Alexis D Synodinos

Bart Haegeman
email: bart.haegeman@sete.cnrs.fr

Arnaud Sentis
email: arnaud.sentis@inrae.fr

José M Montoya

Alexios Synodinos
email: alexios.synodinos@sete.cnrs.fr

Theory of temperature-dependent consumer-resource interactions

Keywords: climate change, consumer-resource dynamics, food webs, community stability, biomass distributions, interaction strength, temperature dependence

Changes in temperature affect consumer-resource interactions, which underpin the functioning of ecosystems. However, existing studies report contrasting predictions regarding the impacts of warming on biological rates and community dynamics. To improve prediction accuracy and comparability, we develop an approach that combines sensitivity analysis and aggregate parameters. The former determines which biological parameters impact the community most strongly at a given temperature. The use of aggregate parameters (i.e., maximal energetic efficiency, ρ, and interaction strength, κ), that combine multiple biological parameters, increases explanatory power and reduces the complexity of theoretical analyses.

We illustrate the approach using empirically-derived thermal dependence curves of biological rates and applying it to consumer-resource biomass ratio and community stability. Based on our analyses, we generate four predictions: 1) resource growth rate regulates biomass distributions at mild temperatures, 2) interaction strength alone determines the thermal boundaries of the community, 3) warming destabilises dynamics at low and mild temperatures only, 4) interactions strength must decrease faster than maximal energetic efficiency for warming to stabilise dynamics. We argue for the potential benefits of directly working with the aggregate parameters to increase the accuracy of predictions on warming impacts on food webs and promote cross-system comparisons.

Introduction

Temperature strongly regulates consumer-resource interactions that constitute the fundamental blocks of ecosystems [START_REF] O'connor | Warming and Resource Availability Shift Food Web Structure and Metabolism[END_REF][START_REF] Montoya | Climate change, biotic interactions and ecosystem services[END_REF][START_REF] Petchey | Predicting the effects of temperature on food web connectance[END_REF][START_REF] Rall | Universal temperature and body-mass scaling of feeding rates[END_REF][START_REF] Amarasekare | Effects of Climate Warming on Consumer-Resource Interactions: A Latitudinal Perspective[END_REF], and anthropogenic climate change will, in most cases, increase mean temperatures (IPCC 2013). Therefore, understanding and predicting the impacts of warming on consumer-resource interactions has attracted much interest [START_REF] Vasseur | A mechanistic approach for modeling temperaturedependent consumer-resource dynamics[END_REF][START_REF] Binzer | The dynamics of food chains under climate change and nutrient enrichment[END_REF][START_REF] Thakur | Warming magnifies predation and reduces prey coexistence in a model litter arthropod system[END_REF]. A breakthrough occurred with the postulation that metabolic rate increases exponentially with temperature, with the slope (often referred to as activation energy) conserved across levels of organisation [START_REF] Gilooly | Effects of Size and Temperature on Metabolic Rate[END_REF][START_REF] Brown | TOWARD A METABOLIC THEORY OF ECOLOGY[END_REF]). However, activation energies can vary significantly among organisms and biological rates [START_REF] Dell | Systematic variation in the temperature dependence of physiological and ecological traits[END_REF][START_REF] Réveillon | Repeatable inter individual variation in the thermal sensitivity of metabolic rate[END_REF]. In addition, the thermal response curve of biological rates can decrease at high temperatures, producing a unimodal thermal dependence shape [START_REF] Deutsch | Impacts of climate warming on terrestrial ectotherms across latitude[END_REF][START_REF] Pörtner | Ecology: Physiology and climate change[END_REF][START_REF] Englund | Temperature dependence of the functional response[END_REF][START_REF] Uiterwaal | Functional responses are maximized at intermediate temperatures[END_REF]. This lack of consensus regarding the exact shape of the temperature-dependence of physiological rates (e.g. ingestion rates), behavioural traits (e.g. consumer search or attack rates) or production (carrying capacity) has contributed to diverging, sometimes contradicting, predictions of how consumer-resource interactions will respond to warming (e.g. [START_REF] Vucic-Pestic | Warming up the system: higher predator feeding rates but lower energetic efficiencies[END_REF][START_REF] Sentis | Using functional response modeling to investigate the effect of temperature on predator feeding rate and energetic efficiency[END_REF].

A dual approach to address the divergence in predictions

Two community properties describing important features of the community are consumerresource biomass ratio and stability, defined with respect to the occurrence of oscillations.

Their importance is reflected in their prevalence in the literature on the effects of warming on consumer-resource communities and food webs [START_REF] Rall | Temperature, predator-prey interaction strength and population stability[END_REF][START_REF] Rall | Universal temperature and body-mass scaling of feeding rates[END_REF][START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF][START_REF] Barbier | Pyramids and cascades: a synthesis of food chain functioning and stability[END_REF][START_REF] Bideault | Thermal mismatches in biological rates determine trophic control and biomass distribution under warming[END_REF]. However, predictions vary about the impacts of warming on both these community properties. Biomass ratios have been theorised to increase [START_REF] Rip | Cross-ecosystem differences in stability and the principle of energy flux[END_REF][START_REF] Gilbert | A bioenergetic framework for the temperature dependence of trophic interactions[END_REF] or decrease [START_REF] Vasseur | A mechanistic approach for modeling temperaturedependent consumer-resource dynamics[END_REF] monotonically with warming, though experimentally-derived data have mainly yield unimodal responses [START_REF] Fussmann | Ecological stability in response to warming[END_REF][START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF]. Likewise, the effects of warming on stability remain unclear. Using data on specific rates (e.g. consumer ingestion and metabolism), studies have inferred that stability either increases monotonically (Rall et al. 2010(Rall et al. , 2012;;[START_REF] Vucic-Pestic | Warming up the system: higher predator feeding rates but lower energetic efficiencies[END_REF][START_REF] Fussmann | Ecological stability in response to warming[END_REF] or responds unimodally [START_REF] Sentis | Using functional response modeling to investigate the effect of temperature on predator feeding rate and energetic efficiency[END_REF][START_REF] Betini | Temperature triggers a non linear response in resource-consumer interaction strength[END_REF] to warming. Theoretical work on stability, in particular on the mechanisms causing the onset of oscillations, expands decades [START_REF] Rosenzweig | Graphical Representation and Stability Conditions of Predator-Prey Interactions[END_REF][START_REF] May | Limit cycles in predator-prey communities[END_REF]. [START_REF] Vasseur | A mechanistic approach for modeling temperaturedependent consumer-resource dynamics[END_REF] showed that warming will destabilise consumer-resource communities when the consumer metabolic rate increases slower than the ingestion rate. [START_REF] Johnson | A metric for quantifying the oscillatory tendency of consumer-resource interactions[END_REF] demonstrated the pivotal role of the temperature-dependence of carrying capacity -rather than metabolism and ingestion -in determining warming-stability relationships. All these examples demonstrate that the mixed predictions, whether empirically-derived or theoretical, originate from two distinct sources: the different parameters hypothesised to be driving community responses and the thermal dependence shapes of these parameters. To improve the accuracy of predictions regarding the effects of warming on consumer-resource communities, we need to establish which biological parameters drive community properties (biomass distribution, stability) and to acquire a mechanistic understanding of how their thermal dependence shapes affect community properties.

A dual approach utilising sensitivity analysis and the application of aggregate parameters can address both these issues (Fig. 1). On the one hand, sensitivity analysis establishes the parameters that most strongly influence the community property of interest: it quantifies the incremental increase in a response variable with respect to a small incremental increase in a parameter. It has been extensively used in population ecology and demography [START_REF] Caswell | Sensitivity Analysis: Matrix Methods in Demography and Ecology[END_REF], with important applications in applied ecology [START_REF] Manlik | Applicability and limitations of sensitivity analyses for wildlife management[END_REF]. Since the relative importance of parameters can change along the temperature gradient, a sensitivity analysis allows us to determine the temperatures at which changes in the values of the different parameters have the strongest relative impact [START_REF] Zhao | Effect of water temperature on the dynamic behavior of phytoplankton-zooplankton model[END_REF].

