Anne-Sophie Lafuite

Michel Loreau

Sophie Lafuite

Denise Gonzague

Keywords: Biodiversity, lagged feedback, regime shifts, social-ecological system, social norms, sustainability

In this thesis, I intend to develop theoretical interdisciplinary approaches that provide new insights into the sustainability of coupled social-ecological systems (SESs). Especially, I focus on the long-term consequences of the current biodiversity crisis on human demography. Human-nature interactions occur over many spatial and temporal scales, Yann et Tom, pour ces "kiki" mémorables, les leçons de salsa/rock et les cours de guitare, les cols à vélo avant le passage du tour de France, les bivouacs et toutes ces expériences culinaires (et excursions hospitalières) en votre compagnie. A mes compagnons "outdoor", Alexis, Gauderic, Gougou, Jeff, Jonathan et Kévin, merci de m'avoir sorti le nez de v ma thèse pour aller grimper ou crapahuter en montagne ! Merci à Robinou pour les expériences d'escalade arboricole, les tentative de vol de canoë ratées, et les schoko bons suprise qui redonnent le sourire :) Merci à Anne-So (n°1) pour les pauses thé remplies de conseils, astuces et potins, ainsi qu'à Audrey pour ses nombreux conseils avisés sur mon premier article, et les discussions autour du projet Homme-Nature en général. Et enfin, merci à Arnaud pour son écoute et son soutien sans faille, ses cadeaux étranges, et ses bons petits plats. Et parceque ma vie ne s'est pas limitée à Moulis et son CNRS ces dernières années, merci à ma copine de toujours (ou presque), kikette, et à tous les autres copains niçois, lyonnais ou parisiens, pour leurs encouragements. Merci à Evrard de m'avoir donné la chance de l'accompagner à trois reprises en expédition. Merci pour ces expériences qui ont changé ma vie, m'ont permis de rencontrer des personnes extraordinaires et de tisser des liens d'amitié très forts. En particulier, merci à Aurélie "moineaute" pour tes magnifiques aquarelles qui me rappellent ces moments inoubliables au milieu de la forêt du Sulawesi, et à Tanguy, the spécialiste des crottes d'ours des Pyrénées.

Un merci tout particulier à Yanick, pour ta présence malgré la distance, pour m'avoir transmis un peu de ton énergie inépuisable, et permis de prendre un peu de hauteur.

and mismatches between the scales of human dynamics and ecological processes can be detrimental to sustainability. I use dynamical system modelling to represent humannature interactions through a feedback loop, that includes human economy, demography, and biodiversity dynamics. Human population growth and natural habitat destruction drive species to extinction, thus affecting the provisioning of essential ecosystem services.

In turns, many of these services indirectly or directly affect agricultural productivity, thus threatening the provisioning of food to a growing human population. First, I show that temporal mismatches resulting from a time-delayed loss of biodiversity can generate unsustainable human population cycles. Model analysis enlightens the existence of critical thresholds in the dynamical variables of the system. Remaining within those sustainable boundaries allows avoiding overshoot-and-collapse population crises that greatly reduce human well-being, and thus sustainability. Second, I explore the relationships between human behavioral change and the adaptive capacity of SESs. I show that temporal mismatches postpone desirable behavioral changes and increase the probability of an abrupt shift towards an unsustainable basin of attraction. Policies that help reduce temporal mismatches, i.e., accelerate behavioral changes or reduce extinction debts, are thus crucially needed. The third chapter explores the potential of land-use management directed towards the conservation of natural habitats to reduce temporal mismatches, and shows that i the economic internalization of biodiversity feedbacks can help prevent or mitigate unsustainable crises. This thesis emphasizes the role of feedbacks and scales in human-nature interactions, and highlights the importance of foresight for the long-term sustainability of human societies. Policy implications regarding cross-scale effects are discussed, so as to define integrative approaches fostering adaptability and sustainability. Results call for a better understanding and assessment of both time delays and spatial processes, and urge for the development of integrative management approaches accounting for human demography, socio-economic and long-term ecosystem dynamics, at various spatial and temporal scales.

Résumé général

La perte des habitats naturels et leur fragmentation est responsable d'une crise d'extinction de la biodiversity inédite, qui réduit à son tour l'approvisionnement en services écosystémiques difficilement substituables. La plupart de ces services influencent directement ou indirectement la productivité agricole, menaçant ainsi l'accès à l'alimentation d'une population humaine en pleine expansion. De plus, la destruction des habitats ne mène pas instantanément les espèces à l'extinction. Des espèces vouées à l'extinction peuvent mettre des décennies à s'éteindre, ce qui génère d'importantes dettes d'extinction. Les mécanismes sous-jacents à ces dettes reposent sur des processus écologiques impliquant plusieurs échelles spatiales, et leurs conséquences s'étendent sur des échelles temporelles allant de la décennie au siècle. Les effets d'échelle sont omniprésents dans les dynamiques des systèmes socio-écologiques (SSEs). En particulier, l'interaction entre des variables lentes et rapides peut amplifier des feedbacks non durables et mener à des effets inattendus, comme des transitions abruptes vers des états indésirables. La présente thèse propose d'explorer les conséquences sur la durabilité des SSEs de ces délais temporels. Le travail développe des approches interdisciplinaires et théoriques novatrices, incluant des éléments importants d'économie, de démographie humaine et de dynamique sociale et écologique caractérisant ces systèmes. Les résultats obtenus sont utilisés pour proposer plusieurs pistes de politiques publiques permettant de mieux prendre en compte ces effets d'échelle tout en limitant leurs conséquences négatives et en favorisant l'adaptabilité des SSEs. Un premier modèle couplant de manière stylisée la démographie de la population humaine, l'économie de marché et une érosion différée de la biodiversité affectant la productivité agricole, est d'abord présenté. Son analyse montre l'existence de seuils critiques dans les iii variables dynamiques du système. Maintenir la taille de population humaine et la perte de biodiversité en deçà des seuils durables précédemment identifiés permet d'éviter des effondrements type "overshoot-and-collapse", durant lesquels le bien-être humain est fortement réduit, ainsi que la durabilité du système. Mais la capacité d'adaptation des SSEs n'est pas à sous-estimer et peut permettre de maintenir la durabilité d'un SSE, notamment via des changement de comportement humain. Le chapitre suivant utilise ces seuils de durabilité pour définir une norme de consommation durable, dont l'adoption permet de préserver la durabilité du système. Cependant, lorsque les choix des consommateurs se basent sur leur perception à court terme de l'état de dégradation environnementale, le modèle montre que les délais temporels engendrent de la procrastination vis-à-vis des changements de comportement à effectuer. Cette procrastination peut mener le système à basculer brusquement dans un bassin d'attraction non durable. Dans une telle situation, seule une vision de long terme peut permettre d'éviter un tel changement de régime à large échelle.

Certaines politiques publiques peuvent s'avérer utiles pour accélerer le changement des comportements de consommation en améliorant l'accès à l'information des consommateurs. Les politiques publiques peuvent aussi directement réduire les dettes d'extinction en conservant les habitats naturels, par exemple via la mise en place de taxes foncières limitant la destruction des habitats naturels, ou la création de réserves naturelles. Mais de telles politiques doivent alors prendre en compte les caractéristiques spatiales des SSEs, et notamment les effets d'échelle. Cette thèse démontre l'importance des effets d'échelle temporels sur la durabilité de long terme des SSEs. Une meilleure compréhension de ces effets d'échelles et de leurs conséquences sur les systèmes couplés homme-nature est nécessaire. Le développement d'approches de gestion intégratives doit prendre en compte les feedbacks existant entre la démographie humaine, les systèmes socio-économiques et la dynamique des écosystèmes, à différentes échelles spatiales et temporelles.

Remerciements / Acknowledgments

The work made during this PhD project has benefited from insightful advice, help and support from numerous people I would like to warmly thank here / J'aimerais remercier ici toutes les personnes grâce à qui cette thèse a été rendue possible. Tout d'abord, merci à Michel, mon directeur de thèse, de m'avoir laissée libre dans le choix de mon sujet de thèse, et de m'avoir accordé sa confiance. Merci pour ta patience et ton soutien tout au long de cette (longue) thèse! Merci également à François Salanié pour les discussions et conseils avisés sur les aspects économiques de mon modèle, et pour avoir répondu présent à chacun de mes appels au secours.

I would also like to thank all the members of the "Human Spatial Behavior, Biodiversity and Ecosystem Services" group, for insightful comments and discussions at the beginning of my PhD project; Charles, Ann, Jean, Ingela, and Claire. A special thank to David, for useful comments on my first manuscript, and for the great time we spent crack climbing together in Arizona.

Merci à tous les collègues et amis que j'ai eu la chance de cotoyer durant ces quelques années dans ces belles Pyrénées Ariégeoises. Merci à Maryse et Christian pour leur accueil formidable, et tous ces bons moments partagés dans leur magnifique gîte. Merci à tous les copains de la première heure, Tomas, Shaopeng, Mat, Theresa, Jarad, Aisha, Elie, James,

GENERAL INTRODUCTION Context

Nature and society have long been perceived as separate and independent. This historical view was initiated during the Enlightment era [1], during which developments in science and technology enhanced people's abilities to control or transform nature, "from desert wilderness to cultivated garden" [2]. The forces of industrialization and urbanization further split humans from their environments, apparently decoupling human population growth from natural constraints. At that time, this decoupling between humans and nature was already questioned by Malthus, who enlightened the limits of exponential population growth on a finite planet [3]. However, views of humans as limited by nature remained marginal [4], and change of paradigm started only recently [5]. There is now mounting evidence that humans and nature influence each other through bidirectional interactions [6,7,8]. Yet the coupling of human societies and ecosystem dynamics is not fully recognized nor accounted for in scientific research and public policies [9]. In this coupled human-nature dynamics, human demography and biodiversity play major roles.

Human population growth and scenarios

Human population has been growing tremendously since the 1950s, mainly due to advances in public health, massive consumption of fossil fuels, and increases in agricultural productivity. The rate of population growth has been declining since the 1980s, while the absolute total numbers kept increasing. As of mid-2017, the world population reached 7.6 billion, and its annual growth rate has fallen from 1.24 % to 1.10% between 2005 and 2017, due to fertility declines in most areas of the world. Between 2010 and 2015, countries where fertility is below replacement levels accounted for 46 % of the global population [10].

Under the most likely United Nations population scenarios, the world population is projected to reach 8.6 billion in 2030, and to increase further to 9.8 billion in 2050 and 11.2 billion by 2100 [10]. The same scenarios suggest that global population is likely to continue to rise later in the century, with only little chance (23 %) to stabilize or begin to fall before 2100.

However, these population scenarios are highly sensitive to future fertility and population aging assumptions. Several demographic uncertainties, such as international migration and the structure of families, cast doubts upon the reliability of these population projections [11]. Other demographic models that expand conventional approaches via probabilistic treatments of fertility, mortality, and migration support a peak population forecast at around 2070 followed by a decline [12]. Despite demographic transition and fertility reductions [10,13], a stabilization of the world population seems unlikely this century [14,15]. Moreover, human population projections have poorly accounted for potential ecological feedbacks on human population growth so far, despite the scale of impact of human activities on the environment [16]. Changes in the natural resources and the Earth System (e.g., climate change) have important feedback effects on human societies [9]. However, current models do not incorporate these critical feedbacks, and often take human demography and economic growth as exogenous drivers, using estimates such as the UN population projections.

Accounting for the exhaustion of non-renewable natural resources on the industrial sector, the recent "Limits to Growth" updated report [17] predicts an overshoot and eventual collapse of food production, human population and living standards by 2030-2040, under the business-as-usual scenario. Similarly, another study predicts the decline of agricultural production between 2030-2060 due to water scarcity [18]. More generally, concerns about a potential global collapse of our society [19] due to overshooting Earth's limits, such as biodiversity and essential ecosystem services, are raised in a number of studies [20,7,8,21].

Population growth and natural habitat destruction

Human use of land has transformed ecosystems across most of the terrestrial biosphere for millennia [23]. The conversion of natural lands to croplands, pastures, and urban areas represents the most visible form of human impact on the environment [24], with 40% of Earth's land surface being currently under agriculture [25], and 75% experiencing measurable human pressures [26]. These pressures are rapidly intensifying in biodiversityrich places [27], since most land conversion occurs in the tropics through forest conversion to agriculture [28,29].

In these biodiverse and ecologically fragile areas, high fertility and associated rapid population growth directly contribute to land conversion, such as in Central America [30].

A reason for this high fertility is that children constitute an asset to farm families that are often short on labor [31], and guarantee families a certain number of surviving children despite high rates of child mortality [32]. Such vicious circles of sustained high fertility in the face of declining environmental resources [33] can result in "tragedy of the commons" situations, where overuse of natural resources leads to their collapse [34].

More generally, the relationship between human population dynamics and natural habitat loss is highly dependent on the scale of analysis [30]. At local scales, the effect can be ambiguous [35,36], and population dynamics usually acts in concert with other significant factors such as local institutions, policies, economic globalization, and cultural change [37,38].

At the global level, human population growth and consumption are the two major drivers of humanity's ecological footprint [16]. Between 2000 and 2016, the human pop-GENERAL INTRODUCTION ulation has increased by 23% and the world economy has grown 153%, while the human footprint has increased by just 9%, due to increased resource use efficiency [26]. Building upon such efficiency gains, some growth economists advocate for the potential of economic growth to improve environmental quality, following an initial phase of deterioration [39]. However, the benefits of rising resource use efficiency and the demographic transition [40,41,13] can be undermined by the global dietary transition towards more land demanding diets [42,43], as well as by perverse economic rebound effects due to a higher opportunity cost of land conservation [44,45], both resulting in further habitat destruction, despite a slower human population growth and a higher production efficiency.

Impacts of land conversion on species richness

Domination of ecosystems by humans is increasingly taking over the primary productivity of land (30% of terrestrial productivity), mainly through land conversion and deforestation [46]. Natural habitat loss is threatening more and more species and the ecosystem functions that they provide [47]. Global species extinction rates are about 1000 times the likely background rate of extinction [48,49]. The Living Planet Index shows a 58% decline in populations monitored between 1970 and 2012 (Fig. B). This trend is likely to continue in the 21st century [50], and could result in the loss of two thirds of species populations by 2020 compared to 1970 levels [51,52].

At local spatial scales, the worst impacted habitats show an average 76.5% reduction in local species richness (Fig. C), compared to a 13.6% average reduction globally [22].

Despite recent data syntheses finding mixed evidence for patterns of net species loss at local spatial scales, thus arguing for a no net change in local species richness [53,54], there is growing evidence for a negative relationship between land use changes and local biodiversity loss [55,56].

Additionally to natural habitat loss, habitat fragmentation is causing rapid species loss [57,58] and alteration of ecosystem functioning [59]. Fragmentation of natural habitat amplifies the negative effect of habitat loss on species extinctions [60], and may double biodiversity loss from deforestation in tropical forests [61]. [22].

Biodiversity, ecosystem functioning and agricultural productivity

Species richness, i.e., the numbers of species, is a well-studied surrogate for several other dimensions of biodiversity, such as taxonomic/phylogenetic distinctiveness, and functional diversity [62]. Species extinctions are altering key processes important to the productivity of Earth's ecosystems [63], and further species loss will accelerate change in ecosystem processes [64,65]. Local species richness underpins many ecosystem functions, such as the biogeochemical processes that regulate the Earth system [66]. Thus, changes in biodiversity that alter ecosystem processes can have profound consequences for services that humans derive from ecosystems [46]. The ecosystem consequences of local species loss are as quantitatively significant as the direct effects of several global change stressors, such as ozone, acidification, elevated CO 2 , and nutrient pollution [67].

Quantitative assessments of the relationship between biodiversity and ecosystem process rates show clear evidence of a positive effect of biodiversity for most ecosystem services at the community level [68] and larger spatial scales [69]. Complementarity between species' patterns of resource use can increase average rates of productivity and nutrient retention [63]. Most experiments indicate that the relationship between ecosystem process rates and species richness saturates at higher levels of diversity, suggesting redundancy. However, although species may appear functionally redundant when one function is considered under one set of environmental conditions, many species are needed to maintain multiple functions at multiple times and places in changing environments [70]. Such a complementarity between species and functions across time scales results in a stabilizing effect of diversity on ecosystem processes [71].

Evidence that land conversion and biodiversity loss are threatening the provisioning of essential ecosystem services [6] has contributed to raise concerns about the capacity of the biosphere to provide goods and services in the long term [72], especially regarding the growing demand for food production [73]. Important biodiversity-dependent services to agricultural production include biological control of pests [74], crop pollination [75],

erosion limitation [76], water quality [77] and soil quality [78].

Globally, about 10% of agricultural yields are estimated to be destroyed by animal GENERAL INTRODUCTION pests, and estimated crop losses to pests have significantly increased in the last 40 years, despite a 15-to 20-fold increase in chemical pesticide use [79]. Successful biological control of pests by natural enemies is thus of key economic and ecological importance [80,81],

and is responsible for an estimated 50-90% of the biological pest control occurring in crop fields [82]. Along with biodiversity loss, the simplification of agricultural landscapes in favor of a domination by arable cropland affects the exchange of services between crop and non-crop habitats [83,74]. Landscape complexity has been shown to favor both the diversity and abundance of natural enemies [84], compared to simple and intensely cultivated landscapes [74,83]. However, its effect on service provision compared to simple landscapes can be ambiguous [85], due to changes in trophic interactions between diverse enemy assemblages [86].

Similarly, there is mounting evidence that pollinator decline has significant economic consequences in many agricultural areas, with a total economic value of pollination estimated at €153 billion, or 9.5% of the value of the world agricultural production used for human food in 2005 [75]. Pollinator decline also affects consumers surplus, especially regarding vegetable, fruits, edible oil crops, and nuts, up to €50 billions in each category. Consumption of these goods is predicted to sharply decline compared to current consumption levels. Functional group diversity of bee pollinators increases crop yield through niche complementarity [87]. Natural habitat loss and isolation reduce pollination services despite abundance of honey bees [88], emphasizing the importance of wild pollinator conservation.

The soil organism community also affects land productivity through a wide range of ecosystem services that are essential to the sustainable function of natural and managed ecosystems [78]. Soil organisms have direct and indirect impacts on agricultural productivity, e.g., through carbon and nutrient cycles and decomposition, soil structure modification, and food web interactions [89]. Soil erosion is a major environmental threat to the productive capacity of agriculture, and enhances nutrient loss, which reduces the fertility of remaining soils [90], with important economic consequences [76]. Each year about 10 million ha of cropland are lost due to soil erosion, 10 to 40 times faster than the GENERAL INTRODUCTION rate of soil renewal, thus imperiling future human food security and environmental quality [91]. Enhancing functional diversity in agroecosystems can help secure crop protection and soil fertility [92]. Conservation of biodiversity can also help manage nutrient uptake and storage, and improve water quality by limiting excessive nutrient loading of water bodies [77], since diverse systems capture a greater proportion of biologically available resources such as nitrogen.

Trade-off between land conversion and food production

Land use changes have enabled humans to appropriate an increasing share of the planet's resources, but they have undermined the capacity of ecosystems to sustain food production, maintain freshwater and forest resources, and regulate climate and air quality. Thus, we face the challenge of managing trade-offs between immediate human needs and maintaining the capacity of the biosphere to provide goods and services in the long term [72].

Given the current failure to feed humanity, some authors are pessimistic about the capacity of the planet to make the projected 9.7 billion population food-secure and healthy in 2050 [93]. Indeed, in order to meet the world's future food security and sustainability needs, food production must grow substantially while, at the same time, agriculture's environmental footprint must shrink dramatically [94,95]. Solutions include halting agricultural expansion, closing 'yield gaps' on underperforming lands, increasing cropping efficiency, reducing waste and shifting diets [95,96]. Together, these strategies could double food production while greatly reducing the environmental impacts of agriculture [42].

However, these solutions poorly account for the dependence of agricultural production on biodiversity. Conventional intensification tends to disrupt beneficial functions of biodiversity by assuming that biodiversity in agroecosystems is functionally negligible, and that technological improvements can substitute for ecosystem services [97]. Intensification of agriculture by use of high-yielding crop varieties, fertilization, irrigation, and pesticides has contributed substantially to the tremendous increases in food production over the past 50 years, at the expense of water quality, toxicity from pesticides, nitrous GENERAL INTRODUCTION oxide emissions, and degradation of habitat for biodiversity [98].

Intensification can also exacerbate land-clearing in the absence of appropriate policies and enforcement, as a consequence of economic rebound effects [44]. Thus, a "land sparing" view [99,100,101], i.e., the segregation of land for nature and for production, may not always benefit biodiversity conservation in the long run [102]. Moreover, such a "land sparing" view fails to account for real-world complexity, and especially the dependence of agricultural production on biodiversity-dependent ecosystem services at various spatial scales [103,104]. The preservation of natural habitats and the services they provide, while enhancing food production and facing the looming land scarcity, is a central challenge for sustainability [44] that requires developing holistic ways to conceptualise challenges related to food, biodiversity, and land scarcity [105,106].

Human-nature interactions in the Anthropocene

Nature and society as a coupled system

Until recently, the domains of "society" and "nature" were seen as separate and independent. These distinctions have become difficult to maintain given the intensity and scale of the human activities on Earth. The early 70s have witnessed a rising awareness about the finiteness of non-renewable natural resources and the long term consequences of exponential economic and population growth [107,4]. Although criticized for their relatedness to the Malthusian theory [3], and for underestimating the forward-looking behavior of people

[108] and the potential of economic growth to compensate for declines in environmental quality [109,110,111,112], these early studies paved the way for further investigations, and brought human demography into the debate. The Brundtland report [5] highlighted new environmental concerns, such as deforestation, desertification, biodiversity loss, climate change, as well as the necessity to view economy as a subsystem of the larger ecological system. The report also popularized the idea of "sustainability" and "sustainable development" as a requirement for intra and inter generational justice [113].

GENERAL INTRODUCTION

"Sustainable development is a development that meets the needs of the present without compromising the ability of future generations to meet their own needs."

The Brundtland Report (1987) [5] The Brundtland report challenged many of the fundamental assumptions of neoclassical economics, such as the assumption that human-made capital is a near-perfect substitute for natural resources. However, its general definition of sustainable development focusing on human "needs" was not straightforward to formalize into economic or ecological terms, thus leading to a variety of sustainability definitions, both within economics and ecological research.

Economic sustainability and human well-being

Some of the first formal attempts to analyze the new debate in terms of economics defined sustainability as the "non-declining utility of a representative member of society for millennia into the future" [114]. Following conventional reasoning on sustainability, some authors have argued that internalizing resource market inefficiencies and environmental externalities would allow for a sustainable allocation of human-made and natural capital [115]. This view called "weak sustainability" is seeking to maintain constant the sum of human-made and total natural capital, thus allowing for substitution between them. An alternative approach to sustainable development has focused on natural capital assets, suggesting that they should not decline through time [116]. This "strong sustainability" view seeks to maintain intact natural capital and man-made capital separately, highlighting the limited substitutability between them, and recognizing that natural capital may also provide indirect services to human economies [117].

With the emergence of a sustainability literature following the publication of the Brundtland Report [118,119], mainstream neoclassical economists have started recognizing that economic growth could not in the long term compensate for declines in environmental quality [120], while introducing the idea of a finite, but dynamic, human carrying capacity, including many resource limitations originating from available water, GENERAL INTRODUCTION energy, and other ecosystem goods and services [121,120].

Ecological sustainability and ecosystem resilience

The discovery of multiple basins of attraction in ecosystems in the 1960-1970s inspired environmental scientists to challenge the dominant stable equilibrium view [122]. In the stable equilibrium view, resilience, or "engineering resilience", measures the time it takes the system to recover from a perturbation. Early focus on near-equilibrium stability has led to inappropriate command-and-control resource management, such as maximum sustainable yields, in situations where slowly changing variables can lead to abrupt changes

[123].
Conversely, "ecosystem resilience" has been defined as the propensity of a system to retain its organizational structure following perturbation, and the magnitude of disturbance that can be absorbed before the system shifts towards an alternative equilibrium [123]. This view of ecosystems as resilient and multi stable inspired the definition of ecological sustainability [124], and the mounting evidence of regime shifts in natural ecosystems [125,126] has paved the way for the study of social-ecological regime shifts [127,128,129].

