# A simplified approach for assessing the effects of temperature change on the stability of consumer-resource interactions

Alexis D Synodinos, Jose M Montoya, Arnaud Sentis, Bart Haegeman

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Alexis D Synodinos, Jose M Montoya, Arnaud Sentis, Bart Haegeman. A simplified approach for assessing the effects of temperature change on the stability of consumer-resource interactions. Oikos, 2024, 2025, pp.e10761. 10.1111/oik.10761. hal-05230438

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OIKOS

Research article

# A simplified approach for assessing the effects of temperature change on the stability of consumer-resource interactions

Alexis D. Synodinos<fnref n="1" />,<fnref n="2" />,<fnref n="3" />, Jose M. Montoya<fnref n="1" /> and Arnaud Sentis<fnref n="1" />,<fnref n="4" />, Bart Haegeman<fnref n="1" />,<fnref n="5" />

**Correspondence:** Alexis D. Synodinos (alexis.synodinos@idiv.de)

Oikos

2025: e10761

doi: 10.1111/oik.10761

Subject Editor: Francois Massol

Editor-in-Chief: Pedro Peres-Neto

Accepted 16 December 2024

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Temperature regulates the physiology and behaviour of organisms. Thus, changing temperatures impact the dynamics of species interactions. Considering that consumer-resource interactions underpin ecological communities, the impacts of warming on the stability of consumer-resource interactions have been extensively studied. However, a consensus among empirically determined warming-stability relationships and a clear understanding thereof are lacking. We investigate these relationships systematically by developing a simplified theoretical framework that incorporates empirical data in three steps. We define stability in terms of intrinsic oscillations to avoid comparing disparate stability notions, use a one-dimensional stability metric and convert all empirically-determined thermal dependence parameters into a single function, to directly compare all data. The framework utilises the Rosenzweig-MacArthur model with a saturating consumer functional response, which has been extensively employed to study warming-stability relationships in consumer-resource interactions. The framework is applied on empirical data on the thermal dependence of ectotherm consumer-resource pairs. We find support for four different warming-stability relationships: stability increases, decreases, is hump-shaped or U-shaped with increasing temperature. The diversity of warming-stability relationships, though partly attributable to context-dependence, is fundamentally caused by two factors. First, the relative thermal sensitivities of attack rate and handling time and, second, the scarcity of empirical evidence for the thermal dependence of carrying capacity. The former depends on how these processes are measured, which may not be consistent across studies. The latter necessitates the application of assumptions, which are difficult to verify, yet have a significant impact on warming-stability relationships. We demonstrate how aspects of the data, such as the two aforementioned factors or the range of studied temperatures, can alter the predicted thermal dependence of stability. Thus, we illustrate how our

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<footnote n="1">¹Theoretical and Experimental Ecology Station, CNRS, Moulis, France</footnote>

<footnote n="2">²German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, Leipzig, Germany</footnote>

<footnote n="3">³Department of Physiological Diversity, Helmholtz Centre for Environmental Research (UFZ), Leipzig, Germany</footnote>

<footnote n="4">⁴INRAE, Aix-Marseille Université, UMR RECOVER, Aix-en-Provence, France</footnote>

<footnote n="5">⁵CNRS/Sorbonne Université, UMR Laboratoire d'Océanographie Microbienne, Banyulssur-Mer, France</footnote>
framework facilitates the study of warming-stability relationships in consumer-resource interactions, from producing a concise overview of the empirically determined predictions to analysing the causes of deviation among these.

Keywords: climate change, ectotherms, population fluctuations, predator-prey, theory-data coupling, thermal biology

Introduction

Temperature affects the biology of organisms in many ways. Metabolic rates, for instance, increase exponentially with warming, with this thermal dependence conserved across species, taxa and levels of organisation (Gillooly et al. 2001, Brown et al. 2004, Savage et al. 2004). Metabolic rates do eventually peak and decline, but this occurs at such high temperatures – relative to the species’ temperature range – that it has been ignored in the context of most studies.

Such organism- and population-level responses scale up to ecological communities and ecosystems via interspecific interactions (Thomas et al. 2004, Montoya and Raffaelli 2010, O’Connor et al. 2011). Consumer-resource interactions are particularly important in this regard, as they regulate energy and material flows from primary producers towards higher trophic levels (McCann 2011, Rip and McCann 2011). Hence, studying the impact of temperature change on trophic interactions (Fussmann et al. 2014, Bideault et al. 2021), becomes a fundamental part of the investigation into how increasing ambient temperatures (IPCC 2023) will alter community properties, such as their biomass distributions or dynamics (Petchey et al. 2010, Rao et al. 2024).

