Verónica Ferreira

Eric Chauvet

Cristina Canhoto

Effects of experimental warming, litter species, and presence of macroinvertebrates on litter decomposition and associated decomposers in a temperate mountain stream

Small woodland streams make the majority of water courses in most watersheds. Litter decomposition is a key ecosystem process in these shaded streams, and its response to warming can have profound consequences for food webs and the carbon (C) cycle. However, these responses can be modulated by litter identity and the structure of the detrital food web. Here we report on a manipulative study aiming at evaluating the effects of warming (+2.8 °C), litter identity (chestnut (Castanea sativa) or oak (Quercus robur) litter), and the structure of the detrital food web (presence or absence of macroinvertebrates) on litter decomposition and decomposers in a small, temperate woodland stream. Warming significantly stimulated overall (microbial-+ macroinvertebrate-driven) decomposition of oak and microbial-driven decomposition of chestnut. The similar shredder densities at elevated and ambient temperatures suggest that stimulated overall decomposition of oak resulted from increased activity of macroinvertebrate individuals. Stimulated microbial-driven decomposition of chestnut resulted from higher fungal activity with warming. Stimulation of litter decomposition by warming can lead to increases in the amount of C returned to the atmosphere and to a faster disappearance of litter from the benthos, with consequences to the C cycle and aquatic food webs.

Résumé : Les petits cours d'eau forestiers constituent la majeure partie des cours d'eau dans la plupart des bassins versants. La décomposition de la litière est un processus écosystémique clé dans ces cours d'eau ombragés, et sa réaction au réchauffement peut avoir de profondes répercussions sur les réseaux trophiques et le cycle du carbone (C). Ces réactions peuvent cependant être modulées par la nature de la litière et la structure du réseau trophique détritique. Nous rendons compte d'une étude reposant sur des manipulations visant a `évaluer les effets du réchauffement (+2,8 °C), de la nature de la litière (litière de châtaignier (Castanea sativa) ou de chêne (Quercus robur)) et de la structure du réseau trophique détritique (présence ou absence de macroinvertébrés) sur la décomposition de la litière et les décomposeurs dans un petit cours d'eau forestier de région tempérée. Le réchauffement stimule significativement la décomposition globale (microbienne + par les macroinvertébrés) de la litière de chêne et la décomposition microbienne de la litière de châtaignier. Les densités semblables de déchiqueteurs a `températures élevée et ambiante donnent a `penser que la décomposition globale stimulée de la litière de chêne est le résultat de l'activité individuelle accrue de macroinvertébrés. La décomposition microbienne stimulée de litière de châtaignier résulte d'une augmentation de l'activité des micromycètes associée au réchauffement. La stimulation de la décomposition de la litière par le réchauffement peut mener a `des augmentations de la quantité de C retourné a `l'atmosphère et a `la disparition plus rapide de litière du benthos, ce qui aurait des répercussions sur le cycle du C et les réseaux trophiques aquatiques. [Traduit par la

Introduction

Global mean air temperature is rising at an unprecedented rate, and models considering a doubling in atmospheric CO 2 concentration predict a further increase by 1.5-4.5 °C until the end of this century (IPCC 2013). This warming has the potential to affect species distribution, phenology [START_REF] Parmesan | A globally coherent fingerprint of climate change impacts across natural systems[END_REF], interactions [START_REF] Traill | Mechanisms driving change: altered species interactions and ecosystem function through global warming[END_REF], and biological activities [START_REF] Brown | Toward a metabolic theory of ecology[END_REF]), which can have important effects on biogeochemical cycles (e.g., the carbon (C) cycle). Fresh waters are not protected from this increase in temperature, since there is a direct relationship between water and air temperature [START_REF] Pilgrim | Stream temperature correlations with air temperatures in Minnesota: implications for climate warming[END_REF][START_REF] Morrill | Estimating stream temperature from air temperature: implications for future water quality[END_REF]. The majority of water courses in catchments in temperate regions are small, heterotrophic, woodland streams, where the decomposition of submerged litter is a key ecosystem process [START_REF] Vannote | The river continuum concept[END_REF][START_REF] Allan | Assessing effects of forecasted climate change on the diversity and distribution of European higher plants for 2050[END_REF]. Since recent evidence suggests that fresh waters can contribute large amounts of C to the atmosphere [START_REF] Cole | Plumbing the global carbon cycle: integrating inland waters into the terrestrial carbon budget[END_REF][START_REF] Battin | Biophysical controls on organic carbon fluxes in fluvial networks[END_REF], it becomes urgent to understand how future warming will affect aquatic communities and the decomposition of terrestrial derived organic matter.

The decomposition of submerged organic matter, mostly leaf litter, is carried out by an array of organisms, mostly fungal decomposers and invertebrate shredders [START_REF] Gessner | A perspective on leaf litter breakdown in streams[END_REF]. The response of these key players to warming will partially determine the dynamics of litter processing under future climate scenarios. Laboratory studies have shown that the activity of individual fungal and shredder species is stimulated by increases in temperature, until an optimum temperature is reached [START_REF] Chauvet | Temperature and sporulation of aquatic hyphomycetes[END_REF][START_REF] González | Conversion of leaf litter to secondary production by a shredding caddis-fly[END_REF][START_REF] Dang | Water Framework Directive of the European Parliament and the Council, of 23 October 2000, establishing a framework for Community action in the field of water policy[END_REF], which can translate into accelerated litter decomposition (Ferreira andChauvet 2011a, 2011b). These controlled and replicated experiments give useful insights into the mechanisms governing species activ-V. Ferreira and C. Canhoto. IMAR (Institute of Marine Research), MARE (Marine and Environmental Sciences Centre), Department of Life Sciences, Faculty of Sciences and Technology, University of Coimbra, 3004-517 Coimbra, Portugal. E. Chauvet. Université de Toulouse; UPS, INPT; EcoLab (Laboratoire Ecologie Fonctionnelle et Environnement), 31062 Toulouse, France; CNRS (Centre national de la recherche scientifique); EcoLab; 31062, Toulouse, France. Corresponding author: Verónica Ferreira (e-mail: veronica@ci.uc.pt). ities and ecosystem functioning; however, the generalization of these results to stream ecosystems is hindered by the small-scale and relative simplicity of the conditions simulated [START_REF] Woodward | Climate change and freshwater ecosystems: impacts across multiple levels of organization[END_REF].

Correlative studies over elevation [START_REF] Fabre | Leaf breakdown along an altitudinal stream gradient[END_REF][START_REF] Taylor | Relative influence of shredders and fungi on leaf litter decomposition along a river altitudinal gradient[END_REF], latitudinal [START_REF] Irons | Latitudinal patterns in leaf litter breakdown: is temperature really important?[END_REF][START_REF] Boyero | A global experiment suggests climate warming will not accelerate litter decomposition in streams but might reduce carbon sequestration[END_REF], and geothermal [START_REF] Friberg | Relationships between structure and function in streams contrasting in temperature[END_REF]) gradients have generally shown faster litter decomposition under higher temperatures. However, it is usually difficult in these studies to disentangle the effects of temperature from those associated with water chemistry and biogeography. Also, the temperature range addressed in these studies is often larger than the increase in temperature predicted by climate models (IPCC 2013). Therefore, the use of results observed over large natural thermal gradients to anticipate future effects of warming at a given location is not straightforward.

Our ability to predict the effects of increases in temperature on litter decomposition is complicated further by the possible interactions between warming and other factors (e.g., litter quality) on decomposers and detritivores [START_REF] González | Conversion of leaf litter to secondary production by a shredding caddis-fly[END_REF][START_REF] Fernandes | Higher temperature reduces the effects of litter quality on decomposition by aquatic fungi[END_REF][START_REF] Gonçalves | The effect of temperature on leaf decomposition and diversity of associated aquatic hyphomycetes depends on the substrate[END_REF]. In temperate woodland streams, the autumnal leaf litter input often originates from multiple species that differ in leaf characteristics [START_REF] Buzby | Modeling the potential effects of climate change on leaf pack processing in central Appalachian streams[END_REF]. Leaf litter identity (i.e., its physical and chemical characteristics) is known to affect microbial and invertebrate colonization and activity, with soft, high-quality (low C to nutrients ratio) litter being preferably colonized and faster decomposed than tough, low-quality litter [START_REF] Gessner | Importance of stream microfungi in controlling breakdown rates of leaf litter[END_REF][START_REF] Canhoto | Decomposition of Eucalyptus globulus leaves and three native leaf species (Alnus glutinosa, Castanea sativa and Quercus faginea) in a Portuguese low order stream[END_REF][START_REF] Ferreira | Effects of litter diversity on decomposition and biological colonization of submerged litter in temperate and tropical streams[END_REF]. In oligotrophic streams, warming might stimulate the decomposition of high-quality litter to a larger degree than that of low-quality litter, where microbial activity might be nutrient-limited [START_REF] Thormann | Microcosm tests of the effects of temperature and microbial species number on the decomposition of Carex aquatilis and Sphagnum fuscum litter from southern boreal peatlands[END_REF][START_REF] Berbeco | Soil warming accelerates decomposition of fine woody debris[END_REF]. On the other hand, the decomposition of tough litter might be stimulated by warming to a larger extent than that of soft litter, as it has been shown that the decomposition of lowquality litter is more temperature sensitive than that of highquality litter [START_REF] Conant | Sensitivity of organic matter decomposition to warming varies with its quality[END_REF][START_REF] Gonçalves | The effect of temperature on leaf decomposition and diversity of associated aquatic hyphomycetes depends on the substrate[END_REF].

The experimental manipulation of water temperature under natural field conditions would allow more realistic predictions of the effect of future climate warming on aquatic communities and on the functioning of headwaters, as it would allow to control for possible confounding factors of the temperature effect. This approach has obvious logistical constraints related with the continuous heating of flowing water, which explains why it has been undertaken only once, in early 1990s; at that time, however, only structural aspects of the macroinvertebrate community were assessed [START_REF] Hogg | The consequences of global warming for stream invertebrates: a field simulation[END_REF][START_REF] Hogg | Response of stream invertebrates to a globalwarming thermal regime: an ecosystem-level manipulation[END_REF].