On the other hand, we can aggregate groups of the primary parameters into fewer, biologically meaningful and empirically measurable quantities. The use of such aggregate parameters reduces the complexity of theoretical analyses, provides a mechanistic interpretation for the difference in predictions and facilitates the comparison among predictions [START_REF] Barbier | Pyramids and cascades: a synthesis of food chain functioning and stability[END_REF][START_REF] Bideault | Thermal mismatches in biological rates determine trophic control and biomass distribution under warming[END_REF]. Experimentally, replacing multiple measurements of individual parameters with measurements of the aggregates could also restrict the room for divergent findings. The seminal work of [START_REF] Yodzis | Body size and consumer-resource dynamics[END_REF] reduced the analysis of consumer-resource interactions to two principal aggregate parameters; consumer maximal energetic efficiency and a measure of resource abundance. A variation of maximal energetic efficiency (termed energetic efficiency) has been widely used by empirical studies. However, rather than being measured directly, it has been derived from measurements of its principle components, i.e., feeding and metabolic rates [START_REF] Rall | Temperature, predator-prey interaction strength and population stability[END_REF][START_REF] Vucic-Pestic | Warming up the system: higher predator feeding rates but lower energetic efficiencies[END_REF][START_REF] Sentis | Using functional response modeling to investigate the effect of temperature on predator feeding rate and energetic efficiency[END_REF]. [START_REF] Gilbert | A bioenergetic framework for the temperature dependence of trophic interactions[END_REF] posited that a single aggregate parameter -interaction strength defined as the impact of the consumer on the resource population density -could capture the effects of warming on the stability of consumer-resource interactions. However, their approach was based on a type I (nonsaturating) functional response, whereas most consumer-resource species pairs typically produce type II or III (saturating) functional responses (Jeschke et al. 2004). Moreover, the thermal dependence of interaction strength did not match the impact of warming on stability for type II or III functional responses [START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF], pointing to a more complex relationship between interaction strength, warming and stability. We use two aggregate parameters: the maximal energetic efficiency of the consumer population, defined as the ratio of energetic gains through ingestion with no resource limitation (i.e., maximal energetic gains) over energetic losses associated to metabolic demand [START_REF] Yodzis | Body size and consumer-resource dynamics[END_REF] and interaction strength, measured as the ratio of resource population density without consumers to resource population density with consumers [START_REF] Gilbert | A bioenergetic framework for the temperature dependence of trophic interactions[END_REF]. Thus, our dual approach identifies the parameters causing the divergence in predictions through the sensitivity analysis and simplifies complex theoretical explorations and empirical measurements through the two aggregate parameters (Fig. 1). By expressing the sensitivities and the community dynamics in terms of the aggregate parameters, analyses collapse into two dimensions. This creates a simple and mechanistic tool to increase the accuracy and improve comparability of theoretical and empirically-driven predictions on the impact of temperature changes on consumer-resource communities. The dual approach of sensitivity analysis and parameter aggregation is not tailored to a specific model of consumer-resource interactions as both are often used independently for different ecological models (e.g. Barbier et al., add a ref for sensitivity). Here, we apply this general approach to the Rosenzweig-MacArthur model [START_REF] Rosenzweig | Graphical Representation and Stability Conditions of Predator-Prey Interactions[END_REF], a model frequently used to study the effects of temperature on consumer-resource interactions [START_REF] Fussmann | Ecological stability in response to warming[END_REF][START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF][START_REF] Daugaard | Warming can destabilize predatorprey interactions by shifting the functional response from Type III to Type II[END_REF][START_REF] Dee | Temperature variability alters the stability and thresholds for collapse of interacting species[END_REF]. Moreover, our approach can be applied to both static and dynamic properties of consumer-resource interactions. We illustrate this application using consumer-resource biomass ratio and a stability metric quantifying the proximity to oscillations, respectively; these two variables dominating the literature on the effects of warming on consumer-resource communities [START_REF] Vasseur | A mechanistic approach for modeling temperaturedependent consumer-resource dynamics[END_REF][START_REF] Rall | Food-web connectance and predator interference dampen the paradox of enrichment[END_REF][START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF][START_REF] Betini | Temperature triggers a non linear response in resource-consumer interaction strength[END_REF]. We implement different thermal parameterisations from the literature to elucidate how the relative importance of different parameters and their varying thermal dependence shapes impact predicted effects of temperature on consumerresource interactions. Based on our results, we make four predictions that can be theoretically and empirically tested.

The dual approach

In this study we illustrate the application of the dual approach (i.e. parameter sensitivity and aggregation) using the Rosenzweig-MacArthur model with a type II functional response [START_REF] Rosenzweig | Graphical Representation and Stability Conditions of Predator-Prey Interactions[END_REF]. This model describes the rate of change in resource and consumer biomass densities:

!" !" = 𝑟(1 - ! ! )𝑅 - !" !!!!! 𝐶 (1) !" !" = (𝑒 !" !!!!! -𝑚)𝐶 (2)
R and C are the resource and consumer species biomass densities, respectively. Resource growth is logistic, with an intrinsic growth rate, r, and carrying capacity, K. Resource biomass density is limited by the consumer through a saturating Holling type II functional response with attack rate, a, and handling time, h. Consumer growth is proportional to the assimilated consumed biomass, with e the dimensionless assimilation efficiency; losses occur due to metabolic costs, m. Below we present the formulas most relevant to our study; an extensive analysis of the model is available in the supplementary information (SI 1). We chose a type II response due to its prevalence in many natural consumer-resource interactions (Jeschke et al. 2004), though our approach works for the general form of the functional response (SI 2). Additionally, the functional response can be defined with respect to attack rate and handing time, 𝑓 𝑅 = !"

!!!!! , or maximum consumption rate, J, and half-saturation

density, R 0 , 𝑓 𝑅 = !" ! ! !! (SI 3).

Stability and aggregate parameters

As demonstrated through the 'paradox of enrichment' [START_REF] Rosenzweig | Paradox of Enrichment: Destabilization of Exploitation Ecosystems in Ecological Time[END_REF], the Rosenzweig-MacArthur model produces population cycles (oscillations) with increasing energy fluxes [START_REF] Rip | Cross-ecosystem differences in stability and the principle of energy flux[END_REF]. Therefore, the coexistence equilibrium can be stable or unstable, where dynamics oscillate around the unstable equilibrium (i.e., a limit cycle). The switch from stable to unstable dynamics occurs at a Hopf bifurcation. Theoretical studies have analysed this qualitative change [START_REF] Yodzis | Body size and consumer-resource dynamics[END_REF][START_REF] Vasseur | A mechanistic approach for modeling temperaturedependent consumer-resource dynamics[END_REF][START_REF] Amarasekare | Effects of temperature on consumer-resource interactions[END_REF]) because these distinct stability regimes translate to different temporal dynamics, with oscillations leading to greater variability over time. We applied a stability metric that quantifies the tendency of the dynamics to oscillate [START_REF] Johnson | A metric for quantifying the oscillatory tendency of consumer-resource interactions[END_REF].

For our analyses we assumed dynamics had converged to the stable equilibrium or the limit cycle and determined the coexistence equilibria and the biomass ratio analytically (SI 1).

Equilibrium means zero rate of change for both consumer and resource population biomasses.

For the limit cycle, this yields the unstable equilibrium which is approximately equal to the time-averaged biomass values along the limit cycle. Thus, we set equations ( 1) and (2) to zero, solved to yield the coexistence equilibrium and retrieved the biomass ratio:

ℬ = ! ! ! ! = 𝑒𝑟 !" !!!! !! !"# !!!! (3)
In the model analysis we observed certain repeated parameter groupings (i.e., aggregate parameters) governed the dynamics (SI 1). Such aggregates have been previously used for the analysis of the Rosenzweig-MacArthur model [START_REF] Yodzis | Body size and consumer-resource dynamics[END_REF][START_REF] Vasseur | A mechanistic approach for modeling temperaturedependent consumer-resource dynamics[END_REF]. The aggregates we selected represent ecological mechanisms which can be empirically measured. These are maximal energetic efficiency, 𝜌 = ! !! , and interaction strength,

𝜅 = 𝑎ℎ𝐾 ! !! -1 .
A closer look at ρ and κ elucidates their biological meaning.

! ! is the saturation value of the functional response and thus represents the maximum consumption rate, J. Hence, ρ can be written as:

𝜌 = 𝑒 !"#$%&'( !""#$#%!&#'( !""#$#!$%& * 𝐽 !"#$!%! !"#$%&'()"# !"#$ * 1 𝑚 !"#$%&'( !"#$%&'() !"## = 𝑚𝑎𝑥𝑖𝑚𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑒𝑡𝑖𝑐 𝑔𝑎𝑖𝑛 𝑒𝑛𝑒𝑟𝑔𝑒𝑡𝑖𝑐 𝑙𝑜𝑠𝑠
ρ quantifies the energetic gain-to-loss ratio of the consumer population biomass assuming its maximum feeding rate is realised (i.e. unlimited resources). ρ was introduced by Yodzis and Innes (1992) as a key aggregate parameter to understand food web dynamics. In empirical studies, a variant of ρ termed energetic efficiency, y, has been often applied [START_REF] Rall | Temperature, predator-prey interaction strength and population stability[END_REF][START_REF] Sentis | Using functional response modeling to investigate the effect of temperature on predator feeding rate and energetic efficiency[END_REF][START_REF] Sentis | Temperature-size responses alter food chain persistence across environmental gradients[END_REF]. Unlike ρ, y is a function of the full functional response term and hence also depends on resource density, 𝑦 = ! * !(!)

! , where 𝑓 𝑅 = !" !!!!! at a specified resource density, R.

The second aggregate parameter, κ, can be rewritten in terms of the resource population density:

𝜅 = 𝑎ℎ𝐾 𝑒 𝑚ℎ -1 = 𝐾 𝑅 !
κ is the ratio of the resource equilibrium density without consumers (carrying capacity) to the resource equilibrium density with consumers. κ quantifies the effect of the consumer population on the resource population and measures interaction strength [START_REF] Berlow | Quantifying Variation in the Strengths of Species Interactions[END_REF][START_REF] Gilbert | A bioenergetic framework for the temperature dependence of trophic interactions[END_REF].