However, this view of the resilience and stability of ecosystems is heavily debated within ecological researchers for being inconsistent and one-dimensional, thus unable to capture the multidimensional nature of ecological stability [130].

An ecological economics of sustainability

Towards integrative sustainability approaches

Mounting evidence of the unsustainability of our current development path calls for the development of an integrative theory for long-term ecological, economic and demographic changes, in order to unravel the complexity of coupled human and natural systems [145].

These changes concern rapidly and slowly changing processes, e.g., ecological, social, cultural [133], that vary across space, time, and organizational units, exhibit nonlinear dynamics with thresholds [138], reciprocal feedback loops [9], time lags [139], resilience [133] and surprises [127,129].

Developing theory for sustainable futures requires a model of (1) how human and ecological processes interact across temporal scales, (2) how changes in land use and ecosystems feed back on human demography, and (3) how human societies adjust their behavior in response to perceived changes in ecosystems.

Cross-scale biodiversity feedbacks in coupled SESs

Cross-scale influences have been increasingly studied due to the growing impact of humans on the planet [146]. Scale is usually defined in terms of spatial and temporal dimensions

GENERAL INTRODUCTION

In predator-prey systems, slow-fast dynamics can result in boom-bust cycles, e.g., between insect pests and boreal forests [162]. Interaction between a rapidly growing human population and a slowly regenerating natural resource can lead to similar boombust cycles, and is one of the explanations for the collapse of the civilization on Easter . Further habitat fragmentation then increases relaxation rates, so that extinction debts are generally larger in large habitat fragments [57].

Estimates suggest that 80% of the species extinctions in the Amazon are still pending [170], which may increase the number of 20th-century extinctions in bird, mammal, and amphibian forest-specific species by 120%, i.e., more than 140 forest-specific vertebrates

Breaking the taboo of human demography

The interface of population growth and deterioration of the local resource base has been much neglected by modern demographers and development economists [33], who point to the accumulation of capital and technological progress to discredit potential malthusian overshoot-and-collapse population crises [3,144]. However, accounting for population growth shows that the accumulation of manufactured capital, knowledge and human capital (health and education) has not compensated for the degradation of natural capital in South Asia and sub-Saharan Africa, as well as in the United Kingdom and the United with Easter Island being the most famous and well-studied example of societal collapse [190]. Overexploitation of the main natural resource of this pacific island, palm tree, led to an overshoot-and-collapse population crisis that is well reproduced by a classical predator-prey model [163], with a slowly regenerating resource relative to a fast growing human population.

Among the many extensions to this predator-prey model [191], some have shown that technological progress [192] and human foresight [193] would not have prevented the collapse of Easter Island civilization. These studies suggest that modern societies may be vulnerable to such crises as well, and advocate for the importance of demographic adjustments as a mean to prevent such crises [194]. Other extensions emphasize the role of governance failure, tipping points in ecological systems, and positive feedbacks between the economy and the environment, as preconditions for a crisis [195]. More recently, economic stratification was also highlighted as a potential cause of collapse in unequal societies [196]. However, models that account for biodiversity feedbacks on human demography are still lacking [9].

Accounting for human behavior and perception

Another crucial aspect of human-nature interaction is human behavioral change and adaptive capacity [133,197]. Historically, human responses to crises have allowed societies to prevent or adapt to social ecological traps [198]. Including human behavioral change in coupled ecological-economic models can greatly affect the dynamic of the system [199] and is a central aspect of the adaptive capacity and resilience of coupled human-nature systems [131]. Among the many human behavioral theories available [200], rational economic behavior assumes that human decisions seek to maximize short-term financial returns and GENERAL INTRODUCTION consumption utility. The behavior of the population arises from the aggregation of individual identical behaviors. Prices and foresight are assumed to transmit present and future resource scarcities to human decisions. However, many externalities of human activities that affect public goods, such as biodiversity, are not reflected by market prices.

When not internalized through appropriate economic tools, such externalities can result in the overexploitation of common resources [34].

Early research on the interaction between the environment and human populations have thus emphasized the need for government control in order to prevent the overexploitation of natural resources. However, recent empirical evidence has shown that local communities can achieve sustainable resource use through cooperative self-governance

Objectives, methods and main questions

General objective

Societies and ecosystems interact over many spatial and temporal scales [147] and through several feedbacks [9]. However, social and ecological scales are not always aligned Especially, time delays between ecological and human dynamics, such as extinction debts, generate temporal scale mismatches that affect the feedback loop controlling the dynamics of the system. The aim of this thesis is to investigate theoretically the consequences of such time-delayed biodiversity feedbacks on the sustainability of social-ecological systems.

Theoretical modelling of social-ecological systems

Several approaches have been used to study the sustainability of coupled social-ecological systems [133]. Among them, theoretical modelling allows exploring the long term consequences of the interaction between dynamical variables through abstract simulations.

These simulations allow assessing the qualitative effect of a changing ecological, social, or economic parameters of an SES. Such theoretical approaches can help identify important feedbacks and counter-intuitive effects, and design policies to foster sustainability. However, there is a lack of theoretical SES models that simultaneously account for human demography, behavioral change and biodiversity feedbacks, in a sustainability perspective.

In order to account for the feedback loop between biodiversity loss and human population growth, I build upon the literature on human-nature interactions modeled as dynamical predator-prey systems [163,195,191,194,196]. However, conversely to classical predator-prey models where the natural resource of interest is directly exploited by the predator [163], here I assume an indirect coupling between humans and biodiversity, through the feedback of biodiversity loss on agricultural production. This modelling approach thus challenges classical studies of the trade-off between food production and biodiversity conservation [99,101] that poorly account for the biodiversity feedback on agricultural productivity [6].

Human behavior is modeled both through the rationality of economic decisions, and the conformance to social norms. In the basic model structure (chapter 1), the behavior of the population results from the aggregation of individual economic decisions regarding their agricultural and industrial consumptions [194]. In turns, total demand drives production and land conversion in order to meet the market equilibrium between supply and demand. Since consumption behaviors are not only driven by economic rationality, but are also influenced by others' behaviors [206], I extend this basic model structure to account for the influence of social norms on human decisions (chapter 2).

The ecological compartment of the model includes evidence from theoretical and experimental ecological research. First, the dependence of species extinction on habitat loss is derived from a classical species-area relationship (SAR), one of the best-known patterns SARs only capture, however, the equilibrium number of species of a system, but ignore the transient dynamics of species extinctions, which can be time-delayed [166,169]. Accounting for such extinctions debts can be done through the relaxation rates of ecological communities [170]. Theoretical and experimental evidence suggest that these relaxation rates are proportional to the difference between current and long-term species richness, as captured by SARs [164,166,169]. Time-delayed loss of biodiversity is then assumed to feed back on human dynamics through a reduction in agricultural productivity. Finally, we assume a positive and saturating relationship between biodiversity and ecosystem processes, based upon mounting empirical and theoretical evidence [68,216].

The value of such a minimal conceptual model lies in its analytical tractability and the possibility of fully understanding its dynamics across the parameters' space. Analysis of both its transient, i.e., far-from-equilibrium, and the equilibrium features allows exploring the qualitative effects of the ecological, social, technological or economic parameters.

Main questions

In the following chapters, I aim at investigating the long-term consequences of a timedelayed biodiversity feedback on the sustainability of SESs.

In chapter 1, I seek to derive a condition on parameters' space under which the classical definition of sustainability, i.e., non-decreasing human well-being over time [5,114], is met. Since human well-being depends on biodiversity through the feedback of biodiversitydependent ecosystem services on agricultural consumption, this sustainability condition integrates both economic and ecological constraints. Such an integrative condition allows assessing the qualitative effects of various parameters of the sustainability of the SES, as well as the existence of critical thresholds in the variables of the system.

In chapter 2, I build upon the results of chapter 1 to identify sustainable features of SESs, and investigate conditions under which norm-driven human behavioral change can enforce such sustainable features. I also question the robustness of behavioral changes to time-delayed biodiversity loss, and explore the relationship between time delays and resilience.

Despite the potential of norm-driven behavioral change in enforcing large scale shifts [210], government control remains an important tool to enforce sustainability [34]. However, uncertainty regarding natural processes, such as the precise temporal dynamics of biodiversity loss, can undermine the efficiency of public policies. In chapter 3, I explore the efficiency of a land-use management policy in enforcing sustainability, through the internalization of the externality of biodiversity loss on agricultural production, and despite imperfect information regarding ecological time delays.

CHAPTER 1

Time-delayed biodiversity feedbacks and the sustainability of social-ecological systems

Anne-Sophie Lafuite

Chapter outline

Models capturing the dynamic feedback loop between human population growth and biodiversity loss across several temporal and spatial scales are lacking. Chapter 1 takes on a dynamical system approach in order to capture the interaction between human societies, natural habitat destruction and biodiversity loss. Our modelling approach includes insights from market economics, spatial ecology, and human demography. This work aims at investigating the long-term consequences of time-delayed biodiversity feedbacks on the sustainability of human societies.

Introduction

Current trends in human population growth [1,2] and environmental degradation [3] raise concerns about the long-term sustainability of modern societies, i.e. their capacity to meet their needs "without compromising the ability of future generations to meet their own needs" [4]. Many of the ecosystem services supporting human systems are underpinned by biodiversity [5], and current species extinction rates threaten the Earth's capacity to keep providing these essential services in the long run [6,7]. The long-term ecological feedback of ecosystem services on human societies has been largely ignored by neo-classical economic theory, mainly due to the focus on short-term feedbacks, and the assumption that ecosystem processes can be substituted for by human capital (e.g. tools and knowledge) and labor, thereby releasing ecological checks on human population and economic growth [8,9]. In particular, tremendous increases in agricultural productivity resulted in more than 100% rise in aggregate food supply over the last century [10].

However, the substitution of human capital for natural resources, also called the "technology treadmill" [11], is currently facing important limitations. One such limitation is land scarcity, as the remaining arable land reserve might be exhausted by 2050 [12]. Moreover, recent projections suggest a slowdown in the growth rate of agricultural Total Factor Productivity (TFP), which measures the effect of technological inputs on total output growth relative other inputs [13]. Technological improvements may not compensate for arable land scarcity [14], thus questioning the potential for continued TFP growth in the future [15,16]. Another limitation comes from the loss of many biodiversity-dependent ecosystem services which play a direct or indirect role in agricultural production, such as soil formation [17], nutrient cycling, water retention, biological control of pests [18] and crop pollination [19,20]. Substituting ecological processes with energy and agrochemicals has mixed environmental impacts, with unintended consequences such as water use disturbance, soil degradation, and chemical runoff. These effects are responsible for a slowdown in agricultural yield growth since the mid-1980s [21] and have adverse consequences on biodiversity, human health [22] and the stability of ecosystem processes [23].

CHAPTER 1

Moreover, biodiversity does not respond instantaneously to land-use changes. Habitat fragmentation [24] increases the relaxation time of population dynamics [25] -i.e. how fast a species responds to environmental degradation. As a consequence, species extinctions are delayed in time, which generates an extinction debt [26] and a biodiversity-dependent ecosystem service debt [27]. These ecological debts may persist for more than a century, and increase as species get closer to extinction [28]. The accumulation of these ecological debts may have long-term effects on modern human societies.

Such time-delayed ecological feedbacks have been neglected by the most influential population projection models [29,30,31]. Yet, environmental degradation can have catastrophic consequences for human societies even without any delayed effect [32,33]. A wellknown example is Easter Island, this Polynesian island in which civilization collapsed during the 18th century due to overpopulation, extensive deforestation and overexploitation of its natural resources [34]. In order to investigate the mechanisms behind this collapse, Brander and Taylor [35] modeled the growth of the human population as endogenously driven by the availability of natural resources, the depletion of which was governed by economic constraints. Their model was essentially a Lotka-Volterra predator-prey model, which is familiar to ecologists, with an economic interpretation. It showed that one of Easter Island's ecological characteristics -the particularly low renewal rate of its forests -may be responsible for the famine cycles which brought forward the collapse of this civilization.

Given the unprecedented rates of current biodiversity and ecosystem service loss [6], accounting for their long-term feedback on modern human societies appears crucial. In an attempt to delimit safe thresholds for humanity, a 10% loss in local biodiversity was defined as one of the core "planetary boundaries" which, once transgressed, might drive the Earth system into a new, less desirable state for humans [36,37]. However, in practice, the biodiversity threshold above which ecosystem processes are significantly affected varies among ecosystems [38,39], and is correlated to other thresholds such as land-use change [40]. The definition of integrative thresholds thus requires that we consider the interaction between the economic and ecological components of systems.

CHAPTER 1

In order to explore the long-term consequences of ecological debts for human societies, we build upon Brander and Taylor's framework, but we allow the human population to produce its own resources through land conversion. In our approach, terrestrial natural habitats provide essential ecosystem services to the agricultural lands -which are assumed to be unsuitable to biodiversity. Biodiversity and human population dynamics are coupled through the time-delayed effect of biodiversity-dependent ecosystem services on agricultural production (ecological feedback). It may be viewed as a minimal social-ecological system (SES) model that couples basic insights from market economics and spatial ecology. From market economics, we derive human per capita consumptions and the rate of land conversion. From spatial ecology, we use a classical species-area relationship (SAR) to capture the dependence of species diversity on the remaining area of natural habitat, and account for ecological debts through the relaxation rate of communities following habitat loss.

We investigate the behavior of the system at equilibrium analytically, and then numerically evaluate the trajectories to the equilibrium. We show that the transient dynamics of an SES depends on its ecological, economic, demographic and technological parameters. Some SESs experience large population overshoots followed by reductions in biodiversity, human population size and well-being -which we call "environmental crises". We then analytically derive an integrative sustainability criterion that captures the vulnerability of an SES to such crises. This criterion allows assessing the effects of some parameters on the long-term sustainability of the SES, and deriving integrative land conversion and biodiversity thresholds.

Methods

Coupling human and ecological dynamics

We model the long-term dynamics of three variables: the human population (H), technological efficiency (T) and biodiversity (B).

                   Ḣ = µ(B, T) H Ṫ = σ T(1 -T/T m ) Ḃ = -[B -S(H)] (1)
The human population endogenously grows at a rate µ(B, T), which is explicitly defined as a function of the per capita agricultural and industrial consumptions in section 1.1.1. Technological efficiency is assumed to follow logistic growth at an exogenous rate σ, until a maximum efficiency T m is reached (section 1.1.2). We use an economic general equilibrium framework to derive per capita human consumptions at market equilibrium, i.e. when production supply equals the demand of the human population (section 1.1.3).

Using these consumptions, we derive a proportional relationship between land conversion and the size of the human population (section 1.1.4). Land conversion affects biodiversity through a change in the long-term species richness S(H). Current biodiversity B reaches its long-term level S(H) at a relaxation rate (section 1.2). Biodiversity-dependent ecosystem services then feed back on agricultural production and affect the per capita agricultural consumption and the human growth rate, µ(B, T). Model structure is summarized in Fig. 1.1.

Human demography

The interaction between human population, technology and income has been mainly studied by endogenous growth theory, which distinguishes three phases of economic development [41,42]: (1) a Malthusian regime with low rates of technological change and high rates of population growth preventing per capita income to rise; (2) a Post-Malthusian Regime, where technological progress rises and allows both population and income to grow, and (3) a Modern Growth regime characterized by reduced population growth and sustained income growth [43]. Transition to this third regime results from a demographic transition which reverses the positive relationship between income and population growth.

In order to consider the basic linkages between human demography and economics, the CHAPTER 1

Biodiversity (B)

A 2 y 2 2 ( ( T T ) ) y 1 1 ( ( B B , , T T ) ) A 1 L 1 L 2 f f S S ( (B B) )

S ( S ( H H ) )

Technology (T) Industry Agriculture

Human population (H)

Natural habitat

1-A(H)=A 3 Converted land A(H)=A 1 +A 2 A ( A ( H H ) )
Economic market

Economic market

Figure 1.1: Coupling between human and ecological dynamics Black boxes: production sectors; grey boxes: dynamical variables; white box: auxiliary economic model; dashed lines: production inputs (labor L i , land A i and technology T ); solid lines: per capita outputs (y i ); grey line: biodiversity-dependent ecosystem services (ecological feedback, f S ); black dotted line: effect of land conversion on biodiversity; circle: total land divided into converted land (A 1 + A 2 ) and natural habitat (A 3 ), where

A 1 + A 2 + A 3 = 1.
CHAPTER 1 growth rate of the human population is assumed to depend on the per capita consumptions of agricultural y 1 and industrial goods y 2 [44]:

µ = µ max (1 -exp(y min 1 -y 1 ))exp(-b 2 y 2 ) (2)
where µ max is the maximum human population growth rate, and y min and thus captures the strength of the demographic transition. A higher agricultural goods consumption increases the net human growth rate while a higher industrial goods consumption eventually limits the net growth rate of the human population. Note that both y 1 and y 2 vary with the states of the system (section 1.1.3). The system reaches its long-term equilibrium (y 1 = y min 1

) when further technological change and habitat conversion no longer increase total agricultural production, i.e. no longer compensate for the negative ecological feedback on agricultural production.

Technological change

Technological change is central to explain the transition from a Malthusian to a modern human population growth regime. Technology is often captured through the Total Factor Productivity term (TFP) of a production function, which accounts for effects of technological inputs on total output growth relative to the other inputs, i.e. labor and land in our model. Accelerating TFP growth in recent years partially compensated for the slowing down in input growth (especially land) and allowed total output growth to maintain itself around 2% per year [45]. However, recent reviews suggest that agricultural TFP growth is slowing down in a number of countries [13] and that this trend is likely to continue [15,16].

In our model, technological change increases production efficiency in the agricultural and industrial sectors, leading to higher productions for a given level of inputs (i.e. converted land and labor). We assume a logistic growth of production efficiency towards a maximum efficiency denoted by T m , where σ is the exogenous rate of technological change CHAPTER 1

(system ( 11)). We explore other forms of technological change in Appendix 1.A.

Instantaneous market equilibrium

We assume a closed market where total labor is given by the size of the human population. Consumption and production levels are derived by solving for a general market equilibrium, where prices of the production inputs and consumption goods vary endogenously. As the economic dynamics are much faster than the demographic and ecological dynamics, the market is assumed to reach an equilibrium between supply and demand instantaneously.

On the demand side, the human population is assumed to be homogeneous, i.e. composed of H identical agents. Per capita consumptions of agricultural and industrial goods (y 1 and y 2 , resp.) are derived from the maximization of a utility function, U (y 1 , y 2 ), which is a common economic measure of the satisfaction experienced by consumers (Appendix

1.B.1): U (y 1 , y 2 ) = y η 1 y 1-η 2 (3)
where y 1 is per capita agricultural demand, y 2 is per capita industrial demand, and η is the preference for agricultural goods.

On the supply side, a total quantity Y i of goods is produced by sector i, using two production inputs, labor L i , and converted land A i (Appendix 1.B.2):

Y 1 = TL α 1 1 A 1-α 1 1 f S (B) Y 2 = TL α 2 2 A 1-α 2 2 (4)
where production efficiency is captured by the variable T, α i is the relative use of labor compared to land in sector i, and the function f S (B) measures provisioning of biodiversitydependent ecosystem services to agricultural production (ecological feedback). Such

Cobb-Douglas production functions with constant returns to scale are a common assumption of growth models and allow for the substitution of land by labor. Potential ecological feedbacks on industrial production were ignored.

CHAPTER 1

When demand equals supply, the per capita consumptions are (Appendix 1.B.3):

y 1 (B, T) = γ 1 f S (B) T/T m y 2 (T) = γ 2 T/T m (5)
where γ 1 and γ 2 are explicitly defined as functions of the parameters of the system in Table 1.1.

The per capita consumption of agricultural goods y 1 is thus subject to the trade-off between the negative effect of decreasing biodiversity-dependent ecosystem services, and the positive effect of technological change. Conversely, the per capita consumption of industrial goods y 2 monotonously increases with technological efficiency -as does the strength of the demographic transition.

Land conversion dynamics

Rising agricultural and industrial productions require the conversion and maintenance of land surfaces A 1 and A 2 respectively, at a cost of κ units of labor per unit of converted area. Land conversion reduces the area of natural habitat A 3 , where

A 1 + A 2 + A 3 = 1.
Let us denote by A the converted land, i.e. A = A 1 + A 2 . At the market equilibrium, the relationship between A and the human population size H is (Appendix 1.B.4):

H = φA (6)
where φ represents the density of the human population on converted land, and is explicitly defined as a function of the economic parameters in Table 1.1. The converted area A decreases with the land operating cost κ, since high operating costs reduce the incentive to convert natural habitat. Proportional relationships between human population sizes and converted surfaces are commonly observed in local and regional developing economies, where subsistence agriculture remains strong and the transition to a modern growth regime is not achieved yet [46].

Spatio-temporal dynamics of biodiversity and ecosystem services

We need two types of relationships, which capture (1) the dependence of biodiversity upon the area of natural habitat A 3 and the size of the human population (S(H)), and (

the dependence of ecosystem services upon biodiversity (f S (B)). Since species richness can be related to the area and spatial characteristics of natural habitats [47], we choose species richness as a biodiversity measure in our system.

Species-area relationship (SAR)

SARs are commonly used tools in ecological conservation to assess the effect of natural habitat destruction on species richness [48]. A common way to describe the decrease of species richness with habitat loss is to use a power function S = c A z 3 , where c and z represent the intercept and the slope in log-log scale. By choosing a unitary intercept (c = 1) and under the constraint A 1 + A 2 + A 3 = 1, the number of species S(H) that can be supported in the long run on an area A 3 of natural habitat can be rewritten:

S(H) =        (1 -H/φ) z for H/φ ≤ 1 0 for H/φ > 1 (7)
In terrestrial systems, the slope z typically ranges from 0.1 to 0.4 [49,50], with values depending on ecosystem characteristics and species response to habitat loss. In our model, the long-term species richness S(H) thus decreases when the human population grows, and equals zero if the human population exceeds its maximum viable size φ.

We assume, following empirical [51,52] and theoretical [47,53] expectations, that the rate of community relaxation to this long-term richness is proportional to the difference between current richness, B, and long-term richness S(H) (system (11)), where the relaxation rate is [53].

Biodiversity-ecosystem service relationship (BES)

Species richness has a positive effect on the level of many regulating services [5], among which the pollination and pest regulation services are particularly important to agricultural production -since the production of over 75% of the world's most important crops and 35% of the food produced is dependent upon animal pollination [54].

The relationship between biodiversity and ecosystem services can be captured by a power function [55,56] :

f S (B) = B Ω (8)
where Ω < 1, since the shape of the function f S is mostly concave-down in terrestrial systems [57].

Model summary

System (11) can be rewritten as:

                             Ḣ = µ max H 1 -exp(y min 1 -γ 1 B Ω T/T m ) exp(-b 2 γ 2 T/T m ) Ṫ = σT(1 -T/T m ) Ḃ =          -[B -(1 -H/φ) z ] for H/φ < 1 -B for H/φ > 1 (9)
The aggregate parameters γ 1 , γ 2 , and φ result from the economic derivations of sections In the short term, technological change and human population growth drive land conversion and increase human consumption and well-being. In the long term, the time-delayed biodiversity loss reduces the supply of biodiversity-dependent ecosystem services to agricultural production (ecological feedback). The ecological relaxation rate generates a time lag between human and ecological dynamics, mediated through the ecological feedback.

In the next section, we investigate the consequences of this time lag on the long-term CHAPTER 1

sustainability of an SES.

Results and Discussion

Analysis of the dynamical system

For a given set of parameters and initial conditions, the dynamics of our SES is driven by the interaction between the dynamical variables summarized in eq. ( 9). In section 2.1.1, we characterize the steady states that can potentially be reached by the SES in the long term, and the necessary condition for their stability. We then simulate in section 2.1.2 the transient dynamics of the SES over time, for various time lags between the ecological and human dynamics ( ).