In this study we used a stream manipulative experiment to assess the effect of an increase in water temperature by 2.8 °C on the decomposition of submerged oak (Quercus robur) and chestnut (Castanea sativa) leaf litter, in the presence and absence of invertebrates. We anticipated that (i) warming would stimulate litter decomposition because of the temperature sensitivity of biological activities, (ii) stimulation would be stronger for the tougher litter because of higher temperature sensitivity of decomposition of poor quality litter (initial nitrogen concentration was similar for both litter species), and (iii) stimulation would be stronger in the presence of invertebrates that will take profit from enhanced microbial colonization. This experiment, however, was not replicated at the stream reach scale because of budget and logistical constrains, and, contrary to [START_REF] Hogg | The consequences of global warming for stream invertebrates: a field simulation[END_REF] and [START_REF] Hogg | Response of stream invertebrates to a globalwarming thermal regime: an ecosystem-level manipulation[END_REF], it lacked a before-after control-impact (BACI) design. Thus, this study has low statistical power and it is prone to type II errors.

Materials and methods

Experimental design and study site

Here we studied the individual and interactive effects of changes in water temperature and leaf litter species on microbialdriven and overall (microbial-+ invertebrate-driven) litter decomposition. For this, leaves of oak and chestnut were incubated in fine and coarse mesh bags, over 2 months in winter (27 December 2011-25 February 2012), in both halves of a mountain stream reach that was divided longitudinally. One stream half was experimentally warmed by 2.8 °C above ambient water temperature (experimental half), while the other half served as control (control half). Warming was carried out from 28 October 2011 (i.e., 2 months before the study started) until the study ended. This experiment was not replicated at the stream reach level because of the elevated costs associated with continuously warming flowing water in the field and the difficulty in finding additional appropriate streams to manipulate (see below).

The study reach (ϳ1 m wide and <10 cm deep) was located in Ribeira do Candal (Lousã Mountains, central Portugal; 40°4=44==N, 8°12=10==W, 620 m above sea level), a second-order stream that drains an area of 0.8 km 2 covered by mixed deciduous forest dominated by chestnut and oak and where human activity is low. The bedrock is schist and the stream substrate is composed of cobbles, pebbles, and sand. This stream was selected for this study because it meets a set of criteria needed to carry out the experimental warming: (i) it is a near-pristine stream that (ii) is relatively close to a small mountain village and (iii) is easily accessible.

The stream reach was divided longitudinally in half over a length of 22 m using schist stones, which penetrated the sediment (<10 cm deep) and were cemented to the bedrock, preventing any transfer of water between stream halves. The water provided to the study reach was collected upstream and transported by gravity through pipes into two 260 L stainless steel tanks that delivered the water to both stream halves at 2.3 L•s -1 (Table 1) through two valves inserted in a dam constructed upstream the study reach. One of the tanks was equipped with heating resistors supplied by a constant 42 kW, which warmed the water above ambient temperature. The target increase in water temperature was 3 °C, based on predictions for air temperature in Portugal by the end of this century [START_REF] Miranda | 20th century Portuguese climate and climate scenarios[END_REF] and on the relationship between water and air temperature reported for similar streams [START_REF] Morrill | Estimating stream temperature from air temperature: implications for future water quality[END_REF]. However, owing to electrical shutoffs during thunderstorms, the mean difference in temperature between the stream halves over the study period was 2.8 °C (Table 1). A detailed description of the hydraulic and heating systems can be found in [START_REF] Canhoto | Warming up a stream reach: design of a hydraulic and heating system[END_REF].

Water variables

Water temperature was recorded hourly during the study period in both stream halves using submersed data loggers (Hobo Pendant UA-001-08, Onset Computer Corp., Massachusetts, USA). Hourly temperature records were averaged to produce daily means. Electrical conductivity (LF 330, WTW, Weilheim, Germany), pH (pH 3110, WTW, Weilheim, Germany), and dissolved oxygen (Oxi 3210, WTW, Weilheim, Germany) were recorded weekly in situ in both stream halves. At the same time, 300 mL of water were collected from each stream half, filtered through glass fiber filters (47 mm diameter, pore size 0.7 m; Millipore APFF04700, Millipore Corp., Massachusetts, USA), transported on ice to the laboratory, and frozen until analyzed for nutrient concentrations. Nitrate, nitrite, and ammonium concentrations were determined by ionic chromatography (Dionex DX-120, Sunnyvale, California, USA), and soluble reactive phosphorus concentration was determined by the ascorbic acid method (APHA 1995). Alkalinity was determined by titration of 100 mL of stream water with sulfuric acid 0.01 mol•L -1 to an endpoint of pH 4.2 (APHA 1995). Discharge was determined volumetrically at the output of the valves that delivered water to each stream half [START_REF] Gore | Discharge measurements and streamflow analysis[END_REF].

Leaf litter bags and decomposition

Leaves were collected just after abscission in Lousã Mountains (oak, autumn 2006) and Margaraça forest (chestnut, autumn 2007), central Portugal, air-dried at room temperature, and stored in the dark until used. Batches of air-dried leaves (2.25-2.54 g; approximately five leaves per bag) were moisten with distilled water to render them soft and less susceptible to breakage due to handling and enclosed in fine mesh (10 cm × 12 cm, 0.5 mm mesh) and in coarse mesh bags (10 cm × 12 cm, 10 mm mesh). Fine mesh bags exclude macroinvertebrates, and thus litter decomposition is mainly carried out by the microbial community, while coarse mesh bags allow macroinvertebrates to enter, and thus litter decomposition is driven by the combined activities of macroinvertebrates and microbes. Litter in coarse mesh bags is also susceptible to mass loss owing to physical fragmentation; however, this was likely negligible in our case, as current velocity was low (ϳ0.05 m s -1 ; [START_REF] Ferreira | Role of physical fragmentation and invertebrate activity in the breakdown rate of leaves[END_REF]. Nine litter bags of each litter species and mesh size were deployed in each stream half on 27 December 2011. On the same day, three extra litter bags for each litter species and mesh size were taken to the field, submerged for ϳ10 min, returned to the laboratory, oven-dried at 105 °C for 48 h, weighed (±0.1 mg), and grinded (<500 m powder size; ZM 100, Retsch, Haan, Germany). A subsample of litter powder was oven-dried at 105 °C for 48 h, weighed (±0.1 mg), ignited at 550 °C for 4 h, and reweighed (±0.1 mg) to allow calculation of the ash fraction and the initial ash-free dry mass (AFDM), taking into account mass loss due to handling.

Three replicate litter bags per date were sampled after 11, 20, and 60 days of incubation, placed individually in zip-lock bags, transported to the laboratory in a cooler, and promptly processed. In the laboratory, leaves from fine mesh bags were gently rinsed with distilled water, and two sets of five leaf discs were removed with a cork borer (12 mm diameter) for microbial determinations (see below). Leaves from coarse mesh bags were rinsed with tap water over a 500 m mesh sieve, and the macroinvertebrates retained were recovered (see below). The remaining litter was oven-dried, weighed, and grinded. A subsample of litter powder was oven-dried, weighed, ignited, and reweighed (as above) to allow calculation of the ash fraction and the AFDM remaining (after correction for the mass of discs removed from fine mesh bags). Results were expressed as AFDM remaining at each sampling date (%). Extra subsamples of litter powder from fine mesh bags were used for chemical determinations (see below).

Litter chemical composition and toughness

Oven-dried (105 °C, 48 h) subsamples of litter powder (<500 m size) from day 0 and from fine mesh bags retrieved from the stream were weighed (3.32-4.57 mg) and used for determination of phosphorus (P) concentration. Phosphorus was extracted from litter by incubation with sodium persulphate and sodium hydroxide in the autoclave at 120 °C for 2 h. The pH of the suspension was raised to ϳ4 units with sodium hydroxide, an aliquot was filtered through syringe filters (pore size 0.2 m; Minisart, Sartorius, Goettingen, Germany), and the P concentration in the solution was determined by the ascorbic acid method (APHA 1995). For the same samples, litter powder (0.50-0.72 mg) was used for determination of nitrogen (N) concentrations by an auto analyzer (IRMS Thermo Delta V Advantage with a Flash EA, 1112 series). Results were expressed as %P and %N of dry mass (DM).

Initial leaf toughness was determined for 10 leaves of each species using a penetrometer [START_REF] Graça | Methods to study litter decomposition. A practical guide[END_REF]. Leaves were soaked in tap water for 6 h, immobilized between two acrylic sheets, and the mass required to force a 0.49 mm 2 iron rod through the leaf (avoiding primary and secondary veins and keeping the upper leaf blade facing up) was quantified (three times per leaf). Results were expressed as grams.

Fungal biomass and conidial production

One set of five leaf discs (usually one disc per leaf) taken from fine mesh bags was used for determination of ergosterol concentration, as a proxy of fungal biomass [START_REF] Graça | Methods to study litter decomposition. A practical guide[END_REF]. Leaf discs were frozen at -20 °C until ergosterol extraction, freeze-dried overnight (LY3TTE, Snijders Scientific, Tilburg, Netherlands), weighed (±0.1 mg) to determine disc DM, and placed in tightly closed tubes with alkaline methanol (8 g KOH•L -1 ). Lipids were extracted in a water bath at 80 °C for 30 min. The extract was then purified by solid-phase extraction (Oasis HLB cartridge, barrel size 3 cm 3 , particle size 30 m, pore size 80 Å; Waters Corp., Massachusetts, USA) [START_REF] Graça | Methods to study litter decomposition. A practical guide[END_REF]. Ergosterol was quantified by high-performance liquid chromatography (HPLC) by measuring absorbance at 282 nm. The HPLC system (Dionex DX-120, Sunnyvale, California, USA) was equipped with the LiChroCART 250-4 LiChrospher 100 RP-18 (5 m) column (Merck, Darmstadt, Germany) and maintained at 33 °C. The mobile phase was 100% methanol, flowing at 1.4 mL•min -1 . Ergosterol was converted into mycelial biomass assuming 5.5 g ergosterol•(mg mycelial DM) -1 (Gessner and Chauvet 1993). Discs' DM were converted into AFDM using the ash fraction derived from the discs used for sporulation (see below), and results were expressed as milligrams mycelial biomass per gram leaf litter AFDM.