Using ρ and κ we determine the conditions for positive resource (eq. 4) and consumer (eq. 5)

densities, as well as the Hopf bifurcation (eq. 6) (SI 1):

𝜌 > 1 (4) 𝜅 > 1 (5) 𝜅 -𝜌 -1 = 0 (6)
κ>1 requires that ρ>1 (SI 1). Hence, κ>1 defines the consumer feasibility boundary. We do not consider stochastic extinctions which may occur due to large-amplitude oscillations when population biomass reaches very low values.

To determine stability, we adjusted the metric of [START_REF] Johnson | A metric for quantifying the oscillatory tendency of consumer-resource interactions[END_REF] so that it vanished at the Hopf bifurcation (SI 4). This metric, 𝒮, defines stability solely in relation to the Hopf bifurcation.

𝒮 = - 𝜅 -𝜌 -1 𝜌 -1
𝒮>0 corresponds to a stable equilibrium and 𝒮<0 to oscillations.

Sensitivity analysis

We performed a sensitivity analysis of the biomass ratio (Ɓ) and the stability metric (𝒮) with respect to the original model parameters (i.e., r, a, h, e and m). A sensitivity analysis quantifies the effect of an infinitesimal change in a parameter on the response variable.

Typically, while one parameter is being perturbed, all others are assumed to remain constant and correlations between the parameters are not explicitly considered. However, this approach can be applied in cases of correlated parameter change without a loss of accuracy.

As many environmental conditions (e.g. temperature) induce correlated changes in the parameters , then the sensitivity of the response variable (e.g. Ɓ or 𝒮) with respect to the environmental conditions can be reconstructed by combining the sensitivities of the individual parameters (SI 5.1). Different types of sensitivity indices exist such as simple sensitivity and elasticity [START_REF] Manlik | Applicability and limitations of sensitivity analyses for wildlife management[END_REF][START_REF] Caswell | Sensitivity Analysis: Matrix Methods in Demography and Ecology[END_REF]). Here we used elasticity for biomass ratio (Ɓ) and an adjusted elasticity for stability (𝒮) (SI 5.1), both dimensionless to facilitate direct comparisons between parameter sensitivities.

Elasticity is a proportional sensitivity, quantifying how a relative change in a parameter translates into a relative change in the variable; otherwise known as the log-scaled sensitivity [START_REF] Manlik | Applicability and limitations of sensitivity analyses for wildlife management[END_REF]. Thus, the elasticity of Ɓ with respect to parameter x is given by:

𝜕 ! ℬ = 𝜕ℬ ℬ 𝜕𝑥 𝑥 = 𝜕 ln ℬ 𝜕ln (𝑥) If ∂ x Ɓ=1, a relative increase of 10% in parameter x causes a relative increase of 10% in variable Ɓ. Conversely, ∂ x Ɓ =--1 implies that a relative increase of 10% in parameter x results
in relative decrease of 10% in Ɓ.

For the sensitivity of the stability metric, 𝒮, we used a variation of the elasticity. We defined the sensitivity of 𝒮 as the incremental change in 𝒮 induced by a relative change in parameter x. Our adjustment was possible due to 𝒮 being dimensionless and it prevents sensitivities from diverging to infinity close to the Hopf bifurcation without altering the outcome of our analysis (SI 5.1).

𝜕 ! 𝒮 = 𝜕𝒮 𝜕𝑥 𝑥 = 𝜕𝒮 𝜕ln (𝑥)
∂ x 𝒮=1 implies that a relative increase in parameter x of 10% translates into an absolute increase of 0.1 in 𝒮 and has a stabilising effect. If, conversely, ∂ x 𝒮=--1, the same relative increase in x would lead to decrease of 0.1 in 𝒮 with a destabilising effect.

The magnitude and sign of each sensitivity determine how strongly and in what direction the parameter perturbation impacts the variable, respectively. We used the magnitudes to rank the relative importance of all parameters. The sign provided qualitative information regarding the direction of change (increasing or decreasing the response variable).

All sensitivities could be expressed in terms of ρ and κ. Hence, sensitivities are fully determined in a plane with of ρ and κ as axes (Fig. SI 5.1,5.2). In this ρ-κ plane the sensitivities of all parameters can be ranked by magnitude, splitting the parameter space into regions where different parameters have the highest, second highest, etc. sensitivity magnitude. Here, we present figures where the regions are determined by the top-two ranked sensitivities; we do not portray changes in the rankings of the lowest sensitivities (see Fig. SI 5.4 and SI 5.5 for the complete biomass ratio and stability metric regions, respectively). The biomass ratio and stability metric have different sensitivity expressions and, therefore, produced different regions. For each variable, the regions remain fixed irrespective of the parameterisation used, because the sensitivity expressions stem from the model equations.

The ρ-κ plane provides additional information, such as the feasibility boundary (eq. 5) and the position of the Hopf bifurcation (eq. 6).

Temperature parameterisations

To demonstrate the impacts of different parameter thermal dependencies, we implemented temperature parameterisations from the literature. Maintenance respiration rates, 𝑚, have been shown to increase exponentially with temperature [START_REF] Brown | TOWARD A METABOLIC THEORY OF ECOLOGY[END_REF]. The Arrhenius equation is most often used to describe this thermal dependence [START_REF] Vasseur | A mechanistic approach for modeling temperaturedependent consumer-resource dynamics[END_REF][START_REF] Sentis | Temperature-size responses alter food chain persistence across environmental gradients[END_REF][START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF]. However, the thermal response curves of resource growth rate, r, attack rate, a , and handling time, h, have been represented either through the Arrhenius equation [START_REF] Vasseur | A mechanistic approach for modeling temperaturedependent consumer-resource dynamics[END_REF][START_REF] Binzer | Interactive effects of warming, eutrophication and size structure: impacts on biodiversity and food-web structure[END_REF] or as unimodal functions [START_REF] Amarasekare | Effects of temperature on consumer-resource interactions[END_REF][START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF][START_REF] Zhang | Food-web dynamics under climate change[END_REF][START_REF] Uiterwaal | Functional responses are maximized at intermediate temperatures[END_REF][START_REF] Zhao | Effect of water temperature on the dynamic behavior of phytoplankton-zooplankton model[END_REF]. Carrying capacity, K , and consumer assimilation efficiency, e, have a less clear connection to temperature [START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF][START_REF] Dee | Temperature variability alters the stability and thresholds for collapse of interacting species[END_REF]. We selected two parametrisations related to the ongoing debate surrounding the importance of including the decreasing part of the biological rates beyond the optimal temperature [START_REF] Pawar | Real versus artificial variation in the thermal sensitivity of biological traits[END_REF] and used these as an illustrative comparison. The 'unimodal' model had a unimodal parameterisation for r, a (both hump-shaped) and h (U-shaped), the Arrhenius equation for m (increasing), and constant K and e [START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF]. We compared this to a 'monotonic' parameterisation where all thermal dependencies (r, a, m increasing; h, K decreasing) follow the Arrhenius equation and e is constant [START_REF] Fussmann | Ecological stability in response to warming[END_REF]. Following this comparison, we plotted four additional parameterisations from the literature onto the ρ-κ plane to broaden the comparison and demonstrate the simplicity of applying the approach to empiricallyderived measurements. These consisted of two similar monotonic parameterisations [START_REF] Vucic-Pestic | Warming up the system: higher predator feeding rates but lower energetic efficiencies[END_REF][START_REF] Binzer | Interactive effects of warming, eutrophication and size structure: impacts on biodiversity and food-web structure[END_REF], one where only a was hump-shaped [START_REF] Sentis | Using functional response modeling to investigate the effect of temperature on predator feeding rate and energetic efficiency[END_REF]) and one which though monotonic, included some distinctive thermal dependenciesexponentially increasing K(T) and e(T) and constant h [START_REF] Archer | Consistent temperature dependence of functional response parameters and their use in predicting population abundance[END_REF]. We provide a description of the studies and details of their parameterisations in the supplementary material (SI 6).

We should note that not all parameterisations included the resource growth rate, r, so the biomass ratio could not be calculated in these cases. However, we could calculate ρ and κ and, hence, the biomass ratio elasticities for all parameterisations. Thus, we could determine how biomass ratio sensitivities to individual parameters changed with warming regardless of the actual biomass ratio values. By including studies which had not measured resource growth or estimated the biomass ratio we broadened the scope of the comparison of the biomass ratio sensitivities. Though this does not represent an exhaustive list of parameterisations, we were restricted to parameterisations which could be used to parametrise the Rosenzweig-MacArthur model with a type II response and whose available parameters could yield ρ and κ. 'Mild' and 'extreme' temperatures, as well as 'close' or 'far' from consumer extinction, are defined relative to each parameterisation's temperature range and feasibility boundaries, respectively. The feasible temperature range was determined by interaction strength (κ>1) with the temperature extremes corresponding to the point of consumer extinction. This condition assumes that resources have a broader thermal range than consumers (e.g., [START_REF] Rose | Does low temperature constrain the growth rates of heterotrophic protists? Evidence and implications for algal blooms in cold waters[END_REF][START_REF] West | Impacts of warming revealed by linking resource growth rates with consumer functional responses[END_REF]. If resources go extinct at temperatures consumers could withstand, then the feasibility boundary becomes dependent on resource growth rate, r [START_REF] Amarasekare | Effects of temperature on consumer-resource interactions[END_REF]. The parameterisations we present come from ectotherms, where environmental temperatures correspond to the organisms' temperatures. However, our approach can be transferred to endotherms as it does not depend on a specific thermal parametrisation.