Steady states and stability condition

A steady state is reached by the dynamical variables when the system is at equilibrium, i.e.

when Ḣ = Ṫ = Ḃ = 0 (Appendix 1.C.1). There are two potential steady states (H, T, B)

for our system: (1) an undesirable equilibrium (0, T m , 1), when the parameters do not allow the human population to persist in the long term, and (2) a desirable equilibrium (H * , T m , B * ):

H * = φ 1 -y min 1 /γ 1 1 Ωz B * = y min 1 /γ 1 1 Ω (10)
For a given set of parameters, only one of these equilibria is stable and reached by the

SES (Appendix 1.C.2). The desirable equilibrium (H * , T m , B * ) is stable if condition (11)
is met:

γ 1 > y min 1 (11)
Condition (11) captures the capacity of an SES to persist in the long term, given its technological and economic characteristics ( 11) is (not) met, the SES reaches the desirable (undesirable) equilibrium in the long term. According to eq. ( 10), natural habitat (1 -H/φ) is never entirely converted at equilibrium, since H/φ < 1.

κ/(1 -α 1 η -α 2 (1 -η)) γ 1 ηT m α α 1 1 ((1 -α 1 )/κ) 1-α 1 γ 2 (1 -η)T m α α 2 2 ((1 -α 2 )/κ) 1-α 2 θ 4Ωzy min 1 e -b 2 γ 2 θ θ (γ 1 /y min 1 ) 1 Ωz -1 ∆ -µ max θ Table 1
Condition (11) only guarantees that the desirable equilibrium is reached in the long term, but tells nothing about the trajectories to the equilibrium, and especially about the effect of the relaxation rate .

Transient dynamics for varying relaxation rates

In order to test for the effect of a time-delayed ecological feedback on the transient dynamics of an SES, we numerically evaluate the trajectories to the equilibrium and increase the lag between the human and ecological dynamics by decreasing .

When the relaxation time is negligible (1/ → 0), biodiversity responds instantaneously to habitat conversion (B = S(H)) and the ecological feedback on agricultural production is instantaneous. In this case, Fig. ) and human population declines.

(3) Human population and biodiversity recovery (µ ≥ 0). As the human population declines, the converted surface decreases and allows for the recovery of biodiversity.

When biodiversity is not in deficit anymore (B = B * ), the human population can reach its equilibrium size, H * .

Sustainability sensitivity analysis 2.2.1 Sustainability conditions

Transient environmental crises are undesirable from a sustainability perspective, since these collapses result from a decrease in per capita agricultural consumption, and thus in human utility (eq. ( 3)) -which we use as a proxy for human well-being. Let us define an SES as sustainable when its transient dynamics monotonously converge to the long-term equilibrium (i.e. no crisis) and the system experiences a net growth in human utility.

Condition (12) ensures that technological change is sufficient to compensate for the effect of biodiversity loss on human utility:

T m /T (0) > (γ 1 /y min 1 ) η ( 12
)
where T (0) is the initial technological efficiency. When condition ( 12) is met, human utility at equilibrium is higher than the initial human utility (U * > U (0)), but environ-

Effect of labor intensity

As labor intensity α 1 captures the relative use of labor compared to land in the agricultural sector, a moderate increase in labor intensity (α 1 < 0.7) reduces land conversion requirements and benefits both biodiversity and sustainability (Fig. 1.3.B and E). However, further increasing labor intensity (α 1 > 0.7) increases agricultural productivity to the point where it lowers agricultural prices and increases the demand for natural habitat conversion. This rebound effect thus allows the human population and total human utility to rise again, while biodiversity, sustainability and per capita human utility decrease (Fig. 1.3.B, E and H). A stronger demographic transition positively affects the sustainability criterion for all labor intensities, and thus partially mitigates the negative rebound effect (dashed curve, Fig. 1.3.E). Note that the shape of the relationship between sustainability and labor intensity depends on the land operating cost, κ. If κ = 1, the bell-shaped curve is centered on α 1 = 0.5. If κ > 1 (κ < 1, resp.), the sustainability criterion is maximum at α 1 < 0.5 (α 1 > 0.5, resp.). Indeed, higher operating costs reduce the incentive to convert natural habitat and increase sustainability (Appendix 1.D), thus reducing the sustainability-optimal labor intensity.

Effect of the concavity of ecological relationships

The concavity of the ecological relationships, i.e. the SAR and BES relationships, is captured by their parameters z and Ω, respectively. Since both parameters have similar effects on ∆, we only present the results for Ω. Commonly observed concave-down BES relationships (Ω < 1) lead to highly populated and biodiversity-poor SESs at equilibrium Modern SESs may thus be particularly vulnerable to the current rise in technological efficiency and reduction in labor intensity -unless the demographic transition is strong enough to counteract their negative effects on long-term sustainability. Sustainable SESs are characterized by higher levels of biodiversity, lower human population sizes, industrial consumption and consumption utility at equilibrium, compared to unsustainable SESs. In section 2.3, we derive sustainability thresholds for land conversion, biodiversity and human population size, and explore their efficiency in preserving the long-term sustainability of SESs through numerical simulations.

Application to land-use management 2.3.1 Integrative sustainability threshold for land conversion

Our sustainability condition ∆ > 0 can be rewritten as A < A S (Appendix 1.C.3), where A S is the sustainable land conversion threshold:

A S = 1 - θ µ max + θ µ max ( 14
)
θ is explicitly defined as a function of the economic, technological and ecological parameters of the SES in Table 1.1. A S represents the maximum area of natural habitat that a given SES can convert without becoming unsustainable, and depends on the economic and ecological characteristics of the SES. In particular, A S decreases as the ecological relaxation rate decreases. As a result, the larger the ecological relaxation time (i.e. the lower ), the more natural habitat a SES needs to preserve in order to remain sustainable. Using this land conversion threshold A S , it is also possible to derive a sustainable biodiversity threshold B S = (1 -A S ) z , and a human population threshold H S = φA S .

Land-use scenarios

In this section, we explore the efficiency of the land conversion threshold (eq.( 14)) in preventing environmental crises. To do so, we consider two alternative land-use scenarios:

(1) no restriction on land conversion, and (2) conservation of an area of natural habitat 1 -A S , e.g. through the creation of a protected area. In order to stop land conversion at the sustainable threshold A S without (directly) limiting the dynamics of the human population, we have to define converted land A as a fourth dynamical variable:

Ȧ =        Ḣ/φ for A ≤ A S 0 else (15)

Conclusions

We explore the long-term dynamics of coupled SESs by modelling the main reciprocal feedbacks between human population and biodiversity, and show that the temporal trajectories of the human population and biodiversity are not necessarily monotonous. In particular, for large ecological relaxation times, we observe the emergence of transient environmental crises, i.e. large reductions in biodiversity, human population size and wellbeing. More complex population projection models calibrated with data for the world population, industrialization, pollution, food production and non-renewable resources obtained similar "overshoot and collapse" scenarios [29]. Recent updates even suggest that the business-as-usual trajectory of our societies may be approaching such a global collapse [30,31]. However, neither these population projection models nor neo-classical economic models consider any long-term ecological feedback of biodiversity on agricultural production, which may worsen these crises scenarios.

The present work thus sheds new light upon the importance of accounting for ecological time lags when studying the sustainability of SESs and defining management policies.

CHAPTER 1

Accumulating data on extinction debts [52,58,59,60,61] and on the relationship between biodiversity and the provisioning of ecosystem services [5,56] make it possible to include these time-delayed ecological feedbacks in more complex simulation models, such as existing models coupling the economy and climate [62,63]. The ability of our conceptual model to perform population projections is limited, since many of our economic and ecological parameters do not vary endogenously. For example, the ecological relaxation rate is known to decrease with habitat loss and fragmentation [28], and population density on converted land φ rose between 1993 and 2009 [64], since the rate of land degradation (9%) was lower than that of the human population and economic growth (resp. +23% and +153%).

Although detailed policy-relevant population scenarios would require a simulation model of greater complexity, our toy-model allows assessing the qualitative effects of some parameters of SESs on their long-term sustainability. It provides us with an integrative sustainability criterion that captures the vulnerability of an SES to environmental crises, given its economic, demographic, technological and ecological characteristics. Using this criterion, we show that some of the characteristics common to modern SESs, such as high technological efficiency and low labor intensity, are detrimental to their long-term sustainability. Even though land-use efficiency reduces land conversion in the short term, it also increases the time lag between the dynamics of humans and biodiversity and the vulnerability of the SES to environmental crises. Indeed, a more efficient use of land resources releases ecological checks on human population growth, which eventually increases both land conversion and the extinction debt. Moreover, the commonly observed concave-down SARs and BES relationships exacerbate this effect, since a given amount of habitat conversion translates into a lower loss of biodiversity and ecosystem services with concave-down than with concave-up relationships.

We show that counteracting forces such as the demographic transition and natural habitat conservation can mitigate environmental crises -provided that the ecological relaxation rates and the sensitivity of the human growth rate to industrial consumption are high enough. Indeed, the demographic transition is often presented as the solution to CHAPTER 1 accelerate economic development and reduce environmental impact in developing countries [65,34]. However, recent projections cast doubt upon its stabilizing potential [66].

Our results also show that, in order to efficiently counteract time-delayed biodiversity feedbacks, natural habitat conservation should take place when biodiversity is still abundant. Alternatively, habitat restoration can also help mitigating crises, and century-long extinction debts may be seen as windows of conservation opportunity [52]. However, our results suggest that we should not rely on habitat restoration, since restoration delays are often longer than extinction debts [67,68], which worsens environmental crises.

The lack of such considerations about ecological time lags in current conservation policies calls for a precautionary approach to natural habitat conservation. To this end, we provide an integrative land conversion threshold that captures the long-term ecological dynamics of species confronted with the destruction of their habitat, given the economic, technological and demographic parameters of an SES. The smaller the ecological relaxation rate ( → 0), the more natural habitat should be preserved in order to avoid environmental crises. Since relaxation rates decrease with habitat loss and fragmentation [28], the sustainable amount of natural habitat may increase when more and more natural habitat is lost.

Regarding rates of current biodiversity loss [69], it is thus crucial to determine how much natural habitat we need to preserve in the long run. Recent attempts to define global conservation thresholds for biodiversity and land use [37] neglect the interaction between the social and ecological components of SESs. For instance, as biodiversity moves closer to its own potential threshold, it reduces the land-use change threshold [39].

There is also a lack of context-dependency for the application of these thresholds at local scales, as SESs with different sets of economic, technological and ecological characteristics present different thresholds. Our theoretical model of a coupled SES proposes a way to move beyond these limitations. Our integrative thresholds for land conversion and biodiversity loss are not defined independently from each other, but related through the system's parameters. As a consequence, it is possible to predict how a change in one of the thresholds affects the other, and to define context-dependent land-use policies that CHAPTER 1 prevent environmental crises from occurring.

Despite its simplicity, our model may be seen either as a thought experiment about the sustainability of our planet, or as a representation of a local SES -which could be connected to other SESs by considering trade, human migrations and species dispersal.

Improvements to the model include adding property rights and trade in order to gain realism in land-use change dynamics [70], and internalizing the value of ecosystem services via payments for ecosystem services and taxes on land conversion [71]. Finally, our results emphasize the critical need for a better assessment of ecological time lags and how they feed back on human systems. Reducing uncertainties on ecological time lags effects is crucial to be able to define integrative thresholds and foster sustainability of our planet, which may be seen as a global social-ecological system [7]. The pursuit of such objectives requires taking into account the main internal feedbacks and specificities of coupled socialecological systems, which dynamical system modelling allows.

1.A Endogenous or Exponential Technological Change

In this appendix, the assess the effect of other forms of technological change on the dynamics of the system.

Ṫ = σT (1 -T /T m ) (16)
This function allows us to retain analytical tractability, and for instance, derive analytical sustainability criteria. We could also have modeled technological change as endogenous, e.g. driven by the rate of human population growth :

Ṫ = σ(µ)T (1 -T /T m ) ( 17
)
where σ(µ) is a function of the human growth rate, for instance σ(µ) = µ(B, T ). The behaviour of the system is not qualitatively affected by this assumption. However, the system becomes analytically intractable.

Another possibility is to assume an exponential technological progress, i.e. technological efficiency keeps growing instead of reaching a maximum technological efficiency

T m : Ṫ = σT (18)
In this case, the system has no global equilibrium (H * , T * , B * ) and no analytical results.

However, as technological change increases the strength of the demographic transition, it reduces the growth rate of the human population (µ(T, B) → 0) along with land conversion, so that the human population and biodiversity reach a stationary state in the

1.B Economic derivations

In this section, the aim is to derive the per capita agricultural and industrial consumptions at the market equilibrium, i.e. when the production supply equals the demand of the human population. The human population is assumed to be composed of identical agents, with preferences η. Each agent allocates part of his revenue w to buy agricultural goods, and the rest to buy industrial goods, which generates an aggregate demand for the human population H (section 3.A). On the supply side, firms produce agricultural and industrial goods given the costs of production inputs, i.e. land and labor, which

gives the aggregate supply of the production sectors (section 3.A). Section 3.A solves for the general equilibrium model, and derives the per capita consumptions at the market equilibrium. Section 3.A then derives the rate of land conversion at the economic market equilibrium.

1.B.1 Consumer Optimization

Overall, each agent supplies one unit of labor, so that his revenue is the wage w. Let U (y 1 , y 2 ) be the individual consumer well-being, where y 1 and y 2 are per capita consumption rates of agricultural and industrial goods. Agents maximize their well-being

U (y 1 , y 2 ) = y η 1 y 1-η 2
under the revenue constraint p 1 y 1 + p 2 y 2 ≤ w, where p 1 and p 2 are the prices of agricultural and industrial goods respectively, w the per capita income, and η the preference for agricultural goods. To solve this maximization problem, we define the Lagrangian:

L ≡ U (y 1 , y 2 ) -Λ[p 1 y 1 + p 2 y 2 -w]
First order conditions are:

         ∂L/∂y 1 = ηU (y 1 ,y 2 ) y 1 -Λp 1 = 0 ∂L/∂y 2 = (1-η)U (y 1 ,y 2 ) y 2 -Λp 2 = 0 (19)
Adding both conditions yields U (y 1 , y 2 )/Λ = p 1 y 1 + p 2 y 2 = w, and solving (19) for y 1 and y 2 , and substituting for U (y 1 , y 2 )/Λ yields the aggregate demands:

p 1 Y D 1 = ηwH p 2 Y D 2 = (1 -η)wH (20)

1.B.2 Firms Optimization

Firms in sectors 1 and 2 produce a quantity Y i (i = 1, 2) of output, using labor L i and land A i . A units of land and H units of labor are available. Agricultural firms occupy A 1 units of land, and industrial firms occupy A 2 units of land, such that A = A 1 + A 2 . This comes at an operating cost (including clearance and maintenance) of κ units of labor per unit of land, so that each firm maximizes a profit Π i = p i Y i -wL i -κwA i . The production functions Y i are:

Y 1 = T f S (B) L 1 α 1 A 1 1-α 1 Y 2 = T L 2 α 2 A 2 1-α 2 (21)
where f S (B) is the ecological feedback, T captures production efficiency and α i is labor intensity in sector i. Therefore, first order conditions are:

         ∂Π i /∂L i = α i p i Y i L i -w = 0 ∂Π i /∂A i = (1-α i )p i Y i A i -κw = 0 (22)
Adding both lines of (23) gives the total supply in sector i:

p i Y S i = w(L i + κA i ) (23)

1.B.3 Economic General Equilibrium

Markets' clearing for the agricultural and industrial sectors p i Y D i = p i Y S i (eq. ( 21) and ( 24)) yields:

         ηH = L 1 + κA 1 (1 -η)H = L 2 + κA 2 (24)
Input factors in each sector verify L i /α i = κA i /(1 -α i ) (see eq. ( 23)), so that the equilibrium land allocations are:

A 1 = Hη(1 -α 1 )/κ A 2 = H(1 -η)(1 -α 2 )/κ (25)
and the equilibrium labor allocations are:

L 1 = Hηα 1 L 2 = H(1 -η)α 2 (26)

1.B.4 Land Conversion

The allocation of total labor H between agricultural production (L 1 ), industrial production (L 2 ) and land conversion (κA) writes L 1 + L 2 + κA = H. Replacing L 1 and L 2 by their optimal allocations (eq.( 26)), we deduce the relationship A = H/φ between the human population size H and the converted land A, where the density of the human population per unit of converted land is:

φ = κ 1 -α 1 η -α 2 (1 -η) (27)

1.B.5 Per capita consumptions

Using the equilibrium allocations of labor L i and land A i , the per capita agricultural and industrial consumptions Y /H (eq.( 22)) can be rewritten:

y 1 = γ 1 B Ω T/T m y 2 = γ 2 T/T m ( 28
)
where

γ 1 = ηT m α α 1 1 ((1 -α 1 )/κ) 1-α 1 γ 2 = (1 -η)T m α α 2 2 ((1 -α 2 )/κ) 1-α 2 (29)

1.C Dynamical System Analysis

In this section, we derive the analytical expressions of the equilibria, and the conditions for their stability, as well as a condition for the sustainability of the temporal trajectories, i.e. non-decreasing human well-being over time.

1.C.1 Equilibria and Stability Condition

The Jacobian matrix at the viable equilibrium (H * , T m , B * ) writes:

J * =         0 µ max γ 1 exp(-b 2 γ 2 )ΩH * B Ω-1 * µ max γ 1 exp(-b 2 γ 2 )H * B Ω * -z φ (1 -H * /φ) z-1 - 0 0 0 -σ        
The eigenvalues λ i (i = 1 : 3) of the system are solutions of Det(J * -λI) = 0, where

Det(J * -λI) = -(σ + λ)(λ( + λ) + θµ max /4) ( 30
)
and

θ = 4 z φ (1 -H * /φ) z-1 γ 1 exp(-b 2 γ 2 )ΩH * B Ω-1 * (31)
The first eigenvalue is λ 1 = -σ, and the two others are solution of the characteristic equation λ 2 + λ + θµ max /4 = 0, which discriminant is:

CHAPTER 1 D = ( -θµ max ) ( 32
)
λ 2 = (-- √ D)/2 λ 3 = (-+ √ D)/2 (33)
The viable equilibrium is stable if all the eigenvalues have negative real parts, i.e. √ D < , which gives a stability condition for the viable equilibrium (H * , T m , B * ):

θ > 0 ⇔ H * < φ ⇔ γ 1 > y min 1 (34)

1.C.2 Sustainability Condition

For parameters such that the viability condition γ 1 > y min 1 is met, there are two possible transient dynamics depending on the eigenvalues of the system: (1) for real eigenvalues, a monotonous convergence to equilibrium, and (2) for complex conjugate eigenvalues, damped oscillations. The former case stands for a sustainable system, while the later one stands for an environmental crisis. The eigenvalues are real if and only if the discriminant of the characteristic equation is positive, so that a sustainability condition for our system is:

> θµ max (35)
A sustainability criterion for our system is ∆ = -θµ max , where ∆ > 0 stands for sustainable trajectories.

1.C.3 Sustainability Thresholds

Using the relationship between the biodiversity and human population size at equilibrium H * = φA * , θ (eq.( 31)) can be rewritten as a function of the equilibrium converted area A * . The sustainability criterion ∆ then rewrites:

A * ≤ 1 - θ µ max + θ µ max ( 36
)
where θ = 4Ωzy min 1 e -b 2 γ 2 , and is the ecological relaxation rate. Over a sustainable trajectory (∆ > 0), converted land never exceeds its equilibrium level, i.e. A ≤ A * .

Condition (36) is thus equivalent to A ≤ A S , where A S is the sustainable land conversion threshold of our system. Similarly, the sustainability criterion can be written as a human or a biodiversity threshold as follows:

H * ≤ φA S B * ≥ (1 -A S ) z

1.D Effect of Land Operating Costs on Sustainability

This section presents the qualitative effect of the parameter κ, which captures the cost of natural habitat conversion, on the dynamics of the system.

The exploitation of converted land comes with operating costs at each period of time,

which

Introduction

Early research on the interaction between human populations and their environment emphasized the need for government control in order to prevent the overexploitation of common pool natural resources [1]. However, subsequent research has shown that local communities can achieve sustainable resource use through cooperative self-governance [2].

Successful communities often establish social norms, i.e. rules of shared behavior, that protect common natural resources [3] or help achieve group interests [4]. Such regulatory mechanisms are "bottom-up" processes, as opposed to classical "top-down" government control. Both types of regulation involve, however, sanctioning mechanisms that seek to internalize the externalities of human activities, be they moral incentives (e.g. social exclusion) in the first case, or economic instruments (e.g. taxes, subsidies) and regulatory policies in the second [5,6].

Human behavioral change can significantly affect the dynamics of social-ecological systems (SESs) [7], and is a central aspect of their adaptability and resilience [8,9].

The evolution of social norms affects feedbacks and drivers of SESs, potentially leading to large-scale behavioral shifts [10]. Such shifts may allow escape from social-ecological traps, i.e. persistent mismatches between the responses of people and their ecological conditions, that are undesirable from a sustainability perspective [11].

The establishment of sustainable social norms strongly hinges on the ecological characteristics of SESs. Previous experimental and theoretical studies have emphasized the role of resource productivity and mobility [12] as well as temporal variability [13,14] on the robustness of cooperation. Evidence from the literature on natural resource management shows that the interaction between fast and slow ecosystem processes affects the optimal management strategy [15], while inappropriate management may reinforce undesirable feedbacks and push the SES into a social-ecological trap [16]. However, the consequences of mismatches between slow and fast social-ecological processes on the robustness of cooperation remain an open question.

Extinction debts, i.e. time-delayed loss of species following habitat destruction, can CHAPTER 2 emerge from the spatial dynamics of SESs [17]. Indeed, land conversion and fragmentation alter spatial ecological processes and the rate of relaxation of natural communities, i.e. the rate of change of species richness in response to habitat destruction [18]. Recent evidence suggests increasing ecological relaxation rates, generating large extinction debts [19]. As an example, 80% of the species extinctions in the Amazon are estimated to be pending [20], which may increase the number of 20th-century extinctions in bird, mammal, and amphibian forest-specific species by 120% [21]. In European landscapes, studies find that extinctions lag well behind contemporary levels of socioeconomic pressures [22], the current number of threatened species being better explained by socio-economic indicators from the early or mid-20th century [23].

The accumulation of these extinction debts generates functioning debts [24] that postpone the negative effect of biodiversity loss on ecosystem processes. Since many of the ecosystem services that play a direct or indirect role in agricultural production depend on biodiversity [25,26,27,28], current species extinction rates [29,30,31] do not only threaten the long-term provisioning [32,33] and stability of ecosystem processes [34,35,36], they also generate a time-delayed feedback loop between humans and nature [37]. In the long run, such time-delayed biodiversity feedbacks may result in large environmental crises, i.e. overshoot-and-collapse population cycles [37], similar to the famine cycles that have been observed in extinct societies [38].

Characteristics common to the majority of modern agricultural systems were found to increase the vulnerability of SESs to such crises [37]. Among these characteristics are a high production efficiency and a low labor share per unit of agricultural good, due to the substitution of technology (e.g. machines, fertilizers and pesticides) for human labor and ecosystem services. Recent evidence suggests that land use efficiency has been rising at the global scale [39]. However, it is not clear whether these efficiency gains will help save natural habitats and biodiversity in the long run, due to economic rebound effects [40], i.e. when lower prices stimulate demand and higher yields raise profits, encouraging further agricultural expansion [41]. Such a decoupling between human population growth and ecological dynamics can thus reinforce unsustainable feedbacks [42].

CHAPTER 2

Shifting consumption has, however, been identified as a major strategy that could allow doubling food production while greatly reducing the environmental impacts of agriculture [43,44]. Norm-driven consumption changes towards more environmentally-friendly agricultural goods, whose production relies more on ecosystem services and labor than on technology, may thus play a key role in ensuring the long-term sustainability of SESs at large scales. However, the magnitude of time-delayed ecological feedbacks may postpone the required behavioral changes, and push (or keep) the global SES into a social-ecological trap [42].

The aim of this article is to investigate the effects of time-delayed biodiversity loss on the establishment of sustainable social norms. To this end, we develop a dynamical system model of an endogenously growing human population divided into norm-following and norm-violating consumers, that share a common stock of land and associated biodiversity.

Rising consumption demand of the human population drives production supply and natural habitat conversion through market constraints. The model thus differs from related common-pool resource systems that only consider a constant population of harvesters and a single resource [13]. The present model builds upon previous work [37], where the growth rate of the human population depends on the consumptions of industrial and agricultural goods, as well as on the strength of the demographic transition governed by technological change. The time-delayed loss of biodiversity-dependent ecosystem services then acts as a lagged feedback on agricultural productivity that can push the system into an overshoot-and-collapse crisis [38]. In the following, we present the model structure and show that allowing for consumers' behavioral change generates bistability between sustainable and unsustainable equilibria, and thus the potential for regime shifts. We then explore different scenarios of social pressure and extinction debts, and conclude with a discussion of our results.