The second set of five leaf discs (usually one disc per leaf) taken from fine mesh bags was used to induce sporulation by aquatic hyphomycetes. Leaf discs were incubated in 100 mL Erlenmeyer flasks with 25 mL of filtered stream water from the corresponding stream half (47 mm diameter, pore size 0.7 m; Millipore APFF04700, Millipore Corp., Massachusetts, USA). Incubations took place over 48 h in an orbital shaker (100 r•min -1 ) placed in a room at 15 °C and with a 12 h light : 12 h dark photoperiod. Logistic constraints prevented sporulation from being induced at the temperature observed in each stream half on the sampling date, which would allow assessing the direct effect of temperature on conidial production. However, inducing fungal sporulation from litter retrieved from both stream halves at the same temperature permitted the assessment of differences in conidial production that are due to differences in the number of conidiophores on the surface of decomposing litter induced by differences in temperature during the incubation in the field. The conidial suspensions were poured into 50 mL Falcon tubes, fixed with 2 mL of 37% formalin, and the final volume adjusted to 35 mL with distilled water. The remaining leaf discs were oven-dried, weighed, ignited, and reweighed as above to calculate the discs' ash fraction and AFDM. When preparing filters for conidia counting and identification, 150 L Triton X-100 (0.5%) were added to the suspension and mixed with a magnetic stirring bar to ensure a uniform distribution of conidia, and 1-5 mL of the suspension were filtered (2.5 mm diameter, pore size 5 m; Millipore SMWP, Millipore Corp.). Filters were stained with 0.05% trypan blue in 60% lactic acid, and spores were identified and counted under a compound microscope at 320× magnification [START_REF] Graça | Methods to study litter decomposition. A practical guide[END_REF]. Sporulation rates were expressed as number of conidia per milligram leaf litter AFDM per day, and species richness was expressed as number of taxa per sample.

Macroinvertebrates

Macroinvertebrates (>500 m) retrieved from coarse mesh bags were preserved in 96% ethanol. Individuals were counted, and the shredders were identified to the lowest possible taxonomic level (generally genus or species) following [START_REF] Tachet | Invertébrés d'eau douce. Systématique, biologie, écologie[END_REF]. Abundance was expressed as total number of invertebrates per gram leaf litter AFDM and number of shredders per gram leaf litter AFDM. Shredder richness was expressed as number of taxa per sample.

Data analysis

Water variables were compared between stream halves by twotail paired t tests. Exponential decomposition rates (k, day -1 ) were calculated as the slope of linear regression of the ln-transformed fraction of AFDM remaining over time (days), with the intercept fixed at 0. Proportion of litter mass remaining (ln-transformed) was compared between stream halves (control versus experimental), litter species (oak versus chestnut), and mesh sizes (coarse versus fine) by repeated measures ANOVA (RM ANOVA), with sampling date as the within-subjects factor and stream half, litter species, and mesh size as the between-subjects factors. Post hoc multiple comparisons were made with Fisher's least significant difference test (Statistica 7 software; StatSoft Inc., Tulsa, Oklahoma, USA).

The sensitivities to temperature of decomposition rates were calculated as Q 10-q values: Q 10Ϫq ϭ ͑t l /t h ͒ ͫ10/͑ThϪTl͒ͬ , where t l and t h are the time (days) to decompose 50% AFDM at the lower and higher temperature, respectively, T l is the lower temperature (mean temperature over the incubation period in the control half = 6.4 °C), and T h is the higher temperature (mean temperature over the incubation period in the experimental half = 9.2 °C) [START_REF] Conant | Sensitivity of organic matter decomposition to warming varies with its quality[END_REF]. The time to decompose 50% initial AFDM under 10 °C warming was calculated as follows: time (days) to decompose 50% initial AFDM under ambient temperature/Q 10-q .

Litter initial P and N concentrations and leaf toughness were compared between species by two-tail t tests. Dynamics of nutrients concentration, fungal biomass concentration (log(x)-transformed), sporulation rate by aquatic hyphomycetes, fungal species richness, total macroinvertebrate abundance (log(x + 1)-transformed), shredder abundance (log(x + 1)-transformed), and shredder taxa richness (log(x + 1)-transformed) were compared between stream halves (control versus experimental) and litter species (oak versus chestnut) by RM ANOVA, with sampling date as the withinsubjects factor and stream half and litter species as the betweensubjects factors. Data transformation was carried out when necessary to achieve normality (checked by Shapiro-Wilk's test) and homoscedasticity (checked by the Bartlett 2 test). Fisher's least significant difference test was used for multiple comparisons when the RM ANOVAs detected an effect of stream half, litter species, or their interaction in the above-mentioned variables (Statistica 7 software).

Overall differences in aquatic hyphomycete and shredder communities among treatments were assessed by analyses of similarity (ANOSIM) based on Bray-Curtis similarity matrix of abundance (log(x + 1)-transformed) of conidia and shredder indi-viduals, respectively (Primer 6 software, Primer-E Ltd, Plymouth, UK).

Results

Water variables

The experimental warming successfully increased the water temperature of the experimental half by an average of 2.8 °C above the temperature observed in the control half during the study period (paired t test, P < 0.001; Table 1). This resulted in daily thermal sums of 388 and 560 °C by day 60 for the control and experimental half, respectively.

During the study period, stream water had low conductivity and alkalinity and was well oxygenated, circumneutral, and nutrient-poor (Table 1). Significant differences between stream halves were found only for conductivity (26.3 S•cm -1 in the control half versus 26.4 S•cm -1 in the experimental half; paired t test, P = 0.018) and for dissolved oxygen concentration (12.2 mg•L -1 in the control half versus 11.4 mg•L -1 in the experimental half; P < 0.001), as expected, owing to differences in temperature (Table 1).

Leaf litter decomposition

Litter mass loss over the 60-day incubation period varied between 24% and 47% of initial AFDM across treatments, and the dynamics of litter decomposition fitted the negative exponential model well (0.73 < R 2 < 0.99, P < 0.001 for all treatments). This translated into decomposition rates between 0.0048 and 0.0126 day -1 across treatments (Fig. 1). Chestnut leaf litter decomposed 37%-78% faster than oak leaf litter, which resulted in significant differences in decomposition rates between litter species (RM ANOVA, P < 0.001 for litter species; Table 2) in both stream halves and mesh sizes (Fisher's test, P < 0.030; Fig. 1). Litter enclosed in coarse mesh bags decomposed 29%-59% faster than litter enclosed in fine mesh bags (RM ANOVA, P < 0.001 for mesh size; Table 2), which nevertheless suggests the small contribution of shredders to the overall litter decomposition (23%-37% versus 63%-77% of microbes). This resulted in significant differences in decomposition rates between mesh sizes for oak incubated in the experimental stream half (Fisher's test, P = 0.024) and for chestnut incubated in the control (P = 0.003) and in the experimental halves (P = 0.017); the decomposition rate of oak litter in the control stream half did not significantly differ between mesh sizes (P = 0.120; Fig. 1). Litter incubated in the experimental half decomposed 15%-49% faster than litter incubated in the control half (RM ANOVA, P < 0.001 for stream half; Table 2), but significant differences between stream halves were found only for oak in coarse mesh bags (Fisher's test, P = 0.01) and for chestnut in fine mesh bags (P = 0.024; Fig. 1).

The temperature sensitivity of litter decomposition was higher for oak than for chestnut in both mesh sizes. However, the temperature sensitivity of oak decomposition was higher in coarse than in fine mesh bags, while the opposite pattern was observed for chestnut (Table 3). When considering a 10 °C increase in temperature, however, both litter species decompose at similar rates (Table 3).

Litter chemical composition and toughness

Leaf litter species differed in initial chemical quality and toughness. Initial P concentration was 3.6-fold higher in oak than in chestnut (t test, P < 0.001), although both concentrations were very high (0.45% ± 0.02% DM (mean ± SE) and 0.12% ± 0.01% DM, respectively). Initial N concentration did not significantly differ between litter species (0.98% ± 0.06% for oak and 0.96% ± 0.04% for chestnut; t test, P = 0.825). Initial leaf toughness was significantly higher for oak (95.46 ± 6.60 g) than for chestnut (64.21 ± 4.48 g; t test, P = 0.001).

Once in water, P leached rapidly from the litter, and after 11 days both litter species in both stream halves had 0.05% P, which increased similarly across treatments (50%-77%) over the incubation period (RM ANOVA, P < 0.001 for time; Table 4; Fig. 2a). P concentration did not significantly differ between litter species (RM ANOVA, P = 0.636 for litter species) or temperature regimes (P = 0.237 for stream half; Table 4). N concentration increased across treatments (42%-63%) over the incubation period (RM ANOVA, P < 0.001 for time; Table 4). The N concentration associated with chestnut in the experimental stream half was the highest across treatments (Fisher's test, P < 0.043; Fig. 2b).

Fungal biomass and conidial production

Maximum fungal biomass concentration on decomposing litter was recorded by the last sampling date (130-251 mg•(g AFDM) -1 ) (RM ANOVA, P = 0.001 for time; Table 5; Fig. 3a). Fungal biomass accrual was higher for chestnut than for oak in both stream halves (RM ANOVA, P < 0.001 for litter species), while there was no significant difference between stream halves for any litter species (RM ANOVA, P = 0.516 for stream half; Table 5; Fig. 3a). Sporulation by aquatic hyphomycetes increased until day 20, when the maximum sporulation rate varied between 932 and 3642 conidia•(mg AFDM) -1 •day -1 across treatments, decreasing afterwards (RM ANOVA, P < 0.001 for time; Table 5; Fig. 3b). The sporulation rate associated with chestnut incubated in the experimental half was the highest across treatments (Fisher's test, P < 0.019; Fig. 3b). Aquatic hyphomycetes species richness increased over time (RM ANOVA, P = 0.002 for time; Table 5; Fig. 3c), but no significant difference was found among treatments (RM ANOVA, P = 0.433 for litter species and 0.909 for stream half; Table 5).