In trophic interactions, specific processes contribute to the dynamics (Barbier and Loreau 2019). Changes in the foraging behaviour of consumers, for instance, can fundamentally alter whole food webs. Though warming tends to increase primary production (O’Connor et al. 2009, Shurin et al. 2012), it also alters the make-up of the prey species, which can limit the energy gained through consumption and cause the starvation of predators (Gauzens et al. 2024). In fact, reduced abundances of top-predators have been repeatedly reported (Petchey et al. 1999, Barneche et al. 2021, Gibert et al. 2022). Such changes will impact the stability of trophic interactions, with weaker population fluctuations a possible outcome (Rall et al. 2012, Kratina et al. 2022).

To better understand how temperature's impact on certain processes contributes to changes in stability, we study pairwise consumer-resource interactions. In particular, consumer growth is determined by the consumption rate (i.e. functional response). This commonly increases initially before saturating (i.e. type II; Jeschke et al. 2004); a non-linearity which can generate coupled oscillations in the consumer and resource populations (Rosenzweig and MacArthur 1963). A steeper slope or a lower saturation value increase the likelihood of oscillations (Johnson and Amarasekare 2015), with attack rate determining the former and handling time the latter. Similarly, a higher resource carrying capacity causes oscillations ('paradox of enrichment'; Rosenzweig 1971,

Uszko et al. 2017), as weaker resource self-regulation amplifies the tendency of consumers to overshoot their equilibrium (McCann 2011).

Theory postulates that stability properties of trophic interactions can be deduced from aggregates of such empirically measurable quantities (Yodzis and Innes 1992, Synodinos et al. 2021, Gilbert et al. 2014, Barbier and Loreau 2019). One aggregate is consumer energetic efficiency, which quantifies the energy gained through consumption relative to that lost through metabolic demands (Yodzis and Innes 1992). A second aggregate, interaction strength, quantifies the impact of the consumer on the resource, either in the short-term through the functional response (Rall et al. (2010) or in the long-term via the change in resource equilibrium due to the consumer (Berlow et al. 1999, 2004). Both aggregates are closely linked to various aspects of stability: intrinsic oscillations, consumer persistence and asymptotic resilience. Moreover, the two are interrelated in regulating energy and material transfer through the trophic chain (Nilsson and McCann 2016).

The thermal dependence of energetic efficiency has been found to be decreasing (Vasseur and McCann 2005, Vucic-Pestic et al. 2011, Rall et al. 2012, Archer et al. 2019), increasing (Vasseur and McCann 2005, Csik et al. 2023) or hump-shaped (Sentis et al. 2012). Similarly for interaction strength, evidence exist for a decreasing (Rall et al. 2010, Alvarez-Codesal et al. 2023) and an increasing (Alvarez-Codesal et al. 2023) thermal dependence.

A reduction with warming in either energetic efficiency or interaction strength has been associated with increased stability via the dampening or elimination of oscillations (Vasseur and McCann 2005, Rall et al. 2008). Lower energetic efficiency or weaker interaction strength due to warming have also been argued to increase the risk of consumer extinction due to starvation (Sinervo et al. 2010, Rall et al. 2012, Sentis et al. 2012, Alvarez-Codesal et al. 2023). However, the latter would imply the proximity of the dynamics to the consumer extinction boundary and, therefore, a loss of stability associated to consumer persistence. Hence, decreasing energetic efficiency or interaction strength, induced by warming, could simultaneously increase stability in one sense (weaker oscillations) and reduce it in another (consumer extinction). Such a conflation of stability notions can be common (Donohue et al. 2013).

Therefore, a systematic understanding of how stability changes with temperature has been hampered by a few factors. One, the mixed empirical evidence on certain processes. Second, the conflation of stability measures, a long-standing issue in ecological research (Grimm and Wissel 1997, Donohue et al. 2013). A third issue, not mentioned
yet, regards the thermal dependence parameterisations: thermal dependence data have been fitted with different types of functions across studies (e.g. Arrhenius equation, Gaussian function), which hinders the direct comparison of measurements across systems.

We present a framework that summarises the predictions generated by empirical data, producing a visual overview of warming-stability relationships across systems. Our approach is based on three steps. We define stability solely in terms of the tendency for intrinsic oscillations. We use one-dimensional stability metric developed by Johnson and Amarasekare (2015) precisely for this purpose. We convert temperature dependence parameterisations into a single, canonical form that can accommodate all parameterisations we found in the literature. Thus, we directly compare empirical findings which were parameterised differently across study systems. Based on certain assumptions which we expand on, this framework vastly simplifies the comparison of seemingly disparate data, allowing for the extraction of more general conclusions and the identification of the causes for the divergence in predictions.