Model description 1.Coupling human demography, biodiversity and social dynamics

We model a population of consumers, whose demand for agricultural and industrial goods requires the conversion of their common natural habitat. Our SES model describes the long-term interaction between four dynamical variables (Fig. 2.1): the human population (H), technological efficiency (T), biodiversity (B) and the proportion of sustainable consumers, hereafter "conformers" (q). Conformers, by complying to a sustainable norm prescribing the consumption of environmentally-friendly agricultural goods, reduce their footprint in terms of natural habitat destruction and long-term biodiversity loss. Total habitat is gradually converted towards agricultural and industrial lands. The remaining natural habitat supports a community of species (biodiversity) that provides a range of ecosystem services to agricultural production [35]. Loss of natural habitat leads to time-delayed species extinctions, thus reducing both the common-pool biodiversity and long-term agricultural productivity [45]. Such a lagged feedback on agricultural production can result in long-term environmental crises characterized by overshoot-and-collapse population cycles (Fig. 2.4.c). These crises transiently reduce human well-being [37], thus threatening intergenerational equity and sustainability [46]. Since the vulnerability of SESs to lag effects increases with natural habitat destruction and biodiversity loss [37], a sufficient proportion of conformers reducing their consumption footprint may help limit land conversion while preserving the long-term sustainability of the SES. The following sections present the main features of our dynamical system. Further details about the economic derivations can be found in Lafuite & Loreau (2017) [37], from which the model is extended. ( B ,T ,q B ,T ,q ) )

Technology (T)

Industry Agriculture summary Black boxes: production sectors; grey boxes: dynamical variables; dashed lines: production inputs (labor, land and technology), with α(q) being the share of labor compared to land to produce one unit of agricultural good; solid lines: per capita consumptions of agricultural and industrial goods, y 1 (B, T, q) and y 2 (T); grey dotted lines: ecological feedback; double arrow: social sanctioning (e.g. ostracism); circle: total land divided into converted land A(H, q) and natural habitat, which supports a long-term species richness S(H, q). All functions are explicitly defined in the main text and in Table S2 (electronic supplementary material). (b) Effect of labor elasticity on equilibrium biodiversity. Grey areas represent the amplitude of the transient environmental crises. (c) Effect of labor elasticity on sustainability. The sustainability criterion ∆ is derived in [37]. ∆ > 0 stands for sustainable transient trajectories, i.e. no environmental crises. The sustainability-optimal agricultural labor elasticity α s maximizes both ∆ and the biodiversity at equilibrium, B * . The unsustainable labor elasticity α u is chosen so that α u > α s and ∆(α u ) < 0.

Human population (H)

Natural habitat 1-A(H,q) Converted land A(H,q) Defectors (1-q) Conformers (q) α ( α ( q q ) ) 1 -α ( 1 -α ( q q ) ) L a n d L a n d c o n v e r s i o n c o n v e r s i o n S ( H , q ) B Ω 1 -α 1 -α 2

Human consumption and technological change

Human consumption is related to the production of agricultural and industrial goods through an auxiliary economic model, which is assumed to be at a moving market equilibrium. Following previous work [47,37], the effects of both biodiversity and technology on agricultural (j = 1) and industrial (j = 2) productions, e.g. ecosystem services, chemicals and machines, are captured by the total factor productivity (TFP) term of Cobb-Douglas production functions, with labor L j and land A j as inputs (eq.( 1)).

Y 1 = B Ω T T F P L α 1 A 1-α 1 Y 2 = T T F P L α 2 2 A 1-α 2 2 (1)
where α and α 2 are the labor elasticities of the agricultural and industrial sectors, respectively. The relationship between biodiversity and ecosystem services [35] is captured by a concave-down function of biodiversity, B Ω , with Ω < 1 [48], and the feedback of ecosystem services on industrial production is neglected. For simplicity's sake, technology is taken as exogenous, i.e. independent from the human and ecological dynamics, and technological efficiency is assumed to follow a logistic growth at a rate σ towards a maximum efficiency, T m (eq.( 4)). Such a logistic growth allows reproducing the past rise and current stagnation of the agricultural TFP [49]. Other forms of technological change, e.g. exponential or endogenous, do not qualitatively affect the dynamics of the model [37].

Ṫ = σ T (1 -T/T m ) (2)
Solving for the market equilibrium, i.e. when supply equals demand, gives the per capita industrial and agricultural consumptions as functions of biodiversity and technological efficiency. Industrial consumption, y 2 = γ 2 T/T m , varies with technological efficiency only, while agricultural consumption y 1i = γ 1i B Ω T/T m of conformers (i = s) and defectors (i = u) also depends on biodiversity-dependent ecosystem services. Both industrial and agricultural consumptions increase with technological efficiency, T. By increasing production efficiency (eq.( 4)), technological change helps counterbalancing the feedback CHAPTER 2 of biodiversity loss on agricultural productivity in the short term, thus ensuring that the consumption utility of consumers does not decrease with time [37]. γ 1i and γ 2 are functions of socio-economic parameters that capture the characteristics of agricultural and industrial productions (electronic supplementary material, Table S1). Using these parameters, a norm of sustainable agricultural practices γ 1s is defined in the next section.

A norm of sustainable consumption

The footprint of agricultural goods can be related to the parameters of their production function, and especially to the output elasticity of labor, hereafter denoted as α, and the output elasticity of land, which equals 1 -α (eq.( 1)). In economics, output elasticity captures the percent change in production resulting from a 1% change in an input, and is a proxy for the relative share of inputs used in production. Thus, the higher α, the higher the labor force per unit of land used in agricultural production. Agricultural labor forces have been globally declining with the substitution of machines, fertilizers and pesticides for labor and ecosystem services, and the consequent rise in production efficiency [50] and economies of scale [51]. Conventional industrialized agricultural systems thus have lower labor elasticities α than environmentally-friendly systems, such as small-scale organic farming, where the substitution of labor and ecosystem services for technology is lower.

In previous work, labor elasticity has been related to the sustainability of SESs, in terms of their vulnerability to overshoot-and-collapse crises [37]. To do so, we have captured the transient dynamics of our SES by a sustainability criterion, ∆. This criterion captures the relative rate of change of biodiversity compared to the human population, since it is the difference between the ecological relaxation rate, , and the maximum growth rate of the human population, µ, as ∆ = -θµ. The respective roles of these parameters are detailed in the following sections. θ is a function of assessable ecological and economic parameters of the SES (electronic supplementary material, Table S1), and captures the rate of change of species richness in response to land conversion [18,20]. ∆ > 0 means that the ecological dynamics is fast enough compared to the human dynamics ( > θµ), thus preventing transient overshoot-and-collapse crises. However, ∆ < 0 means that the ecological dynamics is much slower than the human dynamics ( < θµ), so that there is a high probability of experiencing transient crises. A sensitivity analysis of this criterion to the parameters of the system is presented in Appendix 2.D. Using this criterion, we show in Lafuite & Loreau [37] that a low labor elasticity, i.e. a low share of labor in production, or equivalently, a high substitution of human capital for technology, increases the vulnerability of SESs to lag effects, while there exists an intermediate sustainability-optimal labor elasticity that maximizes both long-term biodiversity (Fig. Let us define as α s the sustainability-optimal labor elasticity, and α u < α s an unsustainable labor elasticity chosen such that ∆(α u ) < 0 (Fig. 2.1.a). The expected agricultural labor elasticity then varies with the proportion of conformers as α(q) = qα s + (1 -q)α u .

Through means of eco-labeling, consumers can either buy sustainable agricultural products (y 1s ), or follow their unsustainable consumption habits and buy unsustainable agricultural products (y 1u ). Since economic dynamics are much faster than ecological and demographic dynamics, we assume that agricultural and industrial production instantaneously follows consumers' demand. Such a shift in agricultural production may not be met instantaneously due to inertia and production barriers [52], and farmers' adaptability may have to be supported through adequate policy changes [53]. However, given the large time scales considered here, it seems reasonable to neglect such time delays with respect to the extent of extinction debts. Thus, in our system, a consumption shift towards sustainable goods, which, in turns, drives a shift towards more environmentally-friendly agricultural practices, may prevent environmental crises.

Human demography

The growth rate of human populations can be related to consumption levels [47] by capturing basic linkages between technology and human demography [54,55,56]. Following previous studies [47,56], we assume that the human growth rate endogenously varies with the mean agricultural and industrial consumptions, so as to increase with agricultural consumption, and decrease with industrial consumption, capturing the effect of the demographic transition.

Ḣ = µ H 1 -e y min 1 -y 1 e -b 2 y 2 (3)
µ is the maximum growth rate, y min 1 is the minimum consumption threshold, y 1 = q • y 1s +

(1 -q) • y 1u is the average agricultural consumption, and b 2 is the demographic sensitivity to industrial consumption. The strength of the demographic transition thus gradually increases with industrial consumption and limits human population growth [54].

Dependence of the human growth rate on consumption levels also allows coupling human demography with social changes regarding consumption choices. Indeed, conformers do not only have a lower consumption footprint than defectors, it can be shown that they also have a lower agricultural consumption level, i.e. y 1s < y 1u . As a result, conformers also have a lower reproduction rate compared to defectors. This can be interpreted as a quantity-quality trade-off in both consumption choices and the number of children, a mechanism which has been shown to partly explain the fertility reductions observed during the demographic transition [57]. Under our assumptions, shifting behaviors towards sustainable consumption habits thus reduces the growth rate of the human population, therefore increasing the sustainability of the SES.

Land conversion and biodiversity dynamics

The rate of land conversion is also derived at market equilibrium, as function of the dynamical variables of our system under sustainable and unsustainable labor elasticities, α u and α s (see [37] for more details about the economic derivations). For a given proportion of conformers q and human population H, converted area writes A(H, q) = H/φ, where φ = qφ s + (1 -q)φ u is the mean population density on converted land, and φ u and φ s are explicitly defined as function of the economic parameters of the SES in Table S1 (electronic supplementary material).

Natural habitat conversion results in time-delayed changes in species richness [58], so that the long-term species richness may be reached only after decades [20]. These extinction debts [17] are a result of many mechanisms [59] which lower the relaxation rates of communities [18]. We use a power-law species-area relationship to capture the CHAPTER 2 dependence of long-term species richness on the remaining area of natural habitat [60,61,62,63]. Since A(H, q) ∈ [0; 1], we allow the long-term species richness to vary between 1 (no habitat conversion) and 0 (all habitat is converted) by writing S(H, q) = (1-A(H, q)) z , where the slope z ∈ [0; 1] ensures that the function is concave-down [48]. Following experimental and theoretical results [64,20,65,18], we then assume that the rate of community relaxation is proportional to the difference between current biodiversity B and long-term species richness S(H, q).

Ḃ = -[B -S(H, q)] (4)
where mesures the relaxation rate of the community of species. The inverse of the relaxation coefficient measures the time it takes to lose approximately 63% of the species that are doomed to extinction [18].

Social dynamics

Let us assume that the human population has identified the sustainability-optimal agricultural labor elasticity, α s (Fig. 2.1.b and c). Restricting one's consumption to sustainable agricultural goods has become a social norm, i.e. a shared rule of behavior. Recent studies demonstrate the importance of social norms on eating behaviors [66] and their role in shifting preferences towards healthy food [67,68,69]. The importance of dietary social norms is especially important in young adults [70], whose eating patterns typically become life-long habits [71]. Perception of others' pro-environmental behavior was identified as the first step towards environmentally-friendly behavioral change [72].

Deviance from a social norm can lead to direct or indirect sanctioning from other members of the SES, be they important others or strangers [73]. Ostracism can result in social exclusion or poor reputation [74], thus decreasing the well-being of individuals.

As a consequence, social pressure can reduce the well-being of defectors to the point where it becomes more profitable for them to shift behavior in order to conform to the sustainable norm. A common way to approximate the well-being of consumers is through CHAPTER 2 their consumption utility, which is a function of their per capita agricultural and industrial consumptions, and thus varies with the dynamical variables of the system. Let us denote the utility of a consumer of type i (i = {u, s}) as

U i = y η 1i y 1-η 2
, where η is the preference for agricultural goods. Under our assumption that α s > α u , it can be shown that the consumption utility of defectors in the absence of social pressure, U u , is always higher than the consumption utility of conformers, U s . Therefore, in the absence of social pressure, defectors have no incentive to shift their habits.

Following previous studies [13,14], we assume that social pressure decreases the utility of defectors, U d = U u -w(q) • δ U , so that it may become more profitable for defectors to shift their consumption and comply to the sustainable norm. The severity of the ostracism function, w(q) = w max e t•e r•q , increases with the proportion of conformers in the population, q, and depends on the maximum sanctioning w max , the sanctioning effectiveness threshold t, and the growth rate of the function, r. In addition to depending on the number of conformers in the community, graduated sanctioning and equity considerations leads conformers to act more strongly against defectors which consumption is the most unsustainable [3]. Thus, the lower α u and the larger the difference in consumption utilities between conformers and defectors, δ U = (U u -U s )/U u , the stronger the social pressure.

The proportion of conformers then follows a replicator dynamics [13,14], i.e. varies both with the proportion of conformers q, and the difference between the sustainable consumption utility, U s , and the average consumption utility, U = q

• U s + (1 -q) • U d ,
itself varying with the other dynamical variables of the system through the consumption of agricultural and industrial goods, y 1i and y 2 .

q = q • [U s -U ] = q • (1 -q) • (U u -U s ) • [w(q)/U u -1] (5)
Since U u > U s in our model, a global dietary shift towards sustainable consumption ( q > 0) is only possible if the severity of the social pressure is higher than the utility of defectors in the absence of social pressure, i.e. w(q) > U u (eq.( 5)).

In the following, our focus is on the potential of consumers' behavioral change in preventing unsustainable trajectories, i.e. overshoot-and-collapse population crises leading to biodiversity-poor equilibria in the long run [37]. We first analyze the dynamical system of equations ( 4), ( 2), ( 3) and ( 5), with a negligible ecological relaxation rate ( = 0.1).

The consequences of lag effects are explored in section 2.4.

Results

Social-ecological equilibria

Our SES can have two types of equilibria (H * , B * , T m , q * ), hereafter denoted as viable (H * > 0 and B * < 1) or unviable (H * = 0 and B * = 1), when the economic parameters do not allow the human population to maintain itself in the environment [37]. Let us denote the viable equilibria as (H * i , B * i , T m , q * i ), with i = {u, s, c}. Among the viable equilibria, one is unsustainable (i = u), i.e. only defectors persist (q * u = 0) and the transient dynamics includes overshoot-and-collapse population cycles under large extinction debts (Fig. 2.4.c). The other two types of viable equilibria are either fully sustainable (i = s) when only conformers persist (q * s = 1), or partially sustainable (i = c) when both conformers and defectors coexist (q * c ∈]0; 1[). The coexistence equilibrium satisfies w(q * c ) = U * u (B * c ), for which there is no analytical solution.

A general analytical solution for the unsustainable and fully sustainable equilibria is given in eq. (6), where the population density φ i and γ 1i (i = {u, s}) are explicitly defined as functions of the parameters of the SES in Table S1 (electronic supplementary material).

B * i = y min 1 γ 1i 1 Ω H * i = φ i 1 -B * 1 z i ( 6
)
Under our assumption that α s > α u , it can be shown that γ 1s < γ 1u , so that biodiversity at the sustainable equilibrium is higher than that at the unsustainable equilibrium, i.e

B * s > B * u .
However, population density is also higher at the sustainable equilibrium, i.e.

φ s > φ u , so that the human population size at equilibrium does not necessarily decrease with the proportion of conformers. Compliance to the sustainable consumption norm thus helps preserving biodiversity while not necessarily reducing the size of the human CHAPTER 2 population.

Alternative stable states

A stability analysis of our SES model shows that two of the viable equilibria can be both stable at the same time, depending on the severity of the ostracism function compared to the consumption utility at equilibrium (electronic supplementary material, section 3).

The per capita consumption utilities at the sustainable, unsustainable and coexistence equilibria are equal to U * = (y min

1 ) η γ 1-η 2
. Compliance to the sustainable consumption norm thus does not reduce the long-term consumption utility.

The sustainable equilibrium is stable if the maximum ostracism w( 1) is higher than the consumption utility that the defectors would have at the sustainable equilibrium, For high consumption utilities, w(0) < U * and w(1) < U u (B * s ), ostracism is too weak to allow norm-driven behavioral change, and the unsustainable equilibrium is the only stable equilibrium that the SES can reach ((U) region in Fig. 2.2.b), or both the unsustainable and mixed equilibria are stable ((U/M) region in Fig. 2.2.b). Since there is no analytical expression for the mixed equilibrium, we are not able to derive any stability condition for this bistability region. However, Fig. 2.2.b shows that the shift between the two bistable regions (U/M) and (U/S) depends on the footprint α u of unsustainable consumption.

The larger the footprint of defectors compared to conformers (α u << α s ), the larger the bistability region (U/M) between the mixed and unsustainable equilibria and the smaller the bistability region (U/S). Thus, the larger the required behavioral change to shift from unsustainable habits (α u ) towards sustainable habits (α s ), the more difficult it is to reach sustainability.

Impact of the initial state of the SES

Depending on the parameters of the SES, the size of the human population at the sustainable equilibrium can be either higher (e.g. for T m = 2) or lower (e.g. for T m = 1.8) than at the unsustainable equilibrium. Let us now consider a situation where the human population size at the sustainable equilibrium is lower than that at the unsustainable equilibrium. Fig. 2.3 shows that, when there is bistability ((U/M) and (U/S) panels), the sustainable and mixed equilibria are only reached in the long run when the initial proportion of conformers is high enough. The stronger the ostracism, the lower the minimum proportion of conformers required for sustainability, i.e. the larger the sustainable basin of attraction.

Gradually changing social parameters may thus push an initially unsustainable SES ((U) panel in Fig. 2.3) towards a sustainable path ((U/S) panel in Fig. 2.3), provided that the initial social capital is large enough.

Conformers (q) (U) (0,H * u ) (q * c ,H * c ) (0,H * u ) (0,H * u ) (1,H * s )
Figure 2.3: Effect of the initial conditions on the long-term equilibria, for various ostracism strengths w max . Initial proportions of conformers are q(0) = 0.2 and q(0) = 0.6. Cases (U), (U/M) and (U/S) correspond to the ostracism functions defined in Fig. 2 Under conditions of bistability, the type of equilibrium that will be reached in the long run thus depends on the rate of social change. In the following, we show that the rate of social change also depends on human perception of environmental changes and, in our case, extinction debts.

Impact of extinction debts on the effectiveness of ostracism

We now explore the transient behavior of the SES with varying ecological relaxation rates, , for two of the initial conditions used in Fig. 2.3, corresponding to two initial proportions of conformers q(0) = 0.2 and q(0) = 0.6, with the same human population size H(0) = 0.5. In order to better visualize transient environmental crises, we plot the null-clines and transient trajectories in the human-biodiversity phase plane (Fig. 2.4.a). Ecological relaxation rates slow down the social dynamics by postponing the utility reduction of defectors, U u , and therefore, their consumption shift towards sustainable habits (eq.( 5)).

When the extinction debt is moderate, transient dynamics towards the unsustainable and mixed equilibria show environmental crises, the amplitude of which is lower for the mixed equilibrium (Fig. 2.4.b). The sustainable trajectories do not experience any overshoot-CHAPTER 2 and-collapse behavior, even for high extinction debts ((U/S) panel in Fig. 2.4.c), which confirms the relevance of our sustainability criterion. A high extinction debt leads to very large environmental crises over the unsustainable trajectories (Fig. 2.4.c). Moreover, in the case of bistability between the unsustainable and mixed equilibria, all trajectories now reach the unsustainable equilibrium ((U/M) panel in Fig. 2.4.c). Large ecological relaxation rates thus result in the loss of stability of the mixed equilibrium in favor of the unsustainable equilibrium. This result suggests a shift in the dominant socialecological feedback for increasing relaxation rates. At low relaxation rates, the ecological dynamics is fast enough for the negative effect of environmental degradation on human well-being to result in a fast enough social changes, thus reinforcing sustainable feedbacks through an efficient social ostracism. However, large extinction debts slow down the ecological dynamics and postpone the negative ecological feedback on human well-being. This reduces the efficiency of social ostracism and results in a shift of the dominant feedback towards unsustainable feedbacks, i.e. increasing consumptions and decreasing labor intensities. The stability of the mixed equilibrium appears to be much more sensitive to ecological time-lags than that of the sustainable equilibrium. Thus, moderate behavioral changes leading to a mixed equilibrium may not be robust enough to ecological lag effects. These results suggest that only important behavioral changes allowing to reach the fully sustainable equilibrium may be able to counteract the destabilizing effect of ecological time lags. The extinction debt, by postponing the consequences of environmental degradation on human well-being, thus reduces the robustness of social change and norm-driven sustainability.

Discussion and conclusions

We investigate the robustness of norm-driven sustainability, as measured by a shift towards low-footprint consumption habits. Specifically, we focus on the robustness of SESs to timedelayed biodiversity losses caused by human-driven natural habitat destruction. Timedelayed ecological feedbacks are known to reinforce negative management feedbacks and potentially push SESs into social-ecological traps [15]. However, little research so far has investigated the long-term impacts of extinction debts on the sustainability of coupled SESs. Ecological studies of the anthropogenic impacts on resources or ecosystems often neglect changes in the size and behavior of the human population. Additionally, natural resources are often managed as decoupled from the ecosystems they are part of, and most socio-economic studies overlook the finiteness and physical limits of natural systems.

Modeling sustainability requires accounting for the bidirectional coupling between human and natural systems [75], and especially the feedback loop between human population growth and environmental degradation [37].

CHAPTER 2

In our model, this feedback loop is mediated through biodiversity-dependent ecosystem services to agricultural production. A human population exploits a shared land resource divided into natural habitat and converted agricultural and industrial lands. Natural habitat supports a community of species and provides a range of biodiversity-dependent regulatory services to agricultural production, which can itself be seen as a provisioning service. A norm of sustainable consumption is maintained through social sanctioning of unsustainable consumers. Increasing demand for sustainable consumption translates into more sustainable agricultural practices, which involve the use of a larger proportion of labor compared to land. Finally, human population growth is driven by the interaction between available agricultural resources, technological changes and social changes, thus adding to the growing literature modeling the interaction between human populations and their environments [38,76].

Our approach thus differs from the classical economic literature related to the internalization of intertemporal externalities of agricultural production [77,6]. Though we also consider the pressure of a group over another as a driver of sustainable change, the penalties involved are not chosen optimally. This would require that each of the conformers had access to perfect information regarding the social damage associated with the unsustainable consumption of defectors, as well as the optimal social pressure to apply. The point of this article was not to derive an optimal regulation strategy, but to illustrate the potential of bottom-up consumption changes in driving sustainable shifts in production practices, given the long-term ecological dynamics of the SES.

The sustainable consumption norm is identified following Lafuite and Loreau's (2017) sustainability criterion, which characterizes the vulnerability of an SES to transient "overshootand-collapse" population crises. This criterion captures the difference between the rates of ecological relaxation and human population growth, so that sustainable SESs have high enough ecological relaxation rates compared to the growth rate of their human populations. We verify here the validity of this sustainability criterion, showing that a shift towards more environmentally-friendly agricultural practices, i.e. characterized by a lower substitution of ecosystem services and labor for technology, decreases the vulner-CHAPTER 2 ability of SESs to transient crises. Such a global shift towards sustainable agricultural practices would require reversing current trends of land-intensive and highly mechanized agricultural production towards more labor-intensive productions, e.g. small-scale agroecological farms. Growing evidence suggests that diverse small-scale agro-ecological farms increase carbon sequestration, support biodiversity, rebuild soil fertility and sustain yields over time, thus securing farm livelihoods, while competing with industrial agriculture in terms of total outputs, especially under environmental stress [78].