The aquatic hyphomycete community structure varied greatly over time (ANOSIM, global R = 0.724, P = 0.001; Fig. 4). Early communities were dominated by Tetrachaetum elegans (57%-91%). Flagellospora curvula also had important contribution (3%-37%) until day 20, after which its contribution decreased to <1%. The relative abundances of Anguillospora filiformis, Articulospora tetracladia, and Tricladium chaetocladium increased from <4% at day 11 to 8%-36% at day 60, when they co-dominated with T. elegans (22%-43%; Fig. 4). Litter species also contributed to shape fungal communities (ANOSIM, global R = 0.147, P = 0.013), which can be explained by the presence of Heliscus lugdunensis and Tricladium splendens only in oak and of Alatospora acuminata and Dendrospora erecta only in chestnut. Clavatospora longibrachiata was also more represented in oak than in chestnut. Water temperature did not seem to have any significant role in shaping fungal communities (ANOSIM, global R = 0.038, P = 0.271).

Macroinvertebrates

Maximum abundance of total macroinvertebrates was found at the last sampling date (22-46 individuals•(g AFDM) -1 ), except for oak incubated in the experimental half where abundance peaked at day 20 (16 individuals•(g AFDM) -1 ) (RM ANOVA, P = 0.026 for time; Table 6; Fig. 5a). There was, however, no significant effect of litter species or water temperature on total macroinvertebrate abundance (RM ANOVA, P = 0.142 for litter species and 0.811 for stream half; Table 6).

The percentage of shredder individuals colonizing decomposing litter varied between 0% and 25% across treatments and dates. Shredder abundance was low across treatments (<7 individuals•(g AFDM) -1 ), it did not vary much over time (RM ANOVA, P = 0.441 for time), but it was significantly higher in chestnut incubated in the control half than in the other three treatments (Fisher's test, P < 0.009; Table 6; Fig. 5b). Shredder taxa richness was low (<3 taxa•sample -1 ; total of nine species) and did not vary much over time because of high variability (RM ANOVA, P = 0.530 for time). Shredder richness was significantly higher in chestnut incubated in the control half than in the other three treatments (Fisher's test, P < 0.038; Fig. 5c). Shredder communities were distinct between both litter species (ANOSIM, global R = 0.236, P = 0.026); individuals were more evenly distributed by taxa in chestnut than in oak. Nemoura sp. and the two species of Scirtidae beetles were only present in oak, while Allogamus laureatus and Lepidostoma hirtum only appeared in chestnut (Fig. 6). Neither water temperature (ANOSIM, P = 0.065) nor time (P = 0.296) seemed to play a major role in shaping shredder community structure.

Discussion

Manipulation of warming at the stream level

In this study, we experimentally increased the water temperature by 2.8 °C in one half of a small mountain stream that was divided longitudinally to assess the effects of global warming on the decomposition of two litter species in the presence and absence of macroinvertebrates. This study makes several advances in the assessment of the effects of global warming in stream ecosystems. First, by experimentally manipulating water temperature in a small mountain stream, it addresses the effects of warming at the ecosystem level under realistic environmental conditions while controlling for factors associated with water chemistry and biogeography. The manipulation of temperature of surface flowing waters has been undertaken only once, and it allowed insight into the effects of global warming on aquatic macroinvertebrates' population dynamics [START_REF] Hogg | The consequences of global warming for stream invertebrates: a field simulation[END_REF][START_REF] Hogg | Response of stream invertebrates to a globalwarming thermal regime: an ecosystem-level manipulation[END_REF]. The continuous heating of flowing water presents several challenges, primarily related with the huge amount of energy required (constant supply of 42 kW in this study), which is often not available in remote areas where nearpristine streams can be found, and with the large costs associated with the maintenance of the system. This explains why such an approach has not been attempted more often and the lack of replication when it has been performed (unfortunately, also in this study). One strategy to improve the predictive power of unreplicated study designs is to use a BACI design [START_REF] Smith | BACI design[END_REF]). However, this was not possible in our case, since we were spacelimited (several experiments took place simultaneously in the experimental small stream reach). These design limitations result in low statistical power and thus restricted our ability to detect an effect of warming. Thus, the effects of warming in this study can be seen as highly conservative.

Second, this study evaluates the effects of warming on litter decomposition, together with changes in litter species identity and in shredder presence. Since leaf litter from distinct species differ in physical and chemical characteristics [START_REF] Ostrofsky | Relationship between chemical characteristics of autumnshed leaves and aquatic processing rates[END_REF], [START_REF] Canhoto | Decomposition of Eucalyptus globulus leaves and three native leaf species (Alnus glutinosa, Castanea sativa and Quercus faginea) in a Portuguese low order stream[END_REF][START_REF] Gulis | Are there any substrate preferences in aquatic hyphomycetes?[END_REF][START_REF] Gonçalves | The effect of temperature on leaf decomposition and diversity of associated aquatic hyphomycetes depends on the substrate[END_REF], the decomposition of different litter species might be distinctly susceptible to warming [START_REF] Rouifed | The importance of biotic factors in predicting global change effects on decomposition of temperate forest leaf litter[END_REF][START_REF] Gonçalves | The effect of temperature on leaf decomposition and diversity of associated aquatic hyphomycetes depends on the substrate[END_REF]. The evaluation of the interaction between increases in water temperature and changes in litter species on litter decomposition is increasingly relevant if we take into account the predictions from models simulating future species distribution that suggest a change in the range and area occupied by tree species (Bakkenes et al. 2002), including the common European woodland species Q. robur and C. sativa used in this study [START_REF] Benito Garzón | Effects of climate change on the distribution of Iberian tree species[END_REF]. Also, most shredder species belong to the orders Plecoptera and Trichoptera, which evolve in cold environments, have low upper thermal tolerance limits, and can be therefore negatively affected by increases in temperature [START_REF] Gaufin | Laboratory studies on tolerance of aquatic insects to heated waters[END_REF][START_REF] Pearson | Gradients in regional diversity of freshwater taxa[END_REF]. Additionally, the sensitivity of shredders' activities to temperature might depend on litter species [START_REF] González | Conversion of leaf litter to secondary production by a shredding caddis-fly[END_REF]. The simultaneous evaluation of the effects of warming, changes in litter species, and presence of shredders on litter decomposition increased our ability to foresee future effects of climate change on the functioning of small headwater streams. We found that the increase in water temperature generally stimulated litter decomposition; however, the extent and significance of this stimulation depended on the litter species and presence of macroinvertebrates.

Effect of litter species on litter decomposition

Leaf litter decomposition was faster for chestnut than for oak, which agrees with previous studies [START_REF] Ferreira | Effects of litter diversity on decomposition and biological colonization of submerged litter in temperate and tropical streams[END_REF]). In our case, the difference in decomposition rate between litter species can be attributed to chestnut leaves being softer than oak leaves; leaf toughness has been shown to be a primary determinant of biological colonization and decomposition of leaf litter [START_REF] Li | Effects of leaf toughness and nitrogen content on litter breakdown and macroinvertebrates in a tropical streams[END_REF][START_REF] Graça | Leaf quality of some tropical and temperate tree species as food resource for stream shredders[END_REF]. The initially higher P concentration in oak than in chestnut probably did not have any effect on litter decomposition, since it leached rapidly, and after day 11 (and prob- ably before), a similar P concentration in both litter species was determined. The difference between litter species was the highest for overall litter decomposition in the control half (78% versus 37%-40% in the other three situations), which can be attributed to the higher shredder abundance and species richness associated with chestnut than with oak under ambient temperature. Overall, the magnitude of differences in decomposition between litter species depended on the presence of macroinvertebrates and on water temperature.

Effect of macroinvertebrate presence on litter decomposition

Litter decomposition was generally faster in the presence of macroinvertebrates (coarse mesh bags) than in their absence (fine mesh bags), which was not surprising [START_REF] Cuffney | Experimental evidence quantifying the role of benthic invertebrates in organic matter dynamics of headwater streams[END_REF][START_REF] Stewart | The effect of invertebrates on leaf decomposition rates in two small woodland streams in southern Africa[END_REF]. However, macroinvertebrates were relatively weak contributors to overall litter decomposition; decomposition attributed to invertebrate activity was only 23%-37% of litter decomposition, being highest for chestnut incubated at ambient temperature because of higher colonization by shredders. These results are in agreement with a recent study that also reported a relatively low contribution of invertebrates (9%-29%), compared with microbes, to litter decomposition [START_REF] Taylor | Relative influence of shredders and fungi on leaf litter decomposition along a river altitudinal gradient[END_REF]. In our case, the number of shredders colonizing decomposing litter was globally low, which can explain why litter decomposition was only 29%-59% faster in coarse than in fine mesh bags. These results contrast with those from previous studies in the same stream, where chestnut litter decomposed 218% faster [START_REF] Ferreira | Effects of litter diversity on decomposition and biological colonization of submerged litter in temperate and tropical streams[END_REF]) and oak litter decomposed 166% faster in coarse than in fine mesh bags [START_REF] Gulis | Stimulation of leaf litter decomposition and associated fungi and invertebrates by moderate eutrophication: implications for stream assessment[END_REF]. In both studies, however, litter was incubated further upstream, in the main channel where colonization by invertebrates might have been more effective and mass loss from coarse mesh bags might have been also promoted by physical abrasion, especially during high flow periods; in our study site, flow was continuously kept at 2.3 L•s -1 and physical fragmentation of leaf litter was probably negligible [START_REF] Ferreira | Role of physical fragmentation and invertebrate activity in the breakdown rate of leaves[END_REF].

Effect of warming on litter decomposition

The temperature effect on litter decomposition was similar in both litter species in fine mesh bags (34% in chestnut and 38% in oak), suggesting that stimulation of microbial-driven litter mass loss was independent of litter species, despite sporulation by aquatic hyphomycetes being stimulated in chestnut, but not in oak, under warmer conditions. Warming significantly stimulated decomposition of oak in coarse mesh bags (49%), indicating that invertebrate activity in oak litter was the most responsive to warming, in spite of no significant difference in the number of shredders associated with these leaves between stream halves. Warming also significantly stimulated decomposition of chestnut in fine mesh bags (34%), which can be attributed to the increased fungal activity under elevated temperature. Surprisingly, the decomposition of chestnut litter in coarse mesh bags was the least sensitive to warming (only 15% faster than in the control half), indicating that invertebrate activity in chestnut was inhibited by the increase in temperature, probably as a reflex of the decrease in the number of shredders in chestnut under warmer conditions. These results must, however, be taken conservatively owing to low statistical power. Although temperature sensitivity of litter decomposition was higher for the tough oak, when considering a 10 °C increase in temperature, both litter species decompose at similar rates, because oak decomposes more slowly than chestnut at ambient temperature.