In the methods we expand on the technical aspects of our approach and justify our assumptions. In the results, we present a comprehensive overview of how the empirical evidence suggests warming will alter stability. Our analysis maps warming-stability relationships onto a two-dimensional plane, producing a simple visualisation with four characteristic shapes for the temperature-dependence of stability. The difference in relationships was determined by three factors: the dichotomy between monotonic and non-monotonic thermal dependencies in individual traits, the relative thermal sensitivities of attack rate and handling time and the – most often assumed rather than measured – thermal dependence of carrying capacity. In the discussion, we elaborate on the insights of our results, identify areas where future empirical studies can focus on and clarify the limitations of our approach.

## Material and methods

We use the classical Rosenzweig-MacArthur model with a Holling type II functional response (Eq. 1-2), which will apply to a wide range of consumer-resource interactions (Jeschke et al. 2004). We follow past studies in assuming a direct mapping of environmental temperatures to body temperature (Vasseur and McCann 2005, Amarasekare 2015) which best applies to ectotherm species. Using body temperature rather than ambient temperature as the temperature variable would overcome this constraint for other species without any impact on the qualitative patterns that we will describe. Further, we assume that resources have a wider thermal range of persistence than consumers. This simply guarantees that $\phi$ can be applied to the full temperature gradient, but does not alter the applicability or interpretation of $\phi$.

## The model

The Rosenzweig-MacArthur model:

$$ \frac{dR}{dt} = rR \left(1 - \frac{R}{K}\right) - \frac{aRC}{1 + ahR} \quad (1) $$

$$ \frac{dC}{dt} = e \frac{aRC}{1 + ahR} - mC \quad (2) $$

The equations describe the rate of change of resource, R, and consumer, C, biomass densities ($[R] = [C] = \text{density}$, where $[X]$ denotes the dimension of variable X). The resource growth rate, $r$ ($[r] = 1/\text{time}$), and carrying capacity, K ($[K] = \text{density}$) determine the logistic growth of the resource population. Consumers remove resource biomass through an increasing, saturating functional response (Holling type II). This is expressed as a function of attack rate, $a$ ($[a] = 1/\text{density} \times 1/\text{time}$), and handling time, $h$ ($[h] = \text{time}$). Consumer growth is proportional to the converted consumed biomass,

with assimilation efficiency, $e$ ($[e] = \frac{[C]}{[R]} = \text{dimensionless}$).

Metabolic costs, $m$ ($[m] = 1/\text{time}$), induce losses of consumer biomass. Below we briefly present the dynamics relevant to our analyses and provide only the necessary mathematical formulas; the full mathematical analysis relevant to our study is presented extensively in the Supporting information.

## Model dynamics and stability metric

The model exhibits two qualitatively distinct dynamics (stationary regimes): stable equilibrium and stable limit cycle. In the former case, dynamics converge to the stable equilibrium either directly or via damped oscillations (Supporting information). In a stable limit cycle, populations oscillate periodically around an unstable equilibrium (Supporting information). A Hopf bifurcation occurs at the point where dynamics switch from stable equilibrium to stable limit cycle. We emphasise that this transition is smooth rather than abrupt (Supporting information); this will become relevant for our choice of stability metric. We note here our assumption permeating our analyses; that dynamics have converged to the equilibrium or the limit cycle. This enables us to derive analytical expressions for the coexistence equilibria – whether stable or unstable –, for consumer and resource persistence and for the Hopf bifurcation (Supporting information).

As a measure of (in)stability, we focus on fluctuations in population abundances. Within the Rosenzweig-MacArthur model, this can be quantified with respect to the fluctuations generated by the trophic interaction itself, i.e. the intrinsic oscillations. As we allude to above, the dynamics range from populations converging to a fixed point equilibrium without any fluctuations to self-sustained periodic oscillations around the unstable equilibrium. Therefore, our goal is to quantify the tendency for these oscillations to appear and become ever greater. Johnson and Amarasekare (2015) proposed a metric
to quantify precisely this tendency of dynamics to fluctuate in the Rosenzweig–MacArthur model (Eq. 3).

$$ \text{Tendency for oscillations} \propto \frac{\text{positive feedback}}{\text{negative feedback}} \\ = \frac{ah}{K} = ahK \qquad (3) $$

This metric revolves around a positive and a negative feedback impacting per capita resource growth. The positive feedback manifests via the functional response product $a \times h$. Once the functional response saturates, consumer growth ceases and any additional resource abundance reduces the negative effect of consumption on per capita resource growth, promoting further resource growth. Consumers feed on the additional resources and overshoot their equilibrium. This causes the collapse first in resource and then consumer abundances, thus inducing oscillations. The negative feedback occurs via resource self-regulation, which dampens fluctuations. In this model, this is expressed as the inverse of the carrying capacity ($1/K$). Therefore, $ahK$ can be viewed as the ratio of positive to negative feedbacks in per capita resource growth (Eq. 3).