Under a negligible ecological time delay between natural habitat loss and biodiversity erosion, full sustainability is ensured when both social sanctioning and the proportion of conformers are large enough, and when the required behavioral change to shift from unsustainable to sustainable habits is not too large. Otherwise, a minority of defectors coexists with a majority of conformers at the mixed equilibrium. When social sanctioning and/or the proportion of conformers is too low, only defectors persist at equilibrium. This unsustainable equilibrium is always stable, so that there is bistability between the unsustainable and sustainable or mixed equilibria, when these are stable. These findings echo those of Tavoni et al. [13], who used a similar non-costly social sanctioning to study cooperation in the management of a single natural resource under variable environmental conditions. However, time delays have an opposite effect to resource variability, since temporal variability tends to decrease the mean resource level, thus increasing the probability of a behavioral shift towards norm compliance. Our model differs from Tavoni et al. [13] in many aspects; first, here we focus on the interaction between various ecosystem services, especially provisioning and regulatory services, instead of a single natural resource; second, these services feed back on the dynamics of the human population that uses these services, so that the human population varies endogenously with the state of the environment; lastly, social sanctioning affects consumers' behavior, instead of producers'. The latter feature allows us to focus on the potential of consumers' behavioral changes in establishing sustainability in coupled SESs. We could also have a modeled a reciprocal pressure between conformers and defectors, in which case our result would also depend on the relative strength of these pressures.

Our study provides insights into the consequences of lag effects for norm-driven sustainability. Biodiversity loss acts as a negative feedback on human well-being, through the loss of biodiversity-dependent regulatory services to agricultural production. A timedelayed biodiversity feedback thus maintains a high utility of defectors for a longer period of time. This time lag decreases the efficiency of social ostracism, thus delaying behavioral shift. Postponing the behavioral shift of defectors towards sustainable consumption for too long can make the mixed equilibrium totally unreachable, meaning that a tipping point has been crossed in terms of human population size and habitat destruction. The weaker stability of the mixed equilibrium and its propensity to regime shifts was already observed by Lade et al. [79]. Thus, under large time delays, the only way to reach sustainability is to reach the full-sustainability equilibrium, which requires much larger behavioral changes. However, given the widely observed coexistence of both conformers and defectors in small groups [80], such behavioral changes seem rather unlikely.

Moreover, theory suggests that relaxation rates are not constant, but increase with the extent of habitat destruction and fragmentation [58], thus further delaying the feedback of biodiversity-dependent ecosystem services on human societies [24]. In situations where habitat destruction leads to a strong increase in ecological relaxation rates, we would expect a decrease in sustainability, or a shift towards unsustainable development paths.

An interesting extension to our work would thus be to use a spatially-explicit ecological model, in order to gain more realism regarding the temporal dynamics of ecological relaxation rates under habitat destruction, and study social-ecological regime shifts from a spatial perspective.

The emergence of tipping points and regime shifts in coupled SESs [79] is gaining increasing interest [81], with many implications for the adaptive management of SESs [82].

Regime shifts can lead to social-ecological traps, where unsustainable feedbacks reinforce each others and push the SES into an undesirable state [11]. Some authors suggest that humanity may be locked in a technological innovation pathway that reinforces such unsustainable feedbacks [42]. Time-delayed ecological feedbacks may also affect the human perception of environmental changes, thus worsening the amnesia and shortsightedness CHAPTER 2 observed in conservation science, known as the shifting baseline syndrome [83]. This syndrome refers to a shift over time in the expectation of what a healthy biodiversity baseline is, and can lead to tolerate incremental loss of species through inappropriate management [84]. Time delays can also be related to perceived environmental uncertainty, which has been shown to endanger the establishment of cooperation in SESs with commonpool resources [85].

Our results highlight the importance of accounting for the feedback loop between human demography, environmental degradation and behavioral changes when studying the long-term sustainability of coupled SESs. Especially, the temporal dynamics of coupled social-ecological processes matter, since time-delayed ecological feedbacks alter the human perception of environmental degradation and the pace of behavioral changes. Policies that enhance the adaptive capacity of social-ecological systems may thus benefit from taking both social norms and time delays into account [10]. These insights also point to future research needs regarding the interplay of social, demographic and ecological long-term dynamics.

2.A Functions and aggregate parameters

Functions and aggregate parameters Definition

φ i κ/(1 -α i η -α 2 (1 -η))
Population density on converted land

γ 2 (1 -η) T m (α 2 ) α 2 1-α 2 κ 1-α 2
Max. per capita industrial consumption When available, parameter values are taken from the literature ( , z, Ω, α i , η, t, r). Else, they are calibrated using historical trends, and in order to guarantee the feasibility of the equilibria, i.e. positive human population sizes (µ, b 2 , σ, y min 1 , κ). Finally, ostracism w max is varied over a wide range of values for which the system remains feasible.

γ 1i η T m (α i ) α i 1-α i κ 1-α i Max. per capita agricultural consumption ∆ -4Ωzy min 1 γ 1u y min 1 1 Ωz -1 e -b 2 γ 2 µ Sustainability criterion θ 4Ωzy min 1 γ 1u y min 1 1 Ωz -1 e -b 2 γ 2

2.C Dynamical system analysis

In this appendix, we derive the analytical expressions of the equilibria, and conditions for their stability.

                             Ḣ = µ H 1 -e y min 1 -y 1 e -b 2 y 2 Ḃ = -[B -(1 -H/φ) z ] q = q(1 -q)(U s -U u )(1 -w(q)/U u ) Ṫ = σ T [1 -T/T m ] (7)
Parameters and functions are summarized in Tables 2.1 and 2.2, with y 1 = q y 1s + (1q) y 1u , φ = q φ s + (1 -q) φ u , the consumption utility U i = y η 1i y 1-η 2 (i = {u, s}), and the ostracism function w(q) = w max e te rq .

Solving system (11) for Ḣ = 0, Ḃ = 0, Ṫ = 0 and q = 0 gives five equilibria: (1) a sustainable equilibrium, (H * s , B * s , T m , 1), (2) an unsustainable equilibrium, (

H * u , B * u , T m , 0), (3)
a mixed equilibrium, (H * c , B * c , T m , q * c ), and (4) two unviable equilibria, (0, 1, T m , 0) and (0, 1, T m , 1).

We first evaluate the Jacobian matrix at the viable equilibria, (H * , B * , q * , T m ) where q * = 1 or q * = 0. After simplification, we obtain:

J(H * , B * , q * , T m ) =             0 J * 1 Ωγ * 1 B * J * 1 (γ 1s -γ 1u ) γ * 1 J * 1 Tm - J * 2 φ * - J * 2 φs-φu φ * 2 0 0 0 J * 3 0 0 0 0 -σ             where J * 1 = µe -b 2 γ 2 H * B * Ω , J * 2 = z 1 -H * φ * z-1
, and

J * 3 = (U s (B * ) -U u (B * ))(1 -2q * )(1 - w(q * ) U * u ).
The determinant D of this Jacobian matrix is the product of the four eigenvalues of the system. An equilibrium is locally stable if all its eigenvalues are negative, i.e. D > 0. In order to assess the local stability of the viable equilibria, lets first derive the determinant of J(H * , B * , q * , T m ):

D = -σzΩµy min 1 e -b 2 γ 2 (B * -1 z -1)(1 -2q * )[δ * u w(q * ) + U s (B * ) -U u (B * )] (8)
where

δ u = Uu(B * )-Us(B * ) Uu(B * )
.

We obtain the determinant D s of the Jacobian evaluated at the sustainable equilibrium by taking B * = B * s and q * = 1, so that:

D s = σzΩµy min 1 e -b 2 γ 2 (B * -1 z s -1) δ * [1 -w(1)/U u (B * s )] (9)
where

δ * = (U s (B * s )-U u (B * s )) = y minη 1 γ 1-η 2 (1-(γ 1u /γ 1s ) η ). Since γ 1u > γ 1s and B * s ∈ [0, 1],
we deduce that δ * < 0 and B * -1 z s -1 > 0, so that the sign of D s depends on the last term of eq. ( 9). The sustainable equilibrium (

H * s , B * s , 1, T m ) is thus locally stable (D s > 0) if w(1) > U u (B * s )
where

U u (B * s ) = γ 1-η 2 y min η 1 (γ 1u /γ 1s ) η .
The determinant D u of the Jacobian evaluated at the unsustainable equilibrium (B * = B * u and q * = 0) is:

D u = -σzΩµy min 1 e -b 2 γ 2 (B * -1 z u -1) δ * [1 -w(0)/U u (B * u )] (10)
where

δ * = (U s (B * u )-U u (B * u )) = y minη 1 γ 1-η 2 ((γ 1s /γ 1u ) η -1). Since γ 1u > γ 1s and B * u ∈ [0, 1],
we deduce that δ * < 0 and B * -1 z u -1 > 0, so that the sign of D u depends on the last term of eq. ( 9). The unsustainable equilibrium (

H * u , B * u , 0, T m ) is thus locally stable (D u > 0) CHAPTER 2 if w(0) < U u (B * u )
where

U u (B * u ) = γ 1-η 2 y min η 1 .
Let us now evaluate the Jacobian matrix at the unviable equilibria, (0, 1, q * , T m ) where q * = 1 or q * = 0. After simplification, we obtain:

J(0, 1, q * , T m ) =             J * 0 0 0 0 -z/φ * - z φs-φu φ * 2 0 0 0 J * 3 0 0 0 0 -σ            
where

J * 0 = µe -b 2 y 2 (1 -e y min 1 -γ * 1 ).
The determinant D(0, 1, q * , T m ) writes:

D(0, 1, q * , T m ) = σµe -b 2 γ 2 (1 -e y min 1 -γ * 1 )(1 -2q * )(U s (1) -U u (1))[1 -w(q * )/U u (1)] (11)
where

U s (1) = γ η 1s γ 1-η 2 and U u (1) = γ η 1u γ 1-η 2 , so that U s (1) < U u (1).
Therefore, the determinant D(0, 1, 0, T m ) is:

D(0, 1, 0, T m ) = σµe -b 2 γ 2 (1 -e y min 1 -γ 1u )(U s (1) -U u (1))[1 -w(0)/U u (1)] (12)
Thus, when the viable equilibria are feasible, i.e. when y min 1 < γ 1u < γ 1s , the unviable equilibrium (0, 1, 0, T m ) is stable if w(0) > U u (1). However, since U u (1) > U * , the unviable equilibrium (0, 1, 0, T m ) is only stable when the corresponding viable equilibrium

(H * u , B * u , 0, T m ) is unstable.
Similarly, the determinant D(0, 1, 1, T m ) is:

D(0, 1, 1, T m ) = -σµe -b 2 γ 2 (1 -e y min 1 -γ 1s )(U s (1) -U u (1))[1 -w(1)/U u (1)] (13)
When the viable equilibria are feasible, the unviable equilibrium (0, 1, 1, T m ) is stable if w(1) < U u (1). In this case, both viable (H * s , B * s , 1, T m ) and unviable (0, 1, 1, T m ) equilibria can be stable at the same time, if U u (B * s ) < w(1) < U u (1).

2.D Sensitivity analysis

The sensitivity of the sustainability criterion ∆ to a parameter par (sens∆ par ) can be measured as:

sens∆ par = par ∆ • ∂∆ ∂par ( 14
)
sens∆ par is negative when ∆ and par vary in opposite directions, and positive when they vary in the same direction. The higher the absolute value of sens∆ par , the more sensitive ∆ is to par. We calculate sensitivity over a range of parameter values. For each of the 12 parameters of the model with no social pressure -from which the sustainability criterion is derived (see Lafuite & Loreau (2017) [37]), we use a set of 500 values uniformly distributed in an interval of 20 % around the baseline values used in numerical simulations (Table S2). , the ecological relaxation rate , the cost of land conversion κ and the strength of the demographic transition b 2 , is positive. These parameters thus positively affect sustainability. Indeed, a higher nutritional threshold favors lower population sizes in the long run, thus leading to a lower land conversion pressure that preserves biodiversity and sustainability. A higher ecological relaxation rate means smaller extinction debts, thus a lower probability of population overshoots. A higher conversion cost reduces the incentives to convert natural habitat, while a stronger demographic transition prevents population overshoots by slowing down the growth of the human population. Conversely, the preference for agricultural goods η, the maximum rate of population growth µ and technological efficiency T m negatively affect sustainability (sens∆ < 0), since these parameters exacerbate population overshoots (µ) or land conversion (T m and η).

This sensitivity analysis confirms the sensitivity of the system to extinction debts ( ), but also to nutritional requirements (y min 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 (Table S2).

Chapter outline

In Chapter 2, we have explored the potential of norm-driven behavioral change in enforcing sustainability, through the bottom-up emergence of large-scale behavioral shifts.

A more conventional way to enforce change is through top-down government control.

Chapter 3 aims at investigating the efficiency of evidence-based social planning in mitigating the negative consequences of land conversion on biodiversity and human well-being, despite imperfect information regarding the temporal dynamics of biodiversity loss. This work builds upon the tools developed by environmental and resource economics in order to internalize the negative externalities of human activities.

Introduction

Human use of land has transformed ecosystems across most of the terrestrial biosphere for millennia [1]. The conversion of natural lands to croplands, pastures and urban areas represents the most visible form of human impact on the environment [2], with 40% of Earth's land surface being currently under agriculture [3], and 75% experiencing measurable human pressures [4]. These pressures are rapidly intensifying in biodiversity-rich places, since most land conversion occurs in the tropics through forest conversion to agriculture [5,6]. As a consequence, land use and land cover changes are among major drivers of biodiversity loss, at both local [7] and global scales [8].

In turn, biodiversity loss affects the provisioning of essential ecosystem services, such as pollination, pest control, nutrient cycling and erosion control [9], with consequences on many human activities, and especially for agricultural production [8]. Biodiversity loss is thus a major and underestimated feedback that may affect human population growth in the long run [10], while the magnitude of land use effects on both biodiversity and biodiversity-dependent services is raising concerns about the potential of land-use changes to push terrestrial biodiversity beyond the planetary boundary [11].

These impacts of land-use change on biodiversity are poorly reflected in market prices, and hence have been mostly ignored by decision-makers, despite their large cost for human economies, with an estimated value for the global ecosystem services of $145 trillion in 2011, and up to $20 trillion loss per year between 1997 and 2011 [12]. Loss of biodiversity-dependent ecosystem services thus constitutes a negative externality, that threatens intergenerational equity and sustainability [13].

At the global scale, land conversion is primarily driven by the growth of the human population [14], and arable lands are rapidly shrinking [15]. Recent evidence suggests that land use efficiency has been rising at the global scale [4]. However, such efficiency gains may not help save natural habitats and biodiversity in the long run, due to economic rebound effects, if lower prices stimulate demand and if higher yields raise profits, encouraging agricultural expansion [15]. By increasing the opportunity cost of conserva-tion, these effects undermine the efficiency of regulatory environmental policies, such as government protected forests and natural habitats, in protecting biodiversity [16].

Land-sparing mechanisms that could help overcome these rebound effects include land zoning, incentive-based economic instruments (e.g. land taxes, subsidies and payments), spatially strategic intensification and voluntary standards [16]. Especially, incentive-based mechanisms such as land taxes may allow internalizing the externality of land conversion on biodiversity-dependent ecosystems services and agricultural production [17]. Such mechanisms are based on economic efficiency concepts, so as to achieve the maximum amount of resource protection for a given production level.

During the past decade, the European Union has widely used incentive-based mechanisms to reduce gas emissions from motor fuels and vehicles, but also plastic bags, landfill waste, batteries, pesticides, and fertilizers. Mounting evidence shows that taxes have helped reducing pollution and the consumption of natural resources in many cases, with a higher efficiency and at lower costs than conventional regulatory approaches [18]. However, use of such negative price signals for environmentally damaging activities has been less spread in the US, where tax credits and deductions are favored. More generally, the low level of acceptance of taxes lies in interest group pressures and extensive data requirements (e.g., regarding the external costs of human activities), as well as inadequate sensitivity to issues of sustainability and scientific uncertainty.

Indeed, the efficiency of conventional taxes is limited by available scientific knowledge. This is especially true for the relationship between biodiversity-dependent ecosystem service loss and land use changes, for which there is still a high uncertainty regarding the long-term temporal dynamics of ecosystems in the context of accumulating extinction and functioning debts [19,20,21], i.e. the time-delayed loss of species and services following a change in land use. Moreover, conventional taxes do not necessarily guarantee intergenerational equity and sustainability, i.e. do not prevent the over-use of natural capital and reductions in human well-being over time [13,22].

Some authors have thus proposed to define a broad natural capital depletion tax to ensure that resource inputs from the environment to the economy stay within planetary boundaries and are sustainable [23,24,25]. Implementation of such a tax would raise prices of natural resources, thus encouraging technological advances while slowing down the rate of environmental depletion [18]. Other authors have proposed a corrected version of the net national product that accounts for the effect of agricultural land development on biodiversity, while ensuring a constant social welfare [26,27].

However, these developments have poorly accounted for the temporal dynamics of biodiversity-dependent ecosystem service loss, and do not consider this negative feedback on human demography. Indeed, biodiversity-dependent agricultural consumption affects human demography, resulting in a dynamic feedback loop between land conversion, biodiversity loss and human population growth [28]. Time delays between land conversion and biodiversity loss, mediated through extinction debts [19], result in a lagged feedback on agricultural production [29,21,20]. Such lag effects can result in long-term environmental crises, i.e. overshoot-and-collapse population cycles [28], which reduce human well-being.

In this paper, we propose to assess the efficiency of a natural land depletion tax in securing sustainability and preserving biodiversity, despite uncertainty about the temporal dynamics of biodiversity loss. The paper is organized as follows. In section 1, we present a dynamical system model that couples human demography and technological change to biodiversity loss, through the effect of land conversion on the flow of biodiversitydependent ecosystem services to agricultural production [28]. In section 2, the externality of land conversion on biodiversity is internalized through a natural land depletion tax τ per unit of converted land. We show how this tax affects the consumption levels, the ratio of the production inputs, and the rate of land conversion. In section 3, we analyze the effects of this tax on the long term equilibria and sustainability of the system, as captured by a criterion ensuring a non-decreasing human well-being over time. We show that a land tax can increase both biodiversity and total agricultural production at equilibrium, when the substitution of land for labor and ecosystem services has a net positive effect on total agricultural production. The land tax also reduces the vulnerability of the system to time delays, but its ability to prevent crises depends on its level at equilibrium, and thus on the land conversion policy. Section 4 derives the optimal land conversion policy designed 123 CHAPTER 3 by a foresighted planner, who aims to internalize the externality of land conversion on biodiversity under the assumption that the temporal dynamics of biodiversity is unknown.

We illustrate the efficiency of such a policy in preserving biodiversity, increasing total production, and preventing the unsustainable consequences of time-delayed ecological feedbacks, despite incomplete scientific knowledge regarding their temporal dynamics.

A simple land -biodiversity -demography model 1.Substitution of natural capital for production inputs

We build upon the model of Lafuite & Loreau (2017) [28], that considers a population of consumers whose demand for agricultural (i = 1) and industrial (i = 2) goods requires the conversion of their common natural habitat. The two goods in the model are each produced using labor L i and land A i , and we assume full-employment, i.e. total labor is captured by the size of the human population. Only converted land is capable of producing these goods, while land not converted for production remains as natural habitat capable of supporting species, and provides a range of biodiversity-dependent ecosystem services to agricultural production [9].

Total factor productivity increases with technological efficiency in both sectors, as well as with biodiversity-dependent ecosystem services in the agricultural sector. The ecosystem services provided by this community of species are assumed to increase with biodiversity and saturate at high levels of species richness, through a power-law relationship B Ω , where Ω ∈ [0, 1] [30]. Technological efficiency is also assumed to follow a logistic growth towards a maximum efficiency, T m , in order to reproduce past agricultural productivity rise and current stagnation [31].

By using Cobb-Douglas production functions (eq.( 1)), we allow for the partial substitution between production inputs (labor and land), but also between natural capital (biodiversity-dependent services) and technology.

Y 1 = TB Ω L α 1 1 A 1-α 1 1 Y 2 = TL α 2 2 A 1-α 2 2 (1)
CHAPTER 3

generating an incentive to substitute land for labor.

L i /A i = (κ + τ )α i /(1 -α i ) (7)

Consumption

The total revenue of the tax τ A, where A = A 1 + A 2 is total converted land, is then redistributed among the consumers, who are assumed to maximize their consumption utility U under their revenue constraint p 1 y 1 + p 2 y 2 ≤ w + τ A/H. As a result of this redistribution, the total demand for agricultural and industrial goods (eq.( 8)) increases with the land tax.

p 1 Y D 1 = η w + τ A H H p 2 Y D 2 = (1 -η) w + τ A H H (8)
At the equilibrium between supply and demand, the optimal allocations of labor and land (Appendix 3.A) provide the equilibrium consumptions in the agricultural and industrial sectors:

y 1τ = φ φ τ κ + τ κ α 1 y 1 y 2τ = φ φ τ κ + τ κ α 2 y 2 (9)
where y 1 and y 2 are the business-as-usual consumptions (eq.( 5)), and φ and φ τ are the densities of the human population on converted land, in the business-as-usual and the regulated cases respectively.

Land conversion

The density of the human population on converted land, φ τ = H/A, also derives from the labor market equilibrium L 1 + L 2 + κA = H.

φ τ = φ + τ φ -κ κ ( 10
)
where φ is the population density in the absence of regulation (Table 2). Note that φ > κ since α i ∈ [0, 1] and η ∈ [0, 1], so that φ τ > φ. A tax τ per unit of converted land thus

Functions and aggregate parameters Definition

φ κ 1-α 1 η-α 2 (1-η)
Population density without regulation

φ τ φ + τ φ-κ κ
Population density with regulation

γ 2τ T m (1 -η)α α 2 2 1-α 2 κ 1-α 2 φ φτ κ+τ κ α 2
Max. per capita industrial consumption

γ 1τ T m ηα α 1 1 1-α 1 κ 1-α 1 φ φτ κ+τ κ α 1 Max. per capita agricultural consumption ∆ τ -4Ωzy min 1 γ 1τ y min 1 1 Ωz
-1 e -b 2 γ 2τ µ Sustainability criterion Table 3.2: Functions and aggregate parameters expression and definition. The expressions in the unregulated case are obtained by taking τ = 0, so that

γ 1 = γ 1(τ =0) , γ 2 = γ 2(τ =0) , ∆ = ∆ (τ =0)

Dynamical system analysis

Here, we analyze the effect of a land tax τ on the equilibria of the regulated system:

                   Ḣ = µ H 1 -e y min 1 -y 1τ (B,T) e -b 2 y 2τ (T) Ṫ = σT(1 -T/T m ) Ḃ = -[B -(1 -H/φ τ ) z ] (11)
Parameters and functions are summarized in Table 3.1 and 3.2.

Steady states and sustainability conditions

The equilibrium of the system is reached when technological efficiency is at its maximum level T m ( Ṫ = 0), human consumption is at its equilibrium level y min 1 so that human population cannot grow anymore ( Ḣ = 0), and the extinction debt of biodiversity has been entirely paid, so that B = S(H) ( Ḃ = 0).

There are two possible equilibria: (1) a desirable equilibrium, (H * τ , T m , B * τ ), and (

CHAPTER 3

an undesirable equilibrium, (0, T m , 1).

B * τ = y min 1 γ 1τ 1 Ω H * τ = φ τ (1 -B * 1 z τ )
where γ 1τ and φ τ are explicitly defined in Table 3.2.

The sustainability conditions of the system now depend on the tax τ (Table 3.2):

T m T(0) > B(0) B * τ Ωη ∆ τ > 0 (12)

Effects of the tax on equilibrium features

The effect of the tax τ on the equilibrium features of the model is mediated through the relationships γ 1τ (τ ), i.e. the level of substitution of land for natural capital in the agricultural production, and γ 2τ (τ ), i.e. the per capita level of industrial consumption at equilibrium. Indeed, the level of biodiversity at equilibrium B * τ directly depends on γ 1τ , which in turn determines the level of human population H * τ , while γ 2τ determines the level of industrial consumption, and thus the human well-being at equilibrium u * = (y min

1 ) η γ 1-η 2τ .
The shapes of γ 1τ (τ ) and γ 2τ (τ ) depend on the economic parameters of the system, and especially on the labor elasticities in the agricultural and industrial sectors, α 1 and α 2 . Labor elasticity captures the increase in output resulting from a 1% increase in labor. Since the main effect of the tax is to increase the ratio of labor to land, varying labor elasticities between sectors result in differing effects of the taxation policy on the equilibrium features of the system.