Relative importance of litter species, invertebrate presence, and warming on litter decomposition

According to our results, the three factors under study can be ranked regarding their relative importance on litter decomposition as follows: litter species > invertebrate presence > tempera-ture. This hierarchical arrangement is partially in agreement with that proposed by [START_REF] Aerts | The freezer defrosting: global warming and litter decomposition rates in cold biomes[END_REF] for litter decomposition in terrestrial systems: climate (temperature × moisture) > litter quality > soil invertebrates. The difference in the relative importance of temperature between studies is explained by differences in study scale; our study was performed in a single stream reach, while [START_REF] Aerts | The freezer defrosting: global warming and litter decomposition rates in cold biomes[END_REF] reports results from 34 site-species combinations. The hierarchy of factors found in our study suggests that the effects of an increase in temperature by up to 3 °C on litter decomposition might be smaller than those promoted by changes in riparian vegetation composition or in the activity and (or) abundance of invertebrates [START_REF] Rouifed | The importance of biotic factors in predicting global change effects on decomposition of temperate forest leaf litter[END_REF]). However, changes in these factors might enhance the effects of temperature increase on litter decomposition. The composition of the riparian vegetation and abundance or activity of invertebrates can change as a direct effect of climate change [START_REF] Hogg | Response of stream invertebrates to a globalwarming thermal regime: an ecosystem-level manipulation[END_REF]Bakkenes et al. 2002;[START_REF] Friberg | Relationships between structure and function in streams contrasting in temperature[END_REF] or as a result from anthropogenic activities, which might be intensified under climate change scenarios (e.g., intensification of agricultural practices). The relative importance of temperature on litter decomposition might be expected to rise if increases in temperature go beyond 3 °C or if the availability of dissolved nutrients is high [START_REF] Marcarelli | Temperature and nutrient supply interact to control nitrogen fixation in oligotrophic streams: an experimental examination[END_REF]Ferreira and Chauvet 2011a).

Effect of warming on biological colonization

Warming did not affect fungal biomass accumulation in either litter species. This might be attributed to low magnitude of the experimental warming (2.8 °C; Ferreira andChauvet 2011a, 2011b) to nutrient limitation for fungal growth due to the stream oligotrophic status (Ferreira and Chauvet 2011a) or to low statistical power.

Warming distinctly affected sporulation by aquatic hyphomycetes between litter species, with stimulation in chestnut but not in oak. Additionally, warming led to differences between litter species that did not exist in the control half. The stimulation of fungal sporulation rate by 2.8 °C warming was expected, since the ambient water temperature was relatively low (6.4 °C) (Ferreira andChauvet 2011a, 2011b). In our study, conidial production in the tough oak litter was not sensitive to warming, which together with its stimulation in the soft chestnut litter here and alder (Alnus glutinosa) litter in previous studies (Ferreira andChauvet 2011a, 2011b) might suggest that litter toughness is an important determinant of the response of fungal reproductive activity to temperature increase.

Despite the stimulation of fungal conidial production in chestnut incubated in the experimental half, no effect of temperature was found on the structure of aquatic hyphomycete communities for any litter species, suggesting that warming stimulated most fungal species to a similar degree. This contrasts with previous studies that reported marked changes in community structure with warming [START_REF] Bärlocher | Raised water temperature lowers diversity of hyporheic aquatic hyphomycetes[END_REF][START_REF] Dang | Water Framework Directive of the European Parliament and the Council, of 23 October 2000, establishing a framework for Community action in the field of water policy[END_REF]Ferreira andChauvet 2011a, 2011b;[START_REF] Gonçalves | The effect of temperature on leaf decomposition and diversity of associated aquatic hyphomycetes depends on the substrate[END_REF]. The difference in the response of community structure to warming between our study and previous studies can probably be explained by the smaller temperature increase tested here (2.8 versus 4.3-10 °C in previous studies). Larger temperature increases would likely favor warmwater species as Lunulospora curvula (observed in the study stream during spring) and inhibit cold-water species as F. curvula. Incubation time was the most important factor ruling fungal community structure, which is not surprising giving that fungal species colonize decomposing substrata following an ecological succession, which parallels the physical and chemical changes occurring in the submerged litter over the decomposition process (Gessner et al. 1993). Litter species also contributed to shape fungal communities, mostly with respect to the presence and abundance of rare species, which can be attributed to substrate preferences [START_REF] Thomas | Aquatic hyphomycetes from different substrates: substrate preference and seasonal occurrence[END_REF][START_REF] Gulis | Are there any substrate preferences in aquatic hyphomycetes?[END_REF]; but see [START_REF] Nikolcheva | Seasonal and substrate preferences of fungi colonizing leaves in streams: traditional versus molecular evidence[END_REF].

Contrary to the effect of warming on fungal communities, the increase in temperature mitigated the differences in shredders colonization that existed between litter species in the control half through a decrease in the abundance and richness of shredders associated with chestnut litter in the experimental half. Previous studies have reported reduced abundance of benthic invertebrates under warmer conditions [START_REF] Hogg | The consequences of global warming for stream invertebrates: a field simulation[END_REF][START_REF] Hogg | Response of stream invertebrates to a globalwarming thermal regime: an ecosystem-level manipulation[END_REF][START_REF] Durance | Climate change effects on upland stream macroinvertebrates over a 25-year period[END_REF]. However, no effect of temperature was found on the structure of shredder communities associated with decomposing litter.

Relevance for stream bioassessment

The absence of an effect of warming by 2.8 °C on communities structure associated with decomposing litter might suggest that changes in function can occur without major changes in structure. This has strong implications for biomonitoring programs that evaluate ecosystem health based on the assessment of structural aspects of aquatic communities (e.g., European Water Framework Directive (Directive 2000/60/EC 2000), US Clean Water Act (CWA 33 USC Section 1251 et seq. 1972)). If ecosystem processes can be affected by warming without substantial changes to the aquatic communities, then evaluating the effects of climate change on ecosystem integrity based solely on the assessment of structural parameters might be misleading.

Repercussion on global C cycle

The stimulated litter decomposition under future climate change scenarios can lead to increases in the amount of C returned to the atmosphere and to a faster disappearance of litter from the stream bed, which has the potential to lead to a mismatch between consumers and their resources [START_REF] Durant | Climate and the match or mismatch between predator requirements and resource availability[END_REF]). However, oak litter decomposition is naturally slow (484 days to lose 95% initial mass in the presence of invertebrates under ambient temperature estimated in this study), and therefore a 49% stimulation of the decomposition rate might still allow oak litter to remain in the stream over the year, until the next litter fall period. Food shortage might occur, however, in streams that are supplied with fast-decomposing litter species (e.g., alder, maple) or have higher dissolved nutrient concentration, where a stimulation of litter decomposition can lead to the disappearance of litter from the stream bed well before the next autumnal litter fall. Long-term studies are necessary to evaluate possible consequences of faster litter decomposition on stream food webs.

In summary, this study shows that the effect of warming in stream ecosystems is complex and cannot be easily anticipated. The effect of warming on litter decomposition depended on litter species and presence of macroinvertebrates, which suggests that interactions between increases in temperature and changes in other factors must be taken into account when modeling the future effect of global warming on ecosystem functioning.

The relative importance of the three factors under study on litter decomposition can be ranked as follows: litter species > invertebrate presence > temperature. Nevertheless, even small increases in temperature can produce noticeable effects in ecosystem processes if these interact with changes in other factors, as likely occurs under field conditions.

Warming did not significantly affect the community structure of aquatic hyphomycetes or detritivores associated with decomposing litter, which suggests that aquatic communities are more resistant to environmental changes than ecosystem-level processes. Thus, ecosystem-level processes should be considered when evaluating the effect of changes in environmental factors on ecosystem integrity.

Stimulation of litter decomposition under future scenarios may not strongly affect aquatic communities in streams where slowdecomposing litter (e.g., oak) prevails, as enough litter may still remain in the system until the following litter fall. However, fast decomposition of high-quality (low C:N) soft litter may lead to a mismatch between consumers and resources and to food depletion for aquatic communities during part of the year. Thus, the presence of slow-decomposing litter species in the riparian vegetation may contribute to mitigate the effect of warming on stream ecosystem functioning.

Fig. 1 .

 1 Fig. 1. Decomposition rates of oak and chestnut leaf litter, incubated in fine mesh (FM) and coarse mesh bags (CM) in the control (Cont) and experimental (Exp) stream half over 60 days. Values are means ± SE. Treatments were compared by RM ANOVA; treatments with the same letter do not significantly differ (Fisher's test, P > 0.050).

Fig. 2 .

 2 Fig. 2. (a) Phosphorus (P) and (b) nitrogen (N) concentrations in oak and chestnut leaf litter incubated in fine mesh bags in the control and experimental half over 60 days. Values are means ± SE. SE bars may be smaller than the symbols.

Fig. 3 .

 3 Fig. 3. (a) Fungal biomass concentration, (b) sporulation rates, and (c) species richness of aquatic hyphomycetes associated with oak and chestnut leaf litter incubated in fine mesh bags in the control and experimental half over 60 days. Values are means ± SE.

Fig. 4 .

 4 Fig. 4. Relative contribution of aquatic hyphomycete species to the total conidial production associated with (a) oak and (b) chestnut leaf litter incubated in fine mesh bags in the control (Cont) and experimental (Exp) half over 60 days. Species contributing with a minimum of 5% conidia at least once are depicted. The remaining eight rare species are shown together as "Others", with a combined maximum contribution of 5.05%.

Fig. 5 .

 5 Fig. 5. (a) Total macroinvertebrate abundance, (b) shredder abundance, and (c) shredder taxa richness associated with oak and chestnut leaf litter incubated in coarse mesh bags in the control and experimental half over 60 days. Values are means ± SE.

Fig. 6 .

 6 Fig. 6. Relative contribution of shredder taxa to the total shredder abundance associated with (a) oak and (b) chestnut leaf litter incubated in coarse mesh bags in the control (Cont) and experimental (Exp) half over 60 days.

Table 1 .

 1 Water variables in the control and experimental half during the study period(27 December 2011 -25 February 2012). Temperature: mean daily temperature; DO: dissolved oxygen; SRP: soluble reactive phosphorus; n: number of measurements. Nitrite and ammonium were below detection limits (<100 and <50 g•L -1 , respectively). Values are means ± SD. Comparisons between stream halves were made by two-tail paired t tests; P values are given.