$$ \varphi = ahK \qquad (4) $$

An analytical treatment of the Rosenzweig–MacArthur model's dynamics reveals the significance of the product of attack rate, handling time and carrying capacity, i.e. $\varphi$ (Supporting information). More specifically, aspects of the dynamics (e.g. Hopf bifurcation, coexistence) are regulated by $\varphi$ and a second aggregate parameter, energetic efficiency, $\rho$ (Yodzis and Innes 1992, Synodinos et al. 2021). This serves to reduce the complexity of the analysis to two dimensions without a loss of information. $\varphi$ is precisely the stability metric defined above (Eq. 4). Thus, the connection between $\varphi$ and stability becomes explicit (Supporting information). The product of $a$ and $h$ determines at which resource density the functional response saturates. $K$ corresponds to the resource equilibrium density. Thus, $\varphi$ can be interpreted as a measure of resource abundance as perceived from the consumer (Yodzis and Innes 1992). The second aggregate is the ratio of energy gained via consumption to energy lost through metabolic costs, $\rho=e/mh$ (Yodzis and Innes 1992, Synodinos et al. 2021). See the Supporting information for an expanded definition and for its derivation.

We wish to simplify the analysis of stability further, to a single dimension. Reducing the description of the model's dynamics from two dimensions to one – by using only $\varphi$ – will obviously be accompanied by a certain loss of accuracy. However, the choice of $\varphi$ has been justified in Johnson and Amarasekare (2015), specifically as a metric with respect to temperature change. Their argument rests on the fact that changes in temperature will be manifest in $\varphi$ more strongly than in the other terms of the Hopf bifurcation (i.e. where

the steady state switches from equilibrium to a limit cycle, Supporting information).

This argument works well under the assumption that $\rho$ is not ‘too small’. However, as $\rho \to 1$ from above, the Hopf bifurcation and the consumer extinction boundary approach each other (Supporting information). Therefore, applying $\varphi$ to quantify the tendency for intrinsic oscillations should be done with caution. For our analyses, where we can derive the value of $\rho=e/mh$, we apply an artificial threshold of $\rho > 2$. We note that the threshold choice is arbitrary insofar that there is no further justification for it than that it represents a minimal choice of separating ‘$\rho$ close to one’ from ‘$\rho$ far from one’. Practically, it simply serves as a warning that $\varphi$ might not be reliable below it. In our results, we highlight the temperature ranges which satisfy this condition. However, we emphasise that for $\rho < 2$, $\varphi$ does not cease to be instructive of the thermal dependence of stability, however it may not capture the full picture, as dynamics and population densities become very sensitive to the slightest changes in trait values (Synodinos et al. 2021).

It is worth mentioning that no single metric to quantify the oscillations has been shown to be exact. As with stability more generally (Donohue et al. 2013, 2016), oscillations represent a multi-faceted notion, which is difficult to capture with a single metric, even highly complex ones. Our approach is to use a very simple metric, which nevertheless has clear limitations.

## Temperature-dependencies of parameters

We analyse the impacts of temperature on stability via the thermal dependence of $\varphi$, which is the product of three temperature-dependent parameters (Eq 4): attack rate, $a$, handling time, $h$, and carrying capacity, $K$. Our analysis is based on empirically determined thermal dependencies of these three parameters, as reported in the literature. We only considered studies which reported the thermal dependence of all three parameters – or at minimum studied a system for which the thermal dependence of the three parameters was assumed to be known – over a sufficiently large temperature gradient (i.e. not two temperatures).

This led to the following studies as our data sources: Vucic-Pestic et al. (2011), Amarasekare (2015), West and Post (2016), Uszko et al. (2017) and Archer et al. (2019). We note that the data in Amarasekare (2015) were sourced from different empirical studies of the same organisms (Zamani et al. 2006, Parajulee 2007, Gao et al. 2013). Archer et al. (2019) provide data for two different predators on the same prey, hence we get two data points. Uszko et al. (2017) report three different scenarios for the thermal dependence of $K$, leading to three data points. Vucic-Pestic et al. (2011) measured the functional response on two different prey types, yielding two data points. Finally, two different predators produced two data points for West and Post (2016) (see the Supporting information for details). Using these, we illustrate the different ways in which $\varphi$ – and hence stability – depends on temperature.
The calculation of each study's φ and the comparison of the φ's between studies was complicated by the fact that different functional forms were used to parameterise the data on the thermal dependencies of a, b and K (e.g. Arrhenius, exponential quadratic with optimal temperature, etc. see the Supporting information). To improve comparability between measurements, help identify generalisations and facilitate the inclusion of future empirical data into this framework, we converted the parameterisations to a canonical form, the general Gaussian (Eq. 5). This function can produce both unimodal and monotonic thermal dependencies and represents a simple choice used previously in the literature (Amarasekare 2015, Uszko et al. 2017).