Effect on biodiversity, sustainability and population size

The tax has a positive effect on biodiversity if ∂γ 1τ τ < 0. It can be shown that

∂(B * τ -B * ) ∂τ > 0 f or α 1 ≤ α 2 (13)
so that the tax τ always has a positive effect on the long-term level of biodiversity, when the labor elasticity of the industrial sector is higher or equal to that of the agricultural sector (α 2 ≥ α 1 ), i.e. the most common situation in real-world systems.

Under the assumption that α 1 ≤ α 2 , we distinguish two situations: (1) labor elasticity is higher in the industrial than in the agricultural sector (α 2 > α 1 ), so that γ 1τ decreases with τ while γ 2τ increases (Fig. 3.2.A/C/E/G) and ( 2) labor elasticity in the industrial and agricultural sectors are similar (α 2 ≈ α 1 ), so that both γ 1τ and γ 2τ decrease with τ (Fig. 3.2.B/D/F/H).

By increasing the ratio of labor to land, the tax reduces land conversion and allows preserving more biodiversity through the substitution of land for biodiversity, i.e. γ 1τ decreases with τ (Fig. 3.2.C and D). This higher biodiversity level at equilibrium ensures a higher sustainability of the system, which becomes less vulnerable to transient overshootand-collapse crises, as captured by our sustainability criterion ∆ τ > 0 (Fig. 3.

2.G and H).

This reduction of land conversion is not only compensated by a higher natural capital, but also by a larger labor force, which increases the size of the human population (Fig. 3.2.A and B). However, the effect of the tax on the size of the human population at equilibrium is non-linear, since high tax levels reduce the incentive for land conversion to the point where it becomes economically unviable to convert land anymore, thus reducing the size of the human population at high tax levels (Fig. 3.2.A and B).

Effect on industrial consumption and human well-being

Distinction between cases (1) and ( 2) lies in the effect of τ on the consumption of industrial goods at equilibrium, γ 2τ , and thus on human well-being. When labor elasticity is higher in the industrial than in the agricultural sector (α 2 > α 1 ), land taxation increases industrial consumption at equilibrium compared to the business-as-usual case (Fig. 3.2.E). However, if the industrial labor elasticity is lower (α 2 ≈ α 1 ), land taxation reduces industrial consumption (Fig. 3.2.F).

The tax level required to achieve a positive sustainability criterion (Fig. 3.2.G and H) or to maximize human well-being (Fig. 3.2.A and B) is much higher in case (2) than in case (1), so as to compensate for the lower labor to land ratio of the industrial sector.

Thus, the total labor force, i.e. the size of the human population, is also higher compared to case (1) (Fig. 3.2.A and B).

This increase in population size thus reduces both per capita industrial consumption and human-well-being, in case (2) compared to case (1) (Fig. 3.2.E and F). Despite its positive effects on biodiversity and sustainability, a land tax may thus reduce per capita well-being if the initial labor elasticities are too low, through a large increase in labor, i.e. population size.

Optimal tax vs. sustainability

A regulator seeking to maximize total human well-being through an optimal tax τ opt (Fig. 3.2.A and B) thus needs to account for the economic structure of the system, and especially the relative values of α 1 and α 2 . However, such an optimal tax may not prevent unsustainable crises from occurring, since low relaxation rates can reduce the sustainability of the system to the point where it becomes vulnerable to crises, i.e. if ∆ τ =τ opt < 0 (Fig. 3.2.G and H). Thus, the higher the extinction debt, the higher the tax on land conversion should be in order to avoid unsustainable trajectories. Time-delayed biodiversity loss may make the optimal tax level τ opt unsustainable, so that a land policy that does not account for such temporal delays may fail to prevent unsustainable crises.

In the next section, the optimal land conversion policy is derived in the case of a foresighted regulator, and its efficiency in preventing crises is explored.

Optimal land conversion policy

Here, we allow the tax to vary with the dynamical variables of the system, so as to internalize the negative effects of biodiversity loss on agricultural production, while maximizing human-well-being.

Analytical derivations

At each period, and for a population size H, a technological efficiency T, and a biodiversity level B = (1 -H/φ τ ), we assume that a benevolent social planner aims at maximizing the total discounted utility of consumers, U τ (B, T, H) = H • y 1τ (B, T) η y 2τ (T) 1-η using a land tax τ per unit of converted land as control. This land tax limits the conversion of natural habitat so as to internalize the negative externalities on biodiversity and agricultural production, and varies with the size of the dynamical variables of the system.

The objective of the social planner is to maximize the present value of a continuous sum of discounted utilities, at an annual rate δ, subject to the dynamics of the human population, Ḣ (eq.( 2)), technological change Ṫ (eq.( 4)), the consumptions levels y 1τ and y 2τ , as well as to the loss of biodiversity, B = (1 -H/φ τ ). Thus, we assume here that the social planner does not know the temporal dynamics of biodiversity loss, Ḃ (eq.( 3)).

However, he accounts for the long term effects of land conversion on biodiversity, through the use of a species-area relationship, which is one of the best-known patterns in ecology [43].

max ∞ t 0 e -δt U τ (B, T, H) dt

Conclusions and discussion

In this paper, we linked a simple general equilibrium market model with a dynamical system coupling human demography, technological change and time-delayed biodiversity loss, through the consumption of agricultural and industrial goods, and the time-delayed feedback of biodiversity-dependent ecosystem services on agricultural production. We then demonstrated the effects of a natural land depletion tax on long-term biodiversity, human carrying capacity, well-being and sustainability of the system. Because the land tax increases the ratio of labor to land used in the agricultural and industrial productions, it fosters land-use intensification through the substitution of land for labor in both sectors, but also for biodiversity-dependent ecosystem services in the agricultural sector.

Internalization of biodiversity-dependent externalities on agricultural production thus fosters ecologically-intensive agricultural systems, that preserve more biodiversity and can support a larger human population. To the extent that biodiversity is aligned with the sustainability of the system, i.e. its resistance to overshoot-and-collapse population crises resulting from a time-delayed loss of biodiversity-dependent ecosystem services [28],

such a land taxation policy means more stability and sustainability. However, a land tax can have adverse consequences for the per capita human well-being through its effect on industrial consumption, especially in systems with a low industrial labor intensity, in which substitution of land for labor is not compensated by an increase in natural capital.

This result may change when considering a symmetrical effect of biodiversity on both the agricultural and industrial sectors, since many industrial activities rely on natural services. Indeed, deforestation and biodiversity loss can affect regional and global climate [44] with feedbacks on hydrology [45] and other important provisioning services to the industrial sector, such as wood production and clean water.

The stabilizing effect of the tax did not require a precise knowledge of the ecological dynamics of the system, and especially of the temporal dynamics of biodiversity loss.

Our results thus suggest that lack of data and uncertainty about complex ecological dynamics should not prevent land-use management to adopt a precautionary approach to CHAPTER 3 environmental uncertainty [46]. Sustainable land-use management policies should build upon well-known ecological patterns, such as species-area relationships [43], as well as recent advances in ecological research that allow making large-scale predictions up to continental or global scales, ranging from the future distribution of biological diversity to changes in ecosystem functioning and services [47,48].

The model we developed in this paper was kept quite simple in order to make clear the basic logic of how economic incentives can affect land conversion, biodiversity conservation and sustainability. For example, the proportional relationship between human population growth and land conversion, resulting from the assumption of a constant maintenance cost of κ units of labor per unit of converted land, is unrealistic. Drivers of land use and land cover change are abundant, complex and scale-dependent [49]. At global scale, recent evidence that human population and the world economy are growing faster than the human footprint suggests a globally more efficient use of land [4], so that endogenizing efficiency gains and technological change along with economic growth appears essential in order to gain realism in the relationship between human population growth and land conversion.

Our model considered the use of converted land taxation as a way to preserve natural habitats. Property taxes have been used in several other contexts [50], as a source of revenue [51], a way of promoting urbanization [52] or conversely, in order to reduce the use of land for house building [53] and foster land-use efficiency [50]. However, despite the established efficiency of taxes as a way of internalizing the externalities of human activities [54], such as modern agriculture [55], land taxes are rarely used for conservation purposes, especially in rural areas [56]. Reasons include a higher riskiness of net farmer income, costly administration and informational requirements [57], difficulty to administer progressive tax rates based on land holdings, and political acceptability of negative price signals [51]. For these reasons, taxes remain marginal in both research and implementation on the internalization of ecosystem services into economic decisions [54],

in comparison to national governmental payment programs [58], such as the green payments and subsidies implemented by the European Common Agricultural Policy, whose efficiency in preserving biodiversity appears limited [59,60]. Strong political will is thus required to shift the current paradigm and improve the efficiency of agricultural policies, through a better identification and management of the conflicts between agriculture and biodiversity conservation [61] and the development of more integrated approaches to policy, land-use and biodiversity [62].

Moreover, despite mounting evidence of synergies between biodiversity and multiple ecosystem service conservation [63,64,65], scientific understanding of ecosystem production functions remains a limiting factor in incorporating natural capital into economic decisions [66]. Improving scientific knowledge of the multi-layered relationship between biodiversity and ecosystem services [67] is a necessary step towards the reconciliation of conventional conservation management focused on conserving specific species and habitats [68], with the management of multiple ecosystem services [69,65] within an integrative approach [70].

There are a number of other ways that the model could be enriched, both on the economic side and on the ecology side. We assumed that the provisioning of services to productive lands depended on the total area of natural habitat only. In reality, service provisioning is spatially-and distance-dependent, since intermediate habitat heterogeneity and fragmentation is required to provide access to several services, such as pest control and pollination [71]. In turn, habitat fragmentation affects the viability of communities and generates extinction debts [40,21]. Expanding the model in this way requires the spatialization and differentiation of economic incentives, and the distinction between local and regional or global species richness and services. Second, the assumption that species can only utilize natural habitat is also too restrictive, since certain species can persist on working landscapes (agricultural fields and managed forests). Finally, we have modeled utility as a function of the indirect effect of global species richness on private consumption.

It is equally plausible that alternative measures of biodiversity enter the utility function, such as its cultural, spiritual and aesthetic value. Including these as arguments of utility would require a different objective function, but would not qualitatively alter the results of this paper. tax policies aimed at reducing greenhouse-gas emissions [75]. These more complex models, however, would still contain the basic insights on how time delays and land taxes affect sustainability and biodiversity conservation.

CHAPTER 3

p 1 Y D 1 = η w + τ A H H p 2 Y D 2 = (1 -η) w + τ A H H (21)

Firms Optimization

Firms in sectors 1 and 2 produce a quantity Y i (i = 1, 2) of output, using labor L i and land A i . A units of land and H units of labor are available. Agricultural firms occupy A 1 units of land, and industrial firms occupy A 2 units of land, such that A = A 1 + A 2 .

This comes at an operating cost of κ units of labor per unit of land, so that each firm maximizes a profit Π i = p i Y i -wL i -(κw + τ )A i . The production functions Y i are:

Y 1 = T B Ω L 1 α 1 A 1 1-α 1 Y 2 = T L 2 α 2 A 2 1-α 2 (22)
where B Ω is a concave-down function capturing the ecological feedback of biodiversitydependent ecosystem services on agricultural production, with Ω < 1, T captures the agricultural total factor productivity, i.e. production efficiency, and α i is labor intensity in sector i. Therefore, first order conditions are:

         ∂Π i /∂L i = α i p i Y i L i -w = 0 ∂Π i /∂A i = (1-α i )p i Y i A i -κw -τ = 0 (23)
Adding both lines of (23) gives the total supply in sector i:

p i Y S i = wL i + (κw + τ )A i (24)
Dividing both lines of (23) gives the relationship between input factors in each sector:

L i /α i = (κ + τ )A i /(1 -α i ) (25)
where the density of the human population on converted land φ τ is:

φ τ = φ + τ φ -κ κ ( 31
)
and φ is the population density in the absence of regulation:

φ = κ 1 -α 1 η -α 2 (1 -η) (32)
Note that φ > κ since α i ∈ [0; 1] and η ∈ [0; 1], so that φ τ > φ. A tax τ per unit of converted land thus increases the human population density on converted land.

Per capita consumptions

Using eq.( 30) so as to replace A in the equilibrium allocations of labor L i (eq.( 29)) and land A i (eq.( 28)), we now obtain the inputs' allocations as functions of the dynamical variable H:

L 1 = φ φ τ κ + τ κ α 1 ηH L 2 = φ φ τ κ + τ κ α 2 (1 -η)H (33)
and

A 1 = φ φ τ (1 -α 1 )η κ H A 2 = φ φ τ (1 -α 2 )(1 -η) κ H (34)
The per capita agricultural and industrial consumptions Y i /H (eq.( 22)) can then be rewritten as functions of the dynamical variables of the system B and T as follows:

y 1τ (B, T) = φ φ τ κ + τ κ α 1 y 1 (B, T) y 2τ (T) = φ φ τ κ + τ κ α 2 y 2 (T) (35)
where the per capita consumptions in the absence of regulation are:

y 1 (B, T) = ηα α 1 1 1 -α 1 κ 1-α 1 B Ω T y 2 (T) = (1 -η)α α 2 2 1 -α 2 κ 1-α 2 T ( 36
)
Since φ < φ τ , the effect of a tax τ on the per capita consumptions is not straightforward CHAPTER 3

and will depend on the parameters of the system, and especially on labor intensity α i . In systems with low labor intensities, a tax on converted land will rather reduce agricultural consumption and population growth, while it could have the opposite effect on systems with higher labor intensities.

GENERAL DISCUSSION

Synthesis of the results

In the first chapter of this thesis (Chapter 1), a social-ecological system model including the feedback of biodiversity loss on human demography is described. This simple model is used to perform abstract simulations and qualitatively assess the effects of parameters on the long-term dynamics of the system, and especially the potential This conceptual model allows exploring the long-term consequences of time-delayed biodiversity loss on the dynamics of a human population. The model reproduces classical cyclic dynamics of predator-prey models, when there is a time delay between the dynamics of the predator and that of the prey [98]. Growth of the predator population decimates the slowly regenerating prey, thus resulting in a food shortage and the degrowth of the predator population, until the prey population has regrown large enough for the predator to grow again. A similar mechanism has been invoked to explain the collapse of the human population on Easter Island, where the main variety of palm tree was characterized by a low regeneration rate compared to the varieties found on other Pacific Islands [13].

In a coupled SES, such oscillations result in a decrease in human well-being, and are thus incompatible with sustainability requirements [16]. Given mounting evidence of accumulating extinction and functioning debts, this effect of time delays that generates temporal mismatches between human and ecological timescales should be better recognized and accounted for.

One advantage of the model's simplicity lies in the clear understanding of the effects of the ecological, economic and technological parameters on both transient dynamics and equilibrium features. Analytical results allow identifying parameters' space that make the system more vulnerable to such overshoot-and-collapse crises, as well as thresholds in the dynamical variables beyond which the system starts to oscillate. Due to the coupling of the dynamical variables within the SES model, each threshold is related to the others, so that crossing one of them leads to crossing the others, in a "cascading effect" [54]. Improving knowledge about such thresholds is essential to reduce uncertainty and inform biodiversity conservation, land-use management and demographic policies in an integrative way.

Technological change makes our model SES more vulnerable to crises, when it increases the efficiency of agricultural production. This destabilizing effect of technological change has already been emphasized in typical Easter Island systems, regarding resource harvesting efficiency, carrying capacity and intrinsic growth rate [96]. In our system, a higher production efficiency is equivalent to a higher substitution between human capital and biodiversity-dependent services. This negative effect of technological efficiency thus suggests that conventional technology-intensive agriculture may threaten production systems with overshoot-and-collapse crises in the long run. A higher decoupling from natural constraints thus appears detrimental to sustainability, despite the classical view of growth economists on this topic [11].

Our results thus contrast with the commonly accepted view that technological change Conversely to technological efficiency, labor elasticity reduces the vulnerability of the SES to crises, which is consistent with the effect of technological efficiency, since a higher labor elasticity can be interpreted as a lower reliance on technology, i.e. a lower substitution of technology for natural services and labor. There is mounting evidence that recoupling agricultural systems and natural ecosystem services through ecologically-friendly practices is key to meet future food demand through the sustainable intensification of agricultural production [32,36]. Our results show that this is especially important with regard to long-term ecological dynamics and pending extinction debts [118,5,99].

Our results also emphasize the role of the demographic transition in alleviating transient crises. By slowing down the growth of the human population, the demographic transition reduces the delay between the dynamics of the human population and biodiversity loss, and hence the vulnerability to overshoot-and-collapse crises. However, the observed rapidity and strenght of the demographic transition may not allow counteracting time-delayed feedbacks, since population projections cast doubts upon the efficiency of the demographic transition in halting human population growth this century [40,12].

Interaction between the demographic transition and technological progress results in a non-linear effect of technology on sustainability, since a very high technological efficiency increases industrial consumption and strengthens the demographic transition to the point where the net effect of technological change switches from negative to positive. However, this result is highly dependent upon the assumption that there is no biodiversity feedback on the industrial sector. Deforestation and biodiversity loss can affect regional and global climate [107] with feedbacks on hydrology [59] and other important provisioning services to the industrial sector, such as wood production and clean water. Including a feedback of biodiversity loss on the industrial sector would thus make technological change less likely to have a net positive effect on sustainability.

Adding social interactions to the basic structure presented in Chapter 1 enriches the dynamics of the model, by allowing for alternative stable states to coexist, and for regime shifts between sustainable and unsustainable basins of attraction to occur (Chapter 2).

We consider social mechanisms that foster norm-driven consumption changes, such as disapproval or ostracism, and a consumption norm corresponding to sustainable agricultural practices, as identified in Chapter 1. The demand of norm-following consumers for such sustainable products drives changes in agricultural practices, as a result of market equilibrium between supply and demand. A high enough efficiency of the social pressure and a large enough proportion of norm-following consumers in the population lead to large-scale shifts towards more sustainable consumption and production practices. Such shifts greatly reduce the vulnerability of SESs to unsustainable crises, thus confirming the role of human behavioral change in the adaptive capacity of SESs [37] and in the prevention of social-ecological traps [19]. However, short term perception of environmental ernment control is also essential to prevent the overexploitation of natural resources [46], internalize ecosystem services into economic decisions [94,91,105] and enforce property rights [80]. Especially, foresighted government control can help counteract the negative consequences resulting from shortsighted economic decisions. In our model, foresighted land taxation alleviates the negative consequences of biodiversity loss in the long run, by fostering the substitution of labor, land and technology for biodiversity-dependent ecosystem services, thus preserving more natural habitats and biodiversity than in the business-as-usual scenario (chapter 3). Our results contrast with Easter Island's studies [13] suggesting that an optimal resource management with infinite horizon would not have prevented this historical overshoot-and-collapse population crisis, resulting from the inter-

Perspectives

Overall, our model has a structure upon which future improvements can be built. In the following, we identify potential improvements to the formalization of the main relationships of the model, including the relationship between (1) human consumption and demography, (2) population growth and land conversion, (3) land conversion and biodiversity, and (4) habitat fragmentation and ecosystem services.

Human consumption and demography

Even though various types of technological change did not qualitatively affect our results (chapter 1, Appendix), endogenizing technological change along with economic growth and fertility choices appears necessary in a predictive perspective. In similar models, capital accumulation [1] and economic stratification [72] have destabilizing effects, i.e.

favor population cycles and collapses. Conversely, human foresight regarding natural resources' exploitation has a stabilizing effect, although discounting may reduce this effect [41,24]. Endogenous income and fertility dynamics can be destabilizing in case of a low substitutability between labor and natural resources, driving the economy towards either demographic explosion or collapse [84]. Assuming general equilibrium for the economic sector has major flaws, although most economic models make this assumption. A possible extension of this work would be to develop a non-equilibrium economic sector that does not necessarily converge to an optimal equilibrium. These results highlight the importance of accounting for realistic economic representation when modelling human-nature systems. Accounting for recent theoretical and empirical evidence regarding changing fertility choices and death rates with economic development and accumulation of human capital, such as the child quantity-quality trade-off during the demographic transition [33], or increased death rates due to resource scarcity, would also allow for a more realistic modelling of fertility choices.

More realism can also be added to the relationships between human demography, consumption, and biodiversity. First, in our model, the biodiversity feedback on human population growth only occurs through the level of agricultural consumption. Biodiversity also provides other services than agricultural production, such as carbon sequestration and climate regulation, cultural, recreational and spiritual services [17], all of which affecting human well-being, mortality and fertility choices. Second, accounting for the effect of consumption quantity and quality, on birth and death rates separately, could enrich the behavior of the model [1]. Indeed, standard quantitative consumption metrics are inadequate to capture the nutritional value of agricultural goods, especially regarding the erosion of essential dietary nutrients due to the widespread selection of high-yielding and nutrient-poor cereals, such as wheat, rice, and maize [27]. Nutrient-poor diets, which are higher in refined sugars, refined fats, oils and meats, are detrimental to human health, since they increase incidence of type II diabetes, coronary heart disease and other chronic non-communicable diseases that lower global life expectancies [112].

Thus, agricultural intensification may not allow meeting the nutritional requirements of the global population, and may generate significant feedbacks on human mortality.

Furthermore, alternative diets that offer substantial health benefits, such as pescetarian, mediterranean and vegetarian diets, could help reduce land conversion and resultant species extinctions [112]. Such healthy diets include a larger part of fresh fruits and vegetables, whose production benefits from biodiversity-dependent services, such as pollination.

Less intensive agricultural practices relying more on biodiversity-dependent services than on toxic chemicals, could thus benefit both human health and biodiversity. These complex feedbacks between agricultural practices, biodiversity conservation, food quality and human health may affect the behavior of our conceptual model, and need to be accounted for in order to produce reliable forecasts.

Population growth and land conversion

Drivers of land use and land cover change are abundant, complex and scale-dependent [20]. At global scale, recent evidence that human population and the world economy are growing faster than the human footprint suggests a globally more efficient use of land [117]. However, economic mechanisms such as the displacement, rebound, cascade, and remittance effects that are amplified by economic globalization can accelerate land conversion despite land intensification [57]. Endogenizing efficiency gains and technological change along with economic growth thus appears essential in order to gain realism and predictive power.

At local scales, the effect of human population growth can be ambiguous [64,39], and population dynamics usually acts in concert with other significant factors such as local institutions, policies and cultural change [58,25]. Additionally, economic globalization and rising connectivity between distant SESs can lead to the local specialization in production that drives specialization in ecosystems and their associated biodiversity, leading to significant declines in both local and global biodiversity, especially when endemism is high, i.e. when trading partners contain dissimilar species [92].

In the same vein, human migrations and spatial movements are steadily increasing across regions of the world, with consequences for local labour markets and land-use patterns [75]. Thus, accounting for the spatial human and economic cross-scale effects appears crucial to adequately capture the relationship between human population growth and land conversion.

Land conversion and biodiversity

Another important improvement relates to the spatialization of the ecological compartment of the model. Indeed, habitat loss and fragmentation changes the connectivity and size of fragments of natural habitat, thus affecting the spatial dynamics of populations and making species cross their extinction threshold. Such thresholds depend both on species traits, such as dispersal and colonization, and on the spatial configuration of habitat patches [45]. As a result, habitat fragmentation significantly amplifies the negative effect of habitat loss on species richness, so that accounting for this effect increases the predictive power of species-area curves [100,44].

Fragmentation also has a non-linear effect on the relaxation rate of populations [82].

Habitat fragmentation slows down the transient dynamics of viable populations, i.e. populations that are above their extinction threshold, and accelerates the dynamics of doomed ones, i.e. those that are below their extinction threshold [45]. As a result, species extinctions are expected to occur faster on smaller habitat fragments [34]. Thus, the relaxation rate of communities is not a constant parameter, but endogenously varies with the spatial configuration of habitat fragments.

Lastly, our model assumption that recovery debts are equivalent to extinction debts is unrealistic. First, land degradation can lead to irreversibility of the land conversion process. Soil erosion enhances nutrient loss, which reduces the fertility of soils [89], with important economic consequences [90]. Each year about 10 million ha of cropland are lost due to soil erosion, 10 to 40 times faster than the rate of soil renewal [88]. Second, when recovery is possible, recovering and restored ecosystems have lower species abundances and diversity, and less cycling of carbon and nitrogen than 'undisturbed' ecosystems, due to the accumulation of a recovery debt [70]. Such recovery debts can be much larger than extinction debts, especially due to a time-delayed recolonization e.g. 130-230 years after reforestation in Germany [74]. Recolonization not only depends on the dispersal abilities of species between distant habitat patches, but also on the state of biodiversity at other spatial scales, highlighting the importance of a cross-scale spatial perspective [43].