			Control	Experimental
	Water variable	n	half	half	P
	Temperature (°C)	60	6.4±1.2	9.2±2.4	<0.001
	Discharge (L•s -1 )	9	2.3±0.1	2.3±0.1	0.327
	Conductivity (S•cm -1 )	10	26.3±0.2	26.4±0.2	0.018
	DO (mg•L -1 )	10	12.2±0.7	11.4±0.8	<0.001
	DO (%)	10	105.9±3.4	106.0±3.4	0.846
	pH	10	7.4±0.2	7.3±0.2	0.051
	Alkalinity	6	7.3±0.8	7.8±0.9	0.178
	(mg CaCO 3 •L -1 ) NO 3 (g•L -1 ) SRP (g•L -1 )	11 9	88.2±23.6 2.5±5.6	108.3±17.1 0.8±1.8	0.163 0.277

Note:

Table 2 .

 2 Summary table for RM ANOVA performed on proportion of litter mass remaining (ln-transformed) of oak and chestnut leaf litter incubated in coarse and fine mesh bags in the control and experimental half over 60 days.

	Source of variation	df F	P
	Intercept	1 1252.118 <0.001
	Stream half	1	19.319 <0.001
	Litter species	1	40.772 <0.001
	Mesh size	1	26.473 <0.001
	Stream half × litter species	1	0.044	0.837
	Stream half × mesh size	1	0.001	0.976
	Litter species × mesh size	1	1.007	0.331
	Stream half × litter species × mesh size	1	0.688	0.419
	Error	16
	Time	2 226.417 <0.001
	Time × stream half	2	5.378	0.010
	Time × litter species	2	11.997 <0.001
	Time × mesh size	2	12.718 <0.001
	Time × stream half × litter species	2	0.332	0.720
	Time × stream half × mesh size	2	0.108	0.898
	Time × Litter species × mesh size	2	2.759	0.078
	Time × stream half × litter species × mesh size	2	0.402	0.673
	Error	32

Table 3 .

 3 Temperature sensitivities (Q 10-q ) of litter decomposition and time to decompose 50% initial AFDM under 10 °C warming for oak and chestnut incubated in fine and coarse mesh bags.

	Litter species	Mesh size	Q 10-q	Time (days)
	Oak	Fine	3.18	46
		Coarse	4.15	27
	Chestnut	Fine	2.86	35
		Coarse	1.64	38

Table 4 .

 4 Summary table for RM ANOVAs performed on phosphorus (P) and nitrogen (N) concentrations in oak and chestnut leaf litter incubated in fine mesh bags in the control and experimental half over 60 days.

		P		N
	Source of variation	df F	P	F	P
	Intercept	1 1102.651 <0.001 3794.213 <0.001
	Stream half	1	1.129	0.319	5.529	0.047
	Litter species	1	0.177	0.685	5.194	0.052
	Stream half × litter species 1	0.169	0.691	1.262	0.294
	Error	8
	Time	2	31.481 <0.001	37.015 <0.001
	Time × stream half	2	0.649 0.536	0.059 0.942
	Time × litter species	2	0.332	0.722	2.576	0.107
	Time × stream half × litter	2	2.016	0.166	1.365	0.284
	species
	Error	16

Table 5 .

 5 Summary table for RM ANOVAs performed on fungal biomass concentration (log(x)transformed), sporulation rates, and fungal species richness associated with oak and chestnut leaf litter incubated in fine mesh bags in the control and experimental half over 60 days.

					Sporulation
		Fungal biomass	rate		Species richness
	Source of variation	df F	P		F	P	F	P
	Intercept	1 12 920.456 <0.001 97.442 <0.001 406.125 <0.001
	Stream half	1	0.461	0.516	4.951	0.057	0.014	0.909
	Litter species	1	76.772 <0.001	8.977	0.017	0.681	0.433
	Stream half × litter species	1	2.157	0.180	3.629	0.093	0.347	0.572
	Error	8
	Time	2	11.503	0.001 30.318 <0.001	9.418	0.002
	Time × stream half	2	4.577	0.027	2.363	0.126	0.085	0.919
	Time × litter species	2	3.123	0.072 12.631	0.001	0.012	0.988
	Time × stream half × litter species	2	0.269	0.767	4.517	0.028	0.521	0.604
	Error	16

Table 6 .

 6 Summary table for RM ANOVAs performed on total macroinvertebrate abundance (log(x + 1)-transformed), shredder abundance (log(x + 1)-transformed), and shredder taxa richness (log(x + 1)-transformed) associated with oak and chestnut leaf litter incubated in coarse mesh bags in the control and experimental half over 60 days.

					Shredder
		Total abundance	abundance	Taxa richness
	Source of variation	df F	P		F	P	F	P
	Intercept	1 929.394 <0.001 74.740 <0.001 63.740 <0.001
	Stream half	1	0.061	0.811	9.418	0.015	6.068	0.039
	Litter species	1	2.656	0.142	18.822	0.002 10.527	0.012
	Stream half × litter species	1	0.453	0.520	3.048	0.119	1.074	0.330
	Error	8
	Time	2	4.651	0.026	0.861	0.441	0.661	0.530
	Time × stream half	2	4.890	0.022	1.117	0.351	1.464	0.261
	Time × litter species	2	0.284	0.756	0.334	0.721	1.941	0.176
	Time × stream half × litter species	2	0.070	0.932	0.762	0.483	0.283	0.757
	Error

Acknowledgements

Cristina Docal performed the ion chromatography and HPLC analyses, Ana Lírio and João Rosa helped in the field, Amado and Amado Company built and maintained the warming system, and the Lousã Town Hall provided logistic support. José Paulo Sousa (University of Coimbra) provided useful advice on the statistical analyses, and the comments provided by two anonymous referees greatly improved the manuscript. This study was financed by the IMAR-CMA, the FEDER European Funds through the Program Operational Factors of Competitiveness (COMPETE) and national funds through the Portuguese Science and Technology Foundation (FCT, project PTDC/CLI/67180/2006), and the French EC2CO (Continental and Coastal Ecosphere) CNRS program to the Interhym project. Financial support was also granted by the FCT to VF (postdoctoral fellowships SFRH/BPD/34368/2006 and SFRH/BPD/ 76482/2011, program POPH/FSE) and by the Pessoa exchange program between Portugal and France (PHC 22808TB) from the French Ministry of Foreign and European Affairs.


Hernán Larralde

Pablo Jensen

Margaret Edwards

Two dimensional Hotelling model : analytical results and numerical simulations

We present an analytical solution to a two dimensional Hotelling model with quadratic transportation costs for two stores in a square city. We assume that the consumers choice as to which store to patronize is tempered by a logit function. As in the one-dimensional case, stores are led to aggregate spatially as the disorder introduced by the logit increases. This solution is confirmed by numerical simulations.

I. INTRODUCTION

Hotelling's model [START_REF] Hotelling | Stability and competition[END_REF] is one of the preferred "toy models" of spatial economics. Variations of this model and its "Principle of Minimum Differentiation" have generated a large literature (for reviews, see [START_REF] Anderson | Discrete Choice Theory of Product Differentiation[END_REF][START_REF] Tirole | The theory of Industrial Organization[END_REF][START_REF] Hoover | An Introduction to Regional Economics[END_REF][START_REF] Brown | Stephen Retail Location Theory: The Legacy of Horald Hotelling[END_REF], for recent papers, see [START_REF] Veendorp | Differenciation in a two-dimensional market Regional[END_REF][START_REF] Irmen | Competition in Multi-characteristics Spaces : Hotelling Was Almost Right[END_REF][START_REF] Konishi | Concentration of Competing Retail Stores[END_REF]) allowing researchers to play with elementary mechanisms at will. In essence, this model consists in "consumers" that are distributed in a bounded space and choose to buy at the store which maximizes their utility. The definition of the consumer utility includes store prices and transportation costs. The aim of the model is to determine optimal locations and prices for competing stores trying to maximize their profits.

The main conclusions of the one dimensional Hotelling model for two stores looking for "location then prices" optima are summarized in [START_REF] Anderson | Discrete Choice Theory of Product Differentiation[END_REF]. If the consumers choose systematically the most convenient store, i.e. that which represents the lowest total cost, then stores prefer to locate near the edges of the segment, because this allows them to exert some "market power" on the consumers located close to them. Indeed, when the stores are located near each other, consumers have similar traveling costs for both stores and a slightly lower price is sufficient for one of the stores to collect most of the demand. Therefore, competition drives both stores to minimal prices and zero profits. In contrast, when the stores are located far from each other, consumer's choice is mainly determined by the distance to each store. Therefore the stores can sell at higher prices and obtain a higher profit than in the previous case.

The situation changes when some "disorder" is introduced in the consumer's preferences. This disorder represents additional choice factors, not taken into account explicitly in the analysis, and therefore entering the picture only as a random "noise". Usually, the effect of the random factor in the consumer's choice is introduced through a logit [START_REF] Anderson | Discrete Choice Theory of Product Differentiation[END_REF] function. This randomness in consumer's preferences renders their choice less sensible to stores' characteristics and the advantage of spatial differentiation is lessened. Then stores move towards the centre, which is the "best place in town", i.e. the place which is closest to all consumers. This "minimum differentiation" arises because it is now the random choice factors that give the stores some market power, allowing them to raise the prices above cost prices. The larger the noise, the closer the stores locate to the centre of the "town". Eventually, at very large disorders, both stores locate at the center and are able to ask for high prices.

The original one dimensional Hotelling model has been extended to two dimensional (geographical) spaces by Veendorp and Majeed [START_REF] Veendorp | Differenciation in a two-dimensional market Regional[END_REF] in the absence of disorder. They reach a similar conclusion as other studies [START_REF] Irmen | Competition in Multi-characteristics Spaces : Hotelling Was Almost Right[END_REF]: stores maximally differentiate in one of the dimensions, while adopting an identical location in the other dimension.

However, the case when the consumer choice is influenced by additional (and unobservable) factors, as mimicked by the logit function, has not been studied in two dimensions. This is the goal of the present paper. To the best of our knowledge, this paper presents the first analytical solution to Hotelling model in presence of noise in consumer's choice.

