high-light conditions, but it has negative impact on the others because luxury consumption of N increases the risk of herbivory [START_REF] Tripler | Soil nitrogen availability, plant luxury consumption, and herbivory by white-tailed deer[END_REF].

By allowing a decoupling of nutrient dynamics from biomass growth, luxury consumption affects the control of autotrophs on nutrient cycles. The ability of autotrophs to adapt their stoichiometry depending on nutrient availability can also alter competition and coexistence among species. The resource ratio competition theory [START_REF] Tilman | Resources: a graphical mechanistic approach to competition and predation[END_REF]) was developed to determine the outcomes of competition for inorganic resources among species with different elemental requirements. Variation in the ratio in which resources are supplied will favour some species more than the others, leading to a modification of the elemental composition of the community and thus of the chemical environment. In cases where the growth of autotroph species is proximately limited by a single nutrient, the adaptation of their stoichiometry to that of their resources can lead to colimitation at the community level [START_REF] Danger | Does Liebigâs law of the minimum scale up from species to communities?[END_REF]. These results suggest that the stoichiometry of communities should tend to match that of their resources if the ratio in which resources are available play an important role in the competition between species [START_REF] Schade | A conceptual framework for ecosystem stoichiometry: balancing resource supply and demand[END_REF][START_REF] Danger | Does Liebigâs law of the minimum scale up from species to communities?[END_REF]. Thus variable stoichiometry of autotrophs can change the pattern of nutrient limitation, and hence the consumption and control of resources at the community level.

To account for the ability of organisms to modify their stoichiometry, cell quota models were developed initially for aquatic systems [START_REF] Droop | Vitamin B12 and marine ecology. IV. The kinetics of uptake, growth and inhibition in Monochrysis Lutheri[END_REF]). This approach is now widely used in theoretical ecology (e.g. Klausmeier et al. 2004a;[START_REF] Diehl | Flexible nutrient stoichiometry mediates environmental influences on phytoplankton and its resources[END_REF][START_REF] Bonachela | A model for variable phytoplankton stoichiometry based on cell protein regulation[END_REF]. Models with variable cell quotas thus enable luxury uptake of nutrients by autotrophs, which can have an important impact on nutrient dynamics. For example, a model with fixed phytoplankton stoichiometry and another with variable stoichiometry lead to similar results concerning phytoplankton biomass and distribution in lakes. But the model with variable stoichiometry predicts a much deeper P depletion zone because of luxury uptake of P by phytoplankton (Frassl et al. 2013).

Impact of herbivores and detritivores on nutrient cycles

Variability in autotroph stoichiometry impacts higher trophic levels and detritivores [START_REF] Sterner | Ecological stoichiometry: The biology of elements from molecules to the biosphere[END_REF][START_REF] Thingstad | Nature of phosphorus limitation in the ultraoligotrophic Eastern Mediterranean[END_REF]. Nutrient imbalance, i.e. the dissimilarity in nutrient content between a consumer and its resource, greatly affects the growth and activity of the consumer. Consumers can react in different ways to resources of different elemental compositions, e.g. through changes in production efficiency or food selection (e.g. [START_REF] Strom | Selective feeding by protozoa: model and experimental behaviors and their consequences for population stability[END_REF][START_REF] Gentleman | Functional responses for zooplankton feeding on multiple resources: a review of assumptions and biological dynamics[END_REF][START_REF] Schade | Stoichiometric tracking of soil nutrients by a desert insect herbivore[END_REF]. Food selection of resources by a consumer can be either passive (depending on resource vulnerability, its nutritional or toxic content, e.g. [START_REF] Moore | Eucalyptus foliar chemistry explains selective feeding by koalas[END_REF] or active when selection depends on the relative availability of resources (e.g. rejection of the less abundant resource, [START_REF] Gentleman | Functional responses for zooplankton feeding on multiple resources: a review of assumptions and biological dynamics[END_REF]. Another common mechanism is the adaptation of the growth rate of the consumer by excretion of part of non-assimilated resources in the environment and egestion [START_REF] Sterner | Ecological stoichiometry: The biology of elements from molecules to the biosphere[END_REF][START_REF] Grover | The impact of variable stoichiometry on predatorâprey interactions: A multinutrient approach[END_REF]Anderson et al. 2005). Consumer-driven nutrient recycling (i.e. release of part of the nutrients ingested in the environment) is a major process in biogeochemical cycles.

By affecting the availability of nutrients in the environment, it can modify the limitation conditions of other organisms, especially autotrophs. For example, excretion of nutrients by zooplankton reduces the limitation of phytoplankton growth by P and N through an increase in the recycling of nutrients, both in aquatic and in marine systems [START_REF] Moegenburg | Nutrient regeneration by zooplankton: effects on nutrient limitation of phytoplankton in a eutrophic lake[END_REF][START_REF] Elser | The stoichiometry of consumer-driven nutrient recycling: Theory, observations, and consequences[END_REF][START_REF] Trommer | Zooplankton-mediated nutrient limitation patterns in marine phytoplankton: an experimental approach with natural communities[END_REF]. Zooplankton can also control the availability of nutrients in deep water through vertical migrations and excretion, for example for filter feeders and detritivores [START_REF] Palmer | Production and export in a global ocean ecosystem model[END_REF][START_REF] Thingstad | Nature of phosphorus limitation in the ultraoligotrophic Eastern Mediterranean[END_REF][START_REF] Trommer | Zooplankton-mediated nutrient limitation patterns in marine phytoplankton: an experimental approach with natural communities[END_REF]. By developing a stoichiometrically explicit model of N and P in a plant-herbivore system, [START_REF] Daufresne | Plant-herbivore interactions and ecological stoichiometry: when do herbivores determine plant nutrient limitation?[END_REF] showed that a herbivore with a low N:P ratio can either induce a shift in the limitation of the plant from N to P, or promote N limitation depending on plant affinity for nutrients. Thus consumer-driven nutrient recycling does not necessarily positively impact the growth of autotrophs. Detritivores also affect the availability of inorganic nutrients for autotrophs, as they either immobilize inorganic nutrients or recycle them depending on their stoichiometry and that of inorganic nutrients (Cherif andLoreau 2007, 2009).

Impact of organisms on biogeochemical cycles at global scales

Feedbacks between organisms and their environment are not necessarily the same at local and global scales. For example, proximate and ultimate limiting nutrients are not necessarily the same in a system, as nutrients can affect primary production either directly or indirectly by constraining the biogeochemical cycles of other nutrients over long temporal scales (e.g. [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Moore | Processes and patterns of oceanic nutrient limitation[END_REF]. In this section, we will thus focus on theories and examples of how organisms can affect and regulate their environment at global scales.

Biotic regulation at global scales: the Gaia hypothesis

Biotic regulation of the Earth system has been debated for several decades, mainly concerning the Gaia hypothesis, which postulates that organisms have kept the Earth's surface environment stable and habitable for life through self-regulating feedback mechanisms (e.g. Lovelock andMargulis 1974a, 1974b;[START_REF] Margulis | Biological modulation of the Earthâs atmosphere[END_REF]. The origin of the Gaia hypothesis relates to the detection of the presence of life on a planet through the analysis of its atmospheric composition [START_REF] Lovelock | Planetary atmospheres: compositional and other changes associated with the presence of life[END_REF]. Living organisms are supposed to create a disequilibrium in the composition of atmosphere by using some gases and releasing highly reactive gases like methane and oxygen [START_REF] Lovelock | Planetary atmospheres: compositional and other changes associated with the presence of life[END_REF]Lovelock and Margulis 1974a). Thus a planet without life should be closer to thermodynamic equilibrium in its composition than a planet with living organisms [START_REF] Lovelock | Planetary atmospheres: compositional and other changes associated with the presence of life[END_REF]. However, the atmospheric composition of the Earth seems to be almost stable over a long timescale, leading to the hypothesis that organisms maintain the atmospheric composition near an optimum for their growth and persistence (Lovelock andMargulis 1974a, 1974b;[START_REF] Margulis | Biological modulation of the Earthâs atmosphere[END_REF]. The Gaia hypothesis concerns the regulation by organisms of (1) the atmospheric, oceanic and soil chemical composition, (2) the surface temperature of the Earth between 45 • S and 45 • N latitudes, and (3) the surface pH of the Earth and that of oceans (Lovelock andMargulis 1974a, 1974b;[START_REF] Margulis | Biological modulation of the Earthâs atmosphere[END_REF]. This hypothesis has been strongly controversial, mainly because it seems to imply some conscious foresight by the organisms [START_REF] Doolittle | Is nature really motherly[END_REF]) and because evolution by natural selection does not necessarily promote large-scale environmental regulation [START_REF] Dawkins | The extended phenotype: The long reach of the gene[END_REF]).

To answer these criticisms, a simple theoretical model was built, in which a grey cloudless world is at a similar distance from a star, as the Earth is from the Sun, and which is slowly getting warm [START_REF] Watson | Biological homeostasis of the global environment: the parable of Daisyworld[END_REF]. In the Daisyworld model, life is represented by two types of daisies with the same optimum temperature and temperature tolerance bounds.

The growth rate of daisies is determined only by temperature, which is modified by the daisies as they absorb solar radiation depending on their colour. Black daisies have a warming effect on their local environment, while white daisies have a cooling effect. At low solar forcing, black daisies have an advantage and dominate the community, resulting in a warming of the environment, which increases the ability of life to spread (Figure D1a). As solar forcing increases, white daisies are advantaged and dominate the community. If solar forcing is too high, all daisies go extinct (Figure D1a). The Daisyworld model predicts that the surface temperature of the planet is maintained near daisy optimum temperature for a wide range of solar forcing, thus natural selection and competition at the individual level can promote planetary regulation if the benefit in fitness is superior to the energetic cost (e.g. [START_REF] Watson | Biological homeostasis of the global environment: the parable of Daisyworld[END_REF][START_REF] Lenton | Gaia and natural selection[END_REF]Figure D1b). As organisms both alter and are constrained by their environment, feedback mechanisms at the individual level emerge and can spread to the global level through growth and selection [START_REF] Lenton | Gaia and natural selection[END_REF].

The original Daisyworld model was followed by numerous variants, for instance with mutations [START_REF] Lenton | Gaia and natural selection[END_REF]Lenton andLovelock 2000, 2001;[START_REF] Wood | Daisyworld: A review[END_REF] or spatialization [START_REF] Wood | Daisyworld: A review[END_REF]). The Guild model is a variant of the Daisyworld model to study how organisms are able to regulate the chemical composition of their environment [START_REF] Downing | The simulated evolution of biochemical guilds: Reconciling Gaia theory and natural selection[END_REF][START_REF] Free | Do evolution and ecology need the Gaia hypothesis?[END_REF]. In the Guild model, an initial microorganism species metabolises different chemical forms of a nutrient and excretes byproducts. If these Figure D1: Occupancy of the two different daisies types and average planetary temperature in the Daisyworld model. Simulation were performed using the methodology described in [START_REF] Watson | Biological homeostasis of the global environment: the parable of Daisyworld[END_REF]. (a) Occupancy of black and white daisies against increasing solar energy. The single-daisy occupancies are shown by the black lines, and the red line shows the sum of the two-daisy types. Note that the total amount of life is conserved in the coexistence regime. (b) Mean planetary temperature against increasing solar energy. The red line indicate the system path when the solar energy is steadily increased. The dashed line indicates the system path in the absence of daisies. Adapted from [START_REF] Wood | Daisyworld: A review[END_REF].

microorganisms become so abundant that the excretion of byproducts becomes too important at the global scale, other microorganisms that metabolise these abundant byproducts will evolve [START_REF] Downing | The simulated evolution of biochemical guilds: Reconciling Gaia theory and natural selection[END_REF][START_REF] Lenton | Gaia as a complex adaptive system[END_REF]. The emergence of these recycling loops can result in global regulation of environmental chemical composition [START_REF] Downing | The simulated evolution of biochemical guilds: Reconciling Gaia theory and natural selection[END_REF]. The Flask model (Williams andLenton 2007, 2008) also describes the evolution of microorganisms in a homogeneous environment, but it includes non-nutrient abiotic factors that can be affected by microorganisms. Unlike the Daisyworld and Guild models, the means by which organisms affect their environment are dissociated from their environmental effects that determine their growth [START_REF] Free | Do evolution and ecology need the Gaia hypothesis?[END_REF][START_REF] Williams | The Flask model: emergence of nutrient-recycling microbial ecosystems and their disruption by environment-altering ârebelâ organisms[END_REF]. This model shows that nutrient recycling and regulation of non-nutrient abiotic factors can emerge and be disrupted by evolutionary events, e.g. a 'rebel' organism that shifts the environment outside the preferred conditions of the majority of the community.

Regulation of atmospheric oxygen cycle

The mechanisms underlying regulation of atmospheric oxygen concentration have been debated from the development of the Gaia hypothesis. The proportion in which oxygen is present in the Earth's atmosphere has been relatively constant for 350 Ma. Charcoal records indeed show that fires have occurred all over this period [START_REF] Rowe | Devonian charcoal[END_REF][START_REF] Cressler | Evidence of Earliest Known Wildfires[END_REF], and experiments shows that fires do not occur at oxygen levels lower than 17 % [START_REF] Watson | Methanogenesis, fires and the regulation of atmospheric oxygen[END_REF]. The probability of fires increases rapidly when oxygen level is higher than 21 %, thus oxygen levels lower than 25 % are required to account for the continuous existence of a dense vegetation (e.g. [START_REF] Watson | Methanogenesis, fires and the regulation of atmospheric oxygen[END_REF][START_REF] Lenton | The role of land plants, phosphorus weathering and fire in the rise and regulation of atmospheric oxygen[END_REF][START_REF] Kump | The rise of atmospheric oxygen[END_REF]. As the residence time of oxygen in the atmosphere is much shorter than the period of 350 Ma concerned and oxygen is at the same time a byproduct of autotrophic organisms and a constraint on their growth, regulatory feedbacks were proposed to explain for the near-constancy of atmospheric oxygen concentration (Lovelock and Margulis 1974a;[START_REF] Holland | The chemical evolution of the atmosphere and oceans[END_REF]. Lenton and Watson (2000b) tested several feedback mechanisms that could account for the regulation of atmospheric oxygen concentration using a model of coupled carbon, nitrogen, phosphorus and oxygen global cycles. Oceanic feedbacks rely on anoxia and thus are suppressed when atmospheric oxygen increases (Lenton and Watson 2000b). The increase in fire frequency as a result of increasing oxygen concentration leads to an increase in the supply of P to the ocean, which in turn increases the oceanic primary production and provides a negative feedback on the oxygen concentration [START_REF] Kump | Terrestrial feedback in atmospheric oxygen regulation by fire and phosphorus[END_REF]). However, the efficiency of this negative feedback is low and the feedback can become positive if the marine C:P burial ratio is greater than the terrestrial C:P burial ratio (Lenton and Watson 2000b;[START_REF] Lenton | The role of land plants, phosphorus weathering and fire in the rise and regulation of atmospheric oxygen[END_REF]. Lenton and Watson (2000b) proposed to consider weathering of phosphate rocks by vascular plants as a possible process leading to regulation of atmospheric O 2 concentration.

Weathering by plants occurs for example through the enhancement of the hydrological cycle and thus precipitation, acidification of soils and direct physical action of tree roots on rocks [START_REF] Andrews | Soil CO2 dynamics, acidification, and chemical weathering in a temperate forest with experimental CO2 enrichment[END_REF][START_REF] Hinsinger | Plant-induced weathering of a basaltic rock: experimental evidence[END_REF][START_REF] Lenton | The role of land plants, phosphorus weathering and fire in the rise and regulation of atmospheric oxygen[END_REF]. Rock weathering increases the supply of phosphorus to terrestrial and marine ecosystems, thus enhancing the photosynthetic activity and the efficiency of the various oceanic feedback mechanisms.

When atmospheric oxygen concentration increases, fires reduce the weathering of phosphate rocks, resulting in a decrease in terrestrial and marine primary production, which in turn decreases the atmospheric oxygen concentration. On the contrary, a decrease in atmospheric oxygen concentration enhances rock weathering by plants, leading to an increase in global primary production and thus an increase in atmospheric oxygen concentration. This feedback mechanism predicts a set point of the oxygen level between 18 and 25 % depending on humidity and climate, and could thus provide an explanation for the regulation of atmospheric oxygen concentration (Lenton and Watson 2000b).

Redfield ratios in the ocean

Another example of possible biotic regulation of the environment at the global scale is the Redfield ratios in the ocean. Analysis of the composition of phytoplankton cells showed a mean C:N:P ratio of 106C:16N:1P, which corresponds to a mean requirement of 276 atoms of oxygen for the oxidation of phytoplankton cells [START_REF] Redfield | On the proportions of organic derivations in sea water and their relation to the composition of plankton[END_REF][START_REF] Fleming | The composition of plankton and units for reporting populations and production[END_REF]. [START_REF] Redfield | The biological control of chemical factors in the environment[END_REF] measured the availability of C, N, P and O in water samples collected at several depths in different oceans, as well as the ratio in which these nutrients vary between the samples. The analysis of seawater samples showed that C, N, P and O are available in a ratio of 1000C:15N:1P:200-300O in deep waters, and the ratio in which C, N and P vary in the ocean is 105C:15N:1P. The availability of nutrients in surface waters can differ from the ratio of 1000C:15N:1P:200-300O, as recycling of part of the organic matter in the deep ocean and sedimentation contribute to decrease the availability of nutrients in the surface ocean. [START_REF] Redfield | On the proportions of organic derivations in sea water and their relation to the composition of plankton[END_REF][START_REF] Redfield | The biological control of chemical factors in the environment[END_REF] noticed that (1) the mean C:N:P ratio of phytoplankton are similar to the mean ratio in which these elements vary in seawater, and (2) the mean N:P ratio of phytoplankton cells and the relative amount of O required for their oxidation is similar to the proportions in which N, P and O are available in seawater (Figure E1). The Redfield ratio of 106:16:1 is a cornerstone of marine biogeochemistry and ocean models [START_REF] Falkowski | Rationalizing elemental ratios in unicellular algae[END_REF] and was extended to other elements [START_REF] Ho | The elemental composition of some marine phytoplankton[END_REF][START_REF] Quigg | The evolutionary inheritance of elemental stoichiometry in marine phytoplankton[END_REF]. However, the ratios in which trace metals compose phytoplankton are not similar to those in which they are available in seawater [START_REF] Quigg | The evolutionary inheritance of elemental stoichiometry in marine phytoplankton[END_REF][START_REF] Quigg | Evolutionary inheritance of elemental stoichiometry in phytoplankton[END_REF]. Redfield ratios raise two main issues in the understanding of marine biogeochemistry: (1) Is there a biological significance for the N:P ratio of 16:1 in phytoplankton?, and (2) Which mechanisms can explain the similarity between the mean N:P ratio of phytoplankton and that of seawater? The dashed line is the linear regression (slope of 14.5), which is less than the mean N:P ratio of phytoplankton because nitrate is removed via denitrification in high-nutrient areas and added via remineralization of N-fixer biomass in low-nutrient areas. Extracted from [START_REF] Weber | Oceanic nitrogen reservoir regulated by plankton diversity and ocean circulation[END_REF].

Box E: Redfield ratio in the ocean

The Redfield ratio of 16:1 constitutes a mean value of the ratio in which N and P compose phytoplankton cells, but has sometimes been misinterpreted as a universally constant value of phytoplankton stoichiometry [START_REF] Karl | Nutrient dynamics in the deep blue sea[END_REF][START_REF] Klausmeier | Phytoplankton stoichiometry[END_REF]. Autotrophs are able to adapt their stoichiometry within a certain range, e.g. depending on physiological constraints or the physical structure of ecosystems [START_REF] Sterner | Ecological stoichiometry: The biology of elements from molecules to the biosphere[END_REF][START_REF] Hall | Constraints on primary producer N:P stoichiometry along N:P supply ratio gradients[END_REF]. Phytoplankton can adapt their stoichiometry depending on nutrient availability. For instance, the N:P ratio of cyanobacteria ranges from less than 5:1 when dissolved phosphorus is in excess to more than 100:1 when it is scarce (Geider and LaRoche 2002). Combined with nutrient availability, growth strategies also affect the stoichiometry of phytoplankton (Klausmeier et al. 2004b;[START_REF] Arrigo | Marine microorganisms and global nutrient cycles[END_REF][START_REF] Franz | Dynamics and stoichiometry of nutrients and phytoplankton in waters influenced by the oxygen minimum zone in the eastern tropical Pacific[END_REF][START_REF] Martiny | Strong latitudinal patterns in the elemental ratios of marine plankton and organic matter[END_REF]. [START_REF] Redfield | On the proportions of organic derivations in sea water and their relation to the composition of plankton[END_REF] hypothesized that intracellular properties of phytoplankton could account for the N:P ratio of 16:1. By comparing the stoichiometry of green and red phytoplankton superfamilies and their relative abundance over geological timescales, [START_REF] Quigg | The evolutionary inheritance of elemental stoichiometry in marine phytoplankton[END_REF] proposed that historical changes in the oceanic redox state could have determined the evolutionary trajectory of the elemental stoichiometry of phytoplankton.

However, this result does not solve the issue of the biological significance of the Redfield ratio, which has led to a large number of theoretical studies in the last decade (Klausmeier et al. 2004a(Klausmeier et al. , 2004b[START_REF] Klausmeier | Phytoplankton stoichiometry[END_REF][START_REF] Loladze | The origins of the Redfield nitrogen-to-phosphorus ratio are in a homoeostatic protein-to-rRNA ratio[END_REF][START_REF] Bonachela | A model for variable phytoplankton stoichiometry based on cell protein regulation[END_REF]. Klausmeier et al. (2004b) developed a stoichiometrically explicit model of phytoplankton growth to assess whether or not the Redfield ratio corresponds to an optimal N:P ratio for phytoplankton.

This model describes protein synthesis and relies on the fact that components of the cellular machinery of phytoplankton have different N:P ratios and their relative importance in phytoplankton cells depends on nutrient availability [START_REF] Elser | Organism size, life history, and N:P stoichiometry[END_REF]Geider and LaRoche 2002;[START_REF] Sterner | Ecological stoichiometry: The biology of elements from molecules to the biosphere[END_REF][START_REF] Karpinets | RNA:protein ratio of the unicellular organism as a characteristic of phosphorous and nitrogen stoichiometry and of the cellular requirement of ribosomes for protein synthesis[END_REF]). The assembly machinery, i.e. ribosomes, has a low N:P ratio. In contrast, the resource-acquisition machinery, i.e. pigments and proteins, has a high N:P ratio (Geider and LaRoche 2002;Klausmeier et al. 2004b;[START_REF] Arrigo | Marine microorganisms and global nutrient cycles[END_REF]. During exponential growth, the optimal cellular machinery corresponds to a N:P ratio lower than 9, corresponding to rapidly growing phytoplankton. This growth strategy corresponds to organisms that are able to grow exponentially with fast cell division, as in large diatoms, dinoflagellates and cryptophytes [START_REF] Falkowski | The evolution of modern eukaryotic phytoplankton[END_REF][START_REF] Franz | Dynamics and stoichiometry of nutrients and phytoplankton in waters influenced by the oxygen minimum zone in the eastern tropical Pacific[END_REF]. At competitive equilibrium, the optimal cellular machinery corresponds to an N:P ratio between 35 and 45, depending on whether phytoplankton growth is limited by phosphorus, nitrogen or light (Klausmeier et al. 2004b). This growth strategy corresponds to phytoplankton that are able to grow under low-light or oligotrophic conditions, as the cyanobacteria Prochlorococcus [START_REF] Franz | Dynamics and stoichiometry of nutrients and phytoplankton in waters influenced by the oxygen minimum zone in the eastern tropical Pacific[END_REF]. Thus this model predicts the existence of two optimal growth strategies depending on limitation conditions, but they correspond to N:P ratios either substantially lower (<10) or greater (>30) than the Redfield ratio. The Redfield ratio would then represent an average stoichiometry of phytoplankton considering two optimal strategies, not a universally optimal value of phytoplankton stoichiometry (Klausmeier et al. 2004b[START_REF] Klausmeier | Phytoplankton stoichiometry[END_REF][START_REF] Arrigo | Marine microorganisms and global nutrient cycles[END_REF][START_REF] Franz | Dynamics and stoichiometry of nutrients and phytoplankton in waters influenced by the oxygen minimum zone in the eastern tropical Pacific[END_REF]. A subsequent model was developed that integrates the synthesis of rRNA in addition to that of proteins [START_REF] Loladze | The origins of the Redfield nitrogen-to-phosphorus ratio are in a homoeostatic protein-to-rRNA ratio[END_REF].

Parameterization of this model for the optimal growth of a prokaryote and a eukaryote predicts a homeostatic protein:rRNA ratio (i.e. the ratio of protein mass to rRNA mass in a cell) of 3±0.7:1, which corresponds to an N:P ratio of 16±3:1. The Redfield ratio seems thus to have a biological significance, as it corresponds to the N:P ratio of a microbial cell growing under optimal conditions. However, this N:P ratio of 16 is not universally advantageous for phytoplankton. The optimal N:P ratio is indeed higher when cell growth is limited by phosphorus, and lower when it is limited by nitrogen [START_REF] Loladze | The origins of the Redfield nitrogen-to-phosphorus ratio are in a homoeostatic protein-to-rRNA ratio[END_REF].

Just as in phytoplankton, the N:P ratio of deep-waters can differ from the Redfield ratio due to changes in the N:P ratio of the material that is supplied to the ocean and in microbial activity, e.g. through anaerobic ammonium oxidation (i.e. anammox), denitrification and nitrogen fixation [START_REF] Gruber | Global patterns of marine nitrogen fixation and denitrification[END_REF][START_REF] Karl | A sea of change: Biogeochemical variability in the North Pacific Subtropical Gyre[END_REF][START_REF] Karl | Building the long-term picture: The U.S. JGOFS time-series programs[END_REF][START_REF] Arrigo | Marine microorganisms and global nutrient cycles[END_REF]. The diversity of phytoplankton communities and their spatial distribution can also create regional deviations of the Redfield ratio in deep waters (Weber andDeutsch 2010, 2012). However, the deep-water N:P ratio seems to be almost constant over space and time. The similarity between the ratios in which N, P and O are available in seawater and the requirement of phytoplankton cells to grow and that of microorganisms to oxidise the corresponding organic matter led [START_REF] Redfield | The biological control of chemical factors in the environment[END_REF] to propose three hypotheses: (1) it is a coincidence, (2) organisms have adapted their stoichiometry to that of their environment, or (3) biotic processes control the proportion of N and P in seawater. The first hypothesis is highly improbable, as the similitude concerns nutrients as well as oxygen. The second hypothesis is conceivable as phytoplankton can modify their stoichiometry depending on nutrient availability, but it is much less clear how adaptation could determine the quantity of oxygen required for the oxidation of organic matter [START_REF] Redfield | The biological control of chemical factors in the environment[END_REF]. The third hypothesis was originally favoured.

Nitrogen fixation and denitrification were proposed as mechanisms by which phytoplankton could influence the N:P ratio in deep-waters, while sulfate-reducing bacteria might control the availability of oxygen [START_REF] Redfield | The biological control of chemical factors in the environment[END_REF]. Subsequent studies have mainly focused on the mechanisms that could explain the regulation of the deep-water N:P ratio by phytoplankton (e.g. [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Lenton | Biotic stoichiometric controls on the deep ocean N:P ratio[END_REF]Weber andDeutsch 2010, 2012). The competitive dynamics between N-fixers and non-fixers is often considered as a central feedback mechanism in the determination of the deep-water N:P ratio. If N is scarce in the surface ocean, N-fixers outcompete non-fixers and allow P to enter the system in a ratio corresponding to that of N-fixers. In contrast, when P is scarce, non-fixers outcompete N-fixers, resulting in a decrease in the supply of fixed nitrogen to the surface ocean [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF]Lenton and Watson 2000a;[START_REF] Klausmeier | Phytoplankton stoichiometry[END_REF]. Strong limitation of N-fixation (e.g. proximate limitation of the growth of N-fixers by iron) leads to a decrease in the regulation of the deep-water N:P ratio, thus suggesting that the competition between N-fixers and non-fixers not only affects the deep-water N:P ratio, but also maintains it around the Redfield ratio (Lenton and Watson 2000a). However, in this model, the N:P ratio of both non-fixers and N-fixers is set to the Redfield ratio of 16:1, which could potentially create a bias in the results.

Diversity in the N:P requirements of phytoplankton indeed affects the stoichiometry of the export flux to the deep ocean (Weber andDeutsch 2010, 2012). [START_REF] Lenton | Biotic stoichiometric controls on the deep ocean N:P ratio[END_REF] showed that phytoplankton N:P ratio different from the Redfield ratio does not necessarily affect the ratio in which N and P are available in the deep ocean, even in cases where N-fixers are restricted to a small part of the surface ocean [START_REF] Lenton | Biotic stoichiometric controls on the deep ocean N:P ratio[END_REF][START_REF] Klausmeier | Phytoplankton stoichiometry[END_REF]. The efficiency of regulation of the deep-water N:P ratio by phytoplankton appears to be determined by the N:P threshold for N-fixation, i.e. the N:P ratio of seawater below which N-fixation occurs [START_REF] Lenton | Biotic stoichiometric controls on the deep ocean N:P ratio[END_REF]. However, the N:P threshold for N-fixation is related to the phytoplankton N:P ratio, and thus further understanding of the mechanisms leading to regulation of the N:P ratio of deep waters by phytoplankton are needed.

Resource access limitation

Consumption of inorganic nutrients is a major mechanism leading to biotic regulation of the local, and possibly global, environment. Autotrophic organisms consume inorganic nutrients in their local environment until consumption balances their basal metabolism and mortality, leading to a top-down control of the inorganic nutrient pool by autotrophic organisms [START_REF] Loreau | From populations to ecosystems: Theoretical foundations for a new ecological synthesis[END_REF]). However, top-down resource control only occurs when the resource is accessible to consumers, and autotrophic organisms often access a limited part of the nutrients available in their environment. Resource access limitation is common in natural systems and can occur through either physical or chemical mechanisms.