$$p(T) = p_0 \exp(\alpha_p \tau + \beta_p \tau^2), \qquad (5)$$

where

$$\tau = \frac{T - T_0}{T_w} \qquad (6)$$

$$T_w = T_{\max} - T_{\min} \qquad (7)$$

$$T_0 = \frac{T_{\min} + T_{\max}}{2} \qquad (8)$$

We perform this conversion in two different ways, depending on the availability of the original data. Either we analytically approximate the originally reported function to the canonical form or we refit the data directly with the canonical function to derive the α and β values. The choice between which method to follow was dictated by the availability of data, as reported in the sources. In particular, though the mean values of the fitted parameters in the original studies were always provided, the estimation errors were not. Therefore, we added uncertainty estimates to our converted functions and to the final thermal dependence of φ either by using the reported errors in our conversion formulas or by refitting the data and extracting this information ourselves, where the error estimates were not provided. Below, we describe the analytical approximation approach. At the end of this section we deal with the derivation of the error estimates.

The analytical conversions are based on a Taylor approximation (up to the second order) of the log-transformed temperature-dependent variable around the temperature $T_0 \approx T$. $T_w$ is the width of the temperature range determined by the data (Eq. 7) with $T_0$ the mid-point of this temperature range (Eq. 8). $T_0$ should not be confused with the reference ($T_{ref}$) or optimal ($T_{opt}$) temperature in the original parameterisations. Finally, $p_0$ is the parameter value at $T_0$ and $\alpha_p$ and $\beta_p$ are dimensionless coefficients of the Gaussian function specific to $p(T)$. We refer to these as linear and quadratic sensitivities, respectively.

The approximations of the means for attack rate, handling time and carrying capacity are accurate (Fig. 1). Therefore,

we were able to ignore higher order terms. Moreover, using higher order approximations would produce more complex functional forms (of third degree or higher), which would defeat the purpose of our approach. We note that the original data varied around the mean and hence were not as smooth as the plots portray. We go over each dataset extensively in the appendix where we illustrate this variation in the data (Supporting information). We show that the φ's resulting from the product of the approximations fit the φ's derived from the original parameterisations sufficiently well (Supporting information). We present the transformations for each functional form of the original data extensively in the Supporting information and contend that they can provide a useful tool for comparing empirical findings.

Our convention of using a single, general temperature parameterisation (Eq. 5) facilitates the analysis of the thermal sensitivity of stability. Significantly, the stability metric, φ, becomes the product of three identical exponential functions. Mathematically, this simplifies things drastically: multiplying exponential functions of the same base amounts to adding their exponents. Given that a, b and K all share the same form (Eq. 5), the values of $\alpha_\phi$ and $\beta_\phi$ will be the sum of the respective α's and β's of a, b and K (Eq. 9c). We emphasise that this approach does not account for potential correlations among the three parameters.

$$\ln(\phi) = \ln(a) + \ln(b) + \ln(K) \qquad (9a)$$

$$\text{hence } \alpha_{\phi} \tau + \beta_{\phi} \tau^2 = (\alpha_a + \alpha_b + \alpha_K) \tau + (\beta_a + \beta_b + \beta_K) \tau^2 \qquad (9b)$$

$$\text{so that } \alpha_{\phi} = \alpha_a + \alpha_b + \alpha_K \quad \text{and} \quad \beta_{\phi} = \beta_a + \beta_b + \beta_K \qquad (9c)$$

Each value pair of $\alpha_\phi$ and $\beta_\phi$ determines an exact shape for the thermal dependence of φ, hence stability. Moreover, we can quantify the contribution of each parameter (a, b and K) to the outcome (Eq. 9c). Thus, the temperature-dependent shape of φ will be determined by the signs and relative magnitudes of α and β (Fig. 2). Put simply, whether φ will be monotonic (increasing or decreasing) or non-monotonic (hump-shaped or U-shaped) is determined by the relative magnitudes of α and β: if $|\alpha| > |\beta|$ it will be the former, otherwise the latter. Then, the sign of the 'dominant' coefficient distinguishes between monotonically increasing ($\alpha > 0$, Fig. 2, orange region) or monotonically decreasing ($\alpha < 0$, Fig. 2, yellow region) in the first case or hump-shaped ($\beta < 0$, Fig. 2, blue region) and U-shaped ($\beta > 0$, Fig. 2, green region) in the second case (see the Supporting information for exact conditions). Thus, each study can be mapped as a single point onto the $\alpha - \beta$ plane, determining the shape of the temperature dependence of φ for its study system.