Adoption of a spatially-explicit approach appears crucial in order to account for the complex relationships between habitat loss and biodiversity in a more realistic way.

Habitat fragmentation and ecosystem services

Spatialization is also a crucial aspect of the provisioning of ecosystem services in realworld landscapes [114]. First, landscapes have long been represented as split between natural habitats supporting biodiversity on one side, and inhospitable human-dominated lands on the other [42,86]. However, real-world landscapes are a mix of natural, seminatural and productive lands, with bidirectional interactions between them. Productive land hosts species that are new to the landscape and may propagate to proximate patches of natural habitats, while agricultural chemicals impact wild species and nutrient flows.

Besides, natural patches provide multiple ecosystem services to surrounding fields, such as crop production, pest regulation, decomposition, carbon storage, soil fertility, water quality regulation [66] and pollination [63], but also dis-services such as pest damage and competition for water or pollination between natural habitats and crops [120].

Fragmentation generally has negative effects on ecosystem service supply, through its negative effect on species richness. However, fragmentation can have positive or negative effects on service flow, through an increased proximity between crops and the natural habitat patches providing services [67]. Recent advances on connectivity-ecosystem service provision relationships [68] highlight the potential for highly non-linear effects of fragmentation on service provisioning at local scales [69]. Another cause of non-linearity is the crossing of extinction thresholds beyond a critical amount of habitat loss, that can lead to the collapse of a service in agroecosystems, such as pollination [53]. Such nonlinearities can result in cascading effects when crossing a local threshold results in the crossing of other thresholds at larger spatial scales [54].

Second, spatial aspects can modify the positive relationship between biodiversity and services reported by most experimental studies. For example, pest control usually increases with landscape complexity, which increases both species diversity and the proximity between crops and natural habitats patches hosting natural enemies [9,38]. However, this effect can be ambiguous [111]. Natural enemies usually have a strong positive response to landscape complexity, but this effect does not necessarily translate into pest control, since pest abundances do not always show significant response to landscape complexity [21]. A possible explanation may be that the relationship between pest control and landscape complexity is also driven by changes in negative natural enemy interactions, such as predation from birds. Thus, by altering natural enemy interactions, landscape complexity can provide ecosystem services as well as disservices [62].

In a spatially-explicit context, the relationships between biodiversity, ecosystem services supply and ecosystem services flow are not necessarily linear. Accounting for landscape spatial configuration, distance-dependence in service provisioning, trade offs among multiple ecosystem services, non-linearities and cross-scale interactions thus appears essential in order to adequately capture the complexity of real-world agroecosystems, in relation to the sustainability of SESs relying on them.

Towards predictive social-ecological models

Human demography is a major taboo in public opinion, as well as in studies of the sustainability of modern human societies [24]. Several roots of this taboo include the rise of individualism, deep cultural values surrounding fertility and the idea that progress and 'Providence' will provide for all of our needs [47], which lead to a contradiction with the necessity of women's access to family planning and healthcare as a universal right for women. Ecology thus threatens fundamental and almost religious values [48].

A great part of the uncertainty in our current understanding and projections of human population growth, and hence of the sustainability of human-nature interactions, lies in biodiversity feedbacks on human demography [71]. Incomplete knowledge of the temporal dynamics of biodiversity loss and of the relationships between biodiversity and essential ecosystem services has impaired realistic representation of some processes, which are however critical for a full understanding of these feedbacks. Hence, despite observations of time delays in species extinctions and predictions of increasing species loss and consequences for ecosystem functioning, the feedback of biodiversity on human societies is still represented in a crude way [76]. Surprisingly, the consequences of land use changes on biodiversity appear less studied than those of climate change [113], despite the major threats of future human population growth for natural habitat conversion and massive biodiversity loss driven by land-use changes.

It is critically needed to develop predictive models that include the feedback of biodiversity loss on human demography, especially regarding the potential consequences of time delays highlighted in this thesis. Extinction debts pose a significant but often unrecognized challenge for biodiversity conservation [55,31], while constituting windows of conservation opportunities such as habitat restoration and landscape management [118].

Reducing uncertainty regarding the role and importance of biodiversity is crucial to inform policy-makers, and halt the current extinction crisis. Recent advances in ecological research now allow making large-scale predictions up to continental or global scales, ranging from the future distribution of biological diversity to changes in ecosystem functioning and services [85,52]. Next-generation social-ecological models [93,35] should thus better integrate both human demography and detailed biodiversity feedbacks, along with uncertainty and collective action [2], in order to improve their predictive power in a changing world [73].

Moreover, the development of predictive social-ecological system models appears crucial given the mounting evidence for social-ecological regime shifts [56,101,6]. Increasing risk of such abrupt regime shifts towards often less desirable and more resilient states has fostered the development of various modeling approaches, such as statistical, system dynamics, equilibrium and agent-based modelling. However, these approaches are largely disjunct, and do not necessarily consider the criteria that are empirically studied, thus making testing and application hard [35]. Yet some of these approaches provide useful insights into the management of SESs at risk of regime shifts [23].

Accounting for the feedback of biodiversity on human demography requires transdisciplinary collaboration involving ecological researchers, economists, social and climate scientists, and demographers. The Integrated Valuation of Ecosystem Services and Tradeoffs (InVEST) framework [76] models multiple ecosystem services, biodiversity conservation, commodity production, and tradeoffs at landscape scales, but poorly accounts for the relationships between biodiversity and services, and ignores human demography. Other global Integrated Assessment Modeling frameworks have been developed to specifically address climate change [18] or water availability [49] issues. For example, the Dynamic Integrated Climate-Economy (DICE) model has been extensively used to inform optimal tax policies aimed at reducing greenhouse-gas emissions [77]. Many Integrated Assessment Models (IAMs) have been developed to describe the key processes in the interaction of human development and the natural environment. Biodiversity should be included in existing IAMs, as a dynamic variable coupled to other sectors with feedbacks. Indeed, the Global Change Assessment Model (GCAM) couples the economy, energy sector, land use and water with a climate model, but ignores biodiversity feedbacks [30]; the Integrated Global System Modeling (IGSM) framework simulates the evolution of economic, demographic, trade and technological processes, and the resulting greenhouse gas emissions, conventional air and water pollutants, and land-use/land-cover change, through coupled sub-models of physical, dynamical and chemical processes [108,95]; the Model for Energy Supply Strategy Alternatives and their General Environmental Impact (MESSAGE)

focuses on the global economy and its main sectors (energy, agriculture, forestry), thus neglecting biodiversity feedbacks [65]; the Integrated Model to Assess the Global Environment (IMAGE) represents interactions between society, the biosphere and the climate system, but considers population, economy, policy and technology as external drivers

[110]; Modelling International Relationships in Applied General Equilibrium (MIRAGE) is a multi-sectoral and multi-regional computable general equilibrium model dedicated to trade policy analysis [26]. Future research should build upon the strengths of each of these IAMs in order to fully account for the complexity of coupled social-ecological systems.

Population ethics and inequality

A major difficulty in defining integrative biodiversity-food security policies lies in their ethical consequences. Indeed, a policy seeking to maximize total agricultural production may lead to favor large human populations with a low average well-being [83]. This paradox, known as the "repugnant conclusion", has been extensively studied in population ethics. Solutions to this problem include defining a minimal well-being as a constraint to the maximization problem [10]. Equivalently, in our model we set a minimum consumption level, below which human population starts declining, thus preventing "repugnant" population equilibria in the long run. Yet such a threshold remains arbitrary and may still pose ethical problems in real-world situations.

Another ethical difficulty lies in defining sustainable policies that also guarantee equity and poverty eradication [50]. Income inequality among members of a society, leading to economic stratification, causes social and economic instability [87] and has played a central role in the collapse of past civilizations [29]. Unequal distribution of wealth has grown extremely fast since about 1950, along with population and consumption [15,14].

The average per capita material and energy use in developed countries is higher than in developing countries by a factor of 5 to 10, and about 50% of the world's people live on less than $3 per day, 75% on less than $8.50, and 90% on less than $23 [71]. In agreement with the historical record, models have shown that economic stratification can lead to collapse, whereas collapse can be avoided if resources are distributed equitably [72]. Piketty [87] proposes a global system of progressive wealth taxes to help reduce inequality and avoid the vast majority of wealth coming under the control of a tiny minority.

On uncertainty and risk in a changing world

The complexity and multiplicity of feedbacks that drive the dynamics of SESs make their study and understanding challenging, and the associated uncertainty can undermine decision-making and postpone management shifts. Thus, recognition of the insurance value of biodiversity is crucial to improve decision-making under uncertainty, and foster the adoption of a precautionary approach to this problem. Indeed, in addition to its direct and indirect use or non-use values [17], biodiversity also has an insurance value, since productivity and stability increase with biodiversity under variable environmental conditions [51] and in a spatially heterogeneous environment [60]. This natural insurance value can be seen as a substitute for financial insurance [7].

Ecosystem resilience has also been interpreted as an economic insurance to changes in the provisioning of ecosystem services, the value of which increases with the level of resilience of the system [8]. However, ecosystem resilience does not imply sustainability, since resilience and sustainability are independent concepts [28]. Thus, more criteria than just resilience have to be taken into account when designing policies for the sustainable development of social-ecological systems, and resilience should not be confused with sustainability. Yet the concept of insurance value of both biodiversity and resilience can be useful to foster decision-making under uncertainty. Several precautionary approaches to environmental uncertainty have been proposed, such as flexible environmental assurance bonding systems [22].

Another challenge to decision-making lies in our perception of environmental changes, taking place over spatial and temporal scales far beyond the scales which we directly perceive. Consequences of global changes, such as climate change, are difficult to apprehend and understand without the help of the scientific community. This is even more relevant for time-delayed biodiversity erosion, since it often does not directly affect the individual responsible for its erosion. Such a disconnection between the time and location of an action and its consequences, and the resultant decoupling between human and nature dynamics, is detrimental to the effective regulation of such "tragedy of the commons" situations [46].

However, even if a problem is recognized by experts, the time lag before any societal change in opinion can be significant, due to inertia and hysteresis processes, and especially in presence of a weak central decision-making authority or influential lobbys. Such situations cause abrupt but late shifts in opinion [103]. When consequences of current actions are delayed in time, as is the case with biodiversity loss, opinion shifts may be postponed even more, and come too late to prevent catastrophic outcomes. The consequences of time delays highlighted in this thesis should thus be better recognized and accounted for.

General conclusion

Minimal conceptual models are efficient tools to test hypotheses, develop theory and direct empirical research, and their limited number of parameters allows complete exploration of their behavior [102]. However, many essential aspects are usually neglected in simple models, leading to a trade-off between generality and predictive ability that makes testing and validation very hard. As a consequence, the popularity of abstract models remains marginal among policy makers and ecologists [102]. Conceptual models are, however, an essential step towards understanding the dynamics of larger complex models used to provide quantitative assessments and make predictions [116].

Despite quantitative limitations, minimal conceptual models can provide useful qualitative predictions. For example, despite uncertainty regarding the precise temporal dynamics and strength of biodiversity feedbacks, there is no doubt that time delays increase the vulnerability of human societies to overshoot natural resources. Such quantitative uncertainties should not prevent informed actions from halting current biodiversity loss and natural habitat destruction. A precautionary and farsighted approach is required to foster the necessary shifts in opinion and decision-making, which should be implemented across scales [80] and account for cross-scale interactions [43,52], be they temporal or spatial.

Both modelling exercise and experimental approaches feed knowledge and enlighten choices regarding such a complex adaptive system that represent human-nature interactions.

Figure A :

 A Figure A: Human population growth and global land-use change. Left panel: Population of the world, estimates (1950-2017) and medium-variant projection (2017-2100) with 95% confidence intervals. Extracted from the United Nations' World Population Prospects: the 2017 Revision [10]. Right panel: Projected land-use changes until 2100. Extracted from Newbold et al. (2015) [22].

Figure B :

 B Figure B: The Global Living Planet Index shows a decline of 58 % (range: -48 to -66 %) between 1970 and 2012. Trend in population abundance for 14,152 populations of 3,706 species monitored across the globe between 1970 and 2012. The white line shows the index values and the shaded areas represent the 95 % confidence limits surrounding the trend. Extracted from the WWF report (2016) [51].

Figure C :

 C Figure C: Net change in local richness caused by land use and related pressures by 2000. Extracted from Newbold et al. (2015) [22].

  Despite these criticisms [130], the resilience perspective is increasingly used as an approach for understanding the dynamics of social-ecological systems [131]. It emphasizes nonlinear dynamics, thresholds, how periods of gradual change interplay with periods of rapid change and how such dynamics interact across temporal and spatial scales [132]. Interactional approaches, such as social-ecological systems (SESs), adaptive cycles [133] and political ecology [134, 135], consider ecological and social components as distinct but interacting across spatial and temporal scales, through various feedbacks. Especially, their focus is on the adaptive capacity of human-nature systems [136], and the implications of GENERAL INTRODUCTION non-convexities [137], slow-fast processes [138] and regime shifts [139] for management. However, economic sustainability and social-ecological resilience approaches are still largely disjoint [140], since the efficiency requirements of the economic sustainability often conflict with the stability requirements of the ecological sustainability approach [113, 141]. The development of an ecological economics of sustainability [119] that accounts for biodiversity, non-linearities, the value of ecosystem services [142] and proposes to include natural capital in the concept of wealth [143], has started bridging the gap between economic and ecological sustainability views, though it is still isolated from the main body of contemporary economics and ecological research [144].Despite these recent advances, feedback loops between population growth and the natural capital are still poorly accounted for by modern growth economists.

  , and many of the problems resulting from human-nature interactions can be framed as scale mismatches[148]. Scale mismatches occur when the spatial or temporal reach of management does not align with the spatial or temporal reach of the problem being managed, e.g., biodiversity loss[149]. For example, local habitat destruction can generate strong and non-linear effects on ecosystem service provision [150] and stability [151] at larger spatial scales. Such cross-scale spatial effects can lead to "cascading thresholds", i.e., the tendency of the crossing of one threshold to induce the crossing of other thresholds [152]. Recent studies have focused on spatial scale mismatches, their implications for agroenvironmental policies [153] and the conservation of ecosystem services [154]. Only a few studies, however, have considered the consequences of temporal scale mismatches [155]. Given that the scale of human activities increasingly couples fast and slow processes [156], temporal cross-scale interactions cannot be neglected anymore [157]. Sustainability studies require a shift from an economic short-term perspective (years to decades) focusing on fast moving variables, towards a long-term perspective (decades to centuries) accounting for the interaction between slow-and fast-changing variables. Interaction between slow-and fast-moving processes have been extensively studied in ecology. Slow-fast interactions can lead to sudden shifts in aquatic and terrestrial systems [158]. Studies on lakes [137], coral reefs [138], forests [159] and arid lands [160] have shown that smooth change can be interrupted by sudden drastic switches to a contrasting state [125]. Gradual and slow changes in temperature, nutrient loading, habitat fragmentation or biotic exploitation might have little effect until a threshold is reached at which a large shift occurs [126]. Such shifts can be attributed to alternative stable states [161], and result in large reductions in human well-being that are difficult to reverse [127]. Since alternative states are often undesirable from a sustainability perspective, the consequences of such regime shifts for the management of natural resources has been extensively studied in several systems, including shallow lakes [137] and coral reefs [138]. These studies have emphasized the importance of preserving resilience and building adaptive capacity in systems with slow-fast processes exhibiting potential regime shifts [139].

  Island [163]. Time delays between human and ecological dynamics may thus lead to unexpected overshoot-and-collapse cycles in the long run. Time delays can emerge from land conversion and fragmentation due to a decrease in the relaxation rate of population dynamics [164, 58], or resistant life-cycle stages [165]. As a result, species extinctions are postponed in time (Fig. D.a), which generates extinction debts [166, 167]. Relaxation rates decrease with habitat fragmentation [168], until the extinction threshold of the population is reached [169]

[

  171]. In Europe, such time delays may have led to underestimate population declines of plants and animals facing medium-to-high extinction risks, since these are more closely matched to indicators of socioeconomic pressures from the early or mid-, rather than the late, 20th century [172]. The negative impact of human activities on current biodiversity may not become fully realized until several decades into the future, which constitutes a challenge for biodiversity conservation as well as a window of conservation opportunities [173]. However, restoration measures also have to account for immigration lags, i.e., timedelayed recovery of biodiversity, that can be even larger than extinction debts [174, 175]. As a result of the relationship between biodiversity and ecosystem functioning, these extinction debts translate into time-delayed loss of biodiversity-dependent ecosystem services (Fig. D.b), such as carbon storage [176], nutrient cycling and biomass [59], and pollination service [177]. Given the importance of biodiversity-dependent ecosystem services for humanity and agricultural production, such functioning debts generate a time-delayed feedback loop between biodiversity loss and human societies. Since lag effects in key

Figure D :

 D Figure D: Extinction debt (a) and biodiversity-ecosystem functioning relationship (b). The grey areas capture the magnitude of the extinction and functioning debts, as the difference between current and long-term species richness (a) or ecosystem function (b), for a given level of habitat destruction (a) or species loss (b). Extracted from Isbell et al. (2015) [176].

  States [179]. Looking back at human history shows evidence of repetitions of catastrophic disruptions [180, 181, 182] followed by long periods of reinvention and development [183, 184]. These crises result from a conjunction of social and environmental events, that reinforce persistent mismatches between the responses of people and their social and ecological GENERAL INTRODUCTION conditions [185]. Understanding the history of how humans have interacted with the rest of nature can help clarify the options for managing our increasingly interconnected global system [186]. Some authors even call for the search for organizing principles in secular demographic cycles to transform history into an analytical, predictive science [187]. Study of the collapse of ancient human societies has become a growing field of research [188, 189],

  [201]. Successful communities often establish social norms, i.e., rules of shared behavior, that protect common natural resources [202] or help achieve group interests [203]. The enforcement of cooperation strongly hinges on the ecological characteristics of SESs. Previous experimental and theoretical studies have emphasized the role of resource productivity and mobility [204] as well as temporal variability [205, 206] on the robustness of cooperation. Time-delayed effects can also reinforce procrastination regarding behavioral shifts [207], and lowering of conservation standards, also known as the "shifting baseline syndrome [208, 209]. The evolution of social norms, by affecting feedbacks and drivers of SESs, may thus lead to large-scale behavioral shifts and help reinforce conventional governmental control [210].

  [155], and many of the problems encountered by societies in managing natural resources arise because of a mismatch between the scale of management and the scale(s) of the ecological processes being managed [133]. Such scale mismatches can contribute to a decrease in GENERAL INTRODUCTION social-ecological resilience and human well-being [155].

  in ecology [211]. Power-law SARs (S = cA z ) have received good support from ecological data and theory [212], and have been widely used to predict the change of species richness S with area A, where c and z are empirical constants [213]. One general pattern revealed by SAR studies is that, all else being equal, larger areas are expected to support more species. Underlying mechanisms include the null hypothesis of random placement and passive sampling [214], the area per se hypothesis, i.e., lower extinction probability with larger areas [215], and the habitat diversity hypothesis, i.e., larger areas having greater habitat diversity [211].

1 is> y min 1 , and decreases if y 1 < y min 1 . b 2

 1112 the minimum per capita agricultural goods consumption, such that human population size increases if y 1 is the sensitivity of µ to industrial goods consumption,

. 1 : 1 , b 2 ) 1 (

 1121 Definition and default values of the parameters and dynamical variables. α i and η ∈ [0; 1]. When possible, parameter values have been derived from the literature (Ω, z, , alpha i , η), or calibrated using historical population and land conversion trends (µ max , σ, κ, y min . Other parameter values are chosen so as to keep the system feasible, i.e. with positive population sizes. CHAPTER

( 1 ) 1 ) 1 .( 2 )Figure 1 . 2 :

 111212 Figure 1.2: Transient human population and biodiversity dynamics with varying relaxation coefficients ( ). Parameters are given in Table 3.1, with T m = 1. A: negligible relaxation time ( = 10); B: high relaxation time ( = 0.02); C: differential relaxation times for biodiversity loss ( -= 0.02) and recovery ( + = -/20); D: phase plane trajectories for biodiversity and human population. Grey curve: negligible relaxation time ( = 10); black curve: high relaxation time ( = 0.02) leading to the repetition of phases 1 (thick curve), 2 (dashed curve) and 3 (dotted curve) until the equilibrium (B * , H * ) is reached; double-arrow: maximum amplitude of the transient crisis.

1 (b 2

 12 Fig.1.3.D). Further increasing T m allows the industrial consumption y 2 to reduce the net human growth rate to the point where the system does not experience environmental crises (∆ > 0 for T m > 10). Note that b 2 only affects the transient dynamics of the system, and thus does not prevent high levels of biodiversity loss when technological efficiency rises.

(

  Fig.1.3.C). These SESs are characterized by a higher total human utility (Fig.1.3.I) and a lower sustainability (Fig.1.3.F) than SESs with concave-up BES relationships. A stronger demographic transition increases sustainability for both concave-down and -up BES relationships (dashed curve, Fig.1.3.F).

Figure 1 . 3 :

 13 Figure 1.3: Effect of varying parameter values on the steady states, the sustainability criterion and human well-being at equilibrium. Parameters are given in Table 1.1, with T m = 1.8 (B-E-H) and T m = 1.2 (C-F-I). A to C: Effect of T m , α 1 , and Ω on the human population size (black curve) and biodiversity (grey curve) at equilibrium, when demographic transition is weak (b 2 = 0.2). Grey areas represent the maximum amplitude of the transient crises as defined in Fig.2.D. D to F: Effect of T m , α 1 and Ω on the sustainability criterion (∆ = -θµ max ), when demographic transition is weak (solid curve, b 2 = 0.2) and strong (dashed curve, b 2 = 2). Parameter values for which ∆ is positive correspond to sustainable trajectories. G to I: Effect of T m , α 1 , and Ω on per capita human well-being (U * , black curve) and total human well-being (H * .U * , grey curve) at equilibrium, when demographic transition is weak (b 2 = 0.2). Grey areas represent the maximum amplitude of the transient variations in well-being.

Fig. 1 .

 1 Fig.1.4 shows that, in an unsustainable SES (i.e. ∆ < 0), the first scenario generates transient environmental crises (Fig.1.4.A), while the second scenario prevents environmental crises from occurring (Fig.1.4.B). A precautionary approach to natural habitat conservation appears necessary to prevent biodiversity and human population from exceeding their own sustainable thresholds, thus resulting in environmental crises in the long run.

Figure 1 . 4 :

 14 Figure1.4: Land-use scenarios. For both scenarios, parameters are given in Table1.1, with T m = 5. A: No restriction on land conversion; dynamics of an unsustainable SES (∆ < 0) with no land-use management. B: Implementation of the sustainable land conversion threshold; dynamics of the same SES (∆ < 0) when the sustainable threshold of natural habitat 1 -A S is set aside.

  Technological efficiency T represents the Total Factor Productivity (TFP) of the industrial and agricultural sectors. Following recent concerns regarding the potential for continued TFP growth in the future (Gordon, 2012; Shackelton, 2013), we choose to model technological change as a logistic function, such that technological efficiency grows at an exogenous rate σ, until it reaches a maximum technological efficiency (T m ).

Figure A1. 1 :

 1 Figure A1.1: Effect of endogenous (A) and exponential (B) technological change on long-term sustainability. A: effect of an endogenous technological change (σ = µ(B, T)) on the transient and long-term dynamics of the system, when the maximum technological efficiency T m varies. Grey areas represent the amplitude of the transient environmental crises. Parameters are given in Table 1, with Ω = 2 and b 2 = 0.5.; B: effect of an infinite technological change ( Ṫ = σ) on the transient and long-term dynamics of the system, when the labor intensity α 1 varies. Parameters are given in Table 1, with Ω = 2, b 2 = 0.1 and σ = 0.1.