Physical resource access limitation corresponds to cases where consumers only access part of the nutrients in their environment because of physical barriers. This type of resource access limitation is frequent in terrestrial, aquatic and marine environments. In terrestrial ecosystems, the processes of soil development can make part of the soil inaccessible or inhospitable for the plant rooting system, leading to a physical barrier between plants and an important part of the soil nutrient pool [START_REF] Vitousek | Terrestrial phosphorus limitation: mechanisms, implications, and nitrogen-phosphorus interactions[END_REF]. Pacic horizons (i.e. a thin layer of soil cemented by iron, manganese or organic matter) and spodic horizons (i.e.

a thin layer of soil as a result of the accumulation in the soil of organic or organo-metallic complexes due to water infiltration) are common in high-rainfall areas [START_REF] Vitousek | Terrestrial phosphorus limitation: mechanisms, implications, and nitrogen-phosphorus interactions[END_REF][START_REF] Jien | Pedogenic processes of placic and spodic horizons in subtropical subalpine forest soils with contrasting textures[END_REF]. The vertical discontinuity of the soil results in a restricted drainage, with an accumulation of water above these horizons. Roots cannot develop because of the inefficient drainage and the hard soil layer, which is too difficult to penetrate [START_REF] Ostertag | Effects of nitrogen and phosphorus availability on fine-root dynamics in Hawaiian Montane forests[END_REF][START_REF] Lohse | Contrasting effects of soil development on hydrological properties and flow paths[END_REF]; thus plants are physically separated from the nutrients in deeper parts of the soil. Another example of physical resource access limitation in terrestrial ecosystems is permafrost, which prevent plants to access deep nitrogen pools in Northern peatlands [START_REF] Keuper | A frozen feast: thawing permafrost increases plant-available nitrogen in subarctic peatlands[END_REF]. In marine and aquatic systems, physical resource access limitation is usually due to the presence of a pycnocline, either because of the heating of surface waters when solar radiation is high and vertical exchanges in the water column are low, or because of the cooling of surface waters through precipitation or melting of ice [START_REF] Vallis | Large-Scale Circulation and Production of Stratification: Effects of Wind, Geometry, and Diffusion[END_REF]. The barrier of density limits water exchanges between the surface and deep layers. Photosynthetic activity depletes nutrients in the surface layer [START_REF] Falkowski | Mix and match: how climate selects phytoplankton[END_REF], resulting in the inaccessibility to phytoplankton of most of the nutrients in the water column. Even in a well-mixed water column, phytoplankton only access part of the nutrients in the water column because their growth is limited by light, and thus autotrophic cells cannot grow under a certain depth depending on water turbidity (e.g. [START_REF] Tilzer | Importance of Mixing, Thermal Stratification and Light Adaptation for Phytoplankton Productivity in Lake Tahoe (California-Nevada)[END_REF][START_REF] Mitchell | Light limitation of phytoplankton biomass and macronutrient utilization in the Southern Ocean[END_REF][START_REF] Ruardij | The impact of thermal stratification on phytoplankton and nutrient dynamics in shelf seas: a model study[END_REF]. Note that physical resource access limitation does not only concern autotrophic organisms. It can also affect interactions between a predator and its prey, as part of the prey population can have access to refuges and thus be inaccessible to predators.

Chemical resource access limitation corresponds to cases where one or several chemical forms of a nutrient cannot be assimilated by part of the organisms. It occurs in terrestrial, aquatic, and marine ecosystems. A common example concerns the assimilation of nitrogen.

Only nitrogen-fixing organisms are able to assimilate N 2 , with the result that non-fixing organisms only access part of the nitrogen available in their environment (e.g. [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Menge | Evolutionary tradeoffs can select against nitrogen fixation and thereby maintain nitrogen limitation[END_REF].

Models of resource access limitation were first developed to study competitive interactions between organisms in heterogeneous environments and their implications for ecosystem functioning [START_REF] Huston | Competition and coexistence: The effects of resource transport and supply rates[END_REF][START_REF] Loreau | Coexistence of multiple food chains in a heterogeneous environment: Interactions among community structure, ecosystem functioning, and nutrient dynamics[END_REF][START_REF] Loreau | Biodiversity and ecosystem functioning: A mechanistic model[END_REF]. In these models, each individual plant consumes nutrients in the vicinity of its rooting system, resulting in a resource depletion zone around each plant. But plants do not have a direct access to the shared nutrient pool. The latter is affected by plants only indirectly, through abiotic transport processes (e.g. diffusion, movements of soil water), whose intensity is determined by the chemical properties of the resource, the physical properties of the soil and the gradient of depletion around individual depletion zones. Variations in the intensity of nutrient transport determine the degree of competition between plants through their indirect control of the common nutrient pool [START_REF] Huston | Competition and coexistence: The effects of resource transport and supply rates[END_REF][START_REF] Loreau | From populations to ecosystems: Theoretical foundations for a new ecological synthesis[END_REF]). This model of plant-nutrient dynamics was extended to a system with herbivores and detritivores, where each food chain acts on a given depletion zone [START_REF] Loreau | Coexistence of multiple food chains in a heterogeneous environment: Interactions among community structure, ecosystem functioning, and nutrient dynamics[END_REF]. Detritivores and herbivores impact both the local depletion zone and the shared nutrient pool. The dynamics of the nutrient is determined by the physical properties of the environment, but also by the recycling of organic matter by detritivores, which in turn affects the diversity of plants and herbivores. Resource access limitation applied to depletion of soil nutrients by plants provides a mechanism leading to coexistence of multiple plant species and food chains in heterogeneous environments [START_REF] Huston | Competition and coexistence: The effects of resource transport and supply rates[END_REF][START_REF] Loreau | Coexistence of multiple food chains in a heterogeneous environment: Interactions among community structure, ecosystem functioning, and nutrient dynamics[END_REF][START_REF] Loreau | Biodiversity and ecosystem functioning: A mechanistic model[END_REF][START_REF] Loreau | From populations to ecosystems: Theoretical foundations for a new ecological synthesis[END_REF]. This kind of approach was also used to study nutrient dynamics in streams [START_REF] Deangelis | Modelling nutrient-periphyton dynamics in streams: the importance of transient storage zones[END_REF], where periphyton and detritivores only access nutrient in transient storage zones at the bottom of the streams. As in resource access limitation models applied to terrestrial plants, the nutrient is supplied to transient storage zones only through exchanges with free-flowing water. Resource access limitation is also commonly used in models of local and global phytoplankton-nutrient dynamics in aquatic and marine systems, as phytoplankton only access nutrients in the surface water layer, because of either the depth of the euphotic layer or the pycnocline (e.g. Tyrrell et al. 1999, Lenton andWatson 2000a).

As the consumption of nutrients by organisms is a major process leading to biotic regulation of nutrient cycles, resource access limitation, which often occurs at global scales, is likely to heavily impact feedbacks between organisms and global biogeochemical cycles.

Objectives, methods and main questions

General objective

Understanding the interactions between nutrient cycles and organisms is fundamental to assess and predict the ability of the Earth system to respond to global environmental changes.

The ability of organisms to regulate their environment at global scales has been debated for several decades, whether in relation to the Gaia hypothesis or to the Redfield ratio in the ocean (e.g. [START_REF] Lenton | Gaia and natural selection[END_REF][START_REF] Gruber | Redfieldâs evolving legacy[END_REF]. Strong biotic regulation of global biogeochemical cycles implicitly assumes that organisms have access to all the nutrients in their environment. In reality, most of resources are inaccessible to organisms at global scales because of physical or chemical barriers [START_REF] Loreau | From populations to ecosystems: Theoretical foundations for a new ecological synthesis[END_REF].

During my PhD, I aimed at investigating theoretically the interactions between organisms and nutrient cycles at global scales when access to nutrients is limited. I specifically focused on how autotrophs respond to variations in nutrient supply and how they affect the regulation of both accessible and inaccessible nutrient pools.

Theoretical modelling and application to the global ocean

Theoretical modelling is a useful tool to study ecological interactions at both local and global scales. Simplification of the processes represented in theoretical models allows one to better understand the functioning of the system studied, and the results can be compared with field or experimental data afterwards. In the case of interactions between organisms and nutrient cycles, experimental approaches are often unsuitable at the global scale, thus theoretical models also allow one to study the main drivers of nutrient-organism interactions (e.g. [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF]Lenton andWatson 2000a, 2000b;[START_REF] Yool | Role of diatoms in regulating the oceanâs silicon cycle[END_REF][START_REF] Bouwman | Human alteration of the global nitrogen and phosphorus soil balances for the period 1970-2050[END_REF].

Oceanic primary production represents around 50 % of total primary production [START_REF] Field | Primary production of the biosphere: Integrating terrestrial and oceanic components[END_REF], and thus oceans are a major component of global biogeochemical cycles and their interactions. Anthropogenic activities heavily impact terrestrial, atmospheric and oceanic ecosystems [START_REF] Schlesinger | Biogeochemistry: An analysis of global change[END_REF]. Land-use changes, burning of fossil fuels and the use of fertilizers also indirectly impact oceanic ecosystems through river and atmospheric transport [START_REF] Benitez-Nelson | The biogeochemical cycling of phosphorus in marine systems[END_REF][START_REF] Karl | Nutrient dynamics in the deep blue sea[END_REF][START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF]. Oceanic biogeochemical cycles are thus interesting case studies for the assessment of the ability of organisms to regulate nutrient pools when the supply of nutrients is altered. Furthermore, ocean ecosystems provide a wide range of examples to study the implications of resource access limitation in nutrient-organism interactions. The water column is often separated in two layers by a pycnocline, thus limiting the availability of nutrients for phytoplankton, which provides an example of physical resource access limitation at the global scale [START_REF] Ruardij | The impact of thermal stratification on phytoplankton and nutrient dynamics in shelf seas: a model study[END_REF][START_REF] Gruber | Warming up, turning sour, losing breath: ocean biogeochemistry under global change[END_REF]. Chemical resource access limitation also often occurs, since phytoplankton can only assimilate some chemical forms of nutrients such as iron and nitrogen [START_REF] Menge | Evolutionary tradeoffs can select against nitrogen fixation and thereby maintain nitrogen limitation[END_REF][START_REF] Boyd | The biogeochemical cycle of iron in the ocean[END_REF]. The application of models to oceanic biogeochemical cycles is also facilitated by the fact that oceanic nutrient cycles are well-studied and thus numerous measures of global parameters for the intensity of nutrient flows can be found in the literature (e.g. [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Aumont | An ecosystem model of the global ocean including Fe, Si, P colimitations[END_REF]Treguer and De La Rocha 2013).

Nitrogen and phosphorus are key nutrients for the growth of organisms, and often limit the growth of autotrophs in terrestrial, aquatic and marine ecosystems [START_REF] Vitousek | Nitrogen limitation on land and in the sea: How can it occur?[END_REF][START_REF] Sterner | Ecological stoichiometry: The biology of elements from molecules to the biosphere[END_REF][START_REF] Elser | Global analysis of nitrogen and phosphorus limitation of primary producers in freshwater, marine and terrestrial ecosystems[END_REF][START_REF] Harpole | Nutrient co-limitation of primary producer communities[END_REF]. Oceanic N distribution is controlled by N-fixation, denitrification and annamox processes [START_REF] Gruber | Global patterns of marine nitrogen fixation and denitrification[END_REF][START_REF] Brandes | New developments in the marine nitrogen cycle[END_REF][START_REF] Deutsch | Spatial coupling of nitrogen inputs and losses in the ocean[END_REF]). The oceanic cycle of N is strongly constrained by that of P and Fe, as N-fixation is in most cases limited by the availability of P and/or Fe in seawater [START_REF] Moore | Iron cycling and nutrient-limitation patterns in surface waters of the World Ocean[END_REF][START_REF] Moore | Iron availability limits the ocean nitrogen inventory stabilizing feedbacks between marine denitrification and nitrogen fixation[END_REF][START_REF] Moutin | Phosphate availability and the ultimate control of new nitrogen input by nitrogen fixation in the tropical Pacific Ocean[END_REF][START_REF] Monteiro | Biogeographical controls on the marine nitrogen fixers[END_REF]. Thus the oceanic cycles of N, P and Fe are interesting applications for the study of interactions between nutrient cycles and organisms.

Main questions

My first objective was to assess the potential for biotic regulation of the global biogeochemical cycle of nutrients (Chapter 1). I used a theoretical model based on the theory of resource access limitation, and studied the regulation efficiency of both accessible and inaccessible nutrient concentrations with respect to changes in nutrient supply. I focused on assessing the strength of the regulation of the concentration of a limiting nutrient by autotrophs and the processes that affect it.

My second objective was to better understand how interactions between nutrient cycles could alter their regulation by autotrophs (Chapter 2). First I studied the ability of autotrophs to regulate the concentration of a non-limiting nutrient with respect to changes in its supply. Then I focused on the effects of the interactions between the cycles of a limiting nutrient and that of a non-limiting nutrient on regulation of nutrient concentrations by autotrophs.

My third objective was to understand how competition between functional groups could affect regulation of nutrient cycles (Chapter 3). I focused on the regulation of Redfield ratios in the ocean, which provide a case study of how competition between two functional groups (non-fixers and N-fixers) can affect the stoichiometry of their environment. I adapted the previous model to the N and P cycles in the global ocean, and aimed at assessing the precise mechanisms that control the value of deep-water N:P ratio and the efficiency of its regulation by phytoplankton in the face of increased N and P supplies.

My fourth objective was to study the impact of changes in nutrient supplies on the growth of autotrophs at global scale (Chapter 4). I focused on the oceanic primary production.

With a model of the global oceanic cycles of nitrogen, phosphorus, and iron, I investigated which nutrient limits total oceanic primary production over long timescales, i.e. which is the ultimate limiting nutrient.

Résumé

Les cycles biogéochimiques globaux sont fortement impactés par les activités anthropiques, principalement par la modification des apports de nutriments dans les écosystèmes, que ce soit par l'utilisation d'engrais dans l'agriculture ou l'émission de composés volatiles The issue of the extent to which organisms are able to regulate their environment at large spatial and temporal scales has been debated for several decades [START_REF] Loreau | From populations to ecosystems: Theoretical foundations for a new ecological synthesis[END_REF]. The near constancy of the composition, pH and temperature of the terrestrial atmosphere over long time scales led Lovelock and Margulis (1974) to propose the Gaia hypothesis, which assumes that organisms restrict variations in environmental conditions to a habitable range through feedback mechanisms. This hypothesis has been controversial, in particular because evolution by natural selection does not necessarily promote large-scale environmental regulation (see [START_REF] Lenton | Gaia and natural selection[END_REF][START_REF] Free | Do evolution and ecology need the Gaia hypothesis?[END_REF][START_REF] Tyrrell | On Gaia: A critical investigation of the relationship between life and Earth[END_REF] for reviews). The hypothesis that water chemistry is regulated by phytoplankton in oceans provides another example of regulation by organisms of their environment at global scales. The mean N:P ratio of phytoplankton seems to be relatively constant at large scales [START_REF] Redfield | On the proportions of organic derivations in sea water and their relation to the composition of plankton[END_REF][START_REF] Redfield | The biological control of chemical factors in the environment[END_REF][START_REF] Karl | Total dissolved nitrogen and phosphorus concentrations at US-JGOFS station ALOHA: Redfield reconciliation[END_REF], although phytoplankton stoichiometry can vary depending on local conditions (Klausmeier et al. 2004a;[START_REF] Franz | Dynamics and stoichiometry of nutrients and phytoplankton in waters influenced by the oxygen minimum zone in the eastern tropical Pacific[END_REF]) and phytoplankton growth strategies [START_REF] Arrigo | Marine microorganisms and global nutrient cycles[END_REF]. [START_REF] Redfield | On the proportions of organic derivations in sea water and their relation to the composition of plankton[END_REF] highlighted the similarity between this mean N:P ratio of 16:1 and that of ocean deep waters. He hypothesized that the intracellular content of phytoplankton could be central in this pattern and that the phytoplankton could maintain it through nitrogen fixation, denitrification and recycling. Modelling studies support the hypothesis that Redfield ratios are controlled by the compensatory dynamics between nitrogen-fixing and non-fixing phytoplankton [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF]Lenton and Watson 2000a;[START_REF] Weber | Oceanic nitrogen reservoir regulated by plankton diversity and ocean circulation[END_REF]) and the diversity of metabolic N:P requirements of phytoplankton [START_REF] Weber | Oceanic nitrogen reservoir regulated by plankton diversity and ocean circulation[END_REF].

One key issue with all the hypotheses that assume regulation of global environmental conditions or resources is that organisms typically act on their environment locally, at small space and time scales. At such small scales, organisms are known to modify their environment through a wide range of processes such as resource consumption, metabolism and habitat modification, thereby creating strong feedbacks with, and possibly regulation of, their environment (Kylafis andLoreau 2008, 2011). One example is nitrogen fixers, whose metabolic activities release fixed nitrogen in the environment, thereby increasing nitrogen availability for other organisms. Feedbacks that link organisms and their environment and potentially generate environmental regulation at global scales are increasingly understood regarding the cycles of key nutrients such as oxygen, carbon and nitrogen (Lenton and Watson 2000a, 2000b[START_REF] Gruber | Warming up, turning sour, losing breath: ocean biogeochemistry under global change[END_REF][START_REF] Weber | Oceanic nitrogen reservoir regulated by plankton diversity and ocean circulation[END_REF][START_REF] Canfield | Oxygen: A four billion years history[END_REF]), but a general theoretical framework is still missing. Changes in the stocks or concentrations of inorganic nutrients can have massive effects on the physical characteristics of the global environment and on biodiversity [START_REF] Smith | Eutrophication: impacts of excess nutrient inputs on freshwater, marine, and terrestrial ecosystems[END_REF][START_REF] Diaz | Spreading dead zones and consequences for marine ecosystems[END_REF][START_REF] Smith | Eutrophication science: where do we go from here?[END_REF].

Therefore it is important to clarify the main feedback mechanisms that can contribute to the regulation of inorganic nutrient pools by organisms at global scales.

Resource consumption is the most common biotic process leading to local regulation of inorganic nutrient pools. As the resource is consumed, its abundance tends to decrease to a level where its consumption balances consumers basal metabolism and mortality, leading to top-down control of the resource by the consumers [START_REF] Loreau | From populations to ecosystems: Theoretical foundations for a new ecological synthesis[END_REF]. Top-down resource control, however, occurs only inasmuch as the resource is accessible to consumers. Resource access limitation is common, often because of spatial barriers. For instance, plants have access to soil nutrients only in the vicinity of their rooting system in terrestrial ecosystems [START_REF] Huston | Competition and coexistence: The effects of resource transport and supply rates[END_REF][START_REF] Loreau | Coexistence of multiple food chains in a heterogeneous environment: Interactions among community structure, ecosystem functioning, and nutrient dynamics[END_REF]). Another cause of resource access limitation is the chemical form of nutrients, which can be unusable by some organisms. For instance, only nitrogen fixers are able to assimilate N2, with the result that non-fixers have access only to part of the nitrogen available in their environment. The fact that organisms access only a small part of the resources available in their environment leads to an important question:

Are organisms -in particular autotrophic organisms -able to indirectly regulate nutrient pools to which they do not have direct access? Inaccessible nutrient pools are often much larger than accessible ones, and thus one may expect biotic regulation of these pools to be strongly limited.

Our goal in this work is to elucidate the ability of organisms to regulate the pools of inorganic nutrients in both accessible and inaccessible form, and thereby the biogeochemical cycles of these nutrients at large spatial scales. We first build and analyse a simple generic model of resource regulation with resource access limitation. The model describes the dynamics of a limiting inorganic nutrient in two pools, one accessible and the other inaccessible to autotrophic consumers. We then provide applications of our model to the biogeochemical cycle of three key nutrients in the ocean. The first case study is the oceanic iron cycle, in which we consider that phytoplankton can use only part of the iron in seawater [START_REF] Baker | Atmospheric and marine controls on aerosol iron solubility in seawater[END_REF][START_REF] Boyd | The biogeochemical cycle of iron in the ocean[END_REF]. Assimilable iron is scarce in several oceanic regions, where it drives phytoplankton growth [START_REF] Fung | Iron supply and demand in the upper ocean[END_REF][START_REF] Moore | Iron cycling and nutrient-limitation patterns in surface waters of the World Ocean[END_REF][START_REF] Krishnamurthy | Impacts of atmospheric nutrient inputs on marine biogeochemistry[END_REF]). Thus, it is interesting to assess the extent to which autotrophs can regulate the oceanic cycle of iron in response to increasing atmospheric deposition [START_REF] Jickells | Global iron connections between desert dust, ocean biogeochemistry, and climate[END_REF]Boyd et al. 2010). The second case study is the oceanic phosphorus cycle.

Phosphorus presents an example of physical resource access limitation, as phytoplankton can only access nutrients in the surface ocean, either because of the thermocline or because of the limited depth of the euphotic layer. The interest in studying phosphorus is that it plays a major role in limiting phytoplankton growth in the ocean [START_REF] Sañudo-Wilhelmy | Phosphorus limitation of nitrogen fixation by Trichodesmium in the central Atlantic Ocean[END_REF][START_REF] Elser | Global analysis of nitrogen and phosphorus limitation of primary producers in freshwater, marine and terrestrial ecosystems[END_REF]) and its biogeochemical cycle is strongly affected by agricultural activities [START_REF] Benitez-Nelson | The biogeochemical cycling of phosphorus in marine systems[END_REF][START_REF] Bouwman | Human alteration of the global nitrogen and phosphorus soil balances for the period 1970-2050[END_REF]. The last example is the oceanic silicon cycle.

Just as dissolved phosphorus, only part of silicic acid is accessible to phytoplankton in the ocean due to physical barriers. Silicic acid is a key nutrient in the ocean as it is used by diatoms and influences their distribution [START_REF] Hildebrand | Silicon Metabolism in Diatoms: Implications for Growth[END_REF][START_REF] Yool | Role of diatoms in regulating the oceanâs silicon cycle[END_REF][START_REF] Sarthou | Growth physiology and fate of diatoms in the ocean: a review[END_REF]. This case study is of special interest since it provides an example of the regulation of nutrient pools in response to a decrease in nutrient supply by anthropogenic activities [START_REF] Laruelle | Anthropogenic perturbations of the silicon cycle at the global scale: Key role of the land-ocean transition[END_REF][START_REF] Tréguer | The world ocean silica cycle[END_REF], in contrast to phosphorus and iron, the supply of which increases.

Methods

Our generic model explores the constraints on biotic regulation that arise from the presence of inaccessible nutrient pools [START_REF] Huston | Competition and coexistence: The effects of resource transport and supply rates[END_REF][START_REF] Loreau | Coexistence of multiple food chains in a heterogeneous environment: Interactions among community structure, ecosystem functioning, and nutrient dynamics[END_REF][START_REF] Loreau | From populations to ecosystems: Theoretical foundations for a new ecological synthesis[END_REF]) in global biogeochemical cycles (Figure 1.1, Table 1.1), either because of spatial barriers or because of non-assimilable chemical forms. In this model, we assume that a single inorganic nutrient limits the growth of autotrophic organisms that consume it. The inorganic nutrient occurs in two distinct pools, one that is accessible to organisms, and the other that is inaccessible to them. α represents the fraction of the total volume of the system (i.e. the sum of the volumes of both accessible and inaccessible pools, noted V a + V i ) that is accessible to organisms. In the case of chemical limitation, we consider that the environment is homogeneous, such that the accessible and inaccessible forms of the limiting nutrient occur in the same volume (i.e.

V a = V i ), and thus α = V a /(V a + V i ) = 0.5. N a and N i are nutrient concentrations in the accessible and inaccessible pools, respectively. The two nutrient pools are connected by passive flows governed by the physics or chemistry of the system. We distinguish between the transfer rate from the accessible to the inaccessible pool, k a , and that from the inaccessible to the accessible pool, k i . Both pools have nutrient inflows and outflows from and to the external world. S a and S i refer to the nutrient supply to the accessible and inaccessible pools, respectively, and q a and q i refer to the nutrient turnover rates in the accessible and inaccessible pools, respectively.

Organisms, with concentration B, have a traditional resource-dependent functional response to the concentration of the accessible inorganic nutrient, g(N a ), leading to top-down control at equilibrium. They recycle part of the inorganic nutrient in both accessible and inaccessible forms. m is the turnover rate of nutrient in organisms, λ is the fraction of dead organic matter that is lost from the system, and rec a refers to the fraction of recycling that occurs in the accessible pool. We first use mass balance to obtain a model that tracks nutrient masses. By dividing nutrient mass by the volume of the pool concerned, we then obtain the following model that tracks nutrient concentrations (Figure 1.1):

dN a dt = S a -(k a + q a )N a + 1 -α α k i N i + [mrec a (1 -λ) -g(N a )] B dN i dt = S i + α 1 -α k a N a -(k i + q i )N i + α 1 -α m(1 -rec a )(1 -λ)B dB dt = [g(N a ) -m] B (1.1)
A simple measure of the efficiency with which organisms regulate nutrient concentrations against variations in nutrient supply is one minus the elasticity of the equilibrium nutrient concentration in pool x with respect to nutrient supply to pool y:

ρ x,y = 1 - S y N x * ∂N x ∂S y (1.2)
ρ x,y is defined as the regulation coefficient of the nutrient concentration in pool x with respect to changes in the nutrient supply to pool y. When ρ x,y = 0, there is no regulation (i.e. the proportional variation in nutrient concentration x is equal to that in nutrient supply y). At the other extreme, when ρ x,y = 1, there is perfect regulation (i.e. there is no variation in nutrient concentration x as a result of that in nutrient supply y). Note we calculate the regulation coefficient for the nutrient concentration at equilibrium; thus perfect regulation does not exclude variations in the nutrient concentration during transient dynamics. When 0 < ρ x,y < 1, regulation is partial. Note that biota can sometimes over-regulate the nutrient concentration in pool x, in which case ρ x,y > 1. Some cases where ρ x,y < 0 can also occur; regulation is then negative, i.e. organisms amplify variations in nutrient supply.

The impact of each parameter on each regulation coefficient was determined by the partial derivative of the regulation coefficient with respect to that parameter. When the sign of the partial derivative was not obvious but was constant over the whole interval of parameter values, we determined the values of the regulation coefficients when the parameter is set to

(1) its maximum and (2) its minimum, to emphasize the general impact of the parameter studied on the regulation coefficients (Table 1.2).

To illustrate the general predictions of our model, we apply it to the biogeochemical cycles of iron, phosphorus and silicon in the global ocean. Soluble iron is thus accessible to organisms, while particulate iron is inaccessible. In the case of chemical limitation, α = V a /(V a + V i ) = 0.5. Recycling produces soluble iron, thus in this application rec a = 1. The examples of the phosphorus and silicon cycles correspond to physical limitations. Note that in the phosphorus cycle, there is no supply of dissolved phosphorus to the deep ocean (i.e.

S i = 0, Benitez-Nelson 2000). Thus in this special case, only the regulation of nutrient pools with respect to changes in the supply to the surface ocean is studied.

In numerical simulations, the functional response of phytoplankton to an accessible nutrient is modelled with a Michaelis-Menten function:

g(N a ) = µN a N a + N H (1.3)
where µ is the maximal growth rate of phytoplankton and N H is the half-saturation constant for a nutrient in accessible form.

To perform numerical simulations, we chose parameter values within the range of values found in the literature (Supplementary Tables S1.1, S1.2 and S1.3) in order to be as realistic as possible. We either increased or decreased nutrient supply by 50 % after one third of the simulation time to assess the strength of the regulation of nutrient pools in the present ocean. We increased iron and phosphorus supplies, while we decreased the supply of silicic acid, in agreement with current trends due to anthropogenic activities [START_REF] Bouwman | Human alteration of the global nitrogen and phosphorus soil balances for the period 1970-2050[END_REF][START_REF] Laruelle | Anthropogenic perturbations of the silicon cycle at the global scale: Key role of the land-ocean transition[END_REF]Boyd et al. 2010;[START_REF] Tréguer | The world ocean silica cycle[END_REF].

Results

Model (1.1) has two equilibria, one in the absence and the other in the presence of organisms.

An analysis of the flows between nutrient pools helps to better understand the results (Figure 1.2). The equilibrium concentrations in the absence of organisms (denoted by asuperscript) are:

N - a = S a α(k i + q i ) + k i S i (1 -α) α [q a (k i + q i ) + k a q i ] N - i = S i (1 -α) + k a αN - a (1 -α)(k i + q i ) B -= 0 (1.4)
The equilibrium concentrations in the presence of organisms (denoted by a + superscript) are:

N + a = g -1 (m) N + i = -αk a N + a + S i (1 -α) [1 -rec a (1 -λ)] + α(1 -rec a )(1 -λ) -S a + N + a (k a + q a ) (1 -α) [k i (1 -rec a )(1 -λ) -(k i + q i ) [1 -rec a (1 -λ)]] B + = -S a α(k i + q i ) -k i S i (1 -α) + αN + a [q a (k i + q i ) + k a q i ] αm [k i (1 -rec a )(1 -λ) -(k i + q i ) [1 -rec a (1 -λ)]] = (N + a -N - a ) [q a (k i + q i ) + k a q i ] m [k i (1 -rec a )(1 -λ) -(k i + q i ) [1 -rec a (1 -λ)]] (1.5)
where g -1 refers to the inverse function of g.

Note that the total amount of nutrient in the system is:

N tot = αN a + (1 -α)N i + αB (1.6)
In the presence of organisms, any variation in the supply of nutrient in accessible form is entirely absorbed by organisms because of their top-down control on the accessible nutrient pool, and thus ρ a,a = 1 (Figures 1.3a, 1.4a and 1.5a). Perfect regulation is explained by a strong negative feedback loop between phytoplankton and the accessible pool (path 2-3 in

ρ i,a = -αk a N + a + S i (1 -α) [1 -rec a (1 -λ)] + αN + a (k a + q a )(1 -rec a )(1 -λ) -αk a N + a + S i (1 -α) [1 -rec a (1 -λ)] + α(1 -rec a )(1 -λ) -S a + N + a (k a + q a ) (1.7)
Note that regulation of the inaccessible nutrient concentration with respect to changes in the accessible supply is perfect when there is no recycling to the inaccessible pool (i.e.

rec a = 1), as is the case with the iron cycle in the ocean (Figure 1.3a). In the numerical simulations of the oceanic cycles of silicon and phosphorus, part of the dead organic matter is recycled in the deep ocean (i.e. rec a < 1). Changes in the supply of phosphorus and silicic acid to the surface ocean affect the growth and concentration of autotrophic organisms (Figures 1.4b and 1.5b) and then impact the strength of the recycling flow of organic matter to the deep ocean. Thus regulation of the deep-ocean nutrient concentration with respect to a change in the supply to the surface ocean is partial (ρ i,a = 0.44 and 0.18 in Figures 1.4a and 1.5a, respectively). Recycling to the inaccessible pool has a negative impact on regulation of changes in the accessible nutrient supply (Table 1.2), because any change in this supply alters the growth of organisms and thus the intensity of the recycling flow to the inaccessible pool.

The efficiency of this regulation is also higher when the relative volume of the accessible pool, α, and the transfer rate from the accessible pool to the inaccessible one, k a , are larger (Table 1.2). As changes in nutrient supply are totally absorbed in the accessible pool, variations of these two parameters in the direction indicated contribute to the dilution of changes in biotic flows to the inaccessible pool. Regulation of the inaccessible nutrient concentration with respect to changes in accessible nutrient supply is also more efficient when the turnover rates q a and q i are smaller and larger, respectively (Table 1.2).

There is also perfect regulation of the accessible nutrient concentration with respect to changes in the supply of the nutrient in inaccessible form because of top-down control (negative feedback loop, path 2-3 in Figure 1.2c), and thus ρ a,i = 1 (Figures 1.3c and 1.4c). The concentration of the nutrient in inaccessible form is either partially or negatively regulated with respect to changes its supply (ρ i,i = 2.3.10 -3 and 0.78 in Figures 1.3c and 1.4c, respectively) because there is a positive relationship between the supply and the concentration of the nutrient in inaccessible form (path 1-4-2-5 in Figure 1.2d):

ρ i,i = -αk a N + a [1 -rec a (1 -λ)] + α(1 -rec a )(1 -λ) -S a + N + a (k a + q a ) -αk a N + a + S i (1 -α) [1 -rec a (1 -λ)] + α(1 -rec a )(1 -λ) -S a + N + a (k a + q a ) (1.8)
The efficiency of this regulation is higher when the relative volume of the accessible pool, α, is larger, when the transfer rate from the accessible pool to the inaccessible one, k a , is larger, and when the turnover rates q a and q i are larger and smaller, respectively (Table 1.2). The effect of these four parameters is intuitive as variations of these parameters in the direction indicated contribute to increase the impact of the biotic control of the accessible nutrient relative to that of the independent physical and chemical processes that affect inaccessible nutrient dynamics. We might intuitively expect recycling to the inaccessible pool to have a positive impact on the regulation of changes in the inaccessible nutrient supply, because it increases the control of organisms on the inaccessible pool. However, this occurs only under particular conditions (Table 1.2).