Finally, we quantify the uncertainty in $\alpha_\phi$ and $\beta_\phi$. These uncertainties (or errors) stem from the spread of the points around the fitted temperature dependence functions for a, b and K in the original studies, where the data were sourced from (Supporting information). Thus, the formulas
<figure><img src="image_5.png" /><figcaption>(b) Handling time</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(c) Carrying capacity</figcaption></figure>

<figure><img src="image_7.png" /></figure>

Figure 1. Comparison between original (thick grey curve) and converted (thin red curve) temperature parameterisations for central tendencies of (a) attack rate, (b) handling time and (c) carrying capacity. The variation of the original data around these means is not included in this figure. We show this variation in the Supporting information, where we expand on the conversion of the specific data sets. Further, we account for this uncertainty in the conversion (Eq. 10a–10d) and we illustrate this in the results as confidence regions. All these functions have been normalised by requiring that their maximum over the temperature range is equal to one.
<figure><img src="image_8.png" /><figcaption>Figure 2. Graphical illustration of how the values of α_p (x-axis) and β_p (y-axis) determine the shape of the thermal dependence of p(T) (Eq. 5). α_p and β_p determine the sensitivity of the exponential function (and hence the thermal dependence) on the linear and quadratic parts of the exponent, respectively. A relative dominance of α_p leads to a monotonic shape, while β_p relative dominance produces either hump- or U-shaped functions. Thus, four shapes emerge as defined by four regions, with smooth transitions between adjacent regions: monotonically increasing (red region), monotonically decreasing (yellow region), hump-shaped (blue region) or U-shaped (green region). It is worth pointing out that the boundaries of the coloured regions are illustrative; the shapes change smoothly across the α – β plane.</figcaption></figure>

(Eq. 10a–10d) are only approximate. Without access to the
original data sources, we made the assumption that the differ-
ent parameters are independent, i.e. we neglect correlations
between parameters. Future studies can account for covari-
ance in their data to derive appropriate formulas and main-
tain a satisfactory level of accuracy in the approximation.

$$
EV(\alpha_{\varphi}) = EV(\alpha_{a}) + EV(\alpha_{b}) + EV(\alpha_{K}) \quad (10a)
$$

$$
EV(\beta_{\varphi}) = EV(\beta_{a}) + EV(\beta_{b}) + EV(\beta_{K}) \quad (10b)
$$

$$
ER(\alpha_{\varphi}) = \sqrt{ER(\alpha_{a})^{2} + ER(\alpha_{b})^{2} + ER(\alpha_{K})^{2}} \quad (10c)
$$

$$
ER(\beta_{\varphi}) = \sqrt{ER(\beta_{a})^{2} + ER(\beta_{b})^{2} + ER(\beta_{K})^{2}} \quad (10d)
$$

It is important to note that the temperature ranges var-
ied between the studies in our dataset (x-axes in Fig. 1,
Supporting information). However, we can still compare all
studies within the α – β plane (Fig. 2), because the tempera-
ture expressions in the exponent of the canonical function

(Eq. 5) are normalised around T₀ through the term τ=(T−T₀)/T<sub>w</sub>. In particular, since all T ∈ [T<sub>min</sub>, T<sub>max</sub>], it follows that τ ∈ [−0.5, 0.5] for all parameterisations.

Finally, our focus on the exponent – i.e. the shape of φ – provides the information on the relationship between temperature change and stability (i.e. increasing or decreasing the tendency for oscillations) which was the objective of our qualitative approach in the α – β plane. In the α – β plane, our findings are not sensitive to the value of the intercept φ₀. However, the value of φ₀ is important with respect to the proximity of the community to its coexistence boundaries. As we alluded to above, this should be a consideration when applying φ as a stability metric.

Results

Warming–stability relationships derived from the literature
were distributed across the α – β plane, indicating that our
approach provided a good tool to encapsulate the available
information and compare studies (Fig. 3). Below we describe
our findings, look into the causes of variation and search for
systematic patterns therein.

Scenario 1: stability increases with warming

Four data points produced a monotonically decreasing φ(T) (Fig. 3, Usz2, Usz3, Vuc1, Vuc2). Hence, stability increased with temperature in these cases. Two data points come from an experiment of adult beetles feeding on flightless *Drosophila* adults (Fig. 3, Vuc1) and on beetle larvae (Fig. 3, Vuc2). Both points lie almost on β=0, with Vuc2 having a more negative αφ and being further from the origin. The other two points come from an experiment of *Daphnia hyalina* feeding on the green algae *Monoraphidium minutum* with two different thermal dependencies for the carrying capacity: monotonically decreasing (Fig. 3, Usz2) and hump-shaped (Fig. 3, Usz3). Usz1 and Usz2 are close to the U-shaped and hump-shaped φ boundaries, respectively. This indicates how the shape of φ changes in a continuous rather than abrupt manner between the regions of the α – β plane.