II. OPTIMAL STORE LOCATION IN A TWO DIMENSIONAL CITY

We consider a two-dimensional square city represented by L × L discrete sites occupied uniformly by consumers with unit density.

To model the behavior of the consumers, we assume that each consumer j can assign a "utility" function K js to each store s given by:

K js = R -p s -ad α js (2.1)
where R is the "maximum utility" of buying the product, assumed to be high enough to prevent any negative value for K js . p s is the price of the product at store s, d js the (Euclidian) distance between consumer j and store s, a is the transportation cost coefficient and α an exponent characterizing how transportation costs increase with distance. We take here α = 2 for ease of calculations.

Consumers evaluate K js for the two stores and choose according to these values. A simple way to include the effect of random factors that "smoothen" the consumer's choice among the stores is to use the well-known "logit" weighting factor [START_REF] Anderson | Discrete Choice Theory of Product Differentiation[END_REF]. We assign a probability for consumer j to buy at store s as:

σ js = exp(K js /T ) s exp(K js /T ) (2.2)
T is a parameter which defines how sharply consumers discern between the expected surpluses offered by each site. As T → ∞ consumers do not discriminate between eligible stores, whereas if T → 0, consumers exclusively choose the store with the highest K js .

Note that, the way the problem is posed, we have assumed that the probability that the consumers make a purchase is always 1 and the only decision concerns where the purchase is made.

In this situation, the average market M s for store s is given by

M s = j σ js (2.3)
and the expected profit accrued by this store will be given by

D s = p s M s (2.4) III. ANALYTICAL SOLUTION
For definiteness, we list here the precise rules of the system under consideration:

• First, given any two positions of the stores, S 1 and S 2 , the stores compete in prices until they reach (Nash) equilibrium; that is, until any unilateral change in price for either store leads to a lower profit. We further assume that the stores know the profits accrued by the equilibrium prices for every pair of store positions.

• Knowing the equilibrium prices for each pair of sites and the position of the other store, the stores then compete for optimal (maximal profit) location, until they reach (Nash) equilibrium in positions.

In our analysis we assume S 1 and S 2 to be continuous variables and, as mentioned above, we take α = 2. Thus, M 1 , the average market (number of consumers) attending store 1, is given by:

M 1 = L/2 -L/2 dj x L/2 -L/2 dj y 1 1 + e {p1-p2+a[|J-S1| 2 -|J-S2| 2 ]}/T M 2 = L 2 -M 1 (3.1)
Given locations S 1 and S 2 , price equilibrium is attained when

∂D 1 ∂p 1 = 0 and ∂D 2 ∂p 2 = 0 (3.2)
i.e. any unilateral change in the price offered by each store leads to a reduction of the profit gained by the store (of course, the above equations only reflect that at equilibrium profit is an extremum, this will do for the analysis because the extrema of these functions are maxima). These are two coupled equations for the prices, from which we can determine, in principle, the prices as functions of the positions. Now, equilibrium in the location competition will be achieved when neither store can increase its profit by changing its own position given the other store's location. Specifically, as mentioned above, storeowners know that a change in position will lead to a new set of prices, and they can evaluate the resulting change in profit accrued in the new position at the equilibrium prices corresponding to the new positions.

Thus, equilibrium is achieved when

dD 1 dx 1 , dD 1 dy 1 = 0 and dD 2 dx 2 , dD 2 dy 2 = 0 (3.3)
where x 1 , y 1 and x 2 , y 2 are the components of S 1 and S 2 respectively. Regarding notation we choose to write the above equations as vectors of total derivatives with respect to the components of the positions of each store and reserve the gradient as a vector of partial derivatives to be used below.

From eq.(2.4), we can write the equation for price equilibrium for store 1 as:

p 1 ∂M 1 ∂p 1 + M 1 = 0 (3.4)
then, the condition for location equilibrium becomes

dD 1 dx 1 , dD 1 dy 1 = p 1 ∂M 1 ∂p 2 ∇ 1 p 2 + p 1 ∇ 1 M 1 = 0 (3.5)
where ∇ 1 = (∂/∂x 1 , ∂/∂y 1 ) and we have used eq.(3.4). Note that, as required in the description of the location game, the above expression includes the change in p 2 due to the change of location of store 1. A corresponding equation holds for store 2:

dD 2 dx 2 , dD 2 dy 2 = p 2 ∂M 2 ∂p 1 ∇ 2 p 1 + p 2 ∇ 2 M 2 = 0 (3.6)
From the explicit expression of M 1 , we have:

I ≡ ∂M 1 ∂p 2 = 1 4T L/2 -L/2 dj x L/2 -L/2 dj y 1 cosh 2 ({p 1 -p 2 + a[|J -S 1 | 2 -|J -S 2 | 2 ]}/2T ) (3.7)
whereas

∇ 1 M 1 = a 2T L/2 -L/2 dj x L/2 -L/2 dj y J -S 1 cosh 2 ({p 1 -p 2 + a[|J -S 1 | 2 -|J -S 2 | 2 ]}/2T ) (3.8)
Thus, the location equilibrium equation for store 1 can be written as

∇ 1 p 2 + 2a[ J -S 1 ] = 0 (3.9)
where J = ( j x , j y ) stands for

J = L/2 -L/2 dj x L/2 -L/2 dj y J/cosh 2 {p 1 -p 2 + a[|J -S 1 | 2 -|J -S 2 | 2 ]}/2T L/2 -L/2 dj x L/2 -L/2 dj y 1/cosh 2 ({p 1 -p 2 + a[|J -S 1 | 2 -|J -S 2 | 2 ]}/2T )
Now, ∇ 1 p 2 can be calculated from the conditions of price equilibrium (3.2). Writing them in terms of p 1 , M 1 , etc. (as in eq.(3.4)), and applying ∇ 1 we have:

2 ∂M 1 ∂p 1 + p 1 ∂ 2 M 1 ∂p 2 1 ∇ 1 p 1 + p 1 ∂ 2 M 1 ∂p 1 ∂p 2 + ∂M 1 ∂p 2 ∇ 1 p 2 + p 1 ∇ 1 ∂M 1 ∂p 1 + ∇ 1 M 1 = 0 (3.10) 2 ∂M 2 ∂p 2 + p 2 ∂ 2 M 2 ∂p 2 2 ∇ 1 p 2 + p 2 ∂ 2 M 2 ∂p 1 ∂p 2 + ∂M 2 ∂p 1 ∇ 1 p 1 + p 2 ∇ 1 ∂M 2 ∂p 2 + ∇ 1 M 2 = 0
To simplify the above expressions, we use eq (3.5) and note the following identities:

∂M 2 ∂p 2 = ∂M 1 ∂p 1 = - ∂M 2 ∂p 1 = - ∂M 1 ∂p 2 ≡ -I M 1 + M 2 = L 2 (3.11)
where I is given in eq.(3.7).

Now we limit ourselves to search for equal price symmetric solutions i.e. solutions for which S 1 = -S 2 ; and, by symmetry, p 1 = p 2 . Note that all the derivatives are to be taken before imposing the symmetry conditions. Under these conditons, the terms arising from the second derivatives with respect to the p's vanish (as well as j )

Then, eqs.(3.10) become:

-2I∇ 1 p 1 -p 1 ∇ 1 I = 0 (3.12) -2I∇ 1 p 2 + I∇ 1 p 1 -p 2 ∇ 1 I -∇ 1 M 1 = 0
From these equations we finally get

S = - 3L 2 4aI 2 [∇ 1 I] s2=-s1 (3.13)
Now, writing

S = Lσ γ = 2aL 2 /T (3.14)
we can rewrite equation (3.13) explicitly in terms of the dimensionless vector σ and the parameter γ as

σ = 3 1/2 -1/2 1/2 -1/2 dj x dj y J sinh[γ(J • σ)]/ cosh 3 [γ(J • σ)] 1/2 -1/2 1/2 -1/2 dj x dj y 1/ cosh 2 [γ(J • σ)] 2 (3.15)
These integrals can be performed explicitly:

1/2 -1/2 1/2 -1/2 dj x dj y 1/ cosh 2 [γ(J • σ)] = 2 γ 2 σ x σ y ln cosh[γ(σ x + σ y )/2] cosh[γ(σ x -σ y )/2] (3.16)
and

1/2 -1/2 1/2 -1/2 dj x dj y J sinh[γ(J • σ)]/ cosh 3 [γ(J • σ)] = - 1 2γ ∇ 1/2 -1/2 1/2 -1/2 dj x dj y 1/ cosh 2 [γ(J • σ)]
We further restrict our search to solutions along the diagonal: σ x = σ y , and along the horizontal: σ y = 0. For the horizontal case we obtain the following expression:

σ x = 3 4 sinh(γσ x ) -γσ x cosh(γσ x ) -1 (3.17)
For γ < 4, the only solution for the above equation is σ x = 0. For γ > 4, two other nontrivial solutions σ x = ±σ * h (γ) appear. As γ → ∞ (low T limit), it is easy to see that σ * h → 3/4. Along the diagonal, the equation for

σ x = σ y ≡ σ is σ = 6γσ (4 ln[cosh(γσ)]) 2 [2 ln[cosh(γσ)] -γσ tanh(γσ)] (3.18)
Once again, for γ < 4 the only solution to the above equation corresponds to σ = 0, whereas for γ > 4 we have two nontrivial solutions σ = ±σ * d (γ). Now, when γ → ∞ we find σ * d → 3/8. Thus, for high enough T (γ < 4), the only solutions appear to be at the origin, i.e. at the center of the city. However, at lower T , at least 4 other solutions exist. These solutions move away from the center as T → 0 as far as 3L/4, which is beyond the boundary of the city (which is located at L/2) along the horizontal lines, but only as far as 3L/8 along the diagonal lines.