For both regulation coefficients, note that the regulation efficiency of the inaccessible nutrient concentration is higher when the equilibrium nutrient concentration in the accessible pool, N + a , is larger (Table 1.2). This effect is somewhat counterintuitive as we might expect intuitively that the stronger the top-down biotic control on the accessible nutrient through resource depletion, the stronger the indirect biotic regulation of the inaccessible nutrient.

Thus our generic model of a single nutrient cycle predicts that organisms have only a limited ability to regulate nutrient pools that are not directly accessible to them. Since inaccessible pools are often larger than accessible ones and external forcing often occurs through changes in nutrient supply, regulation of the system as a whole by the biota is expected to be limited.

In the cases of the current oceanic cycles of iron, silicon and phosphorus, numerical simulations performed with realistic parameter values show that accessible nutrient pools are perfectly regulated. In contrast, regulation of inaccessible nutrient pools is in most cases partial, and once even non-existent (ρ i,i = 2.3.10 -3 , Figure 1

Discussion

The importance of physical and chemical processes in the regulation of biogeochemical cycles

Our model highlights the importance of passive physical and chemical processes that govern transfers between accessible and inaccessible nutrient pools, in the regulation of biogeochemical cycles. It is well known that nutrient dynamics in ecosystems are strongly impacted by physical processes (e.g. [START_REF] Karl | Nutrient dynamics in the deep blue sea[END_REF][START_REF] Boyd | The biogeochemical cycle of iron in the ocean[END_REF][START_REF] Franz | Dynamics and stoichiometry of nutrients and phytoplankton in waters influenced by the oxygen minimum zone in the eastern tropical Pacific[END_REF] for marine ecosystems) and chemical reactions (e.g. [START_REF] Falkowski | The global carbon cycle: a test of our knowledge of Earth as a system[END_REF][START_REF] Karl | Nutrient dynamics in the deep blue sea[END_REF]). This phenomenon is revealed in our model by the importance of parameter k a in the regulation coefficients. Thus, physical and chemical processes that govern transfer rates between pools can have a strong influence on the regulation of inaccessible limiting nutrients.

The intensity of chemical and physical flows between the accessible and inaccessible nutrient pools is highly affected by the characteristics of the environment. An example is water column stratification in freshwater and marine systems, which reduces the intensity of physical flows between water layers (i.e. downwellings and upwellings, governed by parameters k a and k i , respectively), as well as the relative volume of the upper accessible layer (α) [START_REF] Riebesell | Sensitivities of marine carbon fluxes to ocean change[END_REF]. Global climate change is expected to increase stratification through increased sea surface temperature and decreased sea surface salinity [START_REF] Gruber | Warming up, turning sour, losing breath: ocean biogeochemistry under global change[END_REF][START_REF] Rees | Pressures on the marine environment and the changing climate of ocean biogeochemistry[END_REF]. By decreasing parameters k a and α, increased water column stratification could strongly reduce the potential for biotic regulation of inaccessible nutrients in lakes and oceans.

At the same time, the decrease in the depth of the upper layer (related to the parameter α) is likely to intensify the recycling flow to the inaccessible nutrient pool because sinking particles will take less time to reach the deep inaccessible layer. As recycling of organic matter by microorganisms consumes oxygen, an increase in oxygen depletion in deep waters is likely to occur [START_REF] Diaz | Spreading dead zones and consequences for marine ecosystems[END_REF]. The increasing eutrophication of deep waters and the spreading of oxygen minimum zones may then have dramatic consequences on freshwater and marine food webs [START_REF] Diaz | Spreading dead zones and consequences for marine ecosystems[END_REF][START_REF] Stramma | Expanding oxygen-minimum zones in the tropical oceans[END_REF][START_REF] Doney | The growing human footprint on coastal and open-ocean biogeochemistry[END_REF].

Biotic regulation on global scales

The potential for biotic regulation of the Earth system has been hotly debated, whether it relates to the Gaia hypothesis (Lovelock and Margulis 1974a;[START_REF] Lenton | Gaia and natural selection[END_REF][START_REF] Free | Do evolution and ecology need the Gaia hypothesis?[END_REF][START_REF] Tyrrell | On Gaia: A critical investigation of the relationship between life and Earth[END_REF] or to Redfield ratios in the ocean [START_REF] Redfield | On the proportions of organic derivations in sea water and their relation to the composition of plankton[END_REF][START_REF] Redfield | The biological control of chemical factors in the environment[END_REF]. Our simple generic model of resource access limitation sheds new light on this long-standing debate. Strong biotic regulation of global biogeochemical cycles implicitly assumes that nutrients are accessible to organisms. But, in reality, resource access limitation is pervasive; massive amounts of nutrients globally are inaccessible to organisms because of physical or chemical barriers. Our model and its applications to the oceanic cycles of iron, silicon and phosphorus predict, as expected intuitively, that autotrophic organisms should be able to strongly regulate the concentrations of limiting nutrients in accessible nutrient pools because of their top-down control on these pools. But our model also predicts that any variation in nutrient supply should be only partially or negatively regulated in inaccessible nutrient pools because organisms have only indirect access to these pools. In the application to the iron cycle however, the absence of recycling in the inaccessible pool lead variations in the supply of accessible, dissolved iron to be entirely absorbed by organisms, and thus regulation of the concentration of inaccessible iron with respect to a change in the supply of accessible iron is perfect (Figure 1.3a).

Our model further shows that either autotrophic organisms have a strong impact on accessible pools and exert weak regulation on the rest of the system, or they have a weak impact on accessible pools and exert moderate regulation on the rest of the system. Thus, strong biotic regulation of global biogeochemical cycles or of the Earth system as a whole seems unlikely because of the inaccessibility to organisms of large amounts of their resources at the global scale. Our predictions focus on the regulation of biogeochemical cycles by autotrophic organisms. Other biotic and abiotic processes could reinforce regulation of biogeochemical cycles by creating negative feedbacks, such as fires in the oxygen cycle (e.g. Lenton and Watson 2000b;[START_REF] Lenton | The role of land plants, phosphorus weathering and fire in the rise and regulation of atmospheric oxygen[END_REF]) and microbial populations that control the denitrification and the anammox processes in the nitrogen cycle [START_REF] Seitzinger | Denitrification across landscapes and waterscapes: a synthesis[END_REF][START_REF] Brandes | New developments in the marine nitrogen cycle[END_REF], and are likely to play an important role in the regulation of global biogeochemical cycles [START_REF] Karl | Nutrient dynamics in the deep blue sea[END_REF]. However, some other processes create positive feedbacks that can destabilize the system, for example the positive climate-CO2 feedback that occurs through thermal stratification and decreased solubility of CO2 in seawater (e.g. [START_REF] Denman | Couplings between changes in the climate system and biogeochemistry[END_REF]).

We assumed top-down control of inorganic nutrients by a single trophic level in our model.

The addition of a second trophic level would lead the ecosystem to be controlled by herbivores instead of autotrophs. Consequently, we may expect decreased regulation of both accessible and inaccessible nutrient pools. Numerical simulations, however, suggest that regulation is little affected by the addition of herbivores in our generic model, except for the regulation of the accessible nutrient concentration with respect to changes in the nutrient supply to the inaccessible pool (results not shown). Although the addition of a second trophic level does have a quantitative impact on nutrient regulation, numerical simulations suggest that it does not qualitatively alter our predictions regarding the regulation of the inaccessible nutrient concentration.

Possible applications of our model

We presented applications of our model to the cycles of iron, silicon and phosphorus in the global ocean. Numerical simulations performed with realistic parameter values illustrate the fact that global biogeochemical cycles are not efficiently regulated by autotrophic organisms (Figures 1.3, 1.4 and 1.5). However, the oceanic cycles of the three key nutrients do not seem to be regulated in the same range. Phytoplankton appears to be completely unable to regulate the concentration of particulate iron with respect to an increase in its supply (ρ i,i = 2.3.10 -3 , Figure 1.3c), while the silicon cycle appears to be quite efficiently regulated with respect to changes in its supply to the deep ocean (ρ i,i = 0.78, Figure 1.4c). This difference could be due to the absence of recycling of organic matter to the inaccessible pool in the iron cycle, which strongly decreases the ability of phytoplankton to impact and possibly regulate the inaccessible nutrient pool.

In the application to the Si cycle, our estimate of the Si standing stock of autotrophs is 60 µmol.m -3 , in agreement with experimental data of 17-217 µmol.m -3 in the upper 120 m of the ocean [START_REF] Adjou | Particulate silica and Si recycling in the surface waters of the Eastern Equatorial Pacific[END_REF][START_REF] Tréguer | The world ocean silica cycle[END_REF]. Likewise, the simulated stock of P in the organisms represents 1.74 % of the total stock of P in the surface ocean, which is comparable with values of 0.03-6.45 % found in the literature [START_REF] Loh | Distribution, partitioning and fluxes of dissolved and particulate organic C, N and P in the eastern North Pacific and Southern Oceans[END_REF]. In the application to the Fe cycle, organisms contain more than 9 times more Fe than dissolved Fe in the water column in our numerical simulations. A possible explanation for this high proportion is that dissolved Fe is scarce in the ocean (e.g. [START_REF] Baker | Atmospheric and marine controls on aerosol iron solubility in seawater[END_REF][START_REF] Boyd | The biogeochemical cycle of iron in the ocean[END_REF]. The simulated concentrations of silicic acid and phosphorus in the surface and deep ocean are in the same range as field measurements, although simulated deep-water concentrations are a little lower than experimental data, for both Si and P [START_REF] Loh | Distribution, partitioning and fluxes of dissolved and particulate organic C, N and P in the eastern North Pacific and Southern Oceans[END_REF][START_REF] Tréguer | The world ocean silica cycle[END_REF]. The concentrations of dissolved phosphorus are also consistent with field measurements [START_REF] Loh | Distribution, partitioning and fluxes of dissolved and particulate organic C, N and P in the eastern North Pacific and Southern Oceans[END_REF]. Numerical simulations also predict a dissolved Fe concentration of 51.4 nmol.m -3 , a high concentration compared to the measured concentration of 1-2 nmol.m -3 in surface waters (e.g. Boyd and Ellwood 2010).

However, these measures are not contradictory, as our estimated concentration of dissolved Fe includes dissolved Fe in the whole water column, and the dissolved Fe concentration is higher in deep waters than in surface waters. Thus the applications of our simple model predict relatively realistic concentrations in both biotic and abiotic pools.

We applied our theory to the example of the biogeochemical cycles of iron, phosphorus and silicon in the ocean, but this theory could also help to study the ability of organisms to regulate nutrient pools at large spatial scales in other systems where access to resources is limited. For instance, oceanic and freshwater systems are often separated in two layers, either by a thermocline or because of the limited depth of the euphotic layer. These systems are heavily impacted by anthropogenic activities, in particular through water eutrophication [START_REF] Smith | Eutrophication: impacts of excess nutrient inputs on freshwater, marine, and terrestrial ecosystems[END_REF][START_REF] Carpenter | Eutrophication of aquatic ecosystems: Bistability and soil phosphorus[END_REF][START_REF] Smith | Eutrophication science: where do we go from here?[END_REF]. Our generic model could be applied to either oceanic or freshwater systems to assess the ability of organisms to regulate the concentration of nutrients such as N, P and Fe that often limit primary production and are increasingly supplied to freshwater (and then marine) ecosystems by human activities.

Another possible application is nutrient dynamics in soils inside and outside plants rooting system and the ability of plants to mitigate the increased supply of nutrients used in fertilizers, such as N and P [START_REF] Bouwman | Human alteration of the global nitrogen and phosphorus soil balances for the period 1970-2050[END_REF][START_REF] Vitousek | Nutrient imbalances in agricultural development[END_REF]). However, the application of our model to the N cycle would require inclusion of a second type of autotrophic organisms, i.e. nitrogen fixers (e.g. [START_REF] Vitousek | Ecosystem constraints to symbiotic nitrogen fixers: A simple model and its implications[END_REF][START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF]. Our model could also be extended to biogeochemical cycles of two nutrients consumed by two groups of autotrophic organisms to assess how competition between two functional groups of autotrophs impacts regulation of nutrient concentrations as well as nutrient ratios. Such a stoichiometric extension could be used to address the issue of the regulation of Redfield ratios in the ocean and determine how organisms can exert a strong control on nutrient ratios in their environment when they are unable to efficiently regulate nutrient concentrations.

Thus, the theory and model of resource access limitation we have started to develop here offer promising tools to resolve long-standing issues and debates over the potential for biotic regulation of various components of the Earth system. Transfer rate from the accessible to the inaccessible pool (k a ) + + Fraction of the system and that is accessible to organisms (α) + + Nutrient turnover rate in the accessible pool (q a ) -+ Nutrient turnover rate in the inaccessible pool (q i ) + -

Figures and Tables

Accessible nutrient concentration at equilibrium (N + a ) + +

Fraction of recycling that occurs in the accessible pool + + when S a -(k a + q a )N + a < (rec a )

-when S a -(k a + q a )N + a >

Fraction of organic matter that is not recycled (λ) + + when S a -(k a + q a )N + a < -when S a -(k a + q a )N + a > ρ i,a is the regulation coefficient of the inaccessible nutrient pool with respect to changes in the accessible nutrient supply. ρ i,i is the regulation coefficient of the inaccessible nutrient pool with respect to changes in the inaccessible nutrient supply. The + and -symbols denote a positive and a negative impact of the parameter on the regulation coefficient, respectively.

S i 9x10 -2 9x10 -2 (Fung et al. 2000) * q a 0
Outflows by scavenging are represented by the parameter k a q i 10 -3

When units in the literature were different from those used here, we used the following assumption to convert them: * The volume of the ocean is 1.346x10 18 m 3 . q a 0 q i 2.53x10 -5 2.53x10 -5 (Treguer and De La Rocha 2013) *

When units in the literature were different from those used here, we used the following assumption to convert them: * The volume of the ocean is 1.346x10 18 m 3 . q a 2.4x10 -5 2.4x10 -5 [START_REF] Slomp | The global marine phosphorus cycle: sensitivity to oceanic circulation[END_REF] q i 2.4x10 -5 2.4x10 -5 [START_REF] Slomp | The global marine phosphorus cycle: sensitivity to oceanic circulation[END_REF] When units in the literature were different from those used here, we used the following assumption to convert them: * The volume of the ocean is 1.346x10 18 m 3 .

Connecting statements

In Chapter 1, we focused on biotic regulation of the concentrations of a limiting nutrient with respect to changes in its supply when the access to nutrients is limited. Our results

suggest that autotrophs cannot efficiently regulate nutrient pools in the whole system due to the inaccessibility of most of the resources. However, limitation conditions of autotrophs are likely to affect the ability of autotrophs to regulate nutrient concentrations in their global environment. Coupling of biogeochemical cycles may also alter the concentration of nutrients in the environment, thereby affecting the efficiency of biotic regulation of global nutrient cycles. In Chapter 2, we will thus focus on the extent to which autotrophs can regulate the concentration of a non-limiting nutrient, and how interactions between the cycle of a limiting nutrient and that of a non-limiting nutrient affect biotic regulation of global biogeochemical cycles.

Chapter 2

Biotic regulation of non-limiting nutrient pools and coupling of nutrient cycles

Anne-Sophie Auguères

Introduction

Biogeochemical cycles are heavily altered by anthropogenic activities at global scales due to climate change, rising atmospheric carbon dioxide and excess nutrient inputs [START_REF] Denman | Couplings between changes in the climate system and biogeochemistry[END_REF][START_REF] Canadell | Interactions of the carbon cycle, human activity, and the climate system: a research portfolio[END_REF][START_REF] Doney | The growing human footprint on coastal and open-ocean biogeochemistry[END_REF][START_REF] Ciais | Carbon and other biogeochemical cycles[END_REF]. Supplies of key nutrients such as carbon, nitrogen, phosphorus and oxygen to terrestrial and marine biogeochemical cycles are heavily affected by agricultural activities, land-use change and burning of fossil fuels [START_REF] Seitzinger | Sources and delivery of carbon, nitrogen, and phosphorus to the coastal zone: An overview of Global Nutrient Export from Watersheds (NEWS) models and their application[END_REF][START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF][START_REF] Bouwman | Human alteration of the global nitrogen and phosphorus soil balances for the period 1970-2050[END_REF]. Given these massive alterations, it is critical to assess the extent to which biotic and abiotic processes can lead to regulation of biogeochemical cycles at global scales.

Biotic regulation of the Earth system is the subject of a long-standing debate, especially concerning the Gaia hypothesis, which assumes that organisms maintain environmental conditions in a habitable range through self-regulating feedback mechanisms (e.g. Lovelock andMargulis 1974a, 1974b;[START_REF] Margulis | Biological modulation of the Earthâs atmosphere[END_REF]. By modifying their environment through resource consumption, metabolism and habitat modification, organisms create strong feedbacks with their local environment (Kylafis andLoreau 2008, 2011). However, theses processes do not necessarily result in regulation of the global environment because resource access is generally limited in space due to physical or chemical barriers (e.g. [START_REF] Ruardij | The impact of thermal stratification on phytoplankton and nutrient dynamics in shelf seas: a model study[END_REF][START_REF] Ostertag | Effects of nitrogen and phosphorus availability on fine-root dynamics in Hawaiian Montane forests[END_REF][START_REF] Menge | Evolutionary tradeoffs can select against nitrogen fixation and thereby maintain nitrogen limitation[END_REF][START_REF] Vitousek | Terrestrial phosphorus limitation: mechanisms, implications, and nitrogen-phosphorus interactions[END_REF]. For instance, in marine and other aquatic systems, physical resource access limitation is usually due to the presence of a pycnocline because of the warming of surface waters when solar radiation is high and vertical exchanges in the water column are low [START_REF] Vallis | Large-Scale Circulation and Production of Stratification: Effects of Wind, Geometry, and Diffusion[END_REF]. As photosynthetic activity depletes nutrients in the surface layer and the barrier of density limits water exchanges with deep waters [START_REF] Falkowski | Mix and match: how climate selects phytoplankton[END_REF], most of the nutrients in the water column are inaccessible to phytoplankton.

Interactions between the geosphere, atmosphere and biosphere result in the coupling of biogeochemical cycles. For example, autotrophs, which use light to assimilate carbon dioxide and inorganic nutrients simultaneously, create a strong coupling between the biogeochemical cycles of key elements such as carbon, nitrogen and phosphorus as well as between these cycles and the global climate [START_REF] Falkowski | The global carbon cycle: a test of our knowledge of Earth as a system[END_REF][START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF]. Energy and food production releases nitrous oxides and ammonia, which spread in the atmosphere and are deposited on the ground or in the water, thereby increasing the growth of plants or phytoplankton and their uptake of atmospheric carbon dioxide [START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF].

Redfield ratios in the ocean provide one potential example of biotic regulation of the global environment that implies interactions between nutrient cycles. Analysis of the composition of phytoplankton cells shows a mean N:P ratio of 16N:1P [START_REF] Redfield | On the proportions of organic derivations in sea water and their relation to the composition of plankton[END_REF][START_REF] Fleming | The composition of plankton and units for reporting populations and production[END_REF], similar to the N:P ratio of 15N:1P in deep waters obtained through the analysis of seawater samples [START_REF] Redfield | On the proportions of organic derivations in sea water and their relation to the composition of plankton[END_REF][START_REF] Redfield | The biological control of chemical factors in the environment[END_REF]. The N:P ratio of deep waters can differ from the Redfield ratio due to changes in the N:P ratio of the material that is supplied to the ocean and in microbial activity, e.g. anammox (i.e. microbial process of anaerobic ammonium oxidation which releases N 2 ), denitrification and nitrogen fixation [START_REF] Gruber | Global patterns of marine nitrogen fixation and denitrification[END_REF][START_REF] Karl | A sea of change: Biogeochemical variability in the North Pacific Subtropical Gyre[END_REF][START_REF] Karl | Building the long-term picture: The U.S. JGOFS time-series programs[END_REF][START_REF] Arrigo | Marine microorganisms and global nutrient cycles[END_REF]. The diversity of phytoplankton communities and their spatial distribution can also create regional deviations of the Redfield ratio in deep waters (Weber andDeutsch 2010, 2012). However, the deep-water N:P ratio seems to be almost constant over space and time, which suggests that biotic processes such as nitrogen fixation and denitrification could control the proportions of N and P in seawater [START_REF] Redfield | The biological control of chemical factors in the environment[END_REF][START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Lenton | Biotic stoichiometric controls on the deep ocean N:P ratio[END_REF]Weber andDeutsch 2010, 2012).

Our goal in this work is to elucidate the ability of organisms to regulate the pools of non-limiting nutrients in both accessible and inaccessible form at large spatial and temporal scales, as well as the interactions between the cycles of a limiting nutrient and a non-limiting one. We first develop and analyse a stoichiometric model of resource regulation with resource access limitation. The model describes the dynamics of a population of autotrophs and two inorganic nutrients -one of which is limiting and the other is non-limiting for the growth of autotrophs -that occur in two pools, one accessible and the other inaccessible to autotrophs.

We then analyse the potential for biotic regulation of the concentrations of both nutrients as well as their ratio with respect to changes in their supply.

Model description

We extend a previous model of resource regulation with resource access limitation (Auguères and Loreau 2015a) to the biogeochemical cycles of two nutrients. In this model, nutrients occur in two pools, one that is accessible to autotrophs, and the other that is inaccessible to them, either because of physical or chemical barriers. N a,1 and N a,2 are the concentrations of nutrients 1 and 2, respectively, in the accessible pool. N i,1 and N i,2 are their concentrations in the inaccessible pool. Autotrophs, whose concentration in the accessible pool is B, consume nutrients in that pool. To differentiate the characteristics of the two nutrient cycles, we add a subscript corresponding to the nutrient considered (i.e. 1 or 2) to all the variables and parameters described in Auguères and Loreau (2015a). Model parameters are described in Table 1. The only parameter that is specific to the present stoichiometric extension of the model is the stoichiometric ratio of autotrophs (R), i.e. the ratio of nutrient 2 to nutrient 1 in autotrophs. For the sake of simplicity, this stoichiometric ratio is supposed to be constant.

The fraction α of the total volume of the system (i.e. the sum of the volumes of both accessible and inaccessible pools, noted V a + V i ) that is accessible to organisms is supposed to be the same for both nutrients. This assumption, which we make for the sake of simplicity, should hold in the case of physical limitation, where physical barriers usually constrain the accessibility of all the nutrients in the same way. In the case of chemical limitation, the accessible and inaccessible forms of each nutrient occur in the same volume (i.e. V a = V i ),

and thus α = V a /(V a + V i ) = 0.5 for both nutrients.

The principle of mass balance is used to build a model that describes nutrient masses in each pool. By dividing nutrient mass by the volume of the pool concerned, we then obtain a model in terms of nutrient concentrations (Figure 2.1):

dN a,1 dt = S a,1 -(k a,1 + q a,1 )N a,1 + 1 -α α k i,1 N i,1 + (mrec 1 (1 -λ 1 ) -G) B dN a,2 dt = S a,2 -(k a,2 + q a,2 )N a,2 + 1 -α α k i,2 N i,2 + (mrec 2 (1 -λ 2 ) -G) RB dN i,1 dt = S i,1 + α 1 -α k a,1 N a,1 -(k i,1 + q i,1 )N i,1 + α 1 -α (1 -rec 1 )(1 -λ 1 )mB dN i,2 dt = S i,2 + α 1 -α k a,2 N a,2 -(k i,2 + q i,2 )N i,2 + α 1 -α (1 -rec 2 )(1 -λ 2 )mRB dB dt = [g(N a ) -m] B (2.1)
We assume that Liebig's (1842) law of the minimum governs the growth of organisms (G):

G = min (g 1 (N a,1 ), g 2 (N a,2 )) (2.2)
where g 1 and g 2 are the functional responses of autotrophs to the concentration of N a,1

and N a,2 , respectively.

We measure the strength of the regulation of the concentrations of either nutrient with respect to changes in the supply of that nutrient by calculating the regulation coefficient ρ jx,jy as one minus the elasticity of the equilibrium nutrient j concentration in pool x with respect to its supply to pool y (Auguères and Loreau 2015a):

ρ jx,jy = 1 - S j,y N j,x * ∂N j,x ∂S j,y (2.3)
ρ jx,jy is defined as the regulation coefficient of the concentration of nutrient j in pool

x with respect to changes its supply to pool y. When ρ jx,jy = 0, there is no regulation.

At the other extreme, when ρ jx,jy = 1, regulation is perfect. Note that we calculate the regulation coefficient for the nutrient concentration at equilibrium; thus perfect regulation does not exclude variations in the nutrient concentration during transient dynamics. When 0 < ρ jx,jy < 1, regulation is partial. Autotrophs can sometimes over-regulate the nutrient i concentration in pool x, in which case ρ jx,jy > 1. Some cases where ρ jx,jy < 0 can also occur; regulation is then negative, i.e. organisms amplify variations in nutrient supply.

We also quantify the effect of changes in the supply of one nutrient on the concentrations of the other using the following equation:

ǫ kx,jy = S j,y N k,x * ∂N k,x ∂S j,y (2.4)
ǫ kx,jy measures the effect of the supply of nutrient N j in pool y on the concentration of nutrient N k in pool x. This effect is positive when the supply of N j in pool y and the concentration of N k in pool x vary in the same direction, negative when they vary in opposite directions, and zero when the supply of N j in pool y does not affect the concentration of

N k , x.
We further quantify the strength of the regulation of the ratio between the two nutrients using the same principle as in equation (2.4). In this case, however, the elasticity of the ratio will change sign depending on whether the nutrient whose supply changes is in the numerator or in the denominator of the ratio. Therefore, to keep signs consistent, we calculate the regulation coefficient of N 1,x : N 2,x ratio when the supply of N 1 is modified in either the accessible or the inaccessible pool, and the regulation coefficient of N 2,x : N 1,x ratio when the supply of N 2 was modified. We obtain the following regulation coefficients for the nutrient ratio in pool x:

ρ (N 1 :N 2 )x,1y = ρ 1x,1y + ǫ 2x,1y ρ (N 2 :N 1 )x,2y = ρ 2x,2y + ǫ 1x,2y
(2.5)

Results

Model (2.1) has three equilibria, depending on the presence or absence of organisms and the nutrient that limits their growth. In the absence of autotrophs, equilibrium concentrations (denoted by asuperscript) are:

N - a,1 = S a,1 α(k i,1 + q i,1 ) + k i,1 S i,1 (1 -α) α[q a,1 (k i,1 + q i,1 ) + k a,1 q i,1 ] N - a,2 = S a,2 α(k i,2 + q i,2 ) + k i,2 S i,2 (1 -α) α[q a,2 (k i,2 + q i,2 ) + k a,2 q i,2 ] N - i,1 = S i,1 (1 -α) + k a,1 αN - a,1 (1 -α)(k i,1 + q i,1 ) N - i,2 = S i,2 (1 -α) + k a,2 αN - a,2 (1 -α)(k i,2 + q i,2 ) B -= 0 (2.6)
In the absence of autotrophs, the concentrations of nutrient i are partially regulated with respect to changes in its supply to both the accessible and inaccessible pools (i.e. 0 < ρ ix,iy < 1) because of chemical and physical processes. As autotrophs are absent, the cycles of nutrients 1 and 2 are independent of each other and thus there is no effect of changes in the supply of nutrient i on the concentrations of nutrient j (i.e. ǫ ix,jy = 0, where i = j).

In the presence of autotrophs, there are two equilibria depending on which nutrient limits the growth of autotrophs. Without loss of generality, we will focus on the case where nutrient 1 is the limiting nutrient and nutrient 2 is the non-limiting one.

N + a,1 = g -1 (m) N + a,2 = αS a,2 (k i,2 + q i,2 ) + k i,2 S i,2 (1 -α) -αmRB + [q i,2 (1 -rec 2 (1 -λ 2 )) + k i,2 λ 2 ] α [(k a,2 + q a2 )(k i,2 + q i,2 ) -k a,2 k i,2 ] N + i,1 = -αk a,1 N + a,1 + S i,1 (1 -α) [1 -rec 1 (1 -λ 1 )] + α(1 -rec 1 )(1 -λ 1 ) [-S a,1 + N a,1 + (k a,1 + q a,1 )] (1 -α)[k i,1 (1 -rec 1 )(1 -λ 1 ) -(k i,1 + q i,1 ) [1 -rec 1 (1 -λ 1 )]] N + i,2 = α -S a,2 + N + a,2 (k a,2 + q a2 ) + mRB + (1 -rec 2 (1 -λ 2 )) k i,2 (1 -α) B + = -S a,1 α(k i,1 + q i,1 ) -k i,1 S i,1 (1 -α) + αN + a,1 [q a,1 (k i,1 + q i,1 ) + k a,1 q i,1 ] αm[k i,1 (1 -rec 1 )(1 -λ 1 ) -(k i,1 + q i,1 ) [1 -rec 1 (1 -λ 1 )]]
(2.7)

The regulation coefficients within a cycle and effects of a cycle on the other are detailed in Appendix S1 for the equilibrium where autotrophs are present. An analysis of the flows between nutrient pools helps to better understand the results (Figure 2.2).

Results for the limiting nutrient are similar to those obtained with the generic model with a single nutrient (Auguères and Loreau 2015a). In the presence of organisms, any variation in the supply of the accessible limiting nutrient is entirely absorbed by organisms because of their top-down control on the accessible pool of the limiting nutrient (negative feedback loop between autotrophs and the accessible pool, path 2-3 in Figure 2.2), and thus ρ 1a,1a = ρ 1a,1i = 1. The concentration of the limiting nutrient in the inaccessible pool is only partially or negatively regulated with respect to changes in its supply because it varies in the same direction as the supply of nutrient in both accessible and inaccessible forms (paths 1a-4, 1a-2-5 and 1b-4-2-5 in Figure 2.2, and thus ρ 1i,1a < 1 and ρ 1i,1i < 1. Changes in the supply of the non-limiting nutrient in either pool have no effect on the limiting nutrient pools because they do not affect biomass (no arrow from the non-limiting nutrient concentrations to biomass in Figure 2.2); thus ǫ 1a,2a = ǫ 1a,2i = ǫ 1i,2a = ǫ 1i,2i = 0.

Changes in the supply of the non-limiting nutrient are either partially or negatively regulated in both pools because concentrations vary in the same direction as supplies (paths 6a-4 and 6b-4 in Figure 2.2), and thus ρ 2a,2a , ρ 2i,2a , ρ 2a,2i and ρ 2i,2i are lower than 1. The absence of a negative feedback loop between autotrophs and the concentration of the non-limiting nutrient in the accessible pool prevents autotrophs from regulating the concentrations of the non-limiting nutrient (Figure 2.2). Changes in the supplies of the limiting nutrient have a negative effect on the concentrations of the non-limiting nutrient, thus ǫ 2a,1a , ǫ 2i,1a , ǫ 2a,1i and ǫ 2i,1i are negative. Concentrations of the non-limiting nutrient and the supply of the limiting nutrient indeed vary in the opposite direction (paths 1a-2-8, 1a-2-8-4, 1b-4-2-8 and 1b-4-2-8-9 in Figure 2.2). This result is intuitive as an increase in the supply of the limiting nutrient enhances the growth of autotrophs, then resulting to an increase in the depletion of the non-limiting nutrient pools.

In this paragraph, we will focus on regulation of the ratio of the limiting nutrient over the non-limiting one with respect to changes in the supplies of each nutrient. When supplies of the limiting nutrient vary, accessible concentration of the limiting nutrient is perfectly regulated (ρ 1a,1a = ρ 1a,1i = 1), while that of the non-limiting nutrient is negatively impacted (ǫ 2a,1a < 0 and ǫ 2a,1i < 0). In that case, ρ (N 1 :N 2 )a,1a and ρ (N 1 :N 2 )a,1i are lower than 1, thus the nutrient ratio is either partially or negatively regulated in the accessible pool. Regulation of the nutrient ratio in the inaccessible pool, however, depends on the relative changes in the concentrations of the two nutrients as a result of variations in the supply of the limiting nutrient. The nutrient ratio in the inaccessible pool is thus either negatively, partially or over-regulated with respect to changes in the supply of the limiting nutrient to both the accessible and the inaccessible pools. Variations in the supplies of the non-limiting nutrient do not affect the concentration of the limiting nutrient (ǫ 1a,2a = ǫ 1a,2i = ǫ 1i,2a = ǫ 1i,2i = 0)

and concentrations of the non-limiting nutrient are partially or negatively regulated (ρ 2a,2a , ρ 2i,2a , ρ 2a,2i and ρ 2i,2i are lower than 1). Thus nutrient ratio is either partially or negatively regulated with respect to changes in the supply of the non-limiting nutrient in both the accessible and the inaccessible pools (i.e. ρ (N 2 :N 1 )a,2a , ρ (N 2 :N 1 )i,2a , ρ (N 2 :N 1 )a,2i and ρ (N 2 :N 1 )i,2i

are lower than 1).

Discussion

Our model predicts that autotrophic organisms should only partially or negatively regulate the concentrations of a non-limiting nutrient. Thus explicit consideration of the cycle of a non-limiting nutrient suggests that autotrophs are not able to efficiently regulate global nutrient cycles. Variations in the supply of a non-limiting nutrient do not affect the growth of autotrophs, and thus they do not affect the concentrations of the limiting nutrient in either the accessible or the inaccessible pool. Variations in the supply of the limiting nutrient, however, affects the growth of autotrophs and their consumption of the non-limiting nutrients. As a result, variations in the supply of the limiting nutrient and concentrations of non-limiting nutrients vary in opposite directions. For example, if the supply of the limiting nutrient increases, depletion of the accessible non-limiting nutrient increases, resulting in a decrease in both the accessible and inaccessible concentrations of the non-limiting nutrient. If the supply of the non-limiting nutrient is also increased, the effects of the increased supply of the two nutrients will partly counterbalance each other, and thus the global regulation of the non-limiting nutrient will be enhanced by interactions between cycles. In contrast, if the supply of the non-limiting nutrient is decreased, it will further amplify the decrease in the concentration of the non-limiting nutrient induced by the increased supply of the limiting nutrient. Our model thus suggests that interactions between nutrient cycles can either (1)

decrease the ability of autotrophs to regulate non-limiting nutrient pools if the supplies of the limiting and the non-limiting nutrients vary in opposite directions, or (2) increase the ability of autotrophs to regulate non-limiting nutrient pools if the supplies of the limiting and the non-limiting nutrients vary in the same direction. For example if we consider the oceanic cycles of Fe and P, both nutrients are increasingly supplied to the surface ocean due to anthropogenic activities (Benitez-Nelson 2000; [START_REF] Krishnamurthy | Impacts of atmospheric nutrient inputs on marine biogeochemistry[END_REF]. Interactions between the two cycles should thus enhance the regulation of concentrations of P or Fe if the oceanic primary production is limited by Fe or P, respectively. In contrast, the supply of some nutrients such as Si is decreased by human activities [START_REF] Laruelle | Anthropogenic perturbations of the silicon cycle at the global scale: Key role of the land-ocean transition[END_REF], and thus the regulation of Si oceanic pools should be decreased by interactions between the oceanic cycles of Si and the nutrient that limits phytoplankton growth, e.g. Fe or P.

In our model, we assumed that Liebig's (1842) law of the minimum governs the growth of autotrophs. However, limitation of the growth of autotrophs by several nutrients simultaneously is common in natural systems [START_REF] Arrigo | Marine microorganisms and global nutrient cycles[END_REF][START_REF] Sterner | On the phosphorus limitation paradigm for lakes[END_REF][START_REF] Harpole | Nutrient co-limitation of primary producer communities[END_REF]. When Liebig's law of the minimum is used, the growth of autotrophs depends on the concentration of the most limiting nutrient. However, colimitation by two nutrients simultaneously results in an alteration of the growth of autotrophs with respect to changes in the concentration of both nutrients [START_REF] Sperfeld | Multiple resource limitation theory applied to herbivorous consumers: Liebigâs minimum rule vs. interactive co-limitation[END_REF]. The increase in the supply of the less-limiting nutrient will enhance the growth of autotrophs, thereby increasing the depletion of both nutrients. As an organism's growth responds to variations in the concentration of the less-limiting nutrient, colimitation is thus likely to enhance regulation of its concentrations of the less-limiting nutrient. However, variations in the growth of autotrophs also impact the consumption of the more-limiting nutrient, which was not the case with limitation by a single nutrient. Thus, colimitation is likely to increase biotic regulation of the less-limiting nutrient but to decrease that of the more-limiting one.

The ability of autotrophs to adapt their stoichiometry depending on nutrient availability can also alter the control of autotrophs on nutrient cycles. Variable stoichiometry of autotrophs can indeed change the patterns of nutrient limitation, and hence the consumption and control of resources [START_REF] Sterner | Ecological stoichiometry: The biology of elements from molecules to the biosphere[END_REF][START_REF] Danger | Does Liebigâs law of the minimum scale up from species to communities?[END_REF]. Consumer-driven nutrient recycling (i.e. the release of part of the nutrients ingested by herbivores in the environment) is a major process affecting the availability of nutrients in the environment, which potentially modifies the limitation conditions of the growth of autotrophs. For example, zooplankton can affect the availability of nutrients in surface and deep waters through vertical migrations and excretion of part of the nutrients ingested [START_REF] Elser | The stoichiometry of consumer-driven nutrient recycling: Theory, observations, and consequences[END_REF][START_REF] Palmer | Production and export in a global ocean ecosystem model[END_REF][START_REF] Thingstad | Nature of phosphorus limitation in the ultraoligotrophic Eastern Mediterranean[END_REF]. This can result in a shift from P or N limitation of autotroph growth to colimitation by the two nutrients simultaneously in freshwater and marine ecosystems [START_REF] Moegenburg | Nutrient regeneration by zooplankton: effects on nutrient limitation of phytoplankton in a eutrophic lake[END_REF][START_REF] Trommer | Zooplankton-mediated nutrient limitation patterns in marine phytoplankton: an experimental approach with natural communities[END_REF]).

However, consumer-driven nutrient recycling does not necessarily have a positive effect on the growth of autotrophs [START_REF] Daufresne | Plant-herbivore interactions and ecological stoichiometry: when do herbivores determine plant nutrient limitation?[END_REF], and thus on biotic regulation of nutrient cycles.

In our model, we considered a single population of autotrophs. Competition between populations of autotrophs with different stoichiometric ratios and/or competition between different functional groups could, however, alter our results. For example, our model predicts that nutrient ratios should be either partially, negatively or over-regulated against changes in the supplies of both the limiting and the non-limiting nutrients. However, observations show that regulation of inaccessible nutrient ratios can be high, as it is the case with Redfield ratios in the ocean [START_REF] Redfield | On the proportions of organic derivations in sea water and their relation to the composition of plankton[END_REF][START_REF] Redfield | The biological control of chemical factors in the environment[END_REF][START_REF] Falkowski | Rationalizing elemental ratios in unicellular algae[END_REF]. To be applied to the oceanic cycles of N and P, our stoichiometric model has to be improved such that it describes the dynamics of two functional groups of autotrophs, i.e. non-fixers and N-fixers (e.g. Tyrrell 1999; Lenton and Watson 2000a). Such a model could then be used to study the regulation of deep-water N:P ratio by autotrophs (Auguères and Loreau 2015b). The compensatory dynamics between both functional groups of autotrophs should then enhance the ability of S a,n Supply of nutrient n to the accessible pool µmol.m -3 .yr -1 S i,n Supply of nutrient n to the inaccessible pool µmol.m -3 .yr -1 q a,n Turnover rate of nutrient n in the accessible pool yr -1 q i,n Turnover rate of nutrient n in the inaccessible pool yr -1 n indices refer to the number of the nutrient concerned, either 1 or 2.

Figures and Tables

Appendix 2.A: Regulation coefficients and interaction between cycles

In this appendix, we detailed calculations of regulation coefficients and effects of a cycle on the other in the case where autotrophs are present. As in the main text, we consider that nutrient N 1 limits the growth of autotrophs.

Equilibrium concentrations are:

N + a,1 = g -1 (m) N + a,2 = αS a,2 (k i,2 + q i,2 ) + k i,2 S i,2 (1 -α) -αmRB + [q i,2 (1 -rec 2 (1 -λ 2 )) + k i,2 λ 2 ] α [(k a,2 + q a2 )(k i,2 + q i,2 ) -k a,2 k i,2 ] N + i,1 = -αk a,1 N + a,1 + S i,1 (1 -α) [1 -rec 1 (1 -λ 1 )] + α(1 -rec 1 )(1 -λ 1 ) [-S a,1 + N a,1 + (k a,1 + q a,1 )] (1 -α)[k i,1 (1 -rec 1 )(1 -λ 1 ) -(k i,1 + q i,1 ) [1 -rec 1 (1 -λ 1 )]] N + i,2 = α -S a,2 + N + a,2 (k a,2 + q a2 ) + mRB + (1 -rec 2 (1 -λ 2 )) k i,2 (1 -α) B + = -S a,1 α(k i,1 + q i,1 ) -k i,1 S i,1 (1 -α) + αN + a,1 [q a,1 (k i,1 + q i,1 ) + k a,1 q i,1 ] αm[k i,1 (1 -rec 1 )(1 -λ 1 ) -(k i,1 + q i,1 ) [1 -rec 1 (1 -λ 1 )]]
(2.8)

Regulation coefficients of the limiting nutrient are:

ρ 1a,1a = 1 ρ 1a,1i = 1 ρ 1i,1a = -αk a,1 N + a,1 + S i,1 (1 -α) [1 -rec 1 (1 -λ 1 )] + α(1 -rec 1 )(1 -λ 1 ) [N a,1 + (k a,1 + q a,1 )] -αk a,1 N + a,1 + S i,1 (1 -α) [1 -rec 1 (1 -λ 1 )] + α(1 -rec 1 )(1 -λ 1 ) [-S a,1 + N a,1 + (k a,1 + q a,1 )] ρ 1i,1i = -αk a,1 N + a,1 [1 -rec 1 (1 -λ 1 )] + α(1 -rec 1 )(1 -λ 1 ) [-S a,1 + N a,1 + (k a,1 + q a,1 )] -αk a,1 N + a,1 + S i,1 (1 -α) [1 -rec 1 (1 -λ 1 )] + α(1 -rec 1 )(1 -λ 1 ) [-S a,1 + N a,1 + (k a,1 + q a,1 )] (2.9)
Effect of the cycle of the limiting nutrient on that of the non-limiting nutrient:

ǫ 2a,1a = -αmRS a,1 ∂B + ∂Sa,1 [q i,2 (1 -rec 2 (1 -λ 2 )) + k i,2 λ 2 ] αS a,2 (k i,2 + q i,2 ) + k i,2 S i,2 (1 -α) -αmRB + [q i,2 (1 -rec 2 (1 -λ 2 )) + k i,2 λ 2 ] ǫ 2a,1i = -αmRS i,1 ∂B + ∂Si,1 [q i,2 (1 -rec 2 (1 -λ 2 )) + k i,2 λ 2 ] αS a,2 (k i,2 + q i,2 ) + k i,2 S i,2 (1 -α) -αmRB + [q i,2 (1 -rec 2 (1 -λ 2 )) + k i,2 λ 2 ] ǫ 2i,1a = αS a,1 ∂N + a,2 ∂Sa,1 (k a,2 + q a2 ) + mR ∂B + ∂Sa,1 (1 -rec 2 (1 -λ 2 )) α -S a,2 + N + a,2 (k a,2 + q a2 ) + mRB + (1 -rec 2 (1 -λ 2 )) ǫ 2i,1i = αS i,1 ∂N + a,2 ∂Si,1 (k a,2 + q a2 ) + mR ∂B + ∂Si,1 (1 -rec 2 (1 -λ 2 )) α -S a,2 + N + a,2 (k a,2 + q a2 ) + mRB + (1 -rec 2 (1 -λ 2 ))
(2.10)

Regulation coefficients of the non-limiting nutrient are:

ρ 2a,2a = αS a,2 (k i,2 + q i,2 ) + k i,2 S i,2 (1 -α) -αmR B + -S a,2 ∂B + ∂Sa,2 [q i,2 (1 -rec 2 (1 -λ 2 )) + k i,2 λ 2 ] αS a,2 (k i,2 + q i,2 ) + k i,2 S i,2 (1 -α) -αmRB + [q i,2 (1 -rec 2 (1 -λ 2 )) + k i,2 λ 2 ] ρ 2a,2i = αS a,2 (k i,2 + q i,2 ) + k i,2 S i,2 (1 -α) -αmR B + -S i,2 ∂B + ∂Si,2 [q i,2 (1 -rec 2 (1 -λ 2 )) + k i,2 λ 2 ] αS a,2 (k i,2 + q i,2 ) + k i,2 S i,2 (1 -α) -αmRB + [q i,2 (1 -rec 2 (1 -λ 2 )) + k i,2 λ 2 ] ρ 2i,2a = α -S a,2 + N + a,2 -S a,2 ∂N + a,2 ∂Sa,2 (k a,2 + q a2 ) + mR B + -S a,2 ∂B + ∂Sa,2 (1 -rec 2 (1 -λ 2 )) α -S a,2 + N + a,2 (k a,2 + q a2 ) + mRB + (1 -rec 2 (1 -λ 2 )) ρ 2i,2i = α -S a,2 + N + a,2 -S i,2 ∂N + a,2 ∂Si,2 (k a,2 + q a2 ) + mR B + -S i,2 ∂B + ∂Si,2 (1 -rec 2 (1 -λ 2 )) α -S a,2 + N + a,2 (k a,2 + q a2 ) + mRB + (1 -rec 2 (1 -λ 2 )) (2.11)
Effect of the cycle of the non-limiting nutrient on that of the limiting nutrient:

ǫ 1a,2a = 0 ǫ 1a,2i = 0 ǫ 1i,2a = 0 ǫ 1i,2i = 0
(2.12)

Connecting statements

In Chapter 2, we focused on biotic regulation of the concentrations of a non-limiting nutrient with respect to changes in its supply and the impact of coupling of nutrient cycles on the efficiency of biotic regulation of both non-limiting and limiting nutrient pools. We showed that regulation of the non-limiting nutrient is weaker than that of the limiting nutrient. Our results also suggest that interactions between nutrient cycles can either enhance or decrease the ability of organisms to regulate nutrient pools, depending on whether the supply of both nutrients vary in the same direction or not. Competition between different populations of autotrophs that consume nutrients in a different ratio could however affect the strength of the coupling of nutrient cycles, and thus potentially the efficiency of their regulation. This is the case for example with Redfield ratios in the ocean, which seem to be maintained by the competition between non-fixing and N-fixing phytoplankton (e.g. [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF]). In Chapter 3, we will thus focus on the extent to which competition between two functional groups (N-fixers and non-fixers) impacts regulation of N and P concentrations in the global ocean as well as their ratio in both the accessible and inaccessible pools.

Chapter 3

Regulation of Redfield ratios in the deep ocean

Anne-Sophie Auguères [START_REF] Redfield | On the proportions of organic derivations in sea water and their relation to the composition of plankton[END_REF][START_REF] Redfield | The biological control of chemical factors in the environment[END_REF].

La régulation du rapport N:P des eaux profondes est un problème majeur dans le domaine de l'écologie marine depuis plusieurs décennies [START_REF] Falkowski | Rationalizing elemental ratios in unicellular algae[END_REF].

Les cycles océaniques du N et du P sont fortement impactés par les activités humaines, principalement par l'augmentation des apports de nutriments dans l'océan de surface Nous avons montré que la valeur du rapport N:P des eaux profondes est déterminée par le rapport N:P des non-fixateurs, ainsi que par le recyclage et la dénitrification. Notre modèle prédit que malgré une régulation inefficace des réservoirs profonds de N et de P, le phytoplancton peut maintenir un rapport N:P quasi constant. La dynamique compensatoire entre les fixateurs et les non-fixateurs permet en effet le contrôle de la composition chimique des eaux profondes par le biais du recyclage de la matière organique (e.g. [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF]Lenton et Watson 2000a). Ce mécanisme pourrait expliquer la quasi constance du rapport N:P des eaux profondes, en accord avec les hypothèses formulées par [START_REF] Redfield | On the proportions of organic derivations in sea water and their relation to the composition of plankton[END_REF][START_REF] Redfield | The biological control of chemical factors in the environment[END_REF]. Le rapport N:P du phytoplancton n'affecte cependant pas la capacité du phytoplancton à réguler le rapport N:P des eaux profondes. Notre modèle suggère que la stratification croissante de la colonne d'eau due à l'augmentation de la température et la diminution de la salinité des eaux de surface pourrait diminuer la stabilité du rapport N:P dans l'océan profond sur de grandes échelles de temps et d'espace.

Introduction

Regulation of environmental conditions by organisms at global scale has been debated

for several decades, especially concerning the controversial Gaia theory. This theory was originally developed to address the issue of the near constancy of physical and chemical properties of the atmosphere over long time scales (Lovelock andMargulis 1974a, 1974b;[START_REF] Margulis | Biological modulation of the Earthâs atmosphere[END_REF]. The Gaia theory proposes that feedback mechanisms between organisms and their environment contribute to the restriction of variations in environmental conditions to a range that is habitable for life. This hypothesis was strongly criticized, as natural selection acts at the individual level to maximize the fitness of organisms in their local environment and does not necessarily promote stability and self-regulation of the global environment (see [START_REF] Lenton | Gaia and natural selection[END_REF][START_REF] Free | Do evolution and ecology need the Gaia hypothesis?[END_REF] for reviews). Redfield ratios in oceans provide another example of possible regulation by organisms of their environment at global scales.

The Redfield ratios are one of the key foundations of ocean biogeochemistry [START_REF] Falkowski | Rationalizing elemental ratios in unicellular algae[END_REF]. Although local limiting conditions and phytoplankton growth strategies can induce local variations in phytoplankton stoichiometry [START_REF] Arrigo | Marine microorganisms and global nutrient cycles[END_REF][START_REF] Franz | Dynamics and stoichiometry of nutrients and phytoplankton in waters influenced by the oxygen minimum zone in the eastern tropical Pacific[END_REF]Martiny et al. 2013), the mean value of phytoplankton C:N:P ratio is considered as relatively constant at large spatial and temporal scales [START_REF] Redfield | On the proportions of organic derivations in sea water and their relation to the composition of plankton[END_REF][START_REF] Redfield | The biological control of chemical factors in the environment[END_REF][START_REF] Karl | Total dissolved nitrogen and phosphorus concentrations at US-JGOFS station ALOHA: Redfield reconciliation[END_REF]Anderson and Sarmiento 1994). The issue of the biological meaning of the mean N:P ratio of 16:1 in phytoplanktonic cells has been a challenge for theoretical ecology in the last decade (Klausmeier et al. 2004a[START_REF] Klausmeier | Phytoplankton stoichiometry[END_REF][START_REF] Loladze | The origins of the Redfield nitrogen-to-phosphorus ratio are in a homoeostatic protein-to-rRNA ratio[END_REF]. It has recently been shown theoretically that the balance between protein and rRNA synthesis leads to a homeostatic protein:rRNA ratio that corresponds to a overall cellular N:P ratio of 16 ± 3 [START_REF] Loladze | The origins of the Redfield nitrogen-to-phosphorus ratio are in a homoeostatic protein-to-rRNA ratio[END_REF]).

Redfield's fundamental insight into the chemistry of marine ecosystems was primarily related to the coupling between the N:P ratio of phytoplankton and that of seawater, regardless of a specific ratio per se. [START_REF] Redfield | On the proportions of organic derivations in sea water and their relation to the composition of plankton[END_REF] highlighted the similarity between the mean N:P ratio of phytoplanktonic cells and that of ocean deep waters. He proposed three hypotheses to explain this similarity: (1) it is a coincidence, (2) phytoplankton can adapt their stoichiometry to environmental conditions, or (3) phytoplankton control the chemical properties of their environment. The first hypothesis seems unlikely and thus was quickly rejected. The second hypothesis is relevant since consumption of N and P in a non-Redfield ratio is common in the ocean, depending on local limiting conditions and phytoplankton growth strategies [START_REF] Geider | Redfield revisited: variability of C:N:P in marine microalgae and its biochemical basis[END_REF][START_REF] Franz | Dynamics and stoichiometry of nutrients and phytoplankton in waters influenced by the oxygen minimum zone in the eastern tropical Pacific[END_REF]. Phytoplankton are thus able to adapt their stoichiometry to nutrient availability to a certain extent, depending on factors such as physiological constraints and the physical structure of ecosystems [START_REF] Hall | Constraints on primary producer N:P stoichiometry along N:P supply ratio gradients[END_REF], which could explain part of the similarity between the mean N:P ratio of phytoplankton and that of deep waters. Adaptation of phytoplankton N:P ratio to environmental conditions should lead to a variability in the deep-water N:P ratio over time, as observed for example in the North-Atlantic Ocean [START_REF] Pahlow | Temporal trends in deep ocean Redfield ratios[END_REF]. However, a recent study showed that the average N:P ratio of 16:1 in phytoplankton can be explained by the balance between protein and rRNA synthesis [START_REF] Loladze | The origins of the Redfield nitrogen-to-phosphorus ratio are in a homoeostatic protein-to-rRNA ratio[END_REF]. Thus the hypothesis that phytoplankton can adapt their stoichiometry to environmental conditions is inconsistent with the fact that phytoplankton N:P ratio may correspond to an optimum cell composition, independently of the composition of the growth medium. [START_REF] Redfield | The biological control of chemical factors in the environment[END_REF] favored the third hypothesis, assuming that the intracellular content of phytoplankton could be central in the similarity observed between phytoplankton and deep-water N:P ratios. Phytoplankton could maintain this pattern through nitrogen fixation, denitrification and recycling. The concentration of fixed inorganic nitrogen is indeed biologically controlled, whereas that of phosphate is set by the riverine inflows from continental sources and by the sedimentary outflows [START_REF] Karl | The role of nitrogen fixation in biogeochemical cycling in the subtropical North Pacific Ocean[END_REF][START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Deutsch | Spatial coupling of nitrogen inputs and losses in the ocean[END_REF]. When fixed inorganic nitrogen becomes limiting for phytoplankton, nitrogen fixation would increase the nitrogen inputs to the seawater [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF]Lenton and Watson 2000a;[START_REF] Schade | A conceptual framework for ecosystem stoichiometry: balancing resource supply and demand[END_REF].

Regulation of the deep-water N:P ratio is a major issue in marine ecology for several decades [START_REF] Falkowski | Rationalizing elemental ratios in unicellular algae[END_REF]. The biological basis of the mean N:P ratio in phytoplankton has received some attention [START_REF] Loladze | The origins of the Redfield nitrogen-to-phosphorus ratio are in a homoeostatic protein-to-rRNA ratio[END_REF], but an important unknown question is whether phytoplankton can be expected to control deep-water composition from an ecological perspective. Nitrogen and phosphorus oceanic cycles are heavily affected by anthropogenic activities, mainly through an increase in N and P supply by rivers [START_REF] Benitez-Nelson | The biogeochemical cycling of phosphorus in marine systems[END_REF][START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF][START_REF] Seitzinger | Global river nutrient export: A scenario analysis of past and future trends[END_REF]) and in the atmospheric deposition of N [START_REF] Galloway | The global nitrogen cycle: changes and consequences[END_REF][START_REF] Duce | Impacts of atmospheric anthropogenic nitrogen on the open ocean[END_REF]. Phytoplankton have access only to the upper layer of the ocean, either because of the limited depth of the euphotic layer or because of the thermocline. Thus, it is critical to understand ( 1) to what extent phytoplankton are able to regulate the N:P ratio of the deep layer in human-altered marine systems, and ( 2) what are the mechanisms that control the value of the deep-water N:P ratio and the efficiency of its regulation.

Our aim in this work is to clarify to what extent and by which way autotrophic organisms are able to control the chemistry of the deep ocean, to which they do not have a direct access.

We address these two issues by building a model for the coupled N and P cycles in the ocean based on [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF], in which phytoplankton access nutrients only in the upper layer, and parametrize our model with existing data. Hereafter, the surface and deep-water N:P ratios refer only to the dissolved nutrients in the water. By performing a sensitivity analysis, we first determine the extent to which the different mechanisms involved in P and N cycles (e.g. denitrification, N-fixation, independent physical flows) control the value of the deep-water N:P ratio in the current ocean. We then analyze the potential for biotic regulation of N and P concentrations as well as of their ratio in the current ocean if either N or P supply is increased by anthropogenic activities (called hereafter "regulation efficiency"). Finally, we perform a sensitivity analysis to assess which mechanisms drive the regulation efficiency of deep-water N:P ratio by autotrophic organisms.

Methods

Model description

Our ocean model describes the biogeochemical cycles of N and P in the ocean (Figure 3.1, Table 3.1), and includes two groups of phytoplankton, N-fixers and non-fixers [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF].

The water column, with depth z tot , is separated in two layers, each of which is considered homogeneous. The upper layer, which corresponds to a fraction p za of the water column, is accessible to phytoplankton. The deep layer is inaccessible to phytoplankton because of light limitation or water column stratification. N concentrations (N a and N i for the upper and deep layers, respectively) include nitrites, nitrates and ammonium, and P concentrations (P a and P i ) correspond to phosphates. The two inorganic pools are connected by physical processes -here, diffusion and water vertical movements (i.e. upwellings and downwellings, governed by parameter K). Both layers have nutrient outflows to unrepresented parts of the Earth system, but only the upper layer has nutrient inflows. N supply to the upper layer (S N ) includes atmospheric and riverine inflows [START_REF] Cornell | Atmospheric inputs of dissolved organic nitrogen to the oceans[END_REF] while P supply (S P ) includes only riverine inflow [START_REF] Benitez-Nelson | The biogeochemical cycling of phosphorus in marine systems[END_REF]. Nutrient outflows correspond to adsorption of inorganic nutrients [START_REF] Benitez-Nelson | The biogeochemical cycling of phosphorus in marine systems[END_REF][START_REF] Morse | Ammonium interaction with coastal marine sediments: influence of redox conditions on K*[END_REF]. We added adsorption to [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF] model to allow P to leave the system in the absence of organisms; this prevents the model from displaying the pathological behavior of indefinite P accumulation in the absence of organisms. Adsorption rates of dissolved P (q P ) and N (q N ) are considered constant in the water column. We assume that Von Liebig's (1842) law of the minimum governs the growth of non-fixers, whose concentration in the upper layer is B. N-fixers, whose concentration in the upper layer is B F , consume phosphates in that layer. Nitrogen fixation is assumed to be the only source of N for N-fixers, whose growth is limited by P because N 2 is in excess in the ocean. Phytoplankton have a traditional resource-dependent functional response to the accessible nutrient concentration. The functional response of non-fixers to P, g(P a ), is distinguished from that of N-fixers, g F (P a ). Their P:N ratio, R, is also assumed to be different from that of N-fixers, R F [START_REF] Lenton | Biotic stoichiometric controls on the deep ocean N:P ratio[END_REF]; both ratios are supposed to be constant. Particle export from the upper to the deep layer is induced by sinking of dead organic matter as well as by grazing and vertical migrations of zooplankton. For the sake of simplicity, grazing is not explicitly taken into consideration and is included in the mortality rate of phytoplankton (m). Part of the organic matter is recycled by microorganisms, leading to a return of nutrients to the water column (Rec tot ) [START_REF] Hood | Pelagic functional group modeling: Progress, challenges and prospects. The US JGOFS Synthesis and Modeling Project: Phase III The US JGOFS Synthesis and Modeling Project[END_REF] , with a fraction p Reca to the upper layer. Lastly, denitrification of organic matter leads to a release of N 2 from the ocean to the atmosphere (D tot ) [START_REF] Hood | Pelagic functional group modeling: Progress, challenges and prospects. The US JGOFS Synthesis and Modeling Project: Phase III The US JGOFS Synthesis and Modeling Project[END_REF], with a fraction p Da from the upper layer. Mass balance is used to build a model that describes the dynamics of N and P masses. By dividing nutrient mass by the volume of the layer concerned, we then obtain a model in terms of nutrient concentrations (Figure 3.1):

dN a dt = S N + K z tot p za (N i -N a ) -q N N a + m(Rec tot p Reca -D tot p Da )(B + B F ) -min (g N (N a ), g P (P a )) B dP a dt = S P + K z tot p za (P i -P a ) -q P P a + mRec tot p Reca (RB + R F B F ) -min (g N (N a ), g P (P a )) RB -g P F (P a )R F B F dN i dt = K z tot (1 -p za ) (N a -N i ) -q N N i + p za 1 -p za m [Rec tot (1 -p Reca ) -D tot (1 -p Da )] (B + B F ) dP i dt = K z tot (1 -p za ) (P a -P i ) -q P P i + p za 1 -p za mRec tot (1 -p Reca )(RB + R F B F ) dB dt = [min (g N (N a ), g P (P a )) -m] B dB F dt = [g P F (P a ) -m] B F (3.1)
When necessary for numerical simulations, functional responses were modelled with a Michaelis-Menten function:

g N (N a ) = µN a N a + N H g P (P a ) = µP a P a + RN H g P F (P a ) = (µ -cost)P a P a + RN H (3.2)
N fixation is energetically more costly than mineral N uptake [START_REF] Vitousek | Ecosystem constraints to symbiotic nitrogen fixers: A simple model and its implications[END_REF][START_REF] Menge | Evolutionary tradeoffs can select against nitrogen fixation and thereby maintain nitrogen limitation[END_REF] and can be limited by iron [START_REF] Mills | Iron and phosphorus co-limit nitrogen fixation in the eastern tropical North Atlantic[END_REF][START_REF] Weber | Oceanic nitrogen reservoir regulated by plankton diversity and ocean circulation[END_REF].

The maximal growth rate of N-fixers is obtained by subtracting a given cost (cost) from the maximal growth rate of non-fixers (µ). N H is the half-saturation constant of non-fixers for N. N-fixers and non-fixers are considered to have the same half-saturation constant for P, which is fixed to R * N H for consistency with the N:P ratio of non-fixers.

Regulation coefficients

We calculated the strength of the regulation of the concentrations of a nutrient A with respect to changes in its supply:

ρ x,y = 1 - S y N x * ∂N x ∂S y (3.3)
ρ x,y is defined as the regulation coefficient of the nutrient A concentration in pool x with respect to changes in the nutrient A supply to pool y. When ρ x,y = 0, there is no regulation (i.e. the proportional variation in nutrient A concentration in pool x is equal to that in nutrient A supply to pool y). At the other extreme, when ρ x,y = 1, there is perfect regulation (i.e. there is no variation in nutrient A concentration in pool x as a result of that in nutrient A supply to pool y). When 0 < ρ x,y < 1, regulation is partial. Note that biota can sometimes over-regulate the nutrient concentration in pool x, in which case ρ x,y > 1.

Some cases where ρ x,y < 0 can also occur; regulation is then negative, i.e. organisms amplify variations in nutrient supply.

We also quantified the effect of changes in the supply of one nutrient A 1 on the concentrations of the other nutrient A 2 using the following equation:

ǫ N 1x,N 2y = S N 2y N 1x * ∂N 1x ∂S N 2y (3.4)
ǫ A1x,A2y measures the effect of the supply of nutrient A 2 in pool y on the concentration of nutrient A 1 in pool x. This effect is positive when the supply of A 2 in pool y and the concentration of A 1 in pool x vary in the same direction, negative when they vary in opposite directions, and zero when the supply of A 2 does not affect the concentration of A 1 .

We further quantified the strength of the regulation of the ratio between the two nutrients using the same principle as in equation (3.2). In this case, however, the elasticity of the ratio will change sign depending on whether the nutrient whose supply changes is in the numerator or in the denominator of the ratio. Therefore, to keep signs consistent, we calculated the regulation coefficient of the deep-water N:P ratio when N supply was modified, and the regulation coefficient of the deep-water P:N ratio when P supply was modified. We obtain the following regulation coefficients for the N:P ratio of deep waters:

ρ (N :P )x,N a = ρ N x,N a + ǫ P x,N a
ρ (P :N )x,P a = ρ P x,P a + ǫ N x,P a

(3.5) Lastly, we performed numerical simulations of the system when N-fixers and non-fixers coexist, as is observed in the ocean. We chose parameter values within the range of values found in the literature (Table 3.1) in order to be as realistic as possible. We increased nutrient supply by 50 % after half of the simulation time to assess the strength of the regulation of nutrient pools in the current ocean.

Sensitivity analysis

The sensitivity of a variable X to the parameter par (sens X,par ) can be measured as:

sens X,par = par X * ∂X ∂par (3.6)
sens X,par is negative when X and par vary in the opposite directions, and positive when they vary in the same direction. The higher the absolute value of sens X,par , the more sensitive X is to par.

Equation (3.6) gives information about local sensitivity. As values found in the literature for parameters often vary between studies (Table 3.1), we chose to calculate sensitivity over a range of parameter values. This avoids conclusions being strongly dependent on the value chosen for numerical simulations. For each of the 17 parameters of the model, we thus used a set of 500 values uniformly distributed in an interval of 20 % around the baseline values used in numerical simulations. When necessary, we adjusted the bounds of the interval to be consistent with the possible values of the parameter. To assess which parameters have the strongest influence on the value and the regulation efficiency of the deep-water N:P ratio,

we measured the sensitivity of (1) N i /P i (first section), ( 2) ρ (P :N )i,P a (third section), and ( 3) ρ (N :P )i,N a (third section) to the set of values for each parameter (Supplementary Table S3.1).

Results

Which mechanisms control the value of the deep-water N:P ratio?

Model (3.1) has six equilibria, depending on the nutrient that limits phytoplankton growth and on the presence or absence of either phytoplankton group. In this study, we focus on the case where N-fixers and non-fixers coexist at equilibrium as it corresponds to the observations in the current ocean. We will first look at parameters and mechanisms that drive the value of the deep-water N:P ratio in the current ocean. In agreement with experimental data, the deep-water N:P ratio is near the one of organisms in our numerical simulations (Figures 3.3c and 3.3g). The sensitivity of the deep-water N:P ratio to R is negative because R represents the P:N ratio of organisms, and thus the two ratios vary in opposite directions. Changes in the deep-water N:P ratio follow almost perfectly those in phytoplankton N:P ratio, with the result that the absolute value of the sensitivity of the deep-water N:P ratio to phytoplankton P:N ratio is almost 1 for all the values of R tested (sensitivity of -1.01 whatever the value of R, Figure 3.2). Although the value of the deep-water N:P ratio is strongly dependent on non-fixer's N:P ratio, it is independent on that of N-fixers (no sensitivity for all the values of R F , Figure 3.2). This difference can be explained by the relative minority of N-fixers at the scale of the global ocean compared to non-fixers (Figures 3.3d and 3.3h), and thus their high N:P ratio has a negligible effect on the mean N:P ratio of phytoplankton in our numerical simulations.

The value of the deep-water N:P ratio depends on parameters related to the recycling of organic matter in the water column (Rec tot , p Reca and m). However, the effect of the fraction of organic matter recycled in the water column (p Reca ) and the mortality rate of phytoplankton (m) on the N:P ratio in the deep ocean strongly depends on the value of these two parameters, with almost no effect for more than half of the values tested (median sensitivities of 0.04 and -0.04 for Rec tot and m, respectively, Figure 3.2). The distribution of sensitivity is particularly spread for Rec tot , with 77 outliers corresponding to a sensitivity greater than 0.38. The deep-water N:P ratio will be higher when the fraction of recycling that occurs in the upper layer is low (median sensitivity of -0.44 for p Reca , Figure 3.2). The intensification of recycling to deep waters (i.e. an increase in Rec tot or a decrease in p Reca ) leads to an increase in their N:P ratio, since the mean N:P ratio of organisms is greater than that of deep waters. The maximal growth rate of non-fixers can also have a positive impact on the value of the deep-water N:P ratio, but only for some values (median sensitivity of 0.06 and 62 outliers above 0.50, Figure 3.2). As expected, the total denitrification has a negative effect on the value of the deep-water N:P ratio (sensitivity of -0.10 for all the values of D tot , Figure 3.

2).

To what extent are phytoplankton able to regulate the deep-water N:P ratio?

We now focus on understanding and quantifying the ability of phytoplankton to regulate the oceanic N and P cycles when nutrient supplies to the surface ocean are increased. Numerical simulations allow the strength of the regulation of N and P pools in the current ocean to be estimated quantitatively (Figure 3.3). An analysis of the flows between nutrient pools helps to better understand the results (Figure 3.4).

When the two phytoplankton groups coexist, P limits N-fixers and N limits non-fixers.

Any increase in the supply of a nutrient is consumed by those organisms whose growth is limited by that nutrient. Therefore regulation in the upper layer is perfect, and there is no effect on the accessible concentration of the other nutrient (Figure 3.3). Perfect regulation is explained by strong negative feedback loops between phytoplankton and the accessible pools (paths 2-3 in Figure 3.4). Thus, regulation of surface N:P ratio by phytoplankton is perfect with respect to changes in both N and P supplies (Figures 3.3c and 3.3g).

Regulation of P concentration in deep waters is either partial or negative because deep P concentration varies in the same direction as P supply (paths 1-2-6 and 1-7, Figure 3.4c).

In numerical simulations, regulation of deep P concentration against changes in P supply was found to be partial (ρ P i,P a = 0.13, Figure 3.3b). Changes in N supply have no effect on accessible P concentration (Figure 3.3f) because they do not affect total biomass (no arrow from accessible N to total biomass in Figure 3.4b).

Changes in N supply affect the competition between the two phytoplankton groups , leading to changes in the mean N:P ratio of organic matter and in N outflow (arrows 5, The deep-water N:P ratio is always over-regulated with respect to changes in N supply, because P concentration is not affected and N concentration is over-regulated. In numerical simulation, regulation of the N:P ratio in the deep layer with respect to changes in N supply was almost perfect, because N concentration is almost perfectly regulated (Figure 3.3f). Regulation of deep-water N:P ratio with respect to changes in P supply cannot be deduced from the analysis of flows between nutrient pools. Even though deep-water nutrient concentrations are affected substantially by changes in P supply (ρ P i,P a = 0.13 and ǫ N i,P a = 0.88), regulation of the N:P ratio in the deep layer was high in numerical simulations (ρ (P :N )i,P a = ρ (N :P )i,N a = 1.01, Figures 3.3c and 3.3g). Phytoplankton thus appear to efficiently regulate the deep-water N:P ratio with respect to changes in both N and P supplies in the current ocean.

Which mechanisms control the regulation efficiency of the deep-water N:P ratio?

Although N and P concentrations in deep waters vary with respect to changes in P supply, numerical simulations show that they vary in a nearly constant ratio (ρ (P :N )i,P a = 1.01, Figure 3.3c). The deep-water N:P ratio is indeed insensitive to both N and P supplies (S P and S N ) over an interval of ± 20 % around the value found in the literature for both supplies (Figure 3.2). Regulation of the deep-water N:P ratio with respect to changes in P supply is thus almost perfect for this set of parameter values. It is important to understand the mechanisms that could explain the almost perfect biotic regulation of deep-water N:P ratio with respect to changes in nutrient supplies, as they can be different from those setting the value of the deep-water N:P ratio itself. In this section, we will no longer focus on the mechanisms driving the value of the deep-water N:P ratio, but in the efficiency of its regulation when nutrient supplies change. The sensitivity analysis is thus performed on the regulation coefficient of the deep-water N:P ratio with respect to changes in P and N supplies.

The efficiency of this regulation appears to mainly be sensitive to recycling parameters.

Regulation will be more efficient when the fraction of organic matter recycled in the water column and the mortality rate of phytoplankton are high (positive sensitivity of ρ (P :N )i,P a to m and most of the values of Rec tot , Figure 3.5a), and when the fraction of recycling that occurs in the upper layer is low (negative sensitivity of ρ (P :N )i,P a to p Reca , Figure 3.5a). The effect of these three parameters in the direction indicated seems intuitive since the intensification of recycling to deep waters leads to an increase in the control of deep-water chemistry by organisms. However, the importance of the sensitivity of ρ (P :N )i,P a to these three parameters, and especially p Reca , strongly depends on the parameter value (difference of 1.07 between the first and the ninth decile, Figure 3.5a). Regulation of the deep-water N:P ratio with respect to changes in P supply also appears to be sensitive to the maximal growth rate of non-fixers, µ, again with important variations depending on the value. Surprisingly, the efficiency of this regulation appears to be insensitive to the N:P ratio of organisms for all the values used in our analysis (Figure 3.5a).

Variations in N supply are well-absorbed in the deep layer, with the result that the deep-water N:P ratio is almost perfectly regulated (ρ (N :P )i,N a = 1.01, Figure 3.3g). The strength of this regulation is insensitive to most of the parameters (no sensitivity of ρ (N :P )i,N a to 15 parameters for the intervals tested, Figure 3.5b). Although most of the values correspond to low sensitivities of ρ (N :P )i,N a , regulation will be more efficient when the fraction of organic matter recycled in the water column and the fraction of recycling that occurs in the upper layer are low (negative sensitivity of ρ (N :P )i,N a to Rec tot and p Reca , Figure 3.5b).

As changes in N supply modify the mean N:P ratio of organic matter and thus N outflow through recycling (Figure 3.4a), an intensification of recycling (i.e. an increase in Rec tot ) will further decrease the ability of organisms to regulate the deep-water N:P ratio, explaining the negative sensitivity of ρ (N :P )i,N a to Rec tot .

Discussion

Our model of coupled N and P biogeochemical cycles in the ocean predicts that phytoplankton should perfectly regulate N and P concentrations in the upper layer because of its top-down control on the upper, accessible nutrient pools. These results are in agreement with chemostat and resource-ratios theories, which predict that organisms consume as much of the limiting resources as possible at equilibrium (e.g. [START_REF] Tilman | Resources: a graphical mechanistic approach to competition and predation[END_REF][START_REF] Smith | The theory of the chemostat, dynamics of microbial competition[END_REF]; as a result, they are expected to absorb any variation in the supply of a limiting nutrient in the accessible pool. As the nutrient supplied to the ocean occurs in the upper layer that is accessible to autotrophic organisms, we might expect perfect regulation of the concentration of P and N concentrations in both the upper and deep layers against changes in their supply. By contrast, we showed that variations in P and N supplies to the surface ocean impact nutrient concentrations in the deep, inaccessible layer. This occurs because biotic recycling of organic matter plays the role of a nutrient supply to the deep-ocean, with the added complexity that the intensity of these biotic inflows depends on the biomass and N:P ratio of phytoplankton in the upper layer. Any change in the supply of either nutrient modifies the competition for P between non-fixers and N-fixers. For example, the addition of P to the surface ocean results in an increase in the growth rate of N-fixers and a decrease in that of non-fixers. The Tyrrell (1999) showed that phytoplankton control the deep-water N:P ratio through the competition between non-fixers and N-fixers. Nitrogen fixation thus adapts the concentration of N to the concentration of P in the surface waters, in a ratio that is transferred to the deep waters through recycling of organic matter. In his model, [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF] set the same N:P ratio for non-fixers and N-fixers, thus the mean N:P ratio of the organic matter remained constant whatever the proportion of N-fixers compared to non-fixers. However, the hypothesis that non-fixers and N-fixers have the same N:P ratio could strongly influence the results as it decreases the ability of phytoplankton to regulate the deep-water N:P ratio. We thus included this differentiation in our model to avoid a possible bias in the quantification of the control of phytoplankton on the deep-water N:P ratio. Our model predicts that the value of the deep-water N:P ratio is mainly controlled by that of non-fixers, as well as, to a lesser extent, by the intensity of recycling of organic matter and denitrification. These predictions are in agreement with previous studies suggesting that the similarity of N:P ratio of both phytoplankton and deep ocean could be related to a balance between denitrification and N-fixation (e.g. [START_REF] Redfield | The biological control of chemical factors in the environment[END_REF][START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Lenton | Biotic stoichiometric controls on the deep ocean N:P ratio[END_REF]. In numerical simulations with realistic data for N and P flows, the N:P ratio of deep waters is slightly lower than that of non-fixing phytoplankton, as observed in the ocean (e.g. [START_REF] Redfield | On the proportions of organic derivations in sea water and their relation to the composition of plankton[END_REF][START_REF] Redfield | The biological control of chemical factors in the environment[END_REF][START_REF] Karl | Total dissolved nitrogen and phosphorus concentrations at US-JGOFS station ALOHA: Redfield reconciliation[END_REF]Anderson and Sarmiento 1994).

We also studied more deeply the parameters, and thus the mechanisms, that are involved in the regulation of deep-water N:P ratio. In the second and third sections of the results, we focused on how the addition of N and P in the surface ocean by human activities affects the deep-water N:P ratio. Our detailed study of the mechanisms that control the regulation efficiency of deep-water N:P ratio allows us to better understand the observed near-constancy of this ratio and to predict its expected changes in the context of global change. Both N and P supplies affect deep N concentration but only P supply affects the deep P concentration.

Competition between the two types of phytoplankton with different N:P ratios keeps the total amount of P stored in phytoplankton constant independently of N supply, and thus P biotic flows are unaffected by variations in N supply. N concentration in deep waters also has a weak response to a 50 % increase in N supply (Figure 3.3). Thus, variations in N supply are well absorbed in the deep layer, contrary to variations in P supply. This difference is due to the compensatory dynamics between N-fixers and non-fixers, which makes the N cycle more adaptable than the P cycle [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Deutsch | Spatial coupling of nitrogen inputs and losses in the ocean[END_REF]. Surprisingly, our model predicts that competition between the two phytoplankton groups sets the efficiency of biotic regulation of the deep-water N:P ratio through recycling of organic matter, with no direct effect of the N:P ratio of phytoplankton and denitrification. This result is counterintuitive since one might expect the stoichiometry of phytoplankton to strongly influence their ability to regulate that of their environment.

In our model, we assume that Von Liebig's (1842) law of the minimum governs the growth of non-fixers, which implies that the growth of organisms is limited by a single nutrient at a given time. However, phytoplankton growth is limited by multiple resources simultaneously in some regions of the world's ocean (e.g. [START_REF] Arrigo | Marine microorganisms and global nutrient cycles[END_REF][START_REF] Elser | Global analysis of nitrogen and phosphorus limitation of primary producers in freshwater, marine and terrestrial ecosystems[END_REF]). In particular, colimitation of phytoplankton growth is commonly observed in oligrotrophic oceanic regions (e.g. [START_REF] Zohary | P-limited bacteria but N and P co-limited phytoplankton in the Eastern Mediterraneanâa microcosm experiment[END_REF][START_REF] Mills | Iron and phosphorus co-limit nitrogen fixation in the eastern tropical North Atlantic[END_REF]). The use of colimitation instead of Liebig's law may alter our results by affecting the dynamics of phytoplankton populations [START_REF] Poggiale | How far details are important in ecosystem modelling: the case of multi-limiting nutrients in phytoplankton-zooplankton interactions[END_REF][START_REF] Sperfeld | Multiple resource limitation theory applied to herbivorous consumers: Liebigâs minimum rule vs. interactive co-limitation[END_REF]). In our model, replacing Liebig's law by N and P colimitation for the growth of non-fixers may increase the strength of the competition with N-fixers, and thus the ability of phytoplankton to regulate nutrient pools might be strengthened.

In our model, we assumed different N:P ratios for the two phytoplankton groups considered (i.e. N-fixers and non-fixers), but fixed ratios within each group. Yet the stoichiometry of phytoplankton can change depending on nutrient limitation conditions (e.g.

Geider and La Roche 2002). Several models of phytoplankton stoichiometry allow N and P cell quotas to be adjusted depending on nutrient availability (Klausmeier et al. 2004a[START_REF] Klausmeier | Phytoplankton stoichiometry[END_REF][START_REF] Diehl | Flexible nutrient stoichiometry mediates environmental influences on phytoplankton and its resources[END_REF]. This plasticity of phytoplankton stoichiometry could alter their ability to control nutrient pools. Incorporating adaptable stoichiometry in the ocean model, however, leads to the same qualitative results as those presented in this paper (see Appendix 3.A), which is why we did not consider this complication here. For the sake of simplicity, our model only considers two groups of phytoplankton. Nonfixers and N-fixers are indeed two key functional groups in the nitrogen cycle. However, taking into account phytoplankton diversity more precisely might influence our predictions regarding the ability of phytoplankton to regulate N and P oceanic pools.

Global climate change is expected to increase water column stratification through increased sea surface temperature and decreased sea surface salinity [START_REF] Riebesell | Sensitivities of marine carbon fluxes to ocean change[END_REF][START_REF] Gruber | Warming up, turning sour, losing breath: ocean biogeochemistry under global change[END_REF][START_REF] Rees | Pressures on the marine environment and the changing climate of ocean biogeochemistry[END_REF]. The degree of stratification is captured by parameter K, i.e. the mixing coefficient between the surface and deep layers. Our sensitivity analyses show that the strength of vertical mixing does not affect the value and the regulation efficiency of the deep-water N:P ratio (Figure 3.5). However, the decrease in the depth of the upper layer induced by increasing stratification is likely to intensify the recycling flow to the deep layer (associated with a decrease in the parameter p Reca ) because sinking particles will take less time to reach the deep layer [START_REF] Riebesell | Sensitivities of marine carbon fluxes to ocean change[END_REF]. Since the sensitivity of regulation coefficients of the deep-water N:P ratio to p Reca is negative, the value of these regulation coefficients is then likely to increase, leading to a further enhancement of the current over-regulation of the deep-water N:P and P:N ratios with respect to changes in N and P supplies, respectively.

Thus, increasing water column stratification will likely result in stronger variability and lower stability of the N:P ratio in the deep ocean over long temporal and spatial scales.

We presented a simple compartment model for the global ocean. This model could also be useful to understand how regulation occurs spatially in the ocean, by integrating the dynamics of interactions between N-fixers and non-fixers in a general circulation model. Although similar results might be expected at the scale of the global ocean, a general circulation model is likely to reveal interesting differences in the regulation of the deep-water N:P ratio among oceanic regions (e.g. [START_REF] Pahlow | Temporal trends in deep ocean Redfield ratios[END_REF].

Figures and Tables

Figure The sensitivity of a variable to a parameter is measured as the elasticity of the variable with respect to the parameter. For each parameter, the local sensitivity is calculated for 500 values uniformly distributed in an interval of ± 20 % around the value used for numerical simulations. z tot is the depth of the water column and p za the fraction that is accessible. S N and S P are supplies of N and P, respectively. K is the vertical mixing coefficient. D tot and Rec tot are the total denitrification and recycling rates, respectively. p Da and p Reca are the fraction of denitrification and recycling in the upper layer, respectively. R and R F are the P:N ratio of non-fixers and N-fixers, respectively. m is the mortality rate and µ the maximum growth rate of non-fixers. cost is the metabolic cost of N-fixation. N H is the half saturation constant for N. q P and q N are adsorption rates of P and N, respectively. (a-d) and N supply in (e-h) are increased by 50% after 100,000 years. Simulations are performed with realistic parameter values. Bold lines are for the upper, accessible layer and dotted lines for the deep, inaccessible layer. (a,e) P concentrations (ρ = 1 and 0.13 in case of a 50%-increase in P supply in the upper and deep layers, respectively, and ǫ = 0 in both layers in case of a 50%-increase in N supply). (b,f) N concentrations (ǫ = 0 and 0.88 in case of a 50%-increase in P supply in the upper and deep layers, respectively, and ρ = 1 and 1.01 in case of a 50%-increase in N supply in the upper and deep layers, respectively). (c,g) N:P ratios (ρ = 1 in the upper layer and ρ = 1.01 in the deep layer in both cases). (d,h) Ratio of non-fixing over N-fixing phytoplankton (B/B F ). The sensitivity of a variable to a parameter is measured as the elasticity of the variable with respect to the parameter. For each parameter, the local sensitivity is calculated for 500 values uniformly distributed in an interval of ± 20 % around the value used for numerical simulations. z tot is the depth of the water column and p za the fraction that is accessible. S N and S P are supplies of N and P, respectively. K is the vertical mixing coefficient. D tot and Rec tot are the total denitrification and recycling rates, respectively. p Da and p Reca are the fraction of denitrification and recycling in the upper layer, respectively. R and R F are the P:N ratio of non-fixers and N-fixers, respectively. m is the mortality rate and µ the maximum growth rate of non-fixers. cost is the metabolic cost of N-fixation. N H is the half saturation constant for N. q P and q N are adsorption rates of P and N, respectively. a -1 4 growth rate of N-fixers: 66-117 [START_REF] Masotti | Photosynthetic characteristics of Trichodesmium in the southwest Pacific Ocean: importance and significance[END_REF])

N H
Half saturation constant of growth of non-fixing phytoplankton for nitrogen µmolN.m -3 1,500 1,429-14,290 [START_REF] Sterner | Algal growth in warm temperate reservoirs: kinetic examination of nitrogen, temperature, light, and other nutrients[END_REF] 1,000 [START_REF] Palmer | Production and export in a global ocean ecosystem model[END_REF] 1,600 [START_REF] Sarthou | Growth physiology and fate of diatoms in the ocean: a review[END_REF] 1,030-2,640 [START_REF] Timmermans | Physiological responses of three species of marine pico-phytoplankton to ammonium, phosphate, iron and light limitation[END_REF]) When units in the literature were different from those used here, we used the following assumptions to convert them: a Molar weights of N and P are 14 g.mol -1 and 31 g.mol -1 , respectively. b The ocean surface is 361x10 12 m 2 . c The volume of the upper and deep layers are 54x10 15 m 3 and 1.292x10 18 m 3 , respectively. d Total oceanic primary production corresponds to 8,800 TgN.a -1 . Table S3.1: Distribution of the sensitivity of the deep-water N:P ratio (N i /P i ), its regulation with respect to changes in P supply (ρ (P :N )i,P a ), and its regulation with respect to changes in N supply (ρ (N :P )i,N a ). a are the first and the ninth deciles, respectively. q 1 and q 3 are the first and the third quartiles, respectively. med is the median value. Parameters without any influence on the variables are not included in the table. D tot and Rec tot are the total denitrification and recycling rates, respectively. p Da and p Reca are the fraction of denitrification and recycling in the upper layer, respectively. R is the P:N ratio of non-fixers. m is the mortality rate and µ the maximum growth rate of non-fixers. cost is the metabolic cost of N-fixation.

Variable

Q N and Q P are the cell quotas of non-fixers. Q N F and Q P F are the cell quotas for N-fixers.

Q N F is supposed to be constant, as N 2 is in excess in the ocean and thus the growth of N-fixers is only limited by P. Q min,N and Q min,P are the minimum quotas at which the growth of non-fixers ceases. Q min,P F is the minimum quota at which the growth of N-fixers ceases. V max,N and V max,P are the maximum uptake efficiencies of non-fixers for N and P, respectively. V max,P F is the maximum uptake efficiency of non-fixers for P. The P:N ratio of

non-fixers is R = Q N /Q P and that of N-fixers is R F = Q P F /Q N F .
Equations for cell quotas are:

dQ N dt = g N (N a ) -GQ N dQ P dt = g P (P a ) -GQ P dQ P F dt = g P F (P a ) -G F Q P F (3.8)
where:

g N (N a ) = V max,N N a N a + N H g P (P a ) = V max,P P a P a + RN H g P F (P a ) = V max,P F P a P a + RN H G = µmin 1 - Q min,P Q P , 1 - Q min,N Q N G F = (µ -cost) 1 - Q min,P F Q P F (3.9)
Model equations are:

dN a dt = S N + K p za z tot (N i -N a ) -q N N a + m(p Reca Rec tot -p Da D tot )(B + B F ) - g N (N a )B Q N dP a dt = S P + K p za z tot (P i -P a ) -q P P a + mp Reca Rec tot (RB + R F B F ) - g P (P a )RB Q P - g P F (P a )R F B F Q P F dN i dt = K (1 -p za )z tot (N a -N i ) -q N N i + m p za 1 -p za ((1 -p Reca )Rec tot -(1 -p Da )D tot )(B + B F ) dP i dt = K (1 -p za )z tot (P a -P i ) -q P P i + mRec tot p za 1 -p za (1 -p Reca )(RB + R F B F ) dB dt = (G -m)B dB F dt = (G F -m)B F (3.10)
For the coexistence equilibrium, nutrient quotas are:

Q N,eq = µQ min,N µ -m Q P,eq = (µ -cost)Q min,P F V max,P V max,P F (µ -cost -m) Q P F,eq = (µ -cost)Q min,P F µ -cost -m (3.11)
For the coexistence equilibrium, concentrations and nutrient quotas are:

P a,eq = mN H Q P F,eq R eq V max,P F -mQ P F,eq N a,eq = mN H Q N,eq V max,N -mQ N,eq P i,eq = -p za Rec tot S P z tot (1 -p Reca ) + P a,eq [K(Rec tot -1) + p za q P Rec tot z tot (1 -p Reca )] K(Rec tot -1) + q P z tot (1 -p za )(p Reca Rec tot -1)

N i,eq = KR F,eq N a,eq + mp za z tot [Rec tot (1 -p Reca ) -D tot (1 -p Da )][B totP,eq + B eq (R F,eq -R eq )] R F,eq [q N z tot (1 -p za ) + K] B eq = mp za z tot B totP,eq {(p Reca Rec tot -p Da D tot )[q N z tot (1 -p za ) + K] + K[Rec tot (1 -p Reca ) -D tot (1 -p Da )]} +p za R F,eq S N z tot [q N z tot (1 -p za ) + K] -q N R F,eq N a,eq [p za q N z 2 tot (1 -p za ) + Kz tot ] mp za z tot {R F,eq [q N z tot (1 -p za ) + K] + (R eq -R F,eq )[K(Rec tot -D tot ) + q N z tot (1 -p za )(p Reca Rec tot -p Da D tot )]} B F,eq =
B totP,eq -R eq B eq R F,eq (3.12)

where B totP,eq = -p za S P z tot [q P z tot (1 -p za ) + K] + q P P a,eq [p za q P z 2 tot (1 -p za ) + Kz tot ] mp za z tot [K(Rec tot -1) + q P z tot (1 -p za )(p Reca Rec tot -1)] The equilibrium values of cell quotas do not depend on nutrient supplies (S N and S P ), thus R eq and R F,eq values are independant of nutrient supplies to the surface ocean. N a,eq and P a,eq have a different formulation in the model with variable stoichiometry, but are still perfectly regulated with respect to changes in N and P supplies. N i,eq and P i,eq have the same formulation as in the model with fixed stoichiometry, except that R and R F are replaced by R eq and R F,eq . As R eq and R F,eq do not depend on nutrient supply, regulation coefficients will be similar to those calculated in the model with fixed stoichiometry. Thus results on regulation coefficients do not qualitatively differ if phytoplankton stoichiometry is fixed or variable.

Results can be quantitatively different, depending on the parameters proper to cell quotas formulation. However, it seems reasonable that R eq and R F,eq will have similar values as those of R and R F in our model, and thus results are likely to be quantitatively similar with both formulations of phytoplankton stoichiometry.

Résumé

La limitation ultime (i.e. sur de grandes échelles de temps) de la production primaire océanique est débattue depuis plusieurs décennies (e.g. [START_REF] Broecker | Tracers in the sea[END_REF]Smith 1984;[START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Moore | Processes and patterns of oceanic nutrient limitation[END_REF].

Les cycles océaniques des nutriments subissent d'importantes modifications dues aux activités anthropiques, principalement par l'augmentation des apports de nutriments dans l'océan de surface (e.g. [START_REF] Schlesinger | Biogeochemistry: An analysis of global change[END_REF][START_REF] Seitzinger | Sources and delivery of carbon, nitrogen, and phosphorus to the coastal zone: An overview of Global Nutrient Export from Watersheds (NEWS) models and their application[END_REF]. Il est donc important d'identifier quel nutriment limite la production primaire océanique sur de grandes échelles de temps, car l'augmentation de ses apports pourrait altérer la structure et le fonctionnement des écosystèmes océaniques à long terme [START_REF] Vitousek | Terrestrial phosphorus limitation: mechanisms, implications, and nitrogen-phosphorus interactions[END_REF].

Dans ce chapitre, nous étudions quel nutriment entre le phosphore (P), l'azote (N) et le fer (Fe) correspond à la limitaton ultime de la production primaire totale océanique. Sperfeld et al. 2012), et (3) une formulation basée sur les modèles de la théorie DEB (Dynamic Energy Budget; [START_REF] Kooijman | The Synthesizing Unit as model for the stoichiometric fusion and branching of metabolic fluxes[END_REF][START_REF] Poggiale | How far details are important in ecosystem modelling: the case of multi-limiting nutrients in phytoplankton-zooplankton interactions[END_REF][START_REF] Sperfeld | Multiple resource limitation theory applied to herbivorous consumers: Liebigâs minimum rule vs. interactive co-limitation[END_REF]. Chaque version du modèle a été paramétrée avec des données issues de la littérature.

Nous avons inclus le cycle du

Lorsque la croissance des non-fixateurs est limitée par le N et que celle des fixateurs est limitée par le P ou le Fe, notre modèle prédit que le nutriment qui limite la croissance des fixateurs est celui qui limite la production primaire totale à de grandes échelles de temps.

Nous avons aussi montré que la production primaire océanique peut être co-limitée par le Fe et le P à de grandes échelles de temps, que la croissance du phytoplancton soit limitée localement par un seul nutriment ou par plusieurs simultanément. Les apports de N n'impactent au contraire que très faiblement la production primaire totale, ce qui peut être expliqué par le fait que la fixation de N 2 permet d'augmenter la quantité de N disponible quand il est peu abondant [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Deutsch | Spatial coupling of nitrogen inputs and losses in the ocean[END_REF]). L'augmentation des apports anthropogéniques de P et de Fe dans l'océan de surface va donc probablement affecter les écosystèmes océaniques à de grandes échelles de temps et d'espace.

Introduction

Nutrient limitation impacts heavily on multiple biological and ecological processes in ecosystems, and thus it is important to assess which processes are limited by which nutrients [START_REF] Arrigo | Marine microorganisms and global nutrient cycles[END_REF][START_REF] Vitousek | Terrestrial phosphorus limitation: mechanisms, implications, and nitrogen-phosphorus interactions[END_REF][START_REF] Moore | Processes and patterns of oceanic nutrient limitation[END_REF]. The limitation of the growth of autotrophic organisms can be studied at different spatial and temporal scales [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Vitousek | Terrestrial phosphorus limitation: mechanisms, implications, and nitrogen-phosphorus interactions[END_REF]. Proximate limiting nutrients (PLNs) limit primary production at small timescales, and their addition stimulates biological processes directly. In contrast, ultimate limiting nutrients (ULNs) control primary production at large timescales, i.e. over hundreds or thousands of years [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Vitousek | Terrestrial phosphorus limitation: mechanisms, implications, and nitrogen-phosphorus interactions[END_REF]. Note that PLNs and ULNs typically concern total primary production in ecosystems, not the growth of single species or functional groups. PLNs and ULNs can be different in the same ecosystem as nutrients can affect primary production indirectly by constraining the biogeochemical cycles of other nutrients at long timescales [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Moore | Processes and patterns of oceanic nutrient limitation[END_REF]. Elucidating the long-term biogeochemical controls and feedbacks on primary production in the global ocean is crucial to understand how marine ecosystems will respond to global environmental changes, as nutrients are increasingly supplied to the surface ocean [START_REF] Smith | Eutrophication: impacts of excess nutrient inputs on freshwater, marine, and terrestrial ecosystems[END_REF].

Proximate limitation of primary production has been studied for more than a century, first in the field of agricultural sciences (Von Liebig 1842). Liebig's law of the minimum states that the growth of organisms is limited by a single nutrient, although proximate colimitation by multiple nutrients also occurs [START_REF] Saito | Some thoughts on the concept of colimitation: Three definitions and the importance of bioavailability[END_REF]). In the ocean, local primary production is often either limited by N, P or Fe [START_REF] Wu | Phosphate depletion in the Western North 197 Atlantic Ocean[END_REF][START_REF] Moore | Iron cycling and nutrient-limitation patterns in surface waters of the World Ocean[END_REF][START_REF] Moore | Relative influence of nitrogen and phosphorus availability on phytoplankton physiology and productivity in the oligotrophic sub-tropical North Atlantic Ocean[END_REF], or colimited by N and P or P and Fe [START_REF] Mills | Iron and phosphorus co-limit nitrogen fixation in the eastern tropical North Atlantic[END_REF][START_REF] Thingstad | Nature of phosphorus limitation in the ultraoligotrophic Eastern Mediterranean[END_REF][START_REF] Moore | Relative influence of nitrogen and phosphorus availability on phytoplankton physiology and productivity in the oligotrophic sub-tropical North Atlantic Ocean[END_REF]). In the global ocean, it is generally accepted that N is the PLN, because it limits the growth of non-fixers and thus it controls competition between N-fixers and non-fixers [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF]Lenton and Watson 2000a).

In contrast to the PLN, determining the ULN in an ecosystem cannot be addressed experimentally due to the large timescale involved; therefore it is usually studied using theoretical models [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF]Lenton and Watson 2000a;[START_REF] Moore | Iron availability limits the ocean nitrogen inventory stabilizing feedbacks between marine denitrification and nitrogen fixation[END_REF]. Until recently, the debate was focused on whether N or P is the ULN in the global ocean [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF]Lenton and Watson 2000a). The traditional view of marine biologists is that N limits oceanic primary production at large spatial and temporal scales, because the N:P ratio of terrestrial inputs to the ocean is low and P regenerates faster than ammonia from the decomposition of organic matter [START_REF] Ryther | Nitrogen, phosphorus, and eutrophication in the coastal marine environment[END_REF]. The classical view of marine geochemists is to consider P as the ULN in the ocean, because P has a longer residence time than N, and there is no atmospheric source of P [START_REF] Redfield | The biological control of chemical factors in the environment[END_REF][START_REF] Broecker | Tracers in the sea[END_REF][START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF]. Recent studies propose that Fe might be the ULN in oceanic systems, because N-fixation can increase the availability of N at large timescales but is often limited by Fe at local and global scales [START_REF] Falkowski | Evolution of the nitrogen cycle and its influence on the biological sequestration of CO2 in the ocean[END_REF][START_REF] Mills | Iron and phosphorus co-limit nitrogen fixation in the eastern tropical North Atlantic[END_REF][START_REF] Moore | Iron availability limits the ocean nitrogen inventory stabilizing feedbacks between marine denitrification and nitrogen fixation[END_REF], and Fe supply to the surface ocean is often low and thus could drive oceanic primary production down at geological timescales [START_REF] Wu | Phosphate depletion in the Western North 197 Atlantic Ocean[END_REF]. Thus, ultimate limitation of oceanic primary production is still an unresolved issue.

Previous modelling studies, however, have a number of limitations. First, they have not considered the hypothesis that colimitation could also operate at large spatial and temporal scales. Second, they have relied on Liebig's law of the minimum to model the growth of organisms. Their results might be different if phytoplankton growth is assumed to be limited by several nutrients simultaneously. Third, few studies consider Fe as a possible ULN for global oceanic primary production [START_REF] Moore | Iron availability limits the ocean nitrogen inventory stabilizing feedbacks between marine denitrification and nitrogen fixation[END_REF]. As Fe is a major element controlling phytoplankton growth [START_REF] Falkowski | Biogeochemical controls and feedbacks on ocean primary production[END_REF][START_REF] Morel | The biogeochemical cycles of trace metals in the oceans[END_REF] just as are P and N [START_REF] Elser | Global analysis of nitrogen and phosphorus limitation of primary producers in freshwater, marine and terrestrial ecosystems[END_REF], it is unclear which of these three nutrients is the ULN in the global ocean. We address this issue by building a model for the coupled N, P and Fe cycles in the ocean. We use several alternative formulations for the growth rate of both N-fixers and non-fixers -i.e., Liebig's law of the minimum (Von Liebig 1842) and proximate colimitation by multiple nutrients [START_REF] Kooijman | The Synthesizing Unit as model for the stoichiometric fusion and branching of metabolic fluxes[END_REF][START_REF] Poggiale | How far details are important in ecosystem modelling: the case of multi-limiting nutrients in phytoplankton-zooplankton interactions[END_REF][START_REF] Sperfeld | Multiple resource limitation theory applied to herbivorous consumers: Liebigâs minimum rule vs. interactive co-limitation[END_REF]) -to check the robustness of our results, and parametrized our model with existing data (Table 4.1).

Methods

Model

Our model describes the biogeochemical cycles of N, P and Fe in the ocean (

dN a dt = S N + K z tot p za (N i -N a ) -q N N a + m(Rec tot p Reca -D tot p Da )(B + B F ) -GB dP a dt = S P + K z tot p za (P i -P a ) -q P P a + mRec tot p Reca (R 1 B + R 1F B F ) -GR 1 B -G F R 1F B F dF e a dt = S F e + K z tot p za (F e i -F e a ) -q f e F e a + mRec tot p Reca (R 2 B + R 2F B F ) -GR 2 B -G F R 2F B F dN i dt = K z tot (1 -p za ) (N a -N i ) -q N N i + p za 1 -p za m [Rec tot (1 -p Reca ) -D tot (1 -p Da )] (B + B F ) dP i dt = K z tot (1 -p za ) (P a -P i ) -q P P i + p za 1 -p za mRec tot (1 -p Reca )(R 1 B + R 1F B F ) dF e i dt = K z tot (1 -p za ) (F e a -F e i ) -q F eF e i + p za 1 -p za mRec tot (1 -p Reca )(R 2 B + R 2F B F ) dB dt = (G -m)B dB F dt = (G F -m)B F (4.1)
Non-fixers, whose concentration in the upper layer is B, consume N, P and Fe in that layer. N-fixers, whose concentration in the upper layer is B F , consume P and Fe in that layer. Nitrogen fixation is assumed to be the only source of N for N-fixers. The growth rate of non-fixers and N-fixers (G and G F , respectively) is modelled with several different formulations to determine if the ULN obtained is dependent on a specific formulation of phytoplankton growth rates. We consider cases where the growth rate of each phytoplankton group is limited by (1) only one nutrient (i.e. Liebig's law of the minimum; Von Liebig 1842), and (2) multiple nutrients (i.e. multiple limitation hypothesis; [START_REF] Kooijman | The Synthesizing Unit as model for the stoichiometric fusion and branching of metabolic fluxes[END_REF][START_REF] Saito | Some thoughts on the concept of colimitation: Three definitions and the importance of bioavailability[END_REF][START_REF] Sperfeld | Multiple resource limitation theory applied to herbivorous consumers: Liebigâs minimum rule vs. interactive co-limitation[END_REF]).

Liebig's law of the minimum

In this section, we focus on cases where the growth rate of each phytoplankton group is limited by a single nutrient. We assume that Liebig's law of the minimum (Von Liebig 1842)

governs the growth of each phytoplankton group. Functional responses were modelled with a Michaelis-Menten function. G Liebig and G F,Liebig are the growth rates of non-fixers and N-fixers, respectively, when Liebig's law of the minimum is used:

G Liebig = µ min P a P a + P H , N a N a + P H /R 1 , F e a F e a + P H R 2 /R 1 G F,Liebig = (µ -cost) min P a P a + P H , F e a F e a + P H R 2F /R 1F (4.2)
N fixation is energetically more costly than mineral N uptake. The maximal growth rate of N-fixers is obtained by subtracting a given cost (cost) from the maximal growth rate of non-fixers (µ). P H is the half-saturation constant of phytoplankton for P, which is assumed to be identical in N-fixers and non-fixers. For consistency with the stoichiometry of non-fixers, the half-saturation constants for N and Fe are fixed to P H /R 1 and P H * R 2 /R 1 , respectively. In the same way, for consistency with the stoichiometry of N-fixers, the half-saturation constant Fe is fixed to P H * R 2F /R 1F .

We can distinguish three cases of limitation by a single nutrient: (1) non-fixers are N-limited and N-fixers are P-limited, (2) non-fixers are N-limited and N-fixers are Fe-limited,

(3) non-fixers are P-limited and N-fixers are Fe-limited. As N-fixers have to assimilate more Fe than do non-fixers, the case where non-fixers are Fe-limited is not studied because both groups would be limited by Fe leading to the extinction of N-fixers.

Proximate colimitation (Multiple limitation hypothesis)

In this section, we consider that multiple nutrients limit phytoplankton growth. The growth of non-fixers is then limited by N, P and Fe simultaneously, whereas that of N-fixers is limited by P and Fe because N 2 is supposed to be their only source of N and is in excess in the ocean. We use two alternative formulations for the functional response of phytoplankton to the availability of nutrients in the environment in the proximate colimitation case: (1) Monod multiplicative form [START_REF] Saito | Some thoughts on the concept of colimitation: Three definitions and the importance of bioavailability[END_REF][START_REF] Sperfeld | Multiple resource limitation theory applied to herbivorous consumers: Liebigâs minimum rule vs. interactive co-limitation[END_REF]) and (2) a formula derived from a synthesizing unit model [START_REF] Kooijman | The Synthesizing Unit as model for the stoichiometric fusion and branching of metabolic fluxes[END_REF][START_REF] Poggiale | How far details are important in ecosystem modelling: the case of multi-limiting nutrients in phytoplankton-zooplankton interactions[END_REF]).

The growth rates of non-fixers and N-fixers modelled with the Monod multiplicative form (G mult and G F,mult , respectively) are:

G mult = µ * P a P a + P H * N a N a + P H /R 1 * F e a F e a + P H R 2 /R 1 G F,mult = (µ -cost) * P a P a + P H * F e a F e a + P H R 2F /R 1F (4.3)
The formulation of the growth rate of the two phytoplankton groups based on the synthesizing unit model is detailed in Appendix 4.A in Supporting Information.

Statistics

The ultimate limiting nutrient of autotrophs is determined by performing a sensitivity analysis of primary production at equilibrium with respect to N, P and Fe supplies. Primary production is obtained by multiplying the growth rate of phytoplankton and their biomass.

For the supply of each nutrient, we used a set of 200 values uniformly distributed between the bounds found in the literature (Table 4.1). By performing numerical simulations until the equilibrium is reached, we then assigned a value of oceanic primary production to each couple of supplies (S N , S P ), (S N , S F e ) and (S P , S F e ) in the cases where Liebig's law is used, and to each triplet of supplies (S N , S P , S F e ) in cases where proximate colimitation is used. To assess the relative influence of each supply on the oceanic primary production at equilibrium, we calculated Sobol first-order and total sensitivity indices [START_REF] Cariboni | The role of sensitivity analysis in ecological modelling[END_REF]). The Sobol first-order sensitivity index measures the effect of variations in a single parameter while the others are fixed (Table 4.2). The Sobol total sensitivity index accounts for all the contributions of a parameter to variations in the output, i.e. the first-order effect plus all its interactions.

Tracking the impact of changes in the supply of each nutrient on the primary production of non-fixers and N-fixers helps to better understand the effects of nutrient supplies on the two functional groups separately and their contributions to variations in total primary production (Figures 4.2 and 4.3,Supplementary Figure S4.1). When proximate colimitation is used, we studied the impact of the supply of each nutrient on primary production by fixing the others, and primary production values were zero-centred to allow a comparison between functional groups and phytoplankton as a whole (Figure 4.3, Supplementary Figure S4.1).

Results

We first consider the results of the model when phytoplankton growth is limited by a single nutrient (Liebig's law of the minimum) as they are easier to interpret. In numerical simulations, a low supply of the nutrient that limits the growth of one phytoplankton group can lead to its extinction (Figures 4.2b,4.2e,4.2g and 4.2h), but we will focus on the cases where non-fixers and N-fixers coexist. When N limits the growth of non-fixers, the increase in N supply slightly increases the growth of non-fixers (Figures 4.2a and 4.2d) and decreases that of , thus total primary production remains constant (Figures 4.2c and 4.2f). In contrast, an increase in either P or Fe supply enhances the growth of N-fixers (Figures 4.2b and 4.2e). The supply of dissolved N induced by the increasing N-fixation then leads to an increase in the growth of non-fixers (Figures 4.2a and 4.2d), resulting in an increase in total primary production (Figures 4.2c and 4.2e). As the impact of an increase in N supply on total primary production is negligible compared with that of an increase in either P or Fe supply (Figures 4.2c and 4.2f), the nutrient that proximately limits the growth of N-fixers (either P or Fe) is the ULN (Table 4.2). In the case where P limits the growth of non-fixers and Fe limits that of N-fixers, an increase in P supply leads to an increase in the growth of non-fixers and a decrease in that of . The opposite occurs when Fe supply is increased (Figures 4.2g and 4.2h). An increased supply of either nutrient results in an increase in total primary production (Figure 4.2i). The external supply of Fe explains 46 % of the variance of total primary production across all numerical simulations, against 44 % for P supply (Table 4.2). The remaining 10 % of the variance are due to interactions between P and Fe supplies. Thus, counterintuitively, proximate limitation of phytoplankton growth by a single nutrient may lead to ultimate colimitation of the oceanic primary production by P and Fe.

Results of simulations when the growth of phytoplankton is proximately limited by several nutrients are similar with both formulations (Figure 4.3, Supplementary Figure 4). When the growth of phytoplankton is proximately colimited by multiple nutrients, increases in N, P and Fe supplies all have a positive impact on the production of non-fixers, but virtually no impact on that of N-fixers (Figure 4.3). As a result, changes in total primary production follow those in the production of non-fixers. Absolute values on the y-axes indicate that changes in total primary production are more important with respect to Fe and P supplies than N supply (Figure 4.3). Sobol indices show that P and Fe have a strong impact on total oceanic primary production at equilibrium in the model when phytoplankton growth is limited by multiple nutrients (Table 4.2). Fe explains 79 % of the variance of total primary production, against 10-11 % for P supply and only 1 % for N supply. The remaining 10 % of the variance of total primary production are explained by interactions between supplies of the three nutrients, mainly between the supplies of P and Fe. Thus, the model with proximate colimition of phytoplankton growth also predicts ultimate colimitation of total primary production of phytoplankton by P and Fe, but with a dominant effect of Fe (Table They also confirm the importance of P [START_REF] Redfield | The biological control of chemical factors in the environment[END_REF][START_REF] Broecker | Tracers in the sea[END_REF][START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF]Lenton and Watson 2000a). By contrast, our model shows that N does not have a notable impact on oceanic primary production in the long run, with Sobol indices close to zero (Table 4.2). Thus N does not seem to play an important role in the ultimate limitation of primary production [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF], which contradicts the classical view of ocean biologists [START_REF] Ryther | Nitrogen, phosphorus, and eutrophication in the coastal marine environment[END_REF]. This result, however, makes sense as N-fixation allows N to enter the system when it is scarce [START_REF] Redfield | The biological control of chemical factors in the environment[END_REF][START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF].

The fact that ultimate colimitation of primary production arises from a model that assumes proximate limitation of autotrophs by a single nutrient is of special interest as previous studies have assumed that oceanic primary production is ultimately limited by a single nutrient [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF]Lenton and Watson 2000a;[START_REF] Moore | Iron availability limits the ocean nitrogen inventory stabilizing feedbacks between marine denitrification and nitrogen fixation[END_REF]. Our model shows that ultimate limitation of the total oceanic primary production is more complex than previously thought, and involves several nutrients simultaneously. Colimitation is generally considered to occur only at small timescales in oceans [START_REF] Mills | Iron and phosphorus co-limit nitrogen fixation in the eastern tropical North Atlantic[END_REF][START_REF] Thingstad | Nature of phosphorus limitation in the ultraoligotrophic Eastern Mediterranean[END_REF][START_REF] Moore | Relative influence of nitrogen and phosphorus availability on phytoplankton physiology and productivity in the oligotrophic sub-tropical North Atlantic Ocean[END_REF][START_REF] Saito | Some thoughts on the concept of colimitation: Three definitions and the importance of bioavailability[END_REF]), but our model predicts that it also operates at large timescales.

The conclusion that both P and Fe have a significant impact on oceanic primary production at large spatial and temporal scales has important implications for the responses of marine ecosystems to global change. The increased supply of Fe and P to the surface ocean due to anthropogenic activities affects ocean ecosystems at small spatial and temporal scales, e.g. through changes in species composition or seawater chemical composition [START_REF] Smith | Eutrophication: impacts of excess nutrient inputs on freshwater, marine, and terrestrial ecosystems[END_REF]), but it is also likely to strongly affect the structure and functioning of ocean ecosystems at large spatial and temporal scales. One important potential consequence of increased primary production is an increased occurrence of dead zones in oceans, which in turn may alter the global oxygen cycle, increase nitrous oxide emissions from denitrification of organic matter [START_REF] Diaz | Spreading dead zones and consequences for marine ecosystems[END_REF], and exacerbate global climate change. q P Adsorption rate of phosphorus a -1 10 -5 2.4x10 -5 [START_REF] Slomp | The global marine phosphorus cycle: sensitivity to oceanic circulation[END_REF] q F e Adsorption rate of iron a -1 10 -4 1.24x10 Half saturation constant of growth of phytoplankton for phosphorus µmolP.m -3 125 240 [START_REF] Sarthou | Growth physiology and fate of diatoms in the ocean: a review[END_REF] 14-94 [START_REF] Timmermans | Physiological responses of three species of marine pico-phytoplankton to ammonium, phosphate, iron and light limitation[END_REF] When units in the literature were different from those used here, we used the following assumptions to convert them: * The ocean surface is 361.10 12 m 2 , † Molar weight of N is 14 g.mol -1 , ‡ The volume of the upper and deep layers are 54x10 15 m 3 and 1.292x10 18 m 3 , respectively, § Total oceanic primary production corresponds to 8,800 TgN.a -1 . S N , S P and S F e are the supplies of nitrogen, phosphorus and iron, respectively, to the surface ocean. Simulations are performed with realistic parameter values. Total Sobol indices are indicated in parentheses. An important difference between first order and total Sobol indices implies that the interactions between the nutrient concerned and the others have an important impact on primary production.

Figures and Tables

Table S4.1: Sobol indices for the sensitivity of the total biomass at equilibrium with respect to nutrient supplies for the model with variable phytoplankton stoichiometry (Appendix 3.B). S N , S P and S F e are the supplies of nitrogen, phosphorus and iron, respectively, to the surface ocean. Simulations are performed with realistic parameter values. When different from the first order indices, total Sobol indices are indicated in parentheses. An important difference between first order and total Sobol indices implies that the interactions between the nutrient concerned and the others have an important impact on primary production. µ -cost = k F Θ tot,F , we obtain the uptake flux of P and Fe (i.e. the growth rate of N-fixers): G F = (µ -cost)Θ P,F e,eq /Θ tot,F .

Growth rate of non-fixers

The growth of non-fixers is limited by N, P and Fe simultaneously. It corresponds to type I colimitation 7 . As in the previous section, we assume that enzymes are synthesizing units and be dissociated, and that the binding rate for each nutrient is constant. where

N H = P H /R 1 , F e H = P H R 2 /R 1
Equations for cell quotas are:

dQ N dt = g N (N a ) -GQ N dQ P dt = g P (P a ) -GQ P dQ P F dt = g P F (P a ) -G F Q P F dQ F e dt = g F e (F e a ) -GQ F e dQ F eF dt = g F eF (F e a ) -G F Q F eF (4.7)
where:

g N (N a ) = V max,N N a N a + P H /R 1 g P (P a ) = V max,P P a P a + P H g F e (F e a ) = V max,F e F e a F e a + P H R 2 /R 1 g P F (P a ) = V max,P F P a P a + P H g F eF (F e a ) = V max,F eF F e a F e a + P H R 2F /R 1F (4.8)
When Liebig's law of the minimum is used, the growth rates of non-fixers and N-fixers are:

G Liebig = µmin 1 - Q min,N Q N , 1 - Q min,P Q P , 1 - Q min,F e Q F e G F,Liebig = (µ -cost)min 1 - Q min,P F Q P F , 1 - Q min,F eF Q F eF (4.9)
The growth rates of non-fixers and N-fixers modelled with the Monod multiplicative form are:

G mult = µ 1 - Q min,N Q N 1 - Q min,P Q P 1 - Q min,F e Q F e G F,mult = (µ -cost) 1 - Q min,P F Q P F 1 - Q min,F eF Q F eF (4.10)
In numerical simulations, we set V max,P = V max,P F =4.5x10 -3 vµmolP.cell -1 .a -1 , Q min,P =1.64x10 -9 µmolP.cell -1 and Q min,P F =1.56x10 -9 µmolP.cell -1 (Klausmeier et al. 2004b). For consistency with the model of fixed stoichiometry, we used Q min,N = Q min,P /R 1 , one of which is accessible to organisms and the other is inaccessible. Our model predicts that accessible nutrient concentration is perfectly regulated with respect to changes in the supplies of the nutrient to both accessible and inaccessible nutrient pools, because organisms exert a top-down control on the accessible pool. This result is in agreement with chemostat and resource-ratio theories, which predict that organisms consume the limiting resource in the greater quantity possible (e.g. [START_REF] Tilman | Resources: a graphical mechanistic approach to competition and predation[END_REF][START_REF] Smith | The theory of the chemostat, dynamics of microbial competition[END_REF]. However, the generic model shows that the concentration of the limiting nutrient in the inaccessible pool is only partially regulated by organisms, and variations in nutrient supply can even be amplified (i.e.

negative regulation). Moreover, organisms cannot at the same time exert a strong control on the accessible nutrient pool and efficiently regulate the inaccessible nutrient pool. These results suggest that organisms cannot exert a strong regulation of nutrient pools in the whole system due to the inaccessibility of part of the resources.

In Chapter 1, we focused on biotic regulation of the concentrations of a limiting nutrient with respect to changes in its supply. However, the ability of autotrophs to regulate nutrient concentrations in their environment should be different for a limiting and a non-limiting nutrient. Interactions between nutrient cycles can also alter the concentration of nutrients in the environment, and thus their regulation by organisms. In Chapter 2, we developed a stoichiometrically explicit model to assess how autotrophs can regulate the concentration of a non-limiting nutrient, and how interactions between the cycle of a limiting nutrient and that of a non-limiting nutrient can affect biotic regulation at global scales. This model of resource access limitation describes the dynamics of an autotroph population and two nutrients, one which limits the growth of autotrophs and the other that is in excess. Our model shows that, intuitively, the regulation of the non-limiting nutrient is weaker than that of the limiting nutrient, in both accessible and inaccessible pools. This result, however, might be quantitatively different if we were to consider colimitation of autotroph growth by both nutrients simultaneously. Regulation of the less limiting nutrient might then be enhanced, as its addition would slightly stimulate the growth of autotrophs (e.g. [START_REF] Sperfeld | Multiple resource limitation theory applied to herbivorous consumers: Liebigâs minimum rule vs. interactive co-limitation[END_REF]. Changes in the supply of the non-limiting nutrient do not affect the concentrations of the limiting one, as the growth of autotrophs is not limited by the nutrient whose supply is modified. Our stoichiometric model also suggests that variations in the supply of the limiting nutrient affect the concentration of the non-limiting one, which alters the ability of organisms to regulate nutrient concentrations. For example, an increase in the supply of the limiting nutrient results in an increase in the growth of autotrophs and thus a decrease in the concentration of the non-limiting nutrient. If the supply of the non-limiting nutrient is also increased, the effect of the increased supply of both nutrients will partly counterbalance and thus the global regulation of the non-limiting nutrient will be enhanced by interactions between cycles. On the contrary, if the supply of the non-limiting nutrient is decreased, it will further amplify the decrease in the concentration of the non-limiting nutrient induced by the increase in the supply of the limiting nutrient. Thus the interactions between nutrient cycles can either enhance or decrease the ability of organisms to regulate nutrient pools, depending on whether the supply of nutrients vary in the same direction or not.

Considering different populations of autotrophs that consume the two resources in a different ratio could also affect the availability of nutrients in the environment, and thus the efficiency of their regulation. For example, N-fixers can increase the concentration of nitrogen available in the environment when it is scarce (e.g. [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Menge | Evolutionary tradeoffs can select against nitrogen fixation and thereby maintain nitrogen limitation[END_REF].

Competition between organisms for resources is thus likely to impact the ability of organisms to regulate nutrient concentrations with respect to changes in nutrient supplies. In Chapter 3, we adapted the stoichiometric model of Chapter 2 to the global cycles of N and P in the ocean to study how competition between two functional groups (N-fixers and non-fixers) impact the regulation of nutrient concentrations as well as their ratio in both the accessible and inaccessible pools. For this Chapter, we studied how model parameters affect the value of deep-water N:P ratio and its regulation with respect to changes in N and P supplies, and thus which mechanisms are important in determining the value of deep-water N:P ratio and its regulation. In agreement with the results of Chapters 1 and 2 and what is observed in the ocean, numerical simulations predicted that regulation of both N and P concentration is only partial in the deep ocean, but that their ratio remains almost constant [START_REF] Redfield | On the proportions of organic derivations in sea water and their relation to the composition of plankton[END_REF][START_REF] Redfield | The biological control of chemical factors in the environment[END_REF][START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Gruber | Redfieldâs evolving legacy[END_REF]. Our model shows that the value of deep-water N:P ratio is controlled by the N:P ratio of non-fixers, and to a lesser extent by the intensity of the recycling and denitrification of organic matter. These results suggest that the similarity of the N:P ratios of phytoplankton and deep waters could be related to a balance between denitrification and N fixation, in agreement with previous studies (e.g. [START_REF] Redfield | The biological control of chemical factors in the environment[END_REF][START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Lenton | Biotic stoichiometric controls on the deep ocean N:P ratio[END_REF][START_REF] Gruber | Redfieldâs evolving legacy[END_REF]. Phytoplankton stoichiometry thus sets the value of the deep-ocean N:P ratio but, surprisingly, it does not affect the efficiency of its regulation.

In Chapter 3, we showed that the competitive dynamics between functional groups can have strong effects on the regulation of nutrients. The increase in the supply of nutrients to ecosystems can impact the total biomass of autotrophs at long temporal scales in different ways. A nutrient whose availability limits the growth of autotrophs at small spatial scales (i.e. proximate limiting nutrient) does not necessarily control the primary production at large timescales, which corresponds to ultimate limitation of primary production [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF][START_REF] Vitousek | Terrestrial phosphorus limitation: mechanisms, implications, and nitrogen-phosphorus interactions[END_REF]. As autotrophs have a major impact on both accessible and inaccessible nutrient pools through resource consumption and recycling, determining the nutrient that limits the total primary production at large spatial and temporal scales is of special interest to elucidate the impact of nutrient fertilisation at global scales. In Chapter 4, we added the iron cycle in the oceanic model of Chapter 3 to elucidate which nutrient between Fe, P and N limits the total oceanic primary production at long temporal scales. We used two alternative hypotheses to describe proximate limitation of N-fixers and non-fixers in the model: (1) Liebig's law of the minimum, and (2) colimitation by multiple nutrients. In each case, we assessed how variations in the supply of Fe, P and N affect the primary production of N-fixers, that of non-fixers, and the total primary production. When the growth of non-fixers is limited by N and that of N-fixers by P, we found that P ultimately limits total primary production, in agreement with previous modelling studies [START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF]Lenton and Watson 2000a). However, our model predicts that the supply of Fe is also a major driver of primary production at long temporal scales, as suggested in recent studies [START_REF] Falkowski | Biogeochemical controls and feedbacks on ocean primary production[END_REF][START_REF] Wu | Phosphate depletion in the Western North 197 Atlantic Ocean[END_REF][START_REF] Mills | Iron and phosphorus co-limit nitrogen fixation in the eastern tropical North Atlantic[END_REF][START_REF] Moore | Iron availability limits the ocean nitrogen inventory stabilizing feedbacks between marine denitrification and nitrogen fixation[END_REF]. In particular, ultimate colimitation of total primary production by P and Fe can arise from models where autotroph growth is proximately limited by either a single or several nutrients. As variations in the supply of nitrogen are buffered by N-fixation, N supply is not important in determining total primary production at large timescales [START_REF] Toggweiler | An ultimate limiting nutrient[END_REF][START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF]. These results suggest that nutrient fertilisation of marine ecosystems by Fe and P may have major effects on primary production and thus on the biogeochemical cycles of other elements at both small and large timescales.

Implications on global scale regulation

We showed that the inaccessibility of part of resources can constrain the efficiency of biotic regulation of nutrient pools at global scales. As the inaccessible pool is often larger than the accessible one, organisms are thus unlikely to be able to regulate nutrient cycles as a whole.

However, the strong regulation of nutrient ratios highlighted by Redfield ratios in the ocean suggest that autotrophic organisms can exert a strong control on chemical properties of their global environment, even in part to which they do not have a direct access.