Sensitivities depend on proximity to thermal boundaries

Biomass ratio: always most sensitive to e and m

We analytically obtained four groups of biomass ratio elasticity magnitudes, ∂ e Ɓ=|∂ m Ɓ|, ∂ K Ɓ=∂ a Ɓ, ∂ r Ɓ and ∂ h Ɓ. e and m always have the largest elasticity and hence the strongest relative impact on the biomass ratio (∂ e Ɓ=|∂ m Ɓ| >∂ K Ɓ, ∂ a Ɓ, ∂ r Ɓ, |∂ h Ɓ|, Table 1). Increasing e increases the biomass ratio (∂ e Ɓ >0), increasing m reduces it (∂ m Ɓ <0). K and a have equal and positive elasticities (∂ K Ɓ=∂ a Ɓ>0), both increasing the biomass ratio. The elasticity of r is constant, ∂ r Ɓ=1; a directly proportional positive effect on Ɓ. Increasing h reduces the biomass ratio, ∂ h Ɓ<0. These are general results, independent of any model parameterisation (with temperature or otherwise) following directly from the Rosenzweig-MacArthur model's equations.

Biomass ratio: high r elasticity far from thermal boundaries

Both the 'unimodal' and 'monotonic' temperature parameterisations produced a unimodal biomass ratio thermal dependence (Fig. 2). The unimodal parameterisation induced thermal boundaries to the community at both low (1℃) and high (33℃) temperatures (Fig. 2a). The biomass ratio exceeded 1 for most temperatures (higher consumer than resource biomass), peaked at 14℃ around Ɓ≈5.5 and decreased rapidly to 0 as it approached both thermal boundaries (low and high temperature extremes). The biomass ratio of the monotonic parameterisation (Fig. 2b) increased with warming from low temperatures, peaked at Ɓ≈0.19, before decreasing to 0 at high temperatures (27.5℃). The two parameterisations are derived from different systems, hence the different temperature ranges.

In both parameterisations, sensitivity to e and m was highest throughout (Fig. 2c,d) -as expected from the analytical findings. Elasticities were split into two groups at mild temperatures: e, m and r had the highest elasticity with ∂ e Ɓ=|∂ m Ɓ|≈∂ r Ɓ=1, while a, h and K elasticities were very low. Approaching the temperature extremes all elasticities besides ∂ r Ɓ diverged; ∂ h Ɓ diverged faster than ∂ K Ɓ=∂ a Ɓ in the unimodal parameterisation (Fig. 2c), while the opposite occurred in the monotonic one (Fig. 2d).

Expressing the elasticities in terms of ρ and κ (Table 1) reduces the sensitivity analysis to two dimensions. Ranking the elasticity magnitudes creates distinct regions in the ρ-κ plane which correspond to different elasticity ranking orderings and provide a mechanistic overview of which elasticities dominate where (Fig. 3a). e and m always have the highest elasticity, so the three regions reflect changes in the second highest-ranked elasticity. Regions adjacent to consumer extinction (κ=1) have high sensitivity to either h (red region) or to a and K (yellow region). r elasticity is highest in the region farthest from consumer extinction (orange region).

The two temperature parameterisations were mapped onto this plane by calculating their ρ and κ values (Fig 3b andc). Despite the two trajectories being markedly different, both occupied the region where r ranked second highest for most temperatures. The unimodal parameterisation produced a unimodal trajectory, crossing the consumer extinction threshold at low and high temperatures (Fig. 3b). The monotonic parameterisation's trajectory converged monotonically towards consumer extinction with increasing temperature (Fig. 3c).

All other parameterisations from the literature also occupied the region of high r elasticity for most temperatures, far from their thermal boundaries (Fig. 4). Three monotonic parameterisations produced monotonic trajectories (Fig. 4a,b,c) which started in the region of high r elasticity and converged monotonically towards consumer extinction (κ=1). With a hump-shaped thermal dependence of attack rate, a unimodal trajectory emerged (Fig. 4d). At low temperatures it occupied the region of high 𝑟 elasticity but moved away from consumer extinction. With further warming, the trajectory switched direction and followed the same path as the monotonic parameterisations, crossing the consumer extinction boundary. A unimodal thermal dependence for attack rate (hump-shaped) and handling time (U-shaped) (Fig. 4e), induced extinctions at low and high temperatures imposing a unimodal trajectory. Unlike all previous parameterisations, the trajectory crossed the extinction threshold in the region of high h elasticity. The final parameterisation, though monotonic, yielded unimodal trajectories (Fig. 4f). A monotonically increasing K(T) (as opposed to decreasing in the other monotonic parameterisations and constant in the unimodal ones) initially forced the trajectory away from consumer extinction, albeit within the region of high r elasticity. Consumer energetic efficiency ρ decreased, pushing consumers towards extinction, thus forcing an abrupt decline towards the consumer boundary.

Stability most sensitive either to 𝑒 and 𝑚 or to 𝑎 and 𝐾

Similarly to the biomass ratio, the analytical approach for the stability sensitivities yielded results conserved independently of the temperature parameterisations (Table 1): equal sensitivity magnitudes pairwise for e and m and for a and K (i.e., |∂ e 𝒮|=∂ m 𝒮 and |∂ K 𝒮|=|∂ a 𝒮|), negative stability sensitivities of e, a and K (∂ e 𝒮, ∂ a 𝒮, ∂ K 𝒮<0) implying they destabilise dynamics, a positive sensitivity of m (∂ m 𝒮>0) indicating a stabilising effect. h can be either stabilising or destabilising and r does not affect the stability regime (∂ r 𝒮=0).

The unimodal temperature parameterisation exhibited oscillations (𝒮<0) over most temperatures (Fig. 5a). Only at low and high thermal extremes did dynamics briefly stabilise prior to consumer extinction. The monotonic temperature parametrisation produced oscillations at low temperatures (𝒮<0), crossed a Hopf bifurcation at 17℃ and dynamics were stable (𝒮>0) thereafter (Fig. 5b). In both cases, stability close to consumer extinction was most sensitive to consumer assimilation efficiency, e, and metabolism, m, (|∂ e 𝒮|=∂ m 𝒮) followed by handling time, h (Fig. 5c,d). Moving away from the thermal boundaries, attack rate, a, and carrying capacity, K, increased in relative importance. Furthest away from the thermal boundaries, stability was most sensitive to changes a and K, followed by h. Even though ∂ h 𝒮 did not rank highest in any temperature range, it was a close second both at the temperature extremes (second to |∂ e 𝒮|=∂ m 𝒮) or furthest away from them (second to |∂ K 𝒮|=|∂ a 𝒮|). Additionally, h switched from destabilising at mild temperatures (∂ h 𝒮<0) to stabilising (∂ h 𝒮>0) close to the temperature extremes (Fig. 5c,d, Fig. S5.2).

The ρ--κ plane for the stability metric was split into four regions; in two regions closest to consumer extinction, |∂ e 𝒮|=∂ m 𝒮 were the largest sensitivities (Fig. 6a, red and yellow regions) and in the two regions furthest from consumer extinction, |∂ K 𝒮|=|∂ a 𝒮|ranked highest (Fig. 6a, green and blue regions). Additionally, the Hopf bifurcation (Fig. 6a, dashed curve) split the plane into stable equilibrium and oscillation regions. Corresponding to the general findings, stability in the two reference ('unimodal' and 'monotonic') parameterisations was most sensitive to changes in e and m at the thermal extremes -close to consumer extinctionand to a and K at milder temperatures -far from consumer extinction (Fig. 6b,c). The unimodal trajectory occupied the region of oscillations for most temperatures, crossing the Hopf bifurcation twice close to consumer extinction, once at low and once at high temperatures (Fig. 6b, blue region). The monotonic trajectory started in the region of oscillations and moved into the stable region with warming, crossing the Hopf bifurcation far from the thermal extreme (Fig. 6c, yellow region).

Parameter thermal dependencies impact warming-stability relationships

Plotting the other temperature parameterisations' trajectories onto the ρ-κ plane reproduced the same patterns with respect to the stability metric's sensitivity (Fig. 7): stability was most sensitive to e and m at the thermal extremes and to a and K far from the extremes.

Significantly, the trajectories revealed the impact of the thermal dependence shape of individual parameters on the warming-stability relationship. In three monotonic parameterisations, warming stabilised the dynamics (Fig. 7 a,b,c). In the cases, when oscillations did take place, these occurred at low temperatures (Fig 7a resident prey,b,c) and dynamics crossed the Hopf bifurcation far from the thermal boundary. In the case with two enrichment levels (Fig. 7c), the high enrichment scenario required higher temperatures to stabilise the dynamics. For the unimodal trajectory with hump-shaped attack rate (Fig. 7d), warming at low temperatures pushed the dynamics towards (low enrichment) or deeper into (high enrichment) the region with oscillations (i.e., destabilised dynamics). Here too, the destabilising impact of enrichment was evident. However, further warming switched the direction of the trajectory. Subsequently, both ρ and κ decreased. κ declined much faster, forcing the dynamics towards the stable region and eventually consumer extinction. Both the switch in the trajectory direction and the Hopf bifurcation (high enrichment scenario) occurred at mild temperatures, in the region of high a and K sensitivity. In the parameterisation with both a (hump-shaped) and h (U-shaped) unimodal (Fig. 7e), the Hopf bifurcation occurred close to the thermal boundaries, where κ increased (low temperatures) or decreased (high temperatures) much faster than ρ. The dynamics were oscillatory for most temperatures, with the switch in the trajectory's direction occurring in the region of highest sensitivity to a and K. The final parameterisation's trajectories were characterised by a negative relationship between ρ and κ (Fig. 7f). Driven by the positive thermal dependence of carrying capacity, warming increased κ and destabilised dynamics which oscillated for most temperatures.