Scenario 2: stability decreases with warming

A single study reported parameters producing an increas-
ing monotonic thermal dependence for φ (T) (Fig. 3, Arc1,
Arc2); hence predicting a monotonic decrease in stability
with warming. This study measured the thermal dependence
of the interaction between two different freshwater predators,
one sedentary (larvae of L. riparia, Arc1) and one mobile (lar-
vae of P. cingulatus, Arc2), feeding on a single prey (blackfly
larvae from the *Simuliidae* family). Carrying capacity, mea-
sured as the prey density in the field, increased with warm-
ing, which drove the strong destabilising pattern. We note
that for part of the gradient, ρ < 2 (hence the dashed curve).
The reduction in ρ = e/mh was driven by increased metabolic
costs, given that handling time showed no thermal sensitivity.
<figure><img src="image_9.png" /></figure>

Figure 3. Computed values for α<sub>φ</sub> and β<sub>φ</sub> pairs based on empirical studies (black points), with confidence regions computed at one standard deviation (grey ellipse). The exact values for α<sub>φ</sub> and β<sub>φ</sub> are provided in the Supporting information. The studies are: Ama (Amarasekare 2015), Arc1 and Arc2 (Archer et al. 2019), Usz1, Usz2 and Usz3 (Uszko et al. 2017), Vuc1 and Vuc2 (Vucic-Pestic et al. 2011), Wes1 and Wes2 (West and Post 2016). In Archer et al. (2019), Arc1 and Arc2 correspond to two different years of measurement; in Uszko et al. (2017), the three points represent three different scenarios for the carrying capacity's thermal dependence; in Vucic-Pestic et al. (2011), Vuc1 and Vuc2 correspond to two different prey types; in West and Post (2016), Wes1 and Wes2 correspond to two different predators. In the panels surrounding the figure, solid black lines of φ(T) correspond to values of ρ > 2, dashed lines to ρ < 2 (see Model dynamics and stability metric for justification of this threshold and the Supporting information for the thermal dependence of ρ in these studies).

The combination of high φ and low ρ imply proximity to the
consumer extinction boundary, where dynamics are particu-
larly sensitive to any temperature change.

Scenario 3: warming–stability relationship U-shaped

Two data points, from a single study, yielded a hump-shaped
φ(T) (Fig. 3, Wes1, Wes2). Hence, stability was minimal
in the mid-temperatures (U-shaped). The study measured
two different *Daphnia* species, the small-bodied *D. ambigua*
(Fig. 3, Wes1) and the large-bodied *D. pulicaria* (Fig. 3,
Wes2) consuming an algae *Scendedesmus obliquus*.

Scenario 4: warming–stability relationship hump-shaped

A U-shaped φ(T), in which stability was maximised around
the temperature optimum, emerged in two cases (Fig. 3,
Ama, Usz1). The one study sourced data from the literature
on the thermal dependence of a cotton aphid *Aphis gossypii*
and its parasite *Aphidius matricatirae* (Fig. 3, Ama). Here, the

minimum of φ(T) occurs close to the upper end of the tem-
perature gradient, driven by a minimal carrying capacity at
very high temperature. In final data point from Uszko et al.
(2017), carrying capacity was considered constant (i.e. no
thermal dependence).

Decomposition of warming–stability outcomes

Here we deconstruct the thermal dependence of φ into that
of its three constituent biological parameters: attack rate, a,
handling time, h, and carrying capacity, K. We, thus, illus-
trate how the different outcomes emerge from the interaction
between these parameters.

Monotonic and non-monotonic thermal
dependencies

Six parameterisations produced a monotonic thermal depen-
dence of φ. In four of these, the temperature-dependent
(i.e. not constant) constituent parameters had a monotonic
thermal dependence (Table 1, Arc1, Arc2, Vuc1, Vuc2).
<table><caption>Table 1. The thermal dependence shapes of attack rate, *a*, handling time, *h* and carrying capacity, *K*, and the resulting shapes in *φ*, in the studies used.</caption><thead><tr><th></th><th>Study</th><th>a</th><th>h</th><th>K</th><th>φ</th></tr></thead><tbody><tr><td>Ama</td><td>Amarasekare 2015</td><td>∩</td><td>∪</td><td>∪</td><td>U</td></tr><tr><td>Arc1</td><td>Archer et al. 2019</td><td>/</td><td>const.</td><td>/</td><td>/</td></tr><tr><td>Arc2</td><td></td><td>/</td><td>const.</td><td>/</td><td>/</td></tr><tr><td>Usz1</td><td>Uszko et al. 2017</td><td>∩</td><td>U</td><td>const.</td><td>U</td></tr><tr><td>Usz2</td><td></td><td>∩</td><td>U</td><td>∩</td><td>U</td></tr><tr><td>Usz3</td><td></td><td>∩</td><td>U</td><td>∩</td><td>U</td></tr><tr><td>Vuc1</td><td>Vucic-Pestic et al. 2011</td><td>/</td><td>U</td><td>U</td><td>U</td></tr><tr><td>Vuc2</td><td></td><td>U</td><td>U</td><td>U</td><td>U</td></tr><tr><td>Wes1</td><td>West and Post 2016</td><td>∩</td><td>U</td><td>U</td><td>∩</td></tr><tr><td>Wes2</td><td></td><td>∩</td><td>U</td><td>U</td><td>∩</td></tr></tbody></table>

In the remaining two cases, attack rate and handling time were non-monotonic, with the carrying capacity monotonically decreasing and hump-shaped (Table 1, Usz2, Usz3, respectively).