To determine the optimal locations we need to evaluate the actual profits obtained by the stores for each of these solutions. This is relatively easy to do. First, we notice that for the equal price symmetric solutions we are considering, we have M 1 = M 2 = L 2 /2. Thus, the equilibrium prices are given by p 1 = p 2 = L 2 /2I (where I is the integral defined in eq.(3.7) and which is trivially related to the integral in eq.(3.16)), and the profits are given by D 1 = D 2 = L 4 /4I. Thus, we need to compare the behavior of I along the horizontal and diagonal solutions:

I(σ * h ) = L 2 2T γσ * h tanh(f racγσ * h 2) ≡ I h (γσ * h ) (3.19)
and

I(σ * d ) = L 2 2T γ 2 σ * d 2 ln[cosh(γσ * d )] ≡ I d (γσ * d ) (3.20)
When solving eqs. (3.17) and (3.18) and using these to evaluate the profits, it appears that for γ > 4 the profit found with the horizontal solution is always higher. For γ ≤ 4 both solutions are at the origin and profits are, of course, equivalent. These results are plotted in Figures 1 and2.

IV. SIMULATION RESULTS

To confirm the previous treatment, we performed numerical simulations under the same hypothesis. Specifically, for each value of γ, we performed a double nested optimization, using brent's minimization technique [START_REF] William | Vetterling Numerical Recipes in Fortran[END_REF]. The town size (L) is taken as 300.

One loop calculates the profits given the stores locations σ 1 and σ 2 .

• Given σ 1 and σ 2 , and an initial value for p 2 , calculate p * 1 so as to maximize D 1 using brent's algorithm • Given σ 1 and σ 2 , and p * 1 , calculate p * 2 so as to maximize D 2 using brent's algorithm • Repeat the two preceding steps until p * 1 and p * 2 converge within a fixed tolerance Numerically, the relative tolerance value for the prize was taken as 10 -6 . To find the optimal stores positions, a similar loop is then used, which includes the first loop at each step, to calculate the profits.

• Given an initial value for σ 2 , calculate σ * 1 so as to maximize D 1 using brent's algorithm • Given σ * 1 , calculate σ * 2 so as to maximize D 2 using brent's algorithm • Repeat the two preceding steps until σ * 1 and σ * 2 converge within a fixed tolerance Numerically, the relative tolerance value for the position was taken as 10 -5 . Optimal locations and corresponding profits as a function of γ are plotted in Figures 1 and2. The agreement with the theoretical predictions is almost perfect, except in the region γ = 4 where discreteness effects seem to play a role. This is confirmed by the results obtained for a larger number of customers (600 x 600 or 1000 x 1000), which are closer to the theoretical results (see Fig. 3).

V. DISCUSSION

In this model, equilibrium configurations result from trying to maximize individual profits, which are the product of price and market. This last quantity, the market of a store, is a decreasing function of the price offered at that store and of distance to the consumers, and a growing function of the price offered by the competition. This combination can give rise to rather unexpected strategies: For example, at low temperatures, when consumers buy almost exclusively at the store which maximizes their utility, a store can find it profitable to reduce its market by increasing its price and moving away from the city center because the other store will now find it profitable to increase its own price, thus returning part of the market to the first store. At this point, the second store might find it even more profitable to move further away from the center and ask for an even higher price, knowing that the first store will also increase its price. This process can go on until the transportation cost becomes too large to be compensated by the other store's price increase, at which point, the off center equilibrium positions are attained.

On the other hand, as the temperature rises, consumers become more indiferent to the travelling cost and stores can gain more by undercutting the competition as long as no consumers are forced to travel too long a distance to make their purchase. This brings the stores closer to the center of the city even though the price competition brings the profits down.

Eventually, the cost of transportation becomes almost irrelevant, and purchases are mostly decided by the price offered at each store. At the higher values of T , even price is relatively unimportant and there is little to win by undercutting the competition so prices and profits grow again.

Another aspect that has drawn a great deal of attention is the fact that the spatial differentiation occurs in one dimension and not in the other [START_REF] Irmen | Competition in Multi-characteristics Spaces : Hotelling Was Almost Right[END_REF]. Clearly this a result of the chosen shape of the city and on a circular city no such thing would happen. Now, admittedly, we did not search analytically for equilibrium positions out of the axes of symmetry, yet we did not find any in our numerical simulations. Thus, given the geometry we considered, the question that remains is why are the equilibria along the paralel axis more profitable than those along the diagonals. A partial explanation for this might be that if one projects the market along the diagonal, it becomes apparent that most of the market is at the middle and diminishes towards the corners. Thus, the loss of market incurred by the first store moving away from the center affects more consumers. Further, to render such a movement profitable, the store must also increase its price. However, this increase must be modest if the combination of movement and price increase is to be compensated by the second store's price hike.

VI. CONCLUSION

In this article we extend the Hotelling model by studying the optimal location of two stores in a two-dimensional city where consumers' choice follows a logit function with parameter T . Starting from an identical position in the center of town when T is large, stores differentiate only along one coordinate as T decreases. Both prices and profits increase as T increases, i.e. as the consumer pay less and less attention to the difference in costs incurred by attending one or the other store. The results are therefore similar to the one dimensional case.

An interesting but much more complicated extension of the model would be to consider the situation in which the consumers, with certain probability, may choose not to make the purchase. For example, one could assume that it is highly improbable that a consumer makes a purchase if the cost incurred in attending either store is too high. Among other things, such an extension would introduce a T dependent monopoly price to the single store system, as well as another distance defining the market ranges of each store; i.e. finite regions around each store beyond which it is high unlikely that consumers will attend the store. This line of research is currently in progress.

FIG. 1 .

 1 FIG. 1. Equilibrium positions as a function of γ.

FIG. 2 .FIG. 3 .

 23 FIG.2. Store profits at the equilibrium positions as a function of γ


- A Drouin

The 2011 FrIAC Event : Comparison MIPAS/REPROBUS/Oslo CTM On the

Anticyclones) events occurring in 2007 and 2011 (#6091).

N. Huret (1) , R. Thiéblemont (1) , F. Jegou (1) , A. Søvde (2) , Y.J. Orsolini (3) , and M.

Active chemistry in the core of the FrIAC : REPROBUS

CONCLUSION :

The 2007 FrIAC event appears to be well represented by the REPROBUS CTM simulation. The N 2 O signal is qualitatively in agreement with the shape of the N 2 O evolution from MLS data, but the abundance of N 2 O in the initial tropical intrusion is smaller in the model. O 3 mixing ratio evolution are very similar. In 15 days, a loss of ozone is observed in the core of the FrIAC. Such loss is being examined by investigating the chemical processes. For the 2011 event, simulations have been made by two different CTM model, and in both cases, there are not able to represent the dynamical signature : N 2 O high tropical values in the core of the FrIAC disappear too quickly (only 50 days). O 3 evolution in the core of the 2011 FrIAC is, as for the 2007 event, in agreement with the measurements. Additional work , has to be made, in particular with higher resolution simulations, to better understand why the CTMs are not able to represent the 2011 event. Allen et al. [2011] has encountered the same type of differences on the 2005 FriAC event. Two weeks after the tropical intrusion stage, the FrIAC is well established in the summer polar circulation.

Both CTM models represent the dynamical signal in N 2 O fields but with lower value than the MIPAS measurements. Ozone fields are in good agreement and 15 days after the beginning of the FrIAC event, high O 3 tropical abundance has disappeared in the core of the FrIAC.

• The N 2 O signal of the FrIAC has disappeared in the REPROBUS simulation on the 25 June, whereas this signal still exits in MLS data.

• The 2011 FrIAC event is centered on the North Pole difficulties to follow the evolution of this event with MLS data.

The location of the core of the FrIAC on the 850K isentropic surface as a function of julian day, has been extracted from the MIPAS data and the results of REPROBUS using the criteria of the maximum value of N 2 O mixing ratio at 850K. The locations find using REPROBUS and MIPAS are in agreement.

MIPAS/MLS REPROBUS

Ozone destruction by the NOx-driven catalytic cycle plays an important role in the summer high latitudes, in continuous daylight [Orsolini et al., 2003].

Further study is underway to quantify its effect By comparing with the 2007 event, strong discrepancies appear; in particular regarding NOy and HOx species. This requires further investigations.

  The 2007 FrIAC Event : Comparison MLS/REPROBUS On the 25 April 2007 : Intrusion Location of the core of the FrIAC : MLS/REPROBUS N 2 O and Ozone evolution in the core of the FrIAC :

  During the first stage of the FrIAC (i.e. tropical intrusion), the REPROBUS model well represents the location of the tropical intrusion. However the maximum value of N 2 O inside the tongue is underestimated due to a smaller extent of the intertropical band simulated by the model. O 3 and HCl appears consistent with the satellite data.The location of the core of the FrIAC, on the 850 K isentropic surface as a function of julian day, has been extracted from the MLS Data and the results of REPROBUS using the criteria of the maximum value of N 2 O mixing ratio at 850 K.The model well represents the evolution of the location (latitude and longitude) of the FrIAC core.The time evolution of N 2 O in the core of the FrIAC obtained by the CTM model is quantitatively different from the MLS measurements. However the shape is similar with a first step associated with constant high value of N 2 O (typical of the tropics) during 40 days and then a decrease during 20 days to typical polar content. The ozone mixing ratio rapidly decreases in the core of the FrIAC during the first 20 days. Model results and MLS data are consistent regarding the ozone evolution. N 2 O time evolution O 3 evolution The active chemistry from REPROBUS CTM model in the core of the FriAc is presented. stage of the FrIAC (i.e. tropical intrusion) both CTM models well represent the location of the tropical intrusion. However the maximum value of N 2 O inside the tongue is underestimated as for 2007 event.

N 2 O

 2 mixing ratio evolutions in the core of the FrIAC measured by MIPAS and simulated by REPROBUS are different. For the 2011 case, the model is not able to conserve the N 2 O signal which starts to decrease from the beginning of the FrIAC. O 3 evolution in the core of the FrIAC is well represented by the -mean N2O (in ppbv) at 850 K • The zonal mean of N 2 O fields shows that the FrIAC signal persists 50 days (more than 3 months) in REPROBUS (in MLS data).

Aknowledgments

This study has been conducted within the framework of the STRAPOLETE and ENRICHED project supported by the "Agence Nationale de la Recherche ANR (STRAPOLETE project ANR 08 BLAN 0300), the "Institut Polaire Paul Emile Victor" (IPEV) and the "Centre National d'Etudes Spatiales (CNES)". We thanks ETHER CNES/INSU data base. For their support. the EOS-MLS instrument science team, and MIPAS-IMK science team for the satellite data and the European Centre for Medium-Range Weather Forecasts for providing the ERA-Interim data.