Discussion

Research on the impacts of warming on consumer-resource interactions has yielded mixed results [START_REF] Vasseur | A mechanistic approach for modeling temperaturedependent consumer-resource dynamics[END_REF][START_REF] Englund | Temperature dependence of the functional response[END_REF][START_REF] Rall | Universal temperature and body-mass scaling of feeding rates[END_REF][START_REF] Gilbert | A bioenergetic framework for the temperature dependence of trophic interactions[END_REF][START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF]. Resolving this debate and improving predictions has become even more pressing as most ecosystems face increased temperatures [START_REF] Easterling | Climate extremes: observations, modeling, and impacts[END_REF][START_REF] Walther | Ecological responses to recent climate change[END_REF][START_REF] Root | Fingerprints of global warming on wild animals and plants[END_REF][START_REF] Parmesan | Ecological and Evolutionary Responses to Recent Climate Change[END_REF]). Here, we developed an approach to improve and simplify predictions on the impacts of warming on consumer-resource interactions. This approach integrates two pathways: (1) a sensitivity analysis to identify the key biological parameters whose variations have the largest relative impact on community properties at a given temperature, and (2) aggregate parameters (maximal energetic efficiency, ρ, and interaction strength, κ) to increase explanatory power. We used the Rosenzweig-MacArthur model with a type II functional response, and applied the approach to consumer-resource biomass ratio and a stability metric quantifying the propensity for oscillations [START_REF] Johnson | A metric for quantifying the oscillatory tendency of consumer-resource interactions[END_REF]. Therefore, our insights apply to study systems well-described by the Rosenzweig-MacArthur model. Our analyses revealed that the relative significance of different parameter groupings is determined by the proximity of the consumer to its thermal boundaries. We, further, elucidated how differences in the shape of the thermal dependence curves of individual parameters qualitatively impact predictions. We used empiricallyderived thermal dependence curves of biological parameters from the literature to illustrate this.

We focus our discussion on the formulation of four testable predictions arising from our results. For each prediction, we present its implications and rationale. Then, we discuss the empirical measurement of the aggregate parameters and present important subtleties and potential extensions of our approach.

Prediction 1: Resource growth rate regulates biomass distribution at mild temperatures Implications: We showed that the relative dominance of consumer assimilation efficiency, metabolism and resource growth rate in driving changes in biomass distributions should manifest itself in any consumer-resource community far from its feasibility boundaries, assuming these communities are well-described by the Rosenzweig-MacArthur model (Fig. 3a). Due to the agreement about the thermal dependence of metabolism [START_REF] Rall | Temperature, predator-prey interaction strength and population stability[END_REF][START_REF] Fussmann | Ecological stability in response to warming[END_REF][START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF]) and the negligible -if any-change of assimilation efficiency with warming [START_REF] Dell | Systematic variation in the temperature dependence of physiological and ecological traits[END_REF], differences in the thermal performance curve of resource growth rate will strongly impact biomass ratio predictions. Therefore, improved predictions about the impacts of warming on biomass distributions at mild temperatures necessitate the accurate description of the thermal dependence of resource growth rate.

Reasoning: Far from the community thermal boundaries, consumer assimilation efficiency, metabolism and resource growth rate always had the greatest elasticity with an almost equal relative impact on biomass ratio (∂ e Ɓ=|∂ m Ɓ|≈∂ r Ɓ=1, Fig. 2c,d and Fig. S5.1). Increasing metabolism reduced biomass ratios (Table 1), which is likely to be a universal response across ecosystems, given the positive exponential dependence of metabolism on temperature across organisms [START_REF] Gilooly | Effects of Size and Temperature on Metabolic Rate[END_REF][START_REF] Brown | TOWARD A METABOLIC THEORY OF ECOLOGY[END_REF][START_REF] Rall | Universal temperature and body-mass scaling of feeding rates[END_REF] but see [START_REF] Ehnes | Phylogenetic grouping, curvature and metabolic scaling in terrestrial invertebrates[END_REF]. Conversely, assimilation efficiency increased biomass ratios but has either been assumed to be unaffected by temperature changes [START_REF] Vasseur | A mechanistic approach for modeling temperaturedependent consumer-resource dynamics[END_REF][START_REF] Sentis | Temperature-size responses alter food chain persistence across environmental gradients[END_REF][START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF] or has yielded a weak temperature-dependence [START_REF] Wootton | Effects of temperature and ration level on the growth and food conversion efficiency of Salmo gairdneri, Richardson[END_REF][START_REF] Handeland | The effect of temperature and fish size on growth, feed intake, food conversion efficiency and stomach evacuation rate of Atlantic salmon post-smolts[END_REF][START_REF] Lang | Temperature and consumer type dependencies of energy flows in natural communities[END_REF][START_REF] Daugaard | Warming can destabilize predatorprey interactions by shifting the functional response from Type III to Type II[END_REF] with negligible change compared to other parameters. Increasing resource growth rate also increased the biomass ratio. However, evidence on the shape of resource growth's thermal response remains inconclusive: it can either increase exponentially with temperature [START_REF] Savage | Effects of body size and temperature on population growth[END_REF] or decrease abruptly beyond the thermal optimum [START_REF] Dannon | Functional response and life history parameters of Apanteles taragamae, a larval parasitoid of Maruca vitrata[END_REF][START_REF] Thomas | A global pattern of thermal adaptation in marine phytoplankton[END_REF]. Since the biomass ratio is directly proportional to the resource growth rate (∂ r Ɓ=1, Table 1), it will be strongly affected by the values and shape of the resource growth rate thermal performance curve. Given the consensus surrounding the temperature-dependence of metabolism and the minor scale of potential change in assimilation efficiency with temperature, our findings emphasise the significance of correctly parameterising the resource growth rate when aiming to predict biomass distribution changes due to warming at mild temperatures.

Prediction 2: Interaction strength determines consumer survival with increasing temperatures.

Implications: If resources have a broader thermal range compared to consumers [START_REF] Rose | Does low temperature constrain the growth rates of heterotrophic protists? Evidence and implications for algal blooms in cold waters[END_REF][START_REF] West | Impacts of warming revealed by linking resource growth rates with consumer functional responses[END_REF], the thermal boundaries of the community can be determined by measuring solely the thermal dependence of interaction strength, κ. This quantity -the ratio of the resource equilibrium density without consumers (carrying capacity) to the resource equilibrium density with consumers -can be determined experimentally [START_REF] Berlow | Interaction strengths in food webs: Issues and opportunities[END_REF] or through observations, facilitating predictions and cross-system comparisons thereof.

Reasoning: Close to the consumer extinction boundary, consumer survival becomes extremely sensitive to all parameters apart from resource growth (Fig. 2,S5.1), making accurate predictions challenging. Currently, consumer survival has been inferred through energetic efficiency -the effective energetic gain of consumers at a certain resource density -which requires determining the thermal dependence of the functional response [START_REF] Vucic-Pestic | Warming up the system: higher predator feeding rates but lower energetic efficiencies[END_REF][START_REF] Archer | Consistent temperature dependence of functional response parameters and their use in predicting population abundance[END_REF]. Not only is the functional response's thermal dependence hotly contested [START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF][START_REF] Uiterwaal | Functional responses are maximized at intermediate temperatures[END_REF], but this uncertainty will be exacerbated by its extremely high sensitivity at the community's thermal boundaries. We showed there exists an alternative, empirically more direct and theoretically more robust metric to determine consumer survival, and hence community feasibility.

Interaction strength -the relative values of resource equilibrium without and with consumers [START_REF] Berlow | Quantifying Variation in the Strengths of Species Interactions[END_REF][START_REF] Berlow | Interaction strengths in food webs: Issues and opportunities[END_REF][START_REF] Gilbert | A bioenergetic framework for the temperature dependence of trophic interactions[END_REF]) -provides the necessary condition for consumer survival (κ>1), when resources are thermal generalists compared to consumers.

This provides an accurate threshold and represents a measurable quantity that can be standardised across experimental designs and study systems [START_REF] Berlow | Interaction strengths in food webs: Issues and opportunities[END_REF].

Prediction 3: Warming reduces community stability at low and mild temperatures

Implications: This prediction rests on important assumptions: that resources have a broader thermal range, that organisms currently experience temperatures below their optima [START_REF] Pawar | Real versus artificial variation in the thermal sensitivity of biological traits[END_REF]) and that the functional response is of type II with a unimodal thermal dependence [START_REF] Rall | Universal temperature and body-mass scaling of feeding rates[END_REF][START_REF] Sentis | Using functional response modeling to investigate the effect of temperature on predator feeding rate and energetic efficiency[END_REF][START_REF] Kuiters | Diversity-stability relationships in plant communities of contrasting habitats[END_REF][START_REF] West | Impacts of warming revealed by linking resource growth rates with consumer functional responses[END_REF][START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF][START_REF] Uiterwaal | Functional responses are maximized at intermediate temperatures[END_REF]. We deem these assumptions realistic based on the literature; therefore, we argue that consumer-resource interactions at low and mild temperatures will be destabilised by warming. At higher temperatures, warming should always enhance stability.