The remaining four parameterisations yielded a non-monotonic $\varphi(T)$ (Table 1, Ama, Usz1, Wes1, Wes2). In these cases attack rate and handling time were non-monotonic, but carrying capacity was monotonic in one study (Table 1, Wes1, Wes2).

The split between monotonic and non-monotonic constituent parameters can provide a first clue for the shape of $\varphi$. However, the cases of Usz2 and Usz3 (Fig. 3, Table 1) illustrate that it is not always a robust criterion. Next we look into a subtler difference that may provide an additional criterion in determining the shape of $\varphi$ and, hence, warming-stability relationships.

## Relative sensitivity of attack rate and handling time

In almost all parameterisations, *a* and *h* had the opposite thermal dependence (Table 1, Vuc2 being the exception). *a* increased monotonically or was hump-shaped, while *h* decreased monotonically or was U-shaped, respectively. However, in each study, either *a* or *h* was more sensitive to temperature (i.e. they did not cancel each other out). This difference in sensitivity will have a significant impact on the warming-stability relationship, particularly in cases where the final parameter, *K*, has a weak thermal dependence. Two different studies of *Daphnia* species feeding on algae help to illustrate the importance of the relative thermal sensitivities of *a* and *h* (Fig. 4).

In both cases, attack rate was hump-shaped and handling time U-shaped (Fig. 4 black and grey vectors, respectively). However, in the first study (Fig. 4 Usz1), *a* had a weaker thermal sensitivity than *h*, so the addition of the two vectors landed $\varphi(T)$ in the U-shaped region (green). Additionally, carrying capacity was assumed constant (Fig. 4, Usz1 white loop), thus not having any further impact on the thermal dependence of $\varphi$. In the second study (Fig. 4 Wes2), the opposite result was reported: *a* had a stronger thermal sensitivity than *h*. Combined with a weak thermal dependence of *K*, the thermal dependence of $\varphi$ remained in the blue region, where $\varphi(T)$ is hump-shaped.

## Thermal dependence of carrying capacity

Our approach helps to illustrate the equal contribution of each parameter to the sign and magnitude of $\alpha_{\varphi}$ and $\beta_{\varphi}$ (Eq. 9c). Further, the thermal dependencies of *a* and *h* most often

<figure><img src="image_10.png" /><figcaption>Figure 4. In these plots we illustrate how each individual parameter contributes to the outcome: starting from the origin (0,0), each parameter can be represented as a vector whose projections on the x- and y-axis are determined by the $\alpha$ and $\beta$ values, respectively. By consecutively adding the three vectors (one for each parameter), we end up with the position of $\varphi$ in the plane. Here, attack rate, *a*, is depicted in black (arrow and point), handling time, *h*, in dark grey and carrying capacity, *K*, in white. In Usz1, we use the scenario where *K* has no thermal dependence (constant *K*) and we included a loop-arrow to illustrate this. *a* is hump-shaped ($\beta_a < 0$) and *h* U-shaped ($\beta_h > 0$) with $\beta_h > |\beta_a|$. Thus, $\varphi$ is U-shaped (i.e. $\beta_{\varphi} > 0$). In Wes2, *a* and *h* are again hump- and U-shaped, respectively. However, in this case, $\beta_h < |\beta_a|$. Moreover, the thermal dependence of *K* is both weak and monotonic, so its contribution in the $\beta$ direction is minimal. Thus, $\varphi$ emerges as hump-shaped.</figcaption></figure>
have opposite signs, and hence weaken each other's impact on the final outcome. Therefore, the thermal dependence of carrying capacity will be significant in determining the thermal dependence of stability.

For instance, Uszko et al. (2017) derived the thermal dependence of carrying capacity based on three scenarios: closed nutrient-limited, open nutrient-limited and open light-limited systems. This yielded three temperature dependencies for *K*: constant (Table 1, Usz1), monotonically decreasing (Table 1, Usz2) and hump-shaped (Table 1, Usz3). Comparing the three resulting $\varphi(T)$ shapes we see the significance of $K(T)$. With $K$ constant, a hump-