1 )Figure A1. 2 :

 12 Figure A1.2: Effect of the land conversion cost (κ). A: Effect of varying κ on the viable equilibria (H * , B * ), with default parameters of Table 3.1 and T m = 1.8. The grey area represents the amplitude of the transient crises. B: Effect of κ on the sustainability criterion (∆). Values of κ for which ∆ is positive correspond to sustainable trajectories. C: Effect of κ on human well-being at equilibrium. The grey area represents the amplitude of the transient variations in well-being.

Figure 2 . 1 :

 21 Figure 2.1: Coupling between human, social and ecological dynamics, and definition of the sustainable and unsustainable consumption norms. (a) Modelsummary Black boxes: production sectors; grey boxes: dynamical variables; dashed lines: production inputs (labor, land and technology), with α(q) being the share of labor compared to land to produce one unit of agricultural good; solid lines: per capita consumptions of agricultural and industrial goods, y 1 (B, T, q) and y 2 (T); grey dotted lines: ecological feedback; double arrow: social sanctioning (e.g. ostracism); circle: total land divided into converted land A(H, q) and natural habitat, which supports a long-term species richness S(H, q). All functions are explicitly defined in the main text and in TableS2(electronic supplementary material). (b) Effect of labor elasticity on equilibrium biodiversity. Grey areas represent the amplitude of the transient environmental crises. (c) Effect of labor elasticity on sustainability. The sustainability criterion ∆ is derived in[37]. ∆ > 0 stands for sustainable transient trajectories, i.e. no environmental crises. The sustainability-optimal agricultural labor elasticity α s maximizes both ∆ and the biodiversity at equilibrium, B * . The unsustainable labor elasticity α u is chosen so that α u > α s and ∆(α u ) < 0.

  2.1.a) and sustainability (Fig.2.1.b).

  i.e. w(1) > U u (B * s ), where U u (B * s ) = (γ 1u /γ 1s ) η U * . Conversely, the unsustainable equilibrium is stable if the minimum ostracism w(0) is lower than the consumption utility at the unsustainable equilibrium, i.e. w(0) < U u (B * u ) where U u (B * u ) = U * (Fig.2.2.a). Therefore, for intermediate consumption utilities, w(0) < U u (B * u ) < U u (B * s ) < w(1), both the unsustainable and sustainable equilibria are stable ((U/S) region in Fig.2.2.b).

Figure 2 . 2 :

 22 Figure 2.2: Combined effect of the ostracism parameters and the difference between sustainable and unsustainable norms on the stability of the equilibria. (a) Shape of the ostracism function for varying maximum ostracism w max . (U) Weak sanctioning (w max = 0.4) and stability of the unsustainable equilibrium only, such that w(0) < w(1) < U u (B * u ) < U u (B * s ); (U/M) intermediate sanctioning (w max = 0.64) and bistability of the unsustainable and the mixed equilibria; (U/S) strong sanctioning (w max = 0.8) and bistability of the unsustainable and the sustainable equilibria, such that w(1) > U u (B * s ) and w(0) > U u (B * u ). See table S1 in the electronic supplementary material for other parameter values. (b) Stable equilibria with varying maximum ostracism w max and unsustainable labor elasticity α u . Regions (U), (U/M) and (U/S) correspond to the red, blue and black curves in (a), respectively.

Fig. 2 .

 2 Fig.2.5 shows the combined impact of lag effects and social ostracism on the basins of attraction of the sustainable, mixed and unsustainable equilibria. Increasing both the initial proportion of conformers and the strength of the ostracism can push an initially unsustainable SES into the basin of attraction of the sustainable or mixed equilibria (Fig.2.5.a). However, decreasing the ecological relaxation rate reduces the basin of attraction of the mixed equilibrium in favor of the unsustainable equilibrium (Fig.2.5.b).

Figure 2 . 4 :

 24 Figure 2.4: Effect of varying extinction debts and ostracism strengths w max on transient dynamics and stability, for two initial proportions of conformers q(0). Cases (U), (U/M) and (U/S) correspond to the ostracism functions defined in Fig. 2.2.a, with similar initial conditions, i.e. H(0) = 0.5, B(0) = (1 -H(0)/φ(q)) z and q(0) = 0.2 or q(0) = 0.6. (a) low extinction debt ( = 0.1); (b) intermediate extinction debt ( = 0.0025); (c) large extinction debt ( = 0.0005); green dot: (stable) sustainable equilibrium; red dot: (stable) unsustainable equilibrium; blue dot: (stable) mixed equilibrium; transient trajectories are represented by the blue curves. See table S1 in the electronic supplementary material for other parameter values.

Figure 2 . 5 :

 25 Figure 2.5: Proportion of conformers at equilibrium (q * ) under the combined effects of the initial proportion of conformers q(0) and maximum ostracism w max , for an increasing extinction debt. Other initial conditions: H(0) = 0.5, and B(0) = (1 -H(0)/φ(q(0))) z . See table S1 in the electronic supplementary material for other parameter values. (a) low extinction debt ( = 0.2); (b) high extinction debt ( = 0.0005). Blue color represents a population of defectors (q * = 0), red color represents a population of conformers (q * = 1) and intermediate colors represent a coexistence of conformers and defectors (q * ∈ [0, 1]).

Fig.A2. 1

 1 Fig.A2.1 shows the distribution of the sensitivity of the sustainability criterion ∆ to each of the parameters of the basic model [37]. The sensitivity of ∆ to parameters such as the minimum nutritional threshold y min 1

1 ) 1 , CHAPTER 2 areFigure A2. 1 :

 1121 Figure A2.1: Distribution of the sensitivity of the sustainability criterion ∆.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 (TableS2).

Figure 3 . 2 :

 32 Figure 3.2: Effect of a land tax τ on the equilibrium features of the model, for various labor elasticities α 1 and α 2 .

CHAPTER 3 Enriching

 3 both the economic and ecological sides of the model could add insights and greater realism to the analysis of the links between land-use management, sustainability and biodiversity conservation. Specifically directing policy on the basis of this work will require going beyond the conceptual model presented here. Including the feedback of biodiversity on human demography into complex existing modeling tools, such as the Integrated Valuation of Ecosystem Services and Tradeoffs (InVEST) model[72], may help assess and design integrative food-biodiversity policies, in a sustainability perspective. InVEST already models multiple ecosystem services, biodiversity conservation, commodity production, and tradeoffs at landscape scales, but poorly accounts for the relationships between biodiversity and services, and ignores human demography. Other global Integrated Assessment Modeling frameworks have been developed to specifically address climate change[73] or water availability[74] issues. For example, the Dynamic Integrated Climate-Economy (DICE) model has been extensively used to inform optimal

  effects of time delays due to extinction debts. It retains the basic structure of a predator-prey model, where human population growth indirectly depletes biodiversity through the destruction of natural habitats. The feedback loop closes through the effect of biodiversity-dependent ecosystem services loss on agricultural production. Human population consumption and land conversion are derived from an economic general equilibrium model, assuming an instantaneous equilibrium between demand and supply of consumption goods. The basic model thus allows to jointly simulate the temporal dynamics of biodiversity loss and human population growth, constrained by economic rationality and demographic processes such as the demographic transition.

  degradation and uncertainty over time-delayed ecological feedbacks can sap the efficiency of human behavioral change in enforcing sustainability. Increasing the time delay between human and ecological dynamics reduces the resilience of the sustainable equilibrium in favor of the resilience of the unsustainable equilibrium, thus favoring abrupt regime shifts towards the unsustainable basin of attraction. Foresighted social change is thus required to counteract large time-delayed feedbacks and prevent social-ecological traps. Not only can bottom-up social change, arising from the aggregation of individual behaviors, help enforce sustainability at large scales [106, 81, 119, 104, 79], top-down gov-

  1 , Michel Loreau1

	CHAPTER 1
	As published in Ecological Modelling
	Volume 351, 2017, pages 96-108

Table 1

 1

		Dynamical variables	Initial value
	H	Human population	10 -3
	T	Technology	10 -3
	B	Biodiversity	1
		Parameters	Default values
		Economic parameters
	η	Agents preference for agricultural goods	0.5
	α 1	Labor intensity in the agricultural sector	0.3
	α 2	Labor intensity in the industrial sector	0.9
		Technological parameters
	T m	Maximum technological efficiency	varies
	σ	Rate of technological change	0.5
	κ	Land operating cost	0.35
		Demographic parameters
	µ max	Maximum growth rate	1
	y min 1	Minimum per capita agricultural consumption	0.5
	b 2	Sensitivity to industrial goods' consumption	0.2
		Ecological parameters
	Ω	Concavity of the BES relationship	1
	z	Concavity of the SAR	0.3
		Relaxation rate	0.6
		Aggregate parameters
	φ

.1). The higher γ 1 , the higher the initial human growth rate and the lower the biodiversity at equilibrium B * . When condition CHAPTER 1

Table 1

 1

.1, with T m = 5. A: No restriction on land conversion; dynamics of an unsustainable SES (∆ < 0) with no land-use management. B: Implementation of the sustainable land conversion threshold; dynamics of the same SES (∆ < 0) when the sustainable threshold of natural habitat 1 -A S is set aside.

Table A2

 A2

.1: Functions and aggregate parameters. i = {u, s} 2.B

Parameters definition, units and defaults values

		Parameters	Default values Units
	η	Agents preference for agricultural goods	0.35	-
	α s	Sustainable agricultural labor elasticity	0.5	-
	α u	Unsustainable agricultural labor elasticity	0.15	-
	α 2	Labor elasticity in the industrial sector	0.9	-
	w max	Maximum ostracism	varies	H -1
	t	Threshold efficiency	-200	-
	r	Rate of social change	-30	-
	T m	Maximum technological efficiency	1.8	H -α
	σ	Rate of technological change	0.1	H -α .t -1
	κ	Land operating cost	1	H
	µ	Maximum growth rate	1	H.t -1
	y min 1	Minimum per capita agricultural consumption	0.3	H -1
	b 2	Sensitivity to industrial goods' consumption	3.5	-
	Ω	Concavity of the BES relationship	0.4	-
	z	Concavity of the SAR	0.2	-
		Ecological relaxation rate	0.1	t -1

Table A2.2: Definition and default values of the parameters. H: units of labor; t: units of time. CHAPTER 2

Centre for Biodiversity Theory and Modelling, Theoretical and Experimental Ecology Station, CNRS and Paul Sabatier University, Moulis, France

Acknowledgements

This work was supported by the TULIP Laboratory of Excellence (ANR-10-LABX-41) and the Midi-Pyrénées Region. We thank Matthieu Barbier, Kirsten Henderson and David Shanafelt for valuable discussions and helpful comments on earlier versions of the manuscript.

Acknowledgements

This work was supported by the TULIP Laboratory of Excellence (ANR-10-LABX-41) and the Midi-Pyrénées Region. We thank Matthieu Barbier, David Shanafelt and François Salanié for valuable discussions.

The PhD was directed by Michel Loreau, and funded by the Midi-Pyrénées region and the TULIP Laboratory of Excellence (ANR-10-LABX-41).

Abstract

The sustainability of coupled social-ecological systems (SESs) hinges on their long-term ecological dynamics. Land conversion generates extinction and functioning debts, i.e. a time-delayed loss of species and associated ecosystem services. Sustainability theory, however, has not so far considered the long-term consequences of these ecological debts on SESs. We investigate this question using a dynamical model that couples human demography, technological change and biodiversity. Human population growth drives land conversion, which in turn reduces biodiversity-dependent ecosystem services to agricultural production (ecological feedback). Technological change brings about a demographic transition leading to a population equilibrium. When the ecological feedback is delayed in time, some SESs experience population overshoots followed by large reductions in biodiversity, human population size and well-being, which we call environmental crises. Using a sustainability criterion that captures the vulnerability of an SES to such crises, we show that some of the characteristics common to modern SESs (e.g. high production efficiency and labor intensity, concave-down ecological relationships) are detrimental to their long-term sustainability. Maintaining sustainability thus requires strong counteracting forces, such as the demographic transition and land-use management. To this end, we provide integrative sustainability thresholds for land conversion, biodiversity loss and human population size -each threshold being related to the others through the economic, technological, demographic and ecological parameters of the SES. Numerical simulations show that remaining within these sustainable boundaries prevents environmental crises from occurring. By capturing the long-term ecological and socio-economic drivers of SESs, our theoretical approach proposes a new way to define integrative conservation objectives that ensure the long-term sustainability of our planet. Keywords: Biodiversity ; ecological economics ; ecosystem services; extinction debt; social-ecological system ; sustainability where θ is explicitly defined as a function of the economic, demographic, technological and ecological parameters of the SES in Table 1.1. Condition (13) implies that a given SES is not vulnerable to environmental crises if its ecological dynamics ( ) is fast enough compared to the growth of its human population (≈ µ max ), for a given set of parameters (θ).

Let us rewrite condition (13) as a sustainability criterion, ∆ = -θµ max . From the expression of θ in Table 1.1, it is straightforward to deduce the effect of some parameters on ∆. In particular, ∆ decreases with the preference for agricultural goods, η, since it increases the global demand for agricultural goods. ∆ increases with the land operating cost κ, since high operating costs reduce the incentive to convert natural habitat (see Appendix 1.D for more details). The strength of the demographic transition (b 2 ) also increases the sustainability criterion ∆, by reducing the net growth rate of the human population.

In the next sections, we explore the effect of the other parameters through numerical simulations. We use the default parameters of Table 1.1 such that conditions (11) and (12) are met, i.e. the SES can reach its desirable equilibrium, and human well-being at equilibrium is higher than the initial well-being. A given SES is sustainable (unsustainable) if its parameters satisfy ∆ > 0 (∆ < 0) and condition (12). Any increase (decrease) in ∆ has a positive (negative) effect on long-term sustainability. We then assess the effect of some characteristics of SESs on their steady states (H * , B * ) and sustainability (∆) by varying a single parameter at a time: the maximum technological efficiency (T m ), the agricultural labor intensity (α 1 ) and the concavity of the BES relationship (Ω). For each of these parameters, we also plot the effect of the strength of the demographic transition b 2 .

Effect of technological change

In

Chapter outline

Considering the lack of models capturing the dynamic feedback loop between human population growth and biodiversity loss, human behavior and social aspects were not considered in Chapter 1. However, human behavioral change underpins the adaptive capacity of social-ecological systems, and can strongly affect their long-term dynamics.

Chapter 2 aims at investigating the potential of social norms in enforcing sustainability, when behavioral change is based on perceived environmental degradation. This work benefits from recent studies on norm-driven cooperation in the use of natural resources.

Authors' contributions

A.-S.L., C. de M. and M.L. jointly designed the study, developed the model and analyzed and interpreted the model results. A.-S.L. drafted the manuscript; M.L. and C. de M. revised it critically. All authors gave final approval for publication.

Appendix CHAPTER 3

Sustainable land-use management under biodiversity lag effects

Anne-Sophie Lafuite

Abstract

The destruction of natural habitats for agricultural production results in local biodiversity loss. However, many biodiversity-dependent ecosystem services directly or indirectly impact agricultural production. Land conversion thus results in a negative externality, mediated through the erosion of biodiversity. When the consequences of this externality are delayed in time, lack of internalization results in overshoot-and-collapse dynamics, which are undesirable from a sustainability perspective. Here, we show that the internalization of this externality through a land tax results in several win-win effects in the long run. First, more biodiversity is preserved at equilibrium, which increases the carrying capacity and total well-being of the human population. Second, an optimal taxation path that maximizes the discounted sum of human utilities prevents or greatly alleviates such crises, thus increasing the sustainability of the system. Especially, this result holds in the case of imperfect information regarding the precise temporal dynamics of biodiversity loss, suggesting that the design of efficient land-use management policies is possible despite incomplete ecological data. This study thus highlights the necessity of internalizing biodiversity-dependent externalities through economic incentives, especially under uncertainty regarding long-term ecological dynamics.

Keywords: Biodiversity ; ecological economics ; ecosystem services; extinction debt; foresight; social-ecological system ; sustainability; land tax

Biodiversity (B)

A 2 y 2 ( T , τ )

S ( H , τ )

Technology (T) Industry Agriculture

Human population (H)

Natural habitat

Economic market

Economic market

Land conversion A(H,τ)

Regulatory agency

Dynamical system

The long-term behavior of the population is captured by a feedback loop between three dynamical variables: the human population H (eq.( 2)), biodiversity B (eq.( 3)), and technological efficiency T (eq.( 4)).

CHAPTER 3

Human demography can be related to the agricultural and industrial consumptions, y 1 (B, T) and y 2 (T) [32,33,34,35]. These consumptions are derived at market equilibrium (Appendix 3.A):

where γ 1 and γ 2 are functions of the parameters of the system (Table 2). A higher agricultural production allows the population to grow at a maximum rate µ while the average per capita consumption is higher than a minimum consumption, y min (1 -H/φ) z , where z is a constant parameter [36,37,38,39], and φ is the density of the human population on converted land at market equilibrium (Table 2).

Human population growth results in land conversion that reduces the number of species S(H) that natural habitat can support, thus leading to species extinction. These extinctions are delayed in time [19,40], hence the long-term species richness may be reached only after decades [41]. In order to account for this temporal dynamics, we build upon theoretical and experimental evidence regarding the relaxation rate of natural communities following habitat loss [42], which is proportional to the difference between current biodiversity B and long-term species richness S(H), and to a relaxation parameter .

Sustainability conditions

Model analysis allows deriving conditions for its stability and sustainability [28], i.e. nondeclining human well-being over time [13], where well-being

is a function of the agricultural and industrial consumptions, and the preference for agricultural goods η. Two necessary sustainability conditions have been identified that involve a sufficiently high level of substitution of natural capital for technology on the one hand, and the resistance to transient overshoot-and-collapse population crises on the other.

Indeed, rising technological efficiency compensates for the negative feedback of biodiversitydependent ecosystem services on agriculture if it is higher than the loss of services in terms CHAPTER 3 of human well-being, i.e.

However, this condition is not sufficient to ensure sustainability, since time delays in the feedback of biodiversity loss on human demography can result in unsustainable overshoot-and-collapse population cycles. This is the case when biodiversity loss is much slower than human population growth, i.e.

< σµ

where σ is a function of the parameters of the system (Table 2).

In the following, we aim at assessing the efficiency of a land tax in preserving the sustainability of this system under a time-delayed biodiversity feedback on human population growth. First, we show how the tax affects the consumption levels, conversion rate and long-term equilibria of the system.

A natural land depletion tax 2.1 Production

A tax τ per unit of converted area is added to the maintenance cost of κ units of labor per unit of land. At each period, the production profit in sector i = {1, 2} is

with L i the human labor, A i the exploited area in sector i, p i the price of the output, w the consumer wage, and α i the elasticity of labor. Profit maximization gives the production supply for each sector (Appendix 3.A), and the relationship between input factors in each sector (eq.( 7)) shows that a tax τ increases the optimal ratio of labor to land, thus CHAPTER 3

increases the human population density on converted land, which affects the converted surface A = H/φ τ , and the long-term number of species that the remaining natural habitat can support, S(H) = (1 -H/φ τ ) z .

Since φ τ increases with τ , the effect of the tax on consumptions (eq.( 9)) is not straightforward, and depends on the economic parameters of the system. Moreover, a land taxation policy will only help preserve more natural habitats and biodiversity compared to a business-as-usual case, if the human population density on converted land φ τ increases faster than the size of the human population, H. The next section explores the conditions under which this objective can be met, by studying the effects of the tax on the equilibrium features of the model.

Parameters Default values Units Economic parameters η

Agents' preference for agricultural goods 0.

The Hamiltonian function for this problem is

where λ H and λ T are adjoint variables. First order conditions are

and transversality conditions are

Solving for the first order condition ∂H ∂τ = 0 gives the optimal tax τ as a solution of the following equation:

Solving for the first order condition ∂H ∂H = δλ H -λ H gives the dynamics of the adjoint variable λ H as a function of the other dynamical variables of the system (B, H, T) and the CHAPTER 3 control τ :

We do not need the last condition, which gives the temporal dynamics λT , since technological efficiency varies exogenously, and thus does not depend on the other variables of the system, nor on the control τ .

We then simulate system (11) along with the dynamics of the adjoint variable λ H , by solving at each time step for τ using eq.( 18). Thus, though the social planner does not account for time-delayed biodiversity loss ( Ḃ), the numerical simulations do include the effect of extinction debts on agricultural production, human consumption and human demography. Simulations allow exploring the effects of the tax on the transient dynamics of the regulated system.

Numerical simulations

Parameters are chosen so as to meet the sustainability condition (6) in the businessas-usual case. This guarantees that the substitution of natural capital for technology is high enough to ensure a non-declining human well-being over time. Since the tax policy necessarily preserves more biodiversity at equilibrium, the condition remains true in the regulated case. Thus, we can focus on the consequences of ecological time delays for sustainability, by comparing their effects on the regulated and business-as-usual scenarios.

In the case of a negligible extinction debt (e.g. = 1), both the regulated and business-as-usual trajectories are sustainable, i.e. do not experience transient overshootand-collapse population crises. The optimal tax increases with the size of the human population, before reaching its equilibrium level, previously denoted as τ opt (Fig. 3.3.E). This regulatory policy increases the carrying capacity of the human population, i.e. the maximum population size that CHAPTER 3

the environment can support (Fig. 3.3.A). Moreover, the distance between the population equilibrium and the carrying capacity is larger in the regulated than in the business-asusual case, a feature which increases the resistance of the system to time delays, i.e. its sustainability [28].

Indeed, for a higher extinction debt (e.g. = 0.005), the regulated system appears much more resistant to transient population crises compared to the business-as-usual scenario (Fig. 3.3.B). The transient dynamics of the optimal taxation path also changes at high extinction debts, since the tax reaches higher levels during the initial growth phase of the human population (Fig. 3.3.F) in order to counteract the faster population growth, and prevent it to overshoot its carrying capacity. The efficiency of the optimal taxation policy also holds for higher extinction debts, thus making the system very resistant to timedelayed feedbacks, despite incomplete knowledge regarding the precise temporal dynamics of biodiversity loss. 1, except for µ = 0.11. Green curves: regulated system; black curves: business-as-usual; dotted curves: = 0.001; continuous curves: = 0.005.

Appendix

151 CHAPTER 3

3.A General Market Equilibrium Consumer Optimization

The number of agents is captured by the population variable, H. Overall, each agent supplies one unit of labor, and his revenue is the wage w. A tax τ per unit of converted land A is implemented by a social planner. Let U (y 1 , y 2 ) be the individual consumer utility, where y 1 and y 2 are per capita consumption rates of agricultural and industrial goods. Agents are assumed to maximize their utility

, where η is the preference for agricultural goods. After redistribution of the total revenue of the tax τ A among the agents, the revenue constraint writes p 1 y 1 + p 2 y 2 ≤ w + τ A/H, where p 1 and p 2 are the prices of agricultural and industrial goods respectively.

To solve this maximization problem, we define the Lagrangian:

First order conditions are:

Adding both conditions yields U (y 1 , y 2 )/Λ = p 1 y 1 + p 2 y 2 = w + τ A/H, and solving (19) for y 1 and y 2 , and substituting for U (y 1 , y 2 )/Λ yields the per capita demands:

Thus, the aggregate demand for agricultural and industrial goods writes:

Economic General Equilibrium

To close the system, let us choose the wage w as the numeraire and set it equal to 1.

Markets' clearing for the agricultural and industrial sectors p i Y D i = p i Y S i (eq. ( 21) and ( 24)) yields:

Dividing both lines of eq.( 26) gives:

Using eq. ( 25) to replace L 1 and L 2 in eq.( 27), and noting that A 1 + A 2 = A, we derive land allocations at market equilibrium, as functions of the area of converted land A:

Using eq.( 25), the labor allocations at the market equilibrium are:

Land Conversion

Adding both lines of (26) gives the labor market equilibrium L 1 +L 2 +κA = H. Replacing L 1 and L 2 by their optimal allocations, we deduce the relationship between the human population size H and the converted land A:

A = H/φ τ (30) action between a fast growing human population and a slowly regenerating resource [41].

The present results thus reinforce recent evidence about the need to develop incentivebased mechanisms for the simultaneous conservation of biodiversity and ecosystem services in order to foster sustainability [4,115,61,78,97,3], and highlight the efficiency of such mechanisms to counteract time-delayed ecological feedbacks, especially in the absence of precise knowledege about the temporal dynamics of these feedbacks.

In the following, some perspectives of this work in social-ecological systems modelling and sustainability research are identified and discussed.