Reasoning: Stability in the context of consumer-resource interactions has predominantly referred to a qualitative distinction between stable and oscillating dynamics [START_REF] Rosenzweig | Graphical Representation and Stability Conditions of Predator-Prey Interactions[END_REF][START_REF] Yodzis | Body size and consumer-resource dynamics[END_REF][START_REF] Vasseur | A mechanistic approach for modeling temperaturedependent consumer-resource dynamics[END_REF]. We based our analysis on an adjusted stability metric which quantifies the tendency of dynamics to oscillate (Johnson & Amarasekare 2015, SI 4). When comparing existing temperature parameterisations, we found that in most monotonic parameterisations (increasing metabolism and attack rate, decreasing handling time and carrying capacity, assimilation efficiency constant), warming always (i.e., monotonically) stabilised dynamics (Fig. 7a,b,c). The single exception arose when warming and carrying capacity increased simultaneously, which destabilised dynamics (Fig. 7f). Carrying capacity has been described as a proxy for enrichment and its destabilising effect has been established whether independently of temperature [START_REF] Rosenzweig | Paradox of Enrichment: Destabilization of Exploitation Ecosystems in Ecological Time[END_REF] or as antagonistic to warming [START_REF] Binzer | Interactive effects of warming, eutrophication and size structure: impacts on biodiversity and food-web structure[END_REF]. When at least one parameter in the functional response had a unimodal thermal dependence (i.e., hump-shaped attack rate or U-shaped handling time), this yielded a unimodal warming-stability relationship (Fig. 7d,e).

Significantly, the divergence between the unimodal and (most) monotonic parameterisations in the predicted effect of warming on stability manifested itself at low or mild, rather than high temperatures (Fig. 6,7). This pattern originates in the impact of the parameters with unimodal thermal dependencies on stability. Attack rate is destabilising (Table 1,McCann 2011). Thus, a hump-shaped thermal dependence of attack rate destabilises dynamics with warming below the thermal optimum and stabilises dynamics beyond it. Handling time is stabilising close to the thermal extremes (Fig. 6c, S5.2). A U-shaped handling time will rapidly decrease with warming from low temperatures, which is strongly destabilising; a corresponding steep increase at high temperatures produces a strong stabilising effect. Thus, warming at high temperatures will always be stabilising. However, at lower temperatures, unimodal and monotonic thermal dependencies produce contrasting warming-stability relationships. Therefore, the thermal dependence shape of the functional response combined with the temperatures currently experienced by communities relative to their optimal temperature will determine the impact of warming on stability [START_REF] Betini | Temperature triggers a non linear response in resource-consumer interaction strength[END_REF].

Prediction 4: Warming stabilises dynamics only when interaction strength decreases faster than maximal energetic efficiency Implications: The combination of ρ -the energetic gain-to-loss ratio of consumers given unlimited resources -and κ -interaction strength -accurately describes the warmingstability relationship with no recourse to the thermal dependence shapes of individual parameters, the current temperatures relative to the thermal optima, or the proximity to the thermal boundaries of the community. Therefore, differential responses of resources and consumers to warming [START_REF] Dell | Temperature dependence of trophic interactions are driven by asymmetry of species responses and foraging strategy[END_REF]) will be encompassed by the thermal dependence of the aggregates -assuming the consumer-resource system is well-described by the Rosenzweig-MacArthur model. The Hopf bifurcation condition (eq. 6) dictates that κ should decrease faster than ρ for warming to stabilise consumer-resource interactions. Thus, measuring ρ and κ directly can increase the accuracy of warming-stability predictions and simplify cross-system comparisons.

Reasoning: Decreasing energetic efficiency or interaction strength have been considered equivalent to increasing stability [START_REF] Rall | Food-web connectance and predator interference dampen the paradox of enrichment[END_REF][START_REF] Rall | Temperature, predator-prey interaction strength and population stability[END_REF][START_REF] Sentis | Using functional response modeling to investigate the effect of temperature on predator feeding rate and energetic efficiency[END_REF]. Thus, estimates of consumer energetic efficiency or interaction strength based on empirically-derived thermal dependence curves of individual rates (e.g. ingestion rate, attack rate, metabolic rate) have been used to infer the impacts of warming on stability [START_REF] Rall | Temperature, predator-prey interaction strength and population stability[END_REF][START_REF] Rall | Universal temperature and body-mass scaling of feeding rates[END_REF][START_REF] Vucic-Pestic | Warming up the system: higher predator feeding rates but lower energetic efficiencies[END_REF][START_REF] Fussmann | Ecological stability in response to warming[END_REF]). However, this raises two significant issues. On the one hand, even subtle changes in the thermal dependence shapes of individual parameters can yield all possible outcomes [START_REF] Amarasekare | Effects of temperature on consumer-resource interactions[END_REF]. On the other hand, reducing the analysis of stability to a single aggregate parameter has limitations. [START_REF] Gilbert | A bioenergetic framework for the temperature dependence of trophic interactions[END_REF] described the warmingstability relationship with a single aggregate, interaction strength, but their approach was based on a type I functional response and its predictions do not work well in type II or III scenarios [START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF]. [START_REF] Johnson | A metric for quantifying the oscillatory tendency of consumer-resource interactions[END_REF] and [START_REF] Amarasekare | Effects of temperature on consumer-resource interactions[END_REF] attained a single aggregate parameter to reduce the complexity of their explorations; however, this lacks descriptive power of the dynamics close to the community's thermal boundaries (SI 4). Our analysis in the ρ-κ plane suggests that stability cannot be reduced to a single aggregate parameter nor does a decrease in either one or both of ρ and κ suffice to stabilise dynamics. In fact, both ρ and κ can decrease with warming while dynamics become destabilised. A stabilising effect of warming requires not only a concurrent reduction in ρ and κ, but also the latter to decrease faster. Critically, both ρ and κ represent biological quantities which can be consistently measured across study systems.

Working with the aggregate parameters

Working directly with the two aggregate parameters, maximal consumer energetic efficiency, ρ, and interaction strength, κ, can simplify empirical measurements and improve the accuracy of theoretical predictions, particularly for field data and experiments, as we argue below. To determine the thermal dependence of maximal consumer energetic efficiency and interaction strength, one can measure consumer population growth given unlimited resources and resource population density in presence and absence of consumers at different temperatures, respectively. These measurements can be performed in the lab and the field. Interaction strength is commonly determined in field experiments where consumers are excluded [START_REF] Berlow | Interaction strengths in food webs: Issues and opportunities[END_REF][START_REF] Wootton | Effects of temperature and ration level on the growth and food conversion efficiency of Salmo gairdneri, Richardson[END_REF][START_REF] Novak | Estimating interaction strengths in nature: experimental support for an observational approach[END_REF][START_REF] Estes | Trophic downgrading of planet earth[END_REF]. Consumer energetic gain-to-loss ratio under effectively unlimited resources is more rarely estimated.

However, it can be derived from consumer population net growth and metabolism and mortality, quantities measured commonly in the field and in the lab [START_REF] Hanson | Empirical prediction of crustacean zooplankton biomass and profundal macrobenthos biomass in lakes[END_REF][START_REF] Stemberger | Body Size, Food Concentration, and Population Growth in Planktonic Rotifers[END_REF][START_REF] Lampert | Phytoplankton control by grazing zooplankton: A study on the spring clear water phase[END_REF]. Moreover, confounding factors in field measurements of the population-level aggregates should generate less uncertainty compared to that of measuring multiple individual parameters, where uncertainty propagates and often generates large uncertainty in model predictions (e.g. [START_REF] Sentis | Thermal acclimation modulates the impacts of temperature and enrichment on trophic interaction strengths and population dynamics[END_REF]. Therefore, working directly with the aggregate parameters can be both simpler and lead to more accurate predictions in the field. On the other hand, measuring the individual parameters in the lab has well-established protocols and a history of reliable outputs, with measurements requiring only short-term experiments as opposed to the aggregates.

The choice between measuring the aggregate or the individual parameters will be informed by the questions and objectives of each study. The aggregates describe population-level mechanisms of consumer-resource interactions, while the individual parameters correspond to physiological or behavioural processes of individual organisms scaled up to the population level. As we argued, the aggregates can provide more accurate predictions for field measurements whereas individual parameters can be accurately measured in the lab. This does raise the question whether measurements in a controlled laboratory environment can represent noisier conditions in the field. It would be useful to compare directly measured aggregate parameters to aggregate parameter values derived from the individual parameters to determine how well predictions based on individual rates capture the dynamics of the system. Regardless of the choice, our approach provides the tools for both pathways: studies working with individual parameters will benefit from identifying the most important parameters to measure, while aggregate parameter datapoints can be directly mapped onto the ρ-κ landscape.