Our results are not necessarily incompatible with previous results showing a strong regulation of elemental content of their global environment by organisms, as it is the case with the regulation of the O 2 concentration in the atmosphere (Lovelock and Margulis 1974a;[START_REF] Holland | The chemical evolution of the atmosphere and oceans[END_REF][START_REF] Lenton | The role of land plants, phosphorus weathering and fire in the rise and regulation of atmospheric oxygen[END_REF]. The atmosphere can indeed be considered as a much more homogeneous environment than terrestrial and oceanic systems. Our generic model (Chapter 1) indeed showed that intense physical and/or chemical flows enhance the efficiency of biotic regulation of the global nutrient pools. Processes other than resource consumption can also increase the ability of organisms to regulate their environment through negative feedback mechanisms, as it is the case with fires that prevent the concentration of atmospheric O 2 to rise above 25 % [START_REF] Watson | Methanogenesis, fires and the regulation of atmospheric oxygen[END_REF][START_REF] Lenton | The role of land plants, phosphorus weathering and fire in the rise and regulation of atmospheric oxygen[END_REF][START_REF] Kump | The rise of atmospheric oxygen[END_REF]. Microbial processes as anammox and denitrification can also contribute to an increase in the regulation of N concentrations in soils and water [START_REF] Seitzinger | Denitrification across landscapes and waterscapes: a synthesis[END_REF][START_REF] Brandes | New developments in the marine nitrogen cycle[END_REF]Deutsch et al. 2007). However, positive feedbacks can also amplify variations in element contents, for example an increase in CO 2 concentration in the atmosphere increases sea surface temperature, thereby decreasing the dissolution of CO 2 in seawater (e.g. [START_REF] Denman | Couplings between changes in the climate system and biogeochemistry[END_REF].

Effects of global change on biotic regulation of biogeochemical cycles

In this thesis, I studied the effects of anthropogenic nutrient fertilisation on global biogeochemical cycles, e.g. through the release of fertilizers in soil and water or the emission of reactive gases and atmospheric dusts [START_REF] Seitzinger | Sources and delivery of carbon, nitrogen, and phosphorus to the coastal zone: An overview of Global Nutrient Export from Watersheds (NEWS) models and their application[END_REF][START_REF] Duce | Impacts of atmospheric anthropogenic nitrogen on the open ocean[END_REF][START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF][START_REF] Raiswell | The iron biogeochemical cycle past and present[END_REF]. However, the various chapters suggest that components of global change other than nutrient fertilisation could further enhance its impact on nutrient cycles. For example, increased temperature due to the emission of greenhouse gases in the atmosphere will likely increase sea surface temperature, and hence stratification of the water column. Stratification should be further enhanced by decreased sea surface salinity due to ice melting and important precipitation at high latitudes [START_REF] Riebesell | Sensitivities of marine carbon fluxes to ocean change[END_REF][START_REF] Gruber | Warming up, turning sour, losing breath: ocean biogeochemistry under global change[END_REF][START_REF] Rees | Pressures on the marine environment and the changing climate of ocean biogeochemistry[END_REF][START_REF] Rhein | Observations: Ocean[END_REF]. In Chapters 1 and 3, our models predict an important effect of vertical mixing and the proportion of the water column that is accessible to organisms. Water column stratification tends to lower the mixing between the surface layer above the density barrier and deep waters. The increase in sea surface temperature also induces a decrease in the depth of the thermocline, which reduces the fraction of the water column that is accessible to phytoplankton [START_REF] Falkowski | Mix and match: how climate selects phytoplankton[END_REF][START_REF] Riebesell | Sensitivities of marine carbon fluxes to ocean change[END_REF]. The decrease in the depth of the surface layer is then likely to increase the recycling flow to the deep, inaccessible layer because sinking particles will take less time to reach the deep layer [START_REF] Riebesell | Sensitivities of marine carbon fluxes to ocean change[END_REF]. Increasing water column stratification could thus decrease the ability of phytoplankton to regulate nutrient concentrations in the deep ocean, but also decrease the strength of the biotic control of deep-water N:P ratio against changes in nutrient supplies. Another major impact of decreasing vertical mixing and increasing export of organic matter to deep waters is the decrease in the concentration of oxygen in seawater.

As primary production is enhanced by the increased supply of nutrients such as Fe and P to the surface ocean, microorganisms will consume more oxygen to mineralize organic matter.

This tight coupling between the oceanic cycles of N, P and Fe and that of oxygen could thus result in a decrease in the oxygen content of seawater [START_REF] Diaz | Spreading dead zones and consequences for marine ecosystems[END_REF][START_REF] Stramma | Expanding oxygen-minimum zones in the tropical oceans[END_REF][START_REF] Doney | The growing human footprint on coastal and open-ocean biogeochemistry[END_REF][START_REF] Rabalais | Dynamics and distribution of natural and human-caused hypoxia[END_REF].

The coupling between terrestrial ecosystems, oceans and the atmosphere is also a major phenomenon that can affect the regulation of global biogeochemical cycles [START_REF] Denman | Couplings between changes in the climate system and biogeochemistry[END_REF][START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF][START_REF] Krishnamurthy | Impacts of increasing anthropogenic soluble iron and nitrogen deposition on ocean biogeochemistry[END_REF][START_REF] Canfield | The evolution and future of Earthâs nitrogen cycle[END_REF]. For the sake of simplicity, the interconnections between terrestrial, atmospheric and oceanic systems were reduced to external inputs and outputs of nutrients in our models. However, processes that allow nutrients to exit the system studied, and thus to possibly regulate the concentration of nutrients with respect to an increase in their supply, often correspond to an increase in the supply of nutrients in other systems. For example, water infiltration in soils and denitrification in soils and water lead to an increase in the supply of N in the atmosphere and in oceans, respectively [START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF][START_REF] Bouwman | Human alteration of the global nitrogen and phosphorus soil balances for the period 1970-2050[END_REF][START_REF] Seitzinger | Denitrification across landscapes and waterscapes: a synthesis[END_REF][START_REF] Seitzinger | Global river nutrient export: A scenario analysis of past and future trends[END_REF]. The concept of meta-ecosystem, i.e. a set of ecosystems connected by spatial flows of energy, materials and organisms (e.g. [START_REF] Loreau | Meta-ecosystems: a theoretical framework for a spatial ecosystem ecology[END_REF][START_REF] Gravel | Source and sink dynamics in meta-ecosystems[END_REF], could thus provide a powerful theoretical tool to better understand mechanisms of global regulation of biogeochemical cycles at the scale of the entire planet (i.e. including its atmospheric, land and oceanic components; e.g. Lenton andWatson 2000a, 2000b).

Effects of variable stoichiometry and biodiversity on regulation of biogeochemical cycles

In Chapters 2 to 4, we considered that the stoichiometry of autotrophs was fixed within each group of autotrophs. However, variable stoichiometry could affect the strength of the coupling between nutrient cycles [START_REF] Sterner | Ecological stoichiometry: The biology of elements from molecules to the biosphere[END_REF], and thus alter the regulation efficiency of nutrient pools., Allowing N, P and Fe quotas of phytoplankton to vary depending on nutrient availability, however, does not seem to impact our results qualitatively (Appendix 3.A and 4.B). The results can be quantitatively different between models with fixed and variable stoichiometry, depending on the value of the parameters that govern cell quotas (i.e. minimum cell quota and maximum uptake efficiency for each nutrient). However, it seems reasonable that the stoichiometry of phytoplankton at equilibrium will be similar in the two types of models, as realistic parameter values are used in numerical simulations.

Thus adaptable cellular quotas should mostly affect the transient dynamics of the system, and lead to similar results at long temporal scales.

We showed in Chapter 3 that competition between organisms with different resource requirements can enhance the regulation of inaccessible nutrient pools. We only considered two functional groups of phytoplankton in our model as competition between them is a major driver of the global nitrogen cycle. Explicit consideration of a larger functional diversity of phytoplankton in terms of resource requirements might thus further enhance the predicted ability of autotrophs to regulate nutrient pools.

Effects of herbivores on biotic regulation of global biogeocemical cycles

A major limitation of the models considered in this thesis is the lack of higher trophic levels. The consumption of autotrophs by herbivores is indeed only indirectly considered by being included in the constant mortality rate of autotrophs. However, consumer-driven nutrient recycling is known to modify the proximate nutrient limitation of autotrophs in some environments by influencing the recycling of inorganic nutrients in the accessible pool and the exportation of part of the nutrients to inaccessible pools [START_REF] Moegenburg | Nutrient regeneration by zooplankton: effects on nutrient limitation of phytoplankton in a eutrophic lake[END_REF][START_REF] Hasset | Ecological stoichiometry of N and P in pelagic ecosystems: Comparison of lakes and oceans with emphasis on the zooplankton-phytoplankton interaction[END_REF][START_REF] Elser | The stoichiometry of consumer-driven nutrient recycling: Theory, observations, and consequences[END_REF][START_REF] Trommer | Zooplankton-mediated nutrient limitation patterns in marine phytoplankton: an experimental approach with natural communities[END_REF]. Thus consumer-driven nutrient recycling is likely to have important effects on regulation of nutrient cycles by organisms.

The explicit consideration of herbivores also leads the system to be controlled by herbivores instead of autotrophs, which could further affect biotic regulation of nutrient cycles.

Box F: Impact of herbivores on regulation of inaccessible nutrient concentrations -part 1

We added herbivory to the generic model described in Chapter 1. Herbivores recycle part of the inorganic nutrient in both accessible and inaccessible forms. m H is the mortality rate of herbivores, λ H is the fraction of herbivore dead organic matter that is lost from the system, and rec H refers to the fraction of recycling that occurs in the accessible pool for herbivore organic matter. u is the rate at which autotrophs consume the limiting nutrient. f (B, H) is the consumption function of autotrophs by herbivores. Contrary to the generic model without herbivory, the mortality rate of autotrophs (m) does not include grazing by herbivores. We obtain the following model: A useful extension of our models could thus be the explicit consideration of herbivores (Box F). The effect of herbivores on the regulation of nutrient concentration in the accessible pool turns out to depend on the type of functional response that is used in the model. Donor-controlled herbivory (i.e. herbivores consume a constant proportion of the autotroph population, independently of their own biomass) predicts perfect regulation of accessible nutrient concentrations with respect to changes in nutrient supply to both the accessible and inaccessible pools. This result is similar to the case without herbivores presented in Chapter 1, as herbivory does not influence the top-down control of autotrophs on the inorganic limiting nutrient. Recipient-controlled herbivory (i.e. the consumption of autotrophs by herbivores depends on the biomass of both populations, leading to top-down control at equilibrium), variations in the supply of the inorganic nutrient to both pools are only partially absorbed in the accessible pool. This result is intuitive as in this case, the system is controlled by herbivores instead of autotrophs, and thus biotic regulation is not as efficient as in the case without herbivory. However, the impact of herbivores on the regulation of the concentration of the inorganic nutrient in the inaccessible pool is much less clear. Preliminary results with donor-controlled herbivory show that the recycling by herbivores can have three different effects on the regulation of the inaccessible nutrient concentration with respect to an increase in the supply of the nutrient to the accessible pool. Depending on the recycling patterns of herbivore organic matter (i.e. the fraction of nutrient that is not recycled and the fraction of recycling that occurs in the accessible pool), the increase in the rate of autotroph consumption by herbivores can either have (1) a positive effect on the regulation efficiency of the inaccessible nutrient concentration (Figure F1a), (2) a negative effect (Figure F1c), or (3) a positive effect until a threshold value of the consumption rate, followed by a negative effect (Figure F1b). Further study of the parameters that allow switching from one case to another will be crucial to understand how herbivores can affect regulation of nutrient concentrations at global scales.

dN a dt = S a -(k a + q a )N a + 1 -α α k i N i + [mrec a (1 -λ) -uN a ] B + m H rec H (1 -λ H )H dN i dt = S i + α 1 -α k a N a -(k i + q i )N i + α 1 -α m(1 -rec a )(1 -λ)B + α 1 -α mH(1 -rec H )(1 -λ H )H dB dt = [uN a -m] B -f (B,
A major issue in assessing the effects of herbivores on the regulation of global biogeochemical cycles is the choice of the functional response used to determine consumption of autotrophs by herbivores [START_REF] Gentleman | Functional responses for zooplankton feeding on multiple resources: a review of assumptions and biological dynamics[END_REF]. The functional response indeed varies across scales, and it is difficult to know which formulation is best adapted to global scales [START_REF] Englund | Scaling up the functional response for spatially heterogeneous systems[END_REF]. A mathematical analysis of a model describing the dynamics of an inorganic nutrient, an autotroph, a herbivore and detritus showed that donor-controlled herbivory and recipient-controlled herbivory lead to similar results regarding primary production at equilibrium [START_REF] De Mazancourt | Grazing optimization and nutrient cycing: When do herbivores enhance plant production?[END_REF]). However it is not clear whether or not the regulation coefficients of inaccessible nutrient concentration will be similar Figure F1: Impact of increasing herbivory on regulation of the inaccessible concentration of an inorganic nutrient with respect to changes in its supply to the accessible pool. Simulations correspond to donor-controlled herbivory. The rate of autotroph consumption by herbivores varies between 0 (no herbivory) and cmax, which corresponds to the threshold value of consumption where autotrophs go extinct. The dashed line represents the value of the regulation coefficient in the absence of herbivores. Simulations are performed with α = q a = q i = k a = 0.01, k i = S i = 0.002, λ = m = mH = 0.1, rec a = 0.4, S a = 0.2 and u = 0.05. (a) Positive impact of herbivory (λ H = 0.11 and rec H = 0.3), (b) Positive and then negative impact of herbivory (λ H = 0.22 and rec H = 0.2), (c) Negative impact of herbivory (λ H = 0.4 and rec H = 0.5).

for both types of herbivory. A possible way to test the impact of herbivores on the regulation of biogeochemical cycles would be to apply the extension of the generic model with herbivores presented in Box F to spatialized zooplankton -phytoplankton -nutrient interactions in the global ocean. With such an application, we could determine if the spatial differences in the intensity of biogeochemical flows lead to similar results compared to the extension of the generic model with herbivores. Such an application could also reveal interesting differences in the impact of herbivores on the regulation of nutrient cycles among oceanic regions.

Figures

  Figures and Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure B1 :

 B1 Figure B1: Time series of changes in large-scale ocean climate properties. From top to bottom: (a) global ocean inventory of anthropogenic carbon dioxide, updated from Khatiwala et al. (2009); (b) global upper ocean heat content anomaly, updated from Domingues et al. (2008); (c) the difference between salinity averaged over regions where the sea surface salinity is greater than the global mean sea surface salinity (High Salinity) and salinity averaged over regions values below the global mean (Low Salinity), from Boyer et al. (2009). Extracted from the Chapter 3 of the fifth IPCC assessment report (Rhein et al. 2013), page 301.

Figure C1 :

 C1 Figure C1: Comparison between two concepts of limitation for two essential resources. (a-b) Liebig's law of the minimum assuming strictly essential resources (Monod function), and (c-d) colimitation assuming interactive-essential resources (Monod multiplicative function). Resource-dependent growth isoclines (solid lines) indicate equal growth at changing resource availabilities. Liebig's law shows right angle corners of growth isoclines, indicating that growth is strictly limited by only one resource at a certain resource availability. Colimitation shows rounded corners of growth isoclines, indicating a smooth transition of limitation by one to the other resource and a range of resource availabilities at which both resources strongly limit growth simultaneously. Adapted from Sperfeld et al. (2012).

Figure E1 :

 E1 Figure E1: Global relationship between nitrate and phosphate concentrations in seawater based on the Global Ocean Data Analysis Project (GLODAP). Colors delineate the Atlantic (red), Pacific (blue), and Indian (yellow) ocean basins. The bold line through the origin has a slope of 16 that corresponds to the Redfield ratio (N:P ratio of 16:1).The dashed line is the linear regression (slope of 14.5), which is less than the mean N:P ratio of phytoplankton because nitrate is removed via denitrification in high-nutrient areas and added via remineralization of N-fixer biomass in low-nutrient areas. Extracted from[START_REF] Weber | Oceanic nitrogen reservoir regulated by plankton diversity and ocean circulation[END_REF].

  dans l'atmosphère[START_REF] Benitez-Nelson | The biogeochemical cycling of phosphorus in marine systems[END_REF][START_REF] Falkowski | The global carbon cycle: a test of our knowledge of Earth as a system[END_REF][START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF][START_REF] Gruber | Warming up, turning sour, losing breath: ocean biogeochemistry under global change[END_REF]. Il est donc crucial de comprendre dans quelle mesure et à quelle échelle spatiale les cycles biogéochimiques peuvent être régulés naturellement. Les organismes autotrophes régulent la quantité de nutriments dans leur environnement à l'échelle locale par la consommation des ressources. Cependant, la plupart des ressources sont inaccessibles à l'échelle globale, que ce soit à cause de barrières physiques ou à cause de formes chimiques d'un nutriment qui ne sont pas assimilables par les organismes.Dans ce chapitre, nous avons étudié la capacité des organismes autotrophes à réguler les réservoirs d'un nutriment inorganique sous sa forme accessible et inaccessible.Dans un premier temps, nous avons développé un modèle générique de régulation d'une ressource lorsque son accessibilité est limitée. Ce modèle décrit la dynamique d'un nutriment inorganique limitant la croissance des autotrophes et qui est réparti en deux réservoirs, l'un accessible et l'autre inaccessible pour les autotrophes. Dans un second temps, j'ai appliqué ce modèle aux cycles biogéochimiques du fer, de la silice et du phosphore dans l'océan global.L'application au cycle du fer constitue un exemple de limitation chimique de l'accessibilité aux ressources, étant donné que le phytoplancton ne peut assimiler qu'une partie du fer disponible dans l'océan[START_REF] Baker | Atmospheric and marine controls on aerosol iron solubility in seawater[END_REF][START_REF] Boyd | The biogeochemical cycle of iron in the ocean[END_REF]. Le fer assimilable par le phytoplancton étant en concentration très faible dans de nombreuses régions océaniques, sa disponibilité influence très fortement la croissance des autotrophes[START_REF] Fung | Iron supply and demand in the upper ocean[END_REF][START_REF] Moore | Iron cycling and nutrient-limitation patterns in surface waters of the World Ocean[END_REF][START_REF] Krishnamurthy | Impacts of atmospheric nutrient inputs on marine biogeochemistry[END_REF]. La deuxième application du modèle générique est le cycle du phosphore dans l'océan global, ce qui est un exemple de limitation physique de l'accessibilité aux ressources car le phytoplancton ne peut accéder aux nutriments que dans la couche de surface, que ce soit à cause d'une thermocline ou de la profondeur limitée de la couche euphotique. La troisième application du modèle est le cycle de la silice dans l'océan. De la même manière que pour le phosphore, les barrières physiques font que le phytoplancton ne peut accéder qu'à une partie de la silice dans leur environnement. Cet exemple présente un intérêt particulier qui est d'étudier comment les autotrophes peuvent réguler les réservoirs d'un nutriment en réponse à la diminution de ses apports par les activités anthropiques[START_REF] Laruelle | Anthropogenic perturbations of the silicon cycle at the global scale: Key role of the land-ocean transition[END_REF][START_REF] Tréguer | The world ocean silica cycle[END_REF], contrairement aux apports de phosphore et de fer qui augmentent.L'étude du modèle générique montre que les autotrophes ne peuvent pas à la fois avoir un impact fort sur le réservoir accessible et réguler de manière importante la concentration du nutriment qui limite leur croissance dans le réservoir inaccessible. La capacité des organismes à réguler les concentrations des nutriments inaccessibles dépend fortement de l'intensité des flux physique et/ou chimiques entre les réservoirs accessibles et inaccessibles, mais aussi de la fraction du système qui est accessible aux organismes. Les conséquences du réchauffement climatique comme l'augmentation de la stratification de la colonne d'eau dans l'océan pourraient alors réduire encore la régulation par les organismes des nutriments inaccessibles dans les écosystèmes marins et lacustres.IntroductionAnthropogenic alteration of biogeochemical cycles is a major component of current global environmental changes[START_REF] Schlesinger | Biogeochemistry: An analysis of global change[END_REF]. Human activities are profoundly modifying the global biogeochemical cycles of key elements such as carbon, nitrogen, phosphorus, and oxygen[START_REF] Benitez-Nelson | The biogeochemical cycling of phosphorus in marine systems[END_REF][START_REF] Falkowski | The global carbon cycle: a test of our knowledge of Earth as a system[END_REF][START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF][START_REF] Gruber | Warming up, turning sour, losing breath: ocean biogeochemistry under global change[END_REF], mostly through changes in nutrient supply. For example, human activities release carbon dioxide to the atmosphere, which in turn increases its dissolution in oceans[START_REF] Gruber | Warming up, turning sour, losing breath: ocean biogeochemistry under global change[END_REF], while fertilizer application increases the supply of nitrogen and phosphorus in terrestrial ecosystems and oceans[START_REF] Benitez-Nelson | The biogeochemical cycling of phosphorus in marine systems[END_REF][START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF].Given these massive alterations, it is critical to understand to what extent and how global biogeochemical cycles are naturally regulated. Part of this regulation is due to physical and chemical processes. For example, in the carbon cycle, an increase in the partial pressure of CO2 in the atmosphere induces increased dissolution of CO2 in the ocean, leading to some regulation of the partial pressure of atmospheric CO2[START_REF] Falkowski | The global carbon cycle: a test of our knowledge of Earth as a system[END_REF]. Organisms also play a significant part in the regulation of the Earth system (e.g.[START_REF] Canfield | Oxygen: A four billion years history[END_REF][START_REF] Lenton | Revolutions that made the Earth[END_REF], but the extent of their regulating ability at global scales is much less clear.

Figure 1 .

 1 Figure 1.2a). Variations in the supply of the accessible nutrient impact organisms' growth, which may alter the intensity of the recycling flow to the inaccessible nutrient pool. Therefore the inaccessible nutrient concentration is only partially or negatively regulated because it varies in the same direction as the supply of nutrient in accessible form (paths 1-4 and 1-2-5 in Figure 1.2b):

  .3c), although we can observe a response in the growth of autotrophic organisms (Figures 1.3d, 1.4d and 1.5b). These three examples suggest that for both chemical and physical nutrient limitations, nutrient cycles are not efficiently regulated by autotrophic organisms at global scales, although the strength of the biotic regulation of the different pools can quantitatively vary depending on the characteristics of each biogeochemical cycle (Figures 1.3, 1.4 and 1.5).

Figure

  Figure 1.1: Generic model of nutrient dynamics with resource access limitation. Boxes and arrows represent stocks and flows, respectively.

  Figure 1.1: Generic model of nutrient dynamics with resource access limitation. Boxes and arrows represent stocks and flows, respectively.

Figure 1 . 2 :

 12 Figure 1.2: Regulation processes in the generic model. (a) Impact of the supply of a limiting nutrient in accessible form on its concentration in the accessible pool. (b) Impact of the supply of a limiting nutrient in accessible form on its concentration in the inaccessible pool. (c) Impact of the supply of a limiting nutrient in inaccessible form on its concentration in the accessible pool. (d) Impact of the supply of a limiting nutrient in inaccessible form on its concentration in the inaccessible pool. Bold arrows indicate a direct relationship (e.g. an increase in the concentration of the accessible limiting nutrient results in an increase in biomass). Dashed arrows indicate an inverse relationship (e.g. an increase in biomass results in a decrease in nutrient concentration in the accessible pool).

  Figure1.3: Regulation of iron concentrations in the ocean. Nutrient supply is increased by 50% after third of the simulation time (dotted vertical line). Simulations are performed from the application of the generic model to the oceanic cycle of iron with realistic parameter values. Bold lines and dotted lines are for the dissolved and particulate iron pools, respectively. (a) Regulation of iron concentrations in case of an increase in the supply of dissolved iron (ρ = 1 for both pools). (b) Impact of an increase in the supply of dissolved iron on the biomass of autotrophic organisms. (c) Regulation of iron concentrations in case of an increase in the supply of particulate iron (ρ = 1 and ρ = 2.3 * 10 -3 for the dissolved and particulate pools, respectively). (d) Impact of an increase in the supply of particulate iron on the biomass of autotrophic organisms.

Figure

  Figure 2.1: Nutrient stocks and flows in the stoichiometric model. Boxes represent stocks. Blue arrows are flows of the limiting nutrient. Red arrows are flows of the non-limiting nutrient. Purple arrows are flows of both the limiting and the non-limiting nutrients.

  Figure 2.1: Nutrient stocks and flows in the stoichiometric model. Boxes represent stocks. Blue arrows are flows of the limiting nutrient. Red arrows are flows of the non-limiting nutrient. Purple arrows are flows of both the limiting and the non-limiting nutrients.

Figure 2 . 2 :

 22 Figure 2.2: Regulation processes in the stoichiometric model. Bold arrows indicate a direct relationship (e.g. an increase in the concentration of the accessible limiting nutrient results in an increase in the biomass of autotrophs). Dashed arrows indicate an inverse relationship (e.g. an increase in the biomass of autotrophs results in a decrease in nutrient concentration in the accessible pool).

(

  [START_REF] Benitez-Nelson | The biogeochemical cycling of phosphorus in marine systems[END_REF][START_REF] Seitzinger | Sources and delivery of carbon, nitrogen, and phosphorus to the coastal zone: An overview of Global Nutrient Export from Watersheds (NEWS) models and their application[END_REF][START_REF] Gruber | An Earth-system perspective of the global nitrogen cycle[END_REF] Mahowald et al. 2008). Dans ce chapitre, nous avons étudié dans quelle mesure et par quels moyens les autotrophes peuvent contrôler la composition chimique des eaux profondes auxquelles ils n'ont pas d'accès direct.Nous avons adapté le modèle stoechiométrique du Chapitre 2 aux cycles océaniques du N et du P, dans lequel le phytoplancton n'accède aux nutriments inorganiques que dans la couche de surface. Le modèle a été paramétré à l'aide de données existantes sur les flux de N et de P dans l'océan global. En faisant une analyse de sensibilité, nous avons dans un premier temps déterminé dans quelle mesure les mécanismes impliqués dans les cycles du N et du P (e.g. dénitrification, fixation, mélange vertical) contrôlent la valeur du rapport N:P des eaux profondes. Dans un second temps, nous avons analysé les capacités du phytoplancton à réguler les concentrations de N et de P ainsi que le rapport N:P dans l'océan actuel, en réponse à l'augmentation des apports de N et de P d'origine anthropique dans l'océan de surface. Enfin, nous avons déterminé par une analyse de sensibilité les principaux mécanismes impliqués dans la régulation du rapport N:P des eaux profondes par le phytoplancton.

Figures 3 .

 3 Figures 3.4a and 3.4d). For example, an increased N supply is advantageous to non-fixers, resulting in a lower mean N:P ratio of organic matter; the N outflow to the deep layer is reduced and the concentration of N in deep waters decreases. Since deep N concentration and N supply vary in opposite directions (path 1-2-5-6 in Figure3.4a), this leads to an over-regulation by organisms. In numerical simulations for the current ocean, this regulation was almost perfect (ρ N i,N a = 1.01, Figure3.3e) because non-fixers are already dominant(Figure 3.3h), such that the mean N:P ratio of organic matter is little affected by the 50 % increase in N supply. Changes in P supply have an opposite effect on the mean N:P ratio (paths 1-2-4-5 and 1-2-5 in Figure3.4d). As a result, they have a positive effect on deep N concentration (Figure3.3a).

  opposite occurs when N is added to the surface ocean, because the addition of N increases the growth rate of non-fixers and thus their absorption of P. The change in biomass induced by changes in the supply of either N or P then affects the intensity of nutrient recycling to the deep layer. Although nutrient concentrations in the deep layer are substantially affected by changes in P supply, as shown by numerical simulations, regulation of the nutrient ratio can be strong (Figure 3.3c).

  Figure 3.1: Nutrient stocks and flows in the model of coupled nitrogen and phosphorus cycles in the global ocean. Boxes represent stocks. Blue arrows are phosphorus flows, red arrows are nitrogen flows, and purple arrows are both nitrogen and phosphorus flows.

Figure 3

 3 Figure3.3: Regulation of N and P concentrations and the N:P ratio in the ocean. P supply in (a-d) and N supply in (e-h) are increased by 50% after 100,000 years. Simulations are performed with realistic parameter values. Bold lines are for the upper, accessible layer and dotted lines for the deep, inaccessible layer. (a,e) P concentrations (ρ = 1 and 0.13 in case of a 50%-increase in P supply in the upper and deep layers, respectively, and ǫ = 0 in both layers in case of a 50%-increase in N supply). (b,f) N concentrations (ǫ = 0 and 0.88 in case of a 50%-increase in P supply in the upper and deep layers, respectively, and ρ = 1 and 1.01 in case of a 50%-increase in N supply in the upper and deep layers, respectively). (c,g) N:P ratios (ρ = 1 in the upper layer and ρ = 1.01 in the deep layer in both cases). (d,h) Ratio of non-fixing over N-fixing phytoplankton (B/B F ).

Figure 3 . 4 :

 34 Figure 3.4: Regulation processes in the ocean model with N-fixing and non-fixing phytoplankton. (a) Impact of N supply on N pools. (b) Impact of N supply on P pools. (c) Impact of P supply on P pools. (d) Impact of P supply on N pools. Bold arrows indicate a direct relationship (e.g. an increase in accessible N concentration results in an increase in the biomass of non-fixers). Dashed arrows indicate an inverse relationship (e.g. an increase in biomass results in a decrease in accessible nutrient concentrations).

Figure 3

 3 Figure3.5: Distribution of the sensitivity of (a) regulation of deep-water N:P ratio with respect to changes in P supply (ρ (P :N )i,P a ), and (b) regulation of deep-water N:P ratio with respect to changes in N supply (ρ (N :P )i,N a ). The sensitivity of a variable to a parameter is measured as the elasticity of the variable with respect to the parameter. For each parameter, the local sensitivity is calculated for 500 values uniformly distributed in an interval of ± 20 % around the value used for numerical simulations. z tot is the depth of the water column and p za the fraction that is accessible. S N and S P are supplies of N and P, respectively. K is the vertical mixing coefficient. D tot and Rec tot are the total denitrification and recycling rates, respectively. p Da and p Reca are the fraction of denitrification and recycling in the upper layer, respectively. R and R F are the P:N ratio of non-fixers and N-fixers, respectively. m is the mortality rate and µ the maximum growth rate of non-fixers. cost is the metabolic cost of N-fixation. N H is the half saturation constant for N. q P and q N are adsorption rates of P and N, respectively.

  Fe dans le modèle développé dans le chapitre 3. Le modèle utilisé dans ce chapitre décrit donc la dynamique des fixateurs et des non-fixateurs dans l'océan global, ainsi que les cycles du N, du P et du Fe. Pour tester la robustesse de nos résultats, nous avons utilisé plusieurs alternatives pour la formulations du taux de croissance des fixateurs et des non-fixateurs en fonction de la disponibilité des trois nutriments: (1) la loi du minimum de Liebig (Von Liebig 1842), (2) la fonction multiplicative de Monod (e.g.

Figure 4. 1 and

 1 Table4.1). It includes two groups of phytoplankton, N-fixers and non-fixers[START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF] Auguères and Loreau 2015b). The water column is separated in two layers, each of which is considered homogeneous, either because of the limited depth of the euphotic layer or because of the thermocline. The upper layer is accessible to phytoplankton, while the deep layer is inaccessible to phytoplankton because of light limitation or water column stratification. N concentrations (N a and N i for the accessible upper layer and the inaccessible deep layer, respectively) include nitrites, nitrates and ammonium, P concentrations (P a and P i ) correspond to phosphates, and Fe concentrations (F e a and F e i ) correspond to dissolved iron. N and Fe supplies to the upper layer (S N and S F e , respectively) include atmospheric and riverine inflows while P supply (S P ) includes only riverine inflow. Mass balance is used to build a model that describes the dynamics of N, P and Fe masses. By dividing nutrient mass by the volume of the layer concerned, we then obtain a model in terms of nutrient concentrations (Figure4.1):

Figure

  Figure 4.1: Nutrient stocks and flows in the model of nutrient cycles in the global ocean. Boxes and arrows represent nutrient stocks and flows, respectively. (a) Nitrogen oceanic cycle. (b) Phosphorus oceanic cycle. (c) Iron oceanic cycle.

  Figure 4.1: Nutrient stocks and flows in the model of nutrient cycles in the global ocean. Boxes and arrows represent nutrient stocks and flows, respectively. (a) Nitrogen oceanic cycle. (b) Phosphorus oceanic cycle. (c) Iron oceanic cycle.

Figure 4

 4 Figure 4.2: Impact of nutrient supplies on primary production (PP) in the long-term when phytoplankton limitation is modelled with Liebig's law of the minimum. S N , S P and S F e are supplies of N, P and Fe, respectively (µmol.m -3 .a -1 ). Simulations are performed with realistic parameter values. (a) Impact of N and P supplies on non-fixing PP. (b) Impact of N and P supplies on N-fixing PP. (c) Impact of N and P supplies on total PP. (d) Impact of N and Fe supplies on non-fixing PP. (e) Impact of N and Fe supplies on N-fixing PP. (f) Impact of N and Fe supplies on total PP. (g) Impact of P and Fe supplies on non-fixing PP. (h) Impact of P and Fe supplies on N-fixing PP. (i) Impact of P and Fe supplies on total PP.

  Two consumption functions can be used to model herbivory: donor-controlled functions (f (B, H) = cB where c is the consumption rate of autotrophs by herbivores) and recipient-controlled functions (e.g. f (B, H) = γBH where γ is the rate of autotroph consumption per unit of herbivore).

Table 1 .

 1 1: Model parameters Table 1.2: Impact of parameters on biotic regulation of the inaccessible nutrient pool

	Symbol Description

Table S1 .

 S1 1: Parameter values for numerical simulations of the oceanic iron cycle

	Symbol Model value Literature values and comments
			The water column is supposed to be
	α	0.5	homogeneous for chemical inaccessibility
			55-321 (Obayashi and Tanoue 2002)
	m	90	73-657 (Sarthou et al. 2005)
			138-256 (Obayashi and Tanoue 2002)
	µ	125	146-1 205 (Sarthou et al. 2005) 124-299 (Timmermans et al. 2005)
	N H	2x10 -2	2-12x10 -2 (Aumont et al. 2003)
	λ	2x10 -3	2x10 -3 (Slomp and Van Cappellen 2006)
	rec a	1
			5x10 -3 (Aumont et al. 2003)
	k a	4x10 -3	4-6x10 -3 (Parekh et al. 2004)
			1x10 -2 (Aumont et al. 2003)
	k i	2x10 -2	1-2x10 -2 (Boyd and Ellwood 2010)
			1x10 -2 xS
	S a	2x10 -2 xS i

i

[START_REF] Aumont | An ecosystem model of the global ocean including Fe, Si, P colimitations[END_REF]

1-2x10 -2 xS i

[START_REF] Boyd | The biogeochemical cycle of iron in the ocean[END_REF]

Table S1 .

 S1 2: Parameter values for numerical simulations of the oceanic silicon cycle

	Symbol Model value Literature values and comments
	α	4x10 -2	4x10 -2 (Slomp and Van Cappellen 2006)
			55-321 (Obayashi and Tanoue 2002)
	m	90	73-657 (Sarthou et al. 2005)
			138-256 (Obayashi and Tanoue 2002)
	µ	125	146-1 205 (Sarthou et al. 2005) 124-299 (Timmermans et al. 2005)
			200-94700 (Martin-Jézéquel et al. 2000)
	N H	3 900	3900 (Sarthou et al. 2005)
	λ	2.6x10 -2 2.6x10 -2 (Treguer and De La Rocha 2013)
	rec a	5.8x10 -1 5.8x10 -1 (Treguer and De La Rocha 2013)
	k a	7.7x10 -2 7.7x10 -2 (Slomp and Van Cappellen 2006)
	k i	3.2x10 -3 3.2x10 -3 (Slomp and Van Cappellen 2006)
	S a	127.8	127.8 (Treguer and De La Rocha 2013) *
	S i	2.1	2.1 (Treguer and De La Rocha 2013) *

Table S1 .

 S1 3: Parameter values for numerical simulations of the oceanic phosphorus cycle

	Symbol Model value Literature values and comments
	α	4x10 -2	4x10 -2 (Slomp and Van Cappellen 2004)
			55-321 (Obayashi and Tanoue 2002)
	m	90	73-657 (Sarthou et al. 2005)
			138-256 (Obayashi and Tanoue 2002)
	µ	125	146-1 205 (Sarthou et al. 2005) 124-299 (Timmermans et al. 2005)
			240 (Sarthou et al. 2005)
	N H	125	14-94 (Timmermans et al. 2005)
	λ	2x10 -3	2x10 -3 (Slomp and Van Cappellen 2006)
	rec a	0.873
	k a	7.7x10 -2 7.7x10 -2 (Slomp and Van Cappellen 2006)
	k i	3.2x10 -3 3.2x10 -3 (Slomp and Van Cappellen 2006)
			0.56-2.77 (Benitez-Nelson 2000) *
	S a	1.7	1.7 (Slomp and Van Cappellen 2006) *
	S i	0

  1, * and Michel Loreau 1 1 Centre for Biodiversity Theory and Modelling, Station dâEcologie Expérimentale du CNRS, 09200 Moulis, France * Corresponding author; anne-sophie.augueres@ecoex-moulis.cnrs.fr Status: in prep. This chapter corresponds to theoretical results. I will further investigate the impact of coupling of biogeochemical cycles by applying the stoichiometric model to the oceanic cycles of P and Fe. Dans le Chapitre 1, nous avons montré que la concentration du nutriment limitant est entièrement régulée dans le réservoir accessible et seulement partiellement régulée ou amplifiée dans le réservoir inaccessible. Dans le chapitre 2, nous avons étudié les capacités des autotrophes à réguler la concentration d'un nutriment non-limitant, dans les réservoirs accessible et inaccessible. Nous avons aussi regardé comment les variations des apports d'un nutriment affectent la concentration du second nutriment dans chaque réservoir. Nous avons développé une extension stoechiométrique du modèle présenté dans le Chapitre Par exemple, si les apports du nutriment limitant sont augmentés, la croissance des autotrophes est favorisée et la consommation du nutriment non-limitant dans le réservoir accessible diminue. Si les apports du nutriment non-limitant diminuent en parallèle, cela amplifie la diminution de la concentration du nutriment non-limitant due aux variations des apports du nutriment limitant. Au contraire, si les apports du nutriment non-limitant augmentent, l'augmentation de la concentration du nutriment non-limitant sera contre-balancée au moins en partie par la diminution due aux variations des apports du nutriment limitant. Ainsi, notre modèle suggère que les interactions entre les cycles d'un nutriment limitant et d'un nutriment non-limitant peuvent augmenter ou diminuer l'efficacité de la régulation des concentrations du nutriment non-limitant, selon que les apports des deux nutriments varient dans le même sens ou non.

	Résumé

1. Le modèle stoechiométrique décrit la dynamique d'une population d'autotrophes et de deux nutriments inorganiques, l'un limitant et l'autre non-limitant pour la croissance des autotrophes. De même que dans le Chapitre 1, chaque nutriment se retrouve dans deux réservoirs distincts, l'un accessible et l'autre inaccessible aux autotrophes. Nous avons étudié le potentiel de régulation biotique des concentrations des deux nutriments ainsi que leur ratio en réponse à l'augmentation des apports de chaque nutriment. L'étude du modèle stoechiométrique prédit que les variations des apports du nutriment non-limitant ne sont que partiellement régulées ou amplifiées par les autotrophes dans les deux réservoirs. Nous avons aussi montré que les variations des apports du nutriment limitant impactent la croissance des autotrophes et donc l'importance de la consommation du nutriment non-limitant. Ainsi, les apports du nutriment limitant et les concentrations du nutriment non-limitant dans les deux réservoirs varient dans des sens opposés.

Table 2 .

 2 1: Parameters of the stoichiometric model

	Symbol Description

Table 3 .

 3 1: Parameter values used in simulations for the model of phosphorus and nitrogen cycles in the global ocean.

	Symbol Description	Units	Model value	Literature values
	z tot	Depth of the water column	m	3,730
		Fraction of the water column
	p za	that corresponds to the upper		4.2x10 -2 4.2x10 -2 (Slomp and Van Cappellen 2006)
		layer
	K	Vertical mixing coefficient	m.a -1	11.5	11.5 (Slomp and Van Cappellen 2006) b
	S N	Nitrogen supply (riverine and atmospheric)	µmolN.m -3 .a -1 135	211 (Codispoti et al. 2001) a,c 132-198 (Brandes et al. 2007) a,c 172 (Gruber and Galloway 2008) a,c
	S P	Phosphorus supply (riverine)	µmolP.m -3 .a -1 1.9	0.56-2.77 (Benitez-Nelson 2000) c 1.7 (Slomp and Van Cappellen 2006) c
	q N	Adsorption rate of nitrogen	a -1	10 -6
	q P	Adsorption rate of phosphorus a -1	10 -5	2.4x10 -5 (Slomp and Van Cappellen 2006)
	R	P:N ratio of non-fixing phytoplankton	molP.molN -1 1:15	1:6-1:14 (Sarthou et al. 2005) 1:5-1:19 (Geider and La Roche 2002)
	R F	P:N ratio of N-fixing phytoplankton	molP.molN -1 1:50	1:5-1:150 (La Roche and Breitbarth 2005)
	m	Mortality rate of phytoplankton (including grazing)	a -1	85	55-321 (Obayashi and Tanoue 2002) 73-657 (Sarthou et al. 2005)
	Rec tot	Fraction of nutrient recycled in the water column		99.8x10 -2 99.8x10 -2 (Slomp and Van Cappellen 2006)
	p Reca	Fraction of total recycling that occurs in the upper layer		87.3x10 -2 87.3x10 -2 (Slomp and Van Cappellen 2006)
	D tot	Denitrification rate in the water column	a -1	1.6x10 -2	1.7x10 -2 (Codispoti et al. 2001) d 1.0-1.8x10 -2 (Seitzinger et al. 2006) d 0.9-1.1x10 -2 (Brandes et al. 2007) d
	p Da	Fraction of total denitrification that occurs in the upper layer	a -1	22.5x10 -2
	µ	Maximum growth rate of non-fixing phytoplankton	a -1	124	138-256 (Obayashi and Tanoue 2002) 146-1 205 (Sarthou et al. 2005) 124-299 (Timmermans et al. 2005)
		Cost associated to N-fixation
	cost	(decrease in the maximal
		growth rate)

Table 4 .

 4 1: Parameter values used in simulations for the model of nitrogen, phosphorus and iron cycles in the global ocean.

	Symbol Description	Units	Model value	Literature values
	z tot	Depth of the water column	m	3,730
		Fraction of the water column
	p za	that corresponds to the upper		4.2x10 -2 4.2x10 -2 (Slomp and Van Cappellen 2006)
		layer
	K	Vertical mixing coefficient	m.a -1	11.5	11.5 (Slomp and Van Cappellen 2006) *
	S N	Nitrogen supply (riverine and atmospheric)	µmolN.m -3 .a -1 132-211	211 (Codispoti et al. 2001) † ‡ 132-198 (Brandes et al. 2007) † ‡ 172 (Wu et al. 2000) † ‡
	S P	Phosphorus supply (riverine)	µmolP.m -3 .a -1 0.6-2.77	0.56-2.77 (Benitez-Nelson 2000) ‡ 1.7 (Slomp and Van Cappellen 2006) ‡
	S F e	Iron supply (riverine and atmospheric)	µmolFe.m -3 .a -1 0.019-0.19 0.019-0.19 (Geider and LaRoche 2002) ‡
	q N	Adsorption rate of nitrogen	a -1	10 -6

  -4 (Moore and Doney 2007)

	R 1	P:N ratio of non-fixing phytoplankton	molP.molN -1 1:15	1:6-1:14 (Sarthou et al. 2005) 1:5-1:19 (Codispoti et al. 2001)
	R 1F	P:N ratio of N-fixing phytoplankton	molP.molN -1 1:50	1:5-1:150 (LaRoche and Breitbarth 2005)
	R 2	Fe:N ratio of non-fixing phytoplankton	molFe.molN -1 1:560	1:342-1:784 (Quigg et al. 2011)
	R 2F	Fe:N ratio of N-fixing phytoplankton	molFe.molN -1 1:280	1:262-1:309 (Quigg et al. 2011)
	m	Mortality rate of phytoplankton (including grazing)	a -1	85	55-321 (Obayashi and Tanoue 2002) 73-657 (Sarthou et al. 2005)
	Rec tot	Fraction of nutrient recycled in the water column		99.8x10 -2 99.8x10 -2 (Slomp and Van Cappellen 2006)
	p Reca	Fraction of total recycling that occurs in the upper layer		87.3x10 -2 87.3x10 -2 (Slomp and Van Cappellen 2006)
	D tot	Denitrification rate in the water column	a -1	1.6x10 -2	1.7x10 -2 (Codispoti et al. 2001) § 1.0-1.8x10 -2 (Seitzinger et al. 2006) § 0.9-1.1x10 -2 (Brandes et al. 2007) §
	p Da	Fraction of total denitrification that occurs in the upper layer	a -1	22.5x10 -2
	µ	Maximum growth rate of non-fixing phytoplankton	a -1	124	138-256 (Obayashi and Tanoue 2002) 146-1 205 (Sarthou et al. 2005) 124-299 (Timmermans et al. 2005)
	cost	Cost associated to N-fixation (decrease in the maximal growth rate)	a -1	4	growth rate of N-fixers: 66-117 (Masotti et al. 2007)
	P H

Table 4 .

 4 2: Sensitivity of primary production at equilibrium with respect to nutrient supplies

	Type of limitation	Limitation of non-fixers	Limitation of N-fixers	S N	S P	S F e
	Liebig	N	P	(0.10) 0.04	(0.96) 0.90	-
	Liebig	N	Fe	0.08 (0.14)	-	0.86 (0.92)
	Liebig	P	Fe	-	0.44 (0.54)	0.46 (0.56)
	Monod multiplicative form	N, P and Fe	P and Fe	0.01 (0.04)	0.11 (0.19)	0.79 (0.86)
	Synthesizing units	N, P and Fe	P and Fe	0.01 (0.05)	0.10 (0.19)	0.79 (0.87)

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Acknowledgements

We thank Claire de Mazancourt for valuable comments on the manuscript. This work was supported by the TULIP Laboratory of Excellence (ANR-10-LABX-41).

References for the whole dissertation are available at the end.

Acknowledgements

This work was supported by the TULIP Laboratory of Excellence (ANR-10-LABX-41).

Status: published in Global Biogeochemical Cycles

Keywords: Redfield ratios, regulation, nitrogen cycle, phosphorus cycle, resource access limitation

Acknowledgements

This work was supported by the TULIP Laboratory of Excellence (ANR-10-LABX-41).

References for the whole dissertation are available at the end.

Acknowledgements

We thank Claire de Mazancourt and Sergio Vallina for valuable comments on the manuscript. This work was supported by the TULIP Laboratory of Excellence (ANR-10-LABX-41).

References for the whole dissertation are available at the end.

Supplementary Figures and Tables .

References Abstract

Global biogeochemical cycles are being profoundly affected by human activities; therefore it is critical to understand the role played by organisms in their regulation. Autotrophic organisms can regulate nutrient abundance at local scales through resource consumption, but most resources are inaccessible to them at global scales, either because of physical barriers or because of the presence of non-assimilable chemical forms of nutrients. Here we present a generic model of resource access limitation and apply it to the oceanic cycles of iron, phosphorus and silicon to examine whether phytoplankton can regulate the concentrations of these key nutrients. Our model predicts that autotrophs cannot at the same time strongly impact accessible nutrients and exert perfect regulation on inaccessible nutrients. We show that the ability of organisms to regulate inaccessible nutrient pools strongly depends on passive physical and chemical flows, and on the fraction of the system that is accessible to organisms. Components of global climate change such as increasing water column stratification might result in a further decrease of the biotic regulation of inaccessible nutrients in freshwater and marine systems.

Supplementary Tables

Keywords: biogeochemical cycles, nutrient cycling, regulation, nutrient limitation, resource access limitation, anthropogenic impacts

Abstract

Anthropogenic activities heavily affect biogeochemical cycles at global scales; thus it is critical to understand the degree to which these cycles can be regulated by organisms. Autotrophs can regulate nutrient abundance through resource consumption, but their growth should not be affected by changes in the supply of non-limiting nutrients. Here we present a model where autotrophs consume two nutrients -one limiting and one non-limiting nutrient -and access only part of the nutrients available in the environment. Our model predicts that autotrophs cannot efficiently regulate concentrations of the non-limiting nutrient. We show that changes in the supply of the limiting nutrient affect the concentrations of the non-limiting nutrient, and that the two nutrients vary in opposite directions. Our results suggest that interactions between nutrient cycles can result either in an increase or in a decrease in the regulation efficiency of nutrient concentrations, depending on whether the supplies of the limiting and non-limiting nutrients vary in the same or opposite directions due to anthropogenic activities. autotrophs to regulate global nutrient cycles.

Abstract

Biotic regulation of the environment at global scales has been debated for several decades. An example is the similarity between deep-ocean and phytoplankton mean N:P ratios. N and P cycles are heavily altered by human activities, mainly through an increase in nutrient supply to the upper ocean. As phytoplankton only access nutrients in the upper ocean, it is critical to understand ( 1) to what extent phytoplankton are able to regulate N and P concentrations as well as their ratio in the deep, inaccessible layer, and ( 2) what mechanisms control the value of the deep-water N:P ratio and the efficiency of its biotic regulation. With a model of N and P cycles in the global ocean separated in two layers, we show that the value of the deep-water N:P ratio is determined by non-fixer's N:P ratio, recycling and denitrification. Our model predicts that, although phytoplankton cannot efficiently regulate deep nutrient pools, they can maintain nearly constant ratios between nutrients because compensatory dynamics between non-fixers and nitrogen-fixers allows a control of deep-water chemistry through nutrient recycling. This mechanism could explain the near-constancy of the deep-water N:P ratio, in agreement with [START_REF] Redfield | On the proportions of organic derivations in sea water and their relation to the composition of plankton[END_REF][START_REF] Redfield | The biological control of chemical factors in the environment[END_REF] classical hypothesis. Surprisingly, N:P ratio of phytoplankton does not affect their ability to regulate the deep-water N:P ratio. Our model suggests that increased water column stratification as a result of global climate change may decrease the stability of the N:P ratio in the deep ocean over long temporal and spatial scales.

Supplementary Tables Appendix 3.A: Comparison of models with fixed and variable stoichiometry

In this appendix we will look at differences in regulation coefficients between the model with fixed stoichiometry and the model with variable cell quotas.

Model with fixed stoichiometry

This section corresponds to the model presented in the main text. The N:P ratio of non-fixers and N-fixers (R and R F , respectively) do not vary with nutrient availability.

For the coexistence equilibrium, concentrations are:

where

Model with variable stoichiometry

This section corresponds to the extension of the previous model, where N and P cell quotas of phytoplankton can vary depending on the availability of both nutrients [Klausmeier et al., 2004[START_REF] Klausmeier | Phytoplankton stoichiometry[END_REF]Christian, 2005]. The quotas include both structural and storage pools.

Connecting statements

In Chapter 3, we focused on the regulation of N and P concentrations as well as their ratio in the global ocean with respect to increase in both N and P supplies to the surface ocean. We showed that the compensatory dynamics between functional groups can have strong effects on the regulation of N and P pools in the surface and deep ocean. In Chapter 3, the growth of non-fixers was limited by N and that of N-fixers by P. As recycling of organic matter heavily impacts the nutrient concentrations in both accessible and inaccessible pools, it is crucial to determine which nutrient limits the total oceanic primary production to elucidate the impact of nutrient fertilisation at global scales. N-fixation is a major process involved in regulation of N and P pools in the ocean, but it is often limited by the availability of iron. In Chapter 4, we will thus focus on determining which of Fe, P or N limits the total primary production in the global ocean at large temporal scales.

Chapter 4

Ultimate colimitation of oceanic primary production

Anne-Sophie Auguères

Abstract

Global biogeochemical cycles in oceans are heavily altered by human activities, in particular through increased nutrient supply. Identifying the nutrient that limits oceanic primary production ultimately -i.e. over long timescales-is critical to understand long-term changes in the structure and functioning of ocean ecosystems, but this issue is currently unresolved.

Here we use a simple model of the global oceanic cycles of nitrogen, phosphorus, and iron to determine which of these nutrients ultimately limits oceanic primary production. Our model predicts that the nutrient that limits the growth of nitrogen fixers is generally the ultimate limiting nutrient. It also predicts ultimate colimitation of oceanic primary production by iron and phosphorus, whether phytoplankton growth is limited by a single nutrient or multiple nutrients. Thus the increase in the anthropogenic supply of iron and phosphorus to the surface ocean is likely to strongly affect ocean ecosystems at large spatial and temporal scales.

Discussion

In our model, we assumed that the P:N and Fe:N ratios of non-fixers and N-fixers are different but constant within each group. However, phytoplankton can adapt their stoichiometry depending on nutrient limitation conditions (Geider and LaRoche 2002), as included in models of adaptable cellular quotas (Klausmeier et al. 2004a(Klausmeier et al. , 2004b;;[START_REF] Diehl | Flexible nutrient stoichiometry mediates environmental influences on phytoplankton and its resources[END_REF]. The plasticity of phytoplankton stoichiometry could alter the strength of the effects of external supplies of N, P and Fe on total oceanic primary production. However, the external supply of N cannot impact heavily on total primary production as N is supplied biologically through N-fixation if scarce [START_REF] Redfield | The biological control of chemical factors in the environment[END_REF][START_REF] Broecker | Tracers in the sea[END_REF][START_REF] Tyrrell | The relative influences of nitrogen and phosphorus on oceanic primary production[END_REF]. Therefore the addition of adaptable cellular quotas in our model may only slightly affect the proportions in which the external supplies of Fe and P control total primary production (Appendix 4.B, Supplementary Table S4.1).

We also assumed that the phytoplankton death rate is constant. Explicit consideration of zooplankton predation might affect ultimate limitation of phytoplankton as zooplankton are known to change the PLN of phytoplankton in some ecosystems by modifying the ratio of elements in the environment [START_REF] Hasset | Ecological stoichiometry of N and P in pelagic ecosystems: Comparison of lakes and oceans with emphasis on the zooplankton-phytoplankton interaction[END_REF]. Decomposers also have an important role in oceanic biogeochemical cycles, and can compete with autotrophs for inorganic resources, thereby altering elemental ratios in the environment [START_REF] Cherif | When microbes and consumers determine the limiting nutrient of autotrophs: a theoretical analysis[END_REF]. However, in models that consider proximate colimitation of phytoplankton growth by several nutrients, the impacts of zooplankton or decomposer addition on ultimate limitation should be more benign. As all the nutrients consumed by phytoplankton are already considered limiting, the addition of zooplankton or decomposers might modify the proportions in which Fe and P supplies determine oceanic primary production, but it cannot generate a shift from one type of limitation to another and hence it is unlikely to change our results qualitatively.Our results highlight the importance of Fe in the ultimate limitation of oceanic primary production, in agreement with recent hypotheses [START_REF] Wu | Phosphate depletion in the Western North 197 Atlantic Ocean[END_REF][START_REF] Mills | Iron and phosphorus co-limit nitrogen fixation in the eastern tropical North Atlantic[END_REF][START_REF] Moore | Iron availability limits the ocean nitrogen inventory stabilizing feedbacks between marine denitrification and nitrogen fixation[END_REF]. 130 140 150 160 170

Supplementary Figures and Tables

Appendix 4.A: Formulation of phytoplankton growth rate based on the models of synthesizing units

In this appendix, we present the models of synthesizing units used to obtain formulations of growth rates of N-fixing and non-fixing phytoplankton [START_REF] Poggiale | How far details are important in ecosystem modelling: the case of multi-limiting nutrients in phytoplankton-zooplankton interactions[END_REF].

Growth rate of N-fixers

The growth of N-fixers is limited by P and Fe simultaneously. It corresponds to type I colimitation 7 , i.e. these two nutrients are biogeochemically mutually exclusive but are both drawn down to equally limiting levels. We assume that enzymes are synthesizing units, thus cannot be dissociated. We also assume that the binding rate for each nutrient is constant.

Enzymes of N-fixers can be in a free state (Θ * , * ), bound with a molecule of either P or Fe (Θ P, * and Θ * ,F e , respectively) or bound with both molecules (Θ P,F e ). The total amount of enzymes in N-fixers, Θ tot,F , is constant over time. k F is the release rate of Θ * , * .

We use mass balance to build a model that describes the dynamics of the enzyme at different states and of the nutrients accessible to N-fixers: where F e HF = P H R 2F /R 1F and Θ * , * + Θ P, * + Θ * ,F e + Θ P,F e = Θ tot,F .

We calculate the equilibrium value of the enzymes {Θ * , * ,eq , Θ P, * ,eq , Θ * ,F e,eq , Θ P,F e,eq }.

Note that at equilibrium, dP a /dt = dF e a /dt = -k F Θ P,F e,eq . By setting

and Θ * , * , * + Θ N, * , * + Θ * ,P, * + Θ * , * ,F e + Θ N,P, * + Θ N, * ,F e + Θ * ,P,F e + Θ N,P,F e = Θ tot .

We calculate the equilibrium value of the enzymes:

{Θ * , * , * ,eq , Θ * ,P, * ,eq , Θ * , * ,F e,eq , Θ N,P, * ,eq , Θ N, * ,F e,eq , Θ * ,P,F e,eq , Θ N,P,F e,eq } Note that at equilibrium, dN a /dt = dP a /dt = dF e a /dt = -kΘ N,P,F e,eq . By setting µ = kΘ tot , we obtain the uptake flux of N, P and Fe (i.e. the growth rate of non-fixers): G = µΘ N,P,F e,eq /Θ tot .

Appendix 4.B: Model of N, P and Fe oceanic cycles with variable stoichiometry

In this appendix we present the extension of our model, where N, P and Fe cell quotas of phytoplankton can vary depending on the availability of each nutrient. The quotas include both structural and storage pools.

Q N , Q P and Q F e are the cell quotas of non-fixers. Q N F , Q P F and Q F eF are the cell quotas for N-fixers. Q N F is supposed to be constant, as N 2 is in excess in the ocean and thus the growth of N-fixers is only limited by P and/or Fe.

Model equations are:

Q min,N and Q min,P are the minimum quotas at which the growth of non-fixers ceases.

Q min,P F is the minimum quota at which the growth of N-fixers ceases. V max,N and V max,P are the maximum uptake efficiencies of non-fixers for N and P, respectively. V max,P F is the maximum uptake efficiency of non-fixers for P.

In the same way, we obtain V max,N = V max,P /R 1 , V max,F e = V max,P R 2 /R 1 , and

The model with variable stoichiometry predicts that the nutrient that limits the growth of N-fixers (either P or Fe) is the ultimate limiting nutrient when N limits the growth of non-fixers (Table S4.1). When the growth of N-fixers is limited by Fe and that of N-fixers is limited by P, our model predicts a colimitation of total primary production by P and Fe, with a dominant effect of Fe. The use of Monod multiplicative colimitation leads to the same result (Table S4.1). These results are similar to those of the model with fixed stoichiometry, although the adaptable stoichiometry slightly decreases the effect of P on total primary production and increases that of Fe (Table 4.2, Table S4.1).

Discussion and perspectives

Synthesis of results

The ability of organisms to regulate their global environment has been debated for several decades, mainly concerning the Gaia hypothesis and the Redfield ratios in the ocean (Lenton andWatson 2000a, 2000b;[START_REF] Arrigo | Marine microorganisms and global nutrient cycles[END_REF][START_REF] Free | Do evolution and ecology need the Gaia hypothesis?[END_REF][START_REF] Tyrrell | On Gaia: A critical investigation of the relationship between life and Earth[END_REF]. Strong biotic control of environmental conditions or resources implicitly assumes that the effects of organisms that occur at local scales also occur at global scales. By consuming locally available resources, organisms can exert a strong control on their local environment [START_REF] Loreau | From populations to ecosystems: Theoretical foundations for a new ecological synthesis[END_REF]. At larger scales, however, resource access limitation is common in natural systems, because of physical or chemical barriers (Menge et al. 2009;[START_REF] Riebesell | Sensitivities of marine carbon fluxes to ocean change[END_REF][START_REF] Boyd | The biogeochemical cycle of iron in the ocean[END_REF][START_REF] Vitousek | Terrestrial phosphorus limitation: mechanisms, implications, and nitrogen-phosphorus interactions[END_REF]. In this thesis, I have studied how organisms respond to changes in the supply of nutrients in ecosystems at global scales and how they can regulate the concentration of these nutrients when their accessibility is limited.

In Chapter 1, I used a theoretical framework based on the theory of resource access limitation [START_REF] Huston | Competition and coexistence: The effects of resource transport and supply rates[END_REF][START_REF] Deangelis | Modelling nutrient-periphyton dynamics in streams: the importance of transient storage zones[END_REF][START_REF] Loreau | Coexistence of multiple food chains in a heterogeneous environment: Interactions among community structure, ecosystem functioning, and nutrient dynamics[END_REF][START_REF] Loreau | Biodiversity and ecosystem functioning: A mechanistic model[END_REF] to assess the extent to which autotrophs can regulate the concentrations of accessible and inaccessible pools of a limiting nutrient with respect to changes in its supply. The model describes the dynamics of an autotrophic population and an inorganic nutrient that occurs in two pools,

General conclusion

The overall result of this thesis is that organisms are only able to partially regulate nutrient pools, as they can only access a small part of nutrients in their environment. Mechanisms such as coupling of nutrient cycles and competition between functional groups with a different stoichiometry can alter the strength of biotic regulation of global biogeochemical cycles, either positively or negatively. An inefficient regulation of inaccessible nutrient concentration, however, does not exclude a strong biotic regulation of nutrient ratios, as is the case with Redfield ratios in oceans. Nutrient fertilisation of oceanic and terrestrial ecosystems is thus likely to have a strong impact on primary production and global nutrient cycles at both small and long timescales [START_REF] Smith | Eutrophication: impacts of excess nutrient inputs on freshwater, marine, and terrestrial ecosystems[END_REF]. Although the various theoretical models developed in this thesis allow a better understanding of the regulation efficiency of nutrient concentrations by autotrophs at the global scale, a great number of other regulatory processes can also potentially affect the efficiency of biotic regulation of global biogeochemical cycles.

A further question is how herbivores and other trophic levels affect the regulation of global nutrient cycles. This issue could be addressed through the explicit description of additional trophic levels in the models. and how they can regulate the concentration of these nutrients when their accessibility of is limited. We further investigated the mechanisms driving the regulation of Redfield ratios in oceans, and the effects of nutrient fertilisation on total oceanic primary production.

We showed that organisms cannot efficiently regulate nutrient pools. Mechanisms such as coupling of nutrient cycles and competition between functional groups can alter the strength of biotic regulation of global biogeochemical cycles, either positively or negatively.

An inefficient regulation of inaccessible nutrient concentration, however, does not exclude a strong biotic regulation of nutrient ratios, as is the case with Redfield ratios in oceans.

Nutrient fertilization of oceanic and terrestrial ecosystems is thus likely to have a strong impact on primary production and global nutrient cycles at both small and long timescales.

KEYWORDS: biogeochemical cycles, Earth system, regulation, nutrient cycling, Redfield ratios, nutrient limitation, resource access limitation, anthropogenic impacts, primary