Subtleties and extensions

The sensitivity analysis quantified the sensitivity of the model variables to infinitesimal parameter changes. Therefore, applying its insights to data should take into consideration the scales of parameters in the temperature range of interest and potential uncertainties in the parameter estimates (Manlik et al. 2018, Fig. Sx). Hence, our argument for the reduced significance of the thermal dependence of assimilation efficiency in driving changes in biomass distributions, despite its high sensitivity.

Regarding the stability of consumer-resource interactions, the ρ-κ plane helped visualise the stabilising effect of a type III functional response (Fig. S2.1), which has both theoretical and empirical support [START_REF] Sarnelle | Type III Functional Response in Daphnia[END_REF][START_REF] Kalinkat | Body masses, functional responses and predator-prey stability[END_REF][START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF][START_REF] Daugaard | Warming can destabilize predatorprey interactions by shifting the functional response from Type III to Type II[END_REF]. For the type II response, the defining role of the functional response (attack rate) and the carrying capacity has been widely documented [START_REF] Rosenzweig | Paradox of Enrichment: Destabilization of Exploitation Ecosystems in Ecological Time[END_REF][START_REF] Amarasekare | Effects of temperature on consumer-resource interactions[END_REF][START_REF] Johnson | A metric for quantifying the oscillatory tendency of consumer-resource interactions[END_REF][START_REF] Binzer | Interactive effects of warming, eutrophication and size structure: impacts on biodiversity and food-web structure[END_REF]; we add the important caveat that this is the case only far from consumer extinction (Fig. 6a).

Finally, relaxing certain assumptions can extend our approach. Considering the scenario where the consumer has a broader thermal niche relative to that of the resource will make the thermal limits of coexistence dependent on resource growth [START_REF] Amarasekare | Effects of temperature on consumer-resource interactions[END_REF].

Considering the extinction of populations with very low abundances to account for stochastic extinctions can define a realisable coexistence range within the feasible parameter space.

Breaking down the original model parameters (e.g. handling time includes the handling and ingestion of prey) could facilitate our understanding of the role of more fundamental physiological processes in the dynamics. Finally, climate change will lead to stronger fluctuations in temperatures (IPCC 2013), which have been shown to alter predictions in consumer-resource dynamics [START_REF] Vasseur | Increased temperature variation poses a greater risk to species than climate warming[END_REF][START_REF] Dee | Temperature variability alters the stability and thresholds for collapse of interacting species[END_REF]. This makes the inclusion of temperature variability an important next step.

Conclusions

Warming will have significant, but as yet uncertain impacts on consumer-resource interactions which underpin the structure and functioning of ecosystems. We presented an approach that will help to improve the accuracy of predictions and reconcile divergent results by facilitating cross-system comparisons. This approach first determines the parameters whose variations have the largest effect on community properties. Second, it simplifies analyses to a two-dimensional plane of mechanistically tractable aggregate parameters; maximal consumer energetic efficiency and interaction strength. Applying it to consumerresource biomass ratio and stability, we showed that close to the consumer extinction boundary (i.e., at temperature extremes) both variables are most sensitive to changes in consumer assimilation efficiency and metabolism. Far from the boundary (i.e., mild temperatures), biomass ratio is most sensitive to resource growth rate, consumer assimilation efficiency and metabolism. This yielded our first prediction, that resource growth rate regulates biomass distributions at mild temperatures. The consensus around the thermal dependence of metabolism and the limited potential impact of warming on assimilation efficiency, underscore the importance of correctly measuring the thermal dependence of resource growth rate. Using the two aggregate parameters also simplified the study of important properties of consumer-resource interactions. From this followed our second prediction, that the thermal boundaries of the community are defined by interaction strength alone. In terms of stability, we demonstrated that a unimodal thermal dependence of attack rate or handling time alters predictions of warming-stability relationships below the thermal optimum, where many organisms may be currently living. Hence our third prediction, that initial increases in mean temperatures will destabilise consumer-resource interactions.

Significantly, our approach elucidates how the thermal dependence of stability can be comprehensively characterised by maximal energetic efficiency and interaction strength values. This produced our fourth prediction; a faster reduction of interaction strength than of maximal energetic efficiency with warming is necessary for dynamics to stabilise. Finally, we demonstrated the potential for targeted experiments to measure the thermal dependencies of maximal energetic efficiency and interaction strength to improve predictions. Ultimately, we show that any temperature parameterisation fitted to the Rosenzweig-MacArthur model can be mapped onto the aggregate parameter plane, revealing its stability landscape, providing a mechanistic interpretation for its predictions and allowing for the cross-system comparison of these predictions.

Tables and Figures

Table 1. the Rosenzweig-MacArthur model [START_REF] Rosenzweig | Graphical Representation and Stability Conditions of Predator-Prey Interactions[END_REF] or its bioenergetic equivalent [START_REF] Yodzis | Body size and consumer-resource dynamics[END_REF] have been used most commonly for ectotherm consumerresource pairs. 2a) The current approach is to experimentally measure the response of parameters along an environmental gradient, e.g. the thermal dependence of the resource population maximal growth rate with critical temperatures, CT min , CT max , and the thermal 1). e and m elasticities always rank first. Close to consumer extinction h ranks second highest at low ρ (ρ < 2, red region) and a and K at higher ρ (ρ > 2, yellow region). r ranks second highest far from consumer extinction (orange region). The plane includes the feasibility boundary (κ=1) and the Hopf bifurcation (dotted curve splitting the plane into stable equilibrium and oscillations). 1). e and m rank first close to consumer extinction; h ranks second highest closest to consumer extinction (red region). The sensitivity of stability to a and K increases moving away from consumer extinction. a and K sensitivity ranks second (yellow region), and then first moving further away. Initially e and m rank second (green region) before h becomes significant (blue region). The plane includes the feasibility boundary (κ=1) and the Hopf bifurcation (dotted curve splitting the plane into stable equilibrium and oscillations).

𝜕 ! ℬ = 𝜕 ln ℬ 𝜕ln (𝑥) 𝜕 ! 𝒮 = 𝜕𝒮 𝜕ln (𝑥) r 1 0 K 1 𝜅 -1 - 𝜅 𝜌 -1 a 1 𝜅 -1 - 𝜅 𝜌 -1 h - 1 𝜌 -1 𝜅 -1 𝜅 1 -𝜌 + 2𝜌 𝜌 -1 ! e 𝜌 𝜌 -1 𝜅 -1 + 1 - 2𝜌 𝜌 -1 ! m - 𝜌 𝜌 -1 𝜅 -1 -1 2𝜌 𝜌 -1 !

  Sensitivities of the biomass ratio (∂ x Ɓ) and of the stability metric (∂ x 𝒮) with respect to the six original model parameters. All sensitivities are expressed in terms of ρ and κ.

Figure 1 .

 1 Figure 1. Illustration of the current and new dual approaches to predict the impact of global

Figure 2 .

 2 Figure 2. Consumer-resource biomass ratios for the (a) unimodal and (b) monotonic

Figure 3 .

 3 Figure 3. (a) Biomass ratio elasticity rankings in the ρ-κ plane. The plane is split into regions

  For the (b) unimodal and (c) monotonic parameterisations from the literature, the thermal dependencies of 𝜌 = ! !! and 𝜅 = ! ! ! were calculated. This yielded a trajectory for each parameterisation (solid black line). The paths of the trajectories demonstrate the elasticity of the biomass ratio along the temperature gradient for each parameterisation.

Figure 4 .

 4 Figure 4. Trajectories of the six empirical temperature parameterisations in the ρ-κ plane: a)

Figure 5 .

 5 Figure 5. The thermal dependence of the stability metric, 𝒮, for the (a) unimodal and (b)

Figure 6 .

 6 Figure 6. (a) Stability metric sensitivity rankings in the ρ-κ plane. The plane is split into

  For the (b) unimodal and (c) monotonic parameterisations from the literature, the thermaldependencies of 𝜌 = ! !! and 𝜅 = ! ! !were calculated. This yielded a trajectory for each parameterisation (solid black line). The paths of the trajectories demonstrate the dynamical regime and the sensitivity of the stability metric along the temperature gradient for each parameterisation.

Figure 7 .

 7 Figure 7. Trajectories of the six empirical temperature parameterisations in the ρ--κ plane: a)

Acknowledgments

We thank three anonymous reviewers and the editor for their constructive feedback. This research is supported by the FRAGCLIM Consolidator Grant, funded by the European Research Council under the European Union's Horizon 2020 research and innovation programme (Grant Agreement Number 726176), and by the "Laboratoires d'Excellences (LABEX)" TULIP (ANR-10-LABX-41).

optimum, T opt . These measurements are used to parameterise the model. Importantly, not all parameters are measured, but rather those which are considered significant (e.g. consumer feeding and metabolic rates for warming-stability relationships). Assuming the remaining parameter values, the model is then used to generate predictions. 2b) Our new dual approach aims to increase the accuracy of predictions and facilitate their comparison. First a sensitivity analysis determines which parameters have the greatest relative impact on the community property of interest along the environmental gradient. Then, aggregate parameters which represent biologically measurable quantities are used to express all sensitivities and determine the dynamics. Collapsing analyses to the two aggregate parameters reduces complexity and increases mechanistic tractability. This facilitates the choice of which parameters need to be measured. 3) Through the empirical determination of the most appropriate parameters (either from the original model parameters of the aggregate parameters themselves) and the reduction in the number of measurements required, prediction accuracy improves. The advantages of the new dual approach are twofold. First, as the sensitivity analysis will have identified the most impactful parameters, the source of divergence in predictions can be isolated. Second, the aggregates represent standardised measurable population-level indicators across systems, making theoretical or empirical predictions directly comparable.


Davide Frey

Rachid Guerraoui

Anne-Marie Kermarrec

Maxime Monod

Vivien Quéma

Heterogeneous Gossiping

Gossip is effective in large-scale systems

The increasing diffusion of low-cost network connectivity has been challenging software designers to build scalable systems spanning large masses of nodes distributed over wide area networks. A number of applications, ranging from content distribution to data storage and retrieval, demand efficient techniques to disseminate and locate data in such challenging environments. Beyond the scale of these systems, developers must manage nodes with limited and heterogeneous resources that may join and leave at arbitrary times, exhibit malicious behaviors and whose characteristics often vary over time as a result of discontinuous usage patterns in the presence of multiple applications.

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Yet, gossip is uniform

Whereas gossip protocols are effectively extremely robust, they usually do not account for heterogeneity. Nodes typically have the same number of gossip targets (fanout) and the same dissemination period. They therefore send exactly the same number of messages and are, in this sense, inherently load-balanced. However, in a large scale heterogeneous setting, this uniform distribution of load is not always appropriate. In practice, nodes frequently exhibit significant differences in their capabilities. These may result from differences in computational power as the same application may involve small devices together with powerful machines. They may also result from differences in network connections, parts of the network may be lossy and alter the communication between certain peers, or from differences in actual host applications with some nodes running network-greedy software while others having full bandwidth available. For instance, while the penetration of the Internet into homes has greatly favored the deployment of p2p collaborative systems, bandwidth capability of edge nodes connected through ADSL links remains extremely heterogeneous. This is particularly true for the upload capacity which is of utmost importance for collaborative systems. For instance, the average upload capacity of ADSL connections in Europe is 1077kbps, while it is 899kbps in North America and only 170kbps in Africa [1].

Requiring two nodes with significant discrepancies in their capabilities to perform the same amount of work might lead to an inefficient protocol that could simply turn out to be useless for applications such as video streaming that require fast dissemination. In such applications, the data is of relatively large size and must be delivered within a given time window: the end user typically wants its stream to start as soon as possible or to be synchronized with the live stream, i.e., the buffer used for synchronization should be the smallest possible, meaning that the dissemination of data should be done as fast as possible. Note that when applied to streaming applications, the gossip protocol is usually used to forward stream-packet identifiers, the content being subsequently pulled whenever necessary. Even for peer-to-peer file sharing, the end user may want its application to deliver selected files as early as possible, i.e., downloading them as fast as possible. Besides inefficiency, a uniform gossip protocol may appear unfair in a heterogeneous setting and can be frustrating even if nodes indeed have the same capabilities but some benefit significantly more than others from the dissemination. For instance, in a newsoriented application, some nodes may subscribe to fewer topics than others and thus should also receive fewer events.1 With a uniform gossip protocol, these nodes may find it unfair to contribute to the system as much as the other nodes and may thus decide to leave the application. Again, this calls for a gossip scheme that accounts for heterogeneity. In fact, many researchers have recognized this need in some form or another [START_REF] Bishop | Considering priority in overlay multicast protocols under heterogeneous environments[END_REF][START_REF] Deshpande | Crew: A gossip-based flash-dissemination system[END_REF][START_REF] Pereira | Low latency probabilistic broadcast in wide area networks[END_REF][START_REF] Sung | Enabling contribution awareness in an overlay broadcasting system[END_REF][START_REF] Venkataraman | Chunkyspread: Heterogeneous unstructured treebased peer to peer multicast[END_REF][START_REF] Verma | Controlling gossip protocol infection pattern using adaptive fanout[END_REF][START_REF] Vishnumurthy | On heterogeneous overlay construction and random node selection in unstructured p2p networks[END_REF][START_REF] Vishnumurthy | A comparison of structured and unstructured p2p approaches to heterogeneous random peer selection[END_REF].

Breaking gossip uniformity is challenging

Intuitively, a gossip protocol could account for heterogeneity by introducing a bias towards wealthy nodes so that they perform more work. Two approaches can be considered:

1. Increasing the fanout of wealthy nodes. This basically means that wealthy nodes will send more messages in a given gossip round. 22. Increasing the probability that a wealthy node is selected in a gossip round. This basically means that wealthy nodes will contribute to more gossip rounds.

these ideas to work is however challenging and a mechanism that accounts for heterogeneity should satisfy a number of non-trivial constraints.

• Adapting real contribution. At first glance, each of the two approaches suggested above effectively ensures that wealthy nodes will contribute more to the dissemination by sending more messages. However, the number of messages might not be a good metric for the real contribution of a node. This is particularly the case in a gossip-based video-streaming application. In such a setting, two kinds of messages are typically considered. Those that simply contain stream ids (fairly cheap) and those that contain actual streams. A node first receives an id and requests the actual stream if it does not have it. Sending the actual stream clearly introduces a higher overhead than sending an id. Hence, a mechanism that accounts for heterogeneity should have wealthy nodes really perform more work and not only send more messages.

• Preserving some randomness. The reliability of gossip protocols stems from randomness of the dissemination process. Any bias toward more powerful nodes might hamper this randomness with the risk of decreasing reliability. Relying on a subset of wealthy nodes to work more may, in fact, have a negative impact in case many of these nodes leave the system or, even worse, behave maliciously.

• Local knowledge of global capability. In a large scale setting, one cannot assume global knowledge of the capabilities of all nodes for the same reason for which membership protocols manage partial views instead of global views [START_REF] Eugster | Lightweight probabilistic broadcast[END_REF]. Global knowledge of capabilities is unattainable because of churn and dynamic capability changes as we discuss below.

On the other hand, nodes cannot adapt their workloads based solely on local information about their own capabilities. For instance, assume that each node systematically decreases its fanout by a certain percentage as soon as its capability decreases by the same percentage, then a sudden decrease in the capability values of a number of nodes is likely to interrupt the dissemination process.

• Adapting to capability changes. Capability usages dynamically change over time as applications use bandwidth depending on user behavior (e.g., reading emails, downloading files or watching online multimedia content) and on the peer-to-peer offer (e.g., changes in the availability of files and uploaders). For example, a network-greedy application for peer-to-peer file sharing stops using download bandwidth as soon as the transfers of requested files have completed, which for many people means the application can be closed resulting in a stop of upload bandwidth usage. Hence, a mechanism that accounts for heterogeneity should itself be dynamic. Furthermore, the time required to account for the change should be smaller than the stability period of capabilities.

Gossip theory to the rescue

In the attempt to address these challenges, we have recently proposed HEAP (HEterogeneity-Aware Gossip Protocol ) [START_REF] Frey | Gossiping Capabilities[END_REF], a protocol for collaborative, high-bandwidth content distribution.

Two observations lie at the heart of HEAP:

1. Mathematical and empirical results [START_REF] Eugster | Epidemic information dissemination in distributed systems[END_REF][START_REF] Pittel | On spreading a rumor[END_REF] convey the fact that reliability is ensured as long as the average of all fanouts is approximately ln(n) (in a system of size n). This suggests a way to tune the contributions of nodes by adapting their fanouts according to their capabilities without impacting the reliability of dissemination.

2. The utility of a message is highest in the first hops of dissemination. Thus the capability of a node intervening at the first gossip rounds can have a significant impact on performance [START_REF] Karp | Randomized rumour spreading[END_REF]. This suggests ways to bias the selection of gossip targets so that the positions of nodes in the data-forwarding process match their capabilities. For example, nodes with a very high bandwidth or that are very responsive should be involved at the earliest stages of dissemination in order to optimize the average latency experienced by data receivers. This is particularly crucial in streaming applications where significant work is performed in the early stages of the dissemination where all streams are required.

HEAP incorporates a gossip-based aggregation protocol [START_REF] Jelasity | Gossip-based aggregation in large dynamic networks[END_REF][START_REF] Van Renesse | Astrolabe: A robust and scalable technology for distributed system monitoring, management, and data mining[END_REF] through which each node may obtain a very good sample of the overall system capabilities. This allows nodes (i) to adapt their fanout values based on their capabilities while maintaining the average fanout constant across the whole system; and (ii) to bias the selection of their gossip targets so that the most capable nodes are chosen at the first stages of dissemination. This allows HEAP to optimize the distribution of nodes in the dissemination chain ultimately increasing the experienced performance.

HEAP also incorporates LIFT (LIghtweight Freerider Tracking), a companion subprotocol able to track selfish nodes that declare high capability values in order to augment their perceived performance, without contributing accordingly. Although interesting in its own right, LIFT is particularly important in the context of HEAP as biasing the selection of gossip targets may render the protocol more vulnerable to the presence of nodes that do not contribute as they should.

Within the sport news system of the Télévision Suisse Romande for instance, some users (nodes) subscribe to results about soccer or hockey, which indeed generate a lot of information, while others only subscribe to results about cycling, which generates less traffic, or in chess, which generates very few events in Switzerland. While appealing, a gossip-based implementation of such a system might lead to a feeling of unfairness if the dissemination of sport results is uniformly spread among all nodes.

Biasing the frequency of dissemination has the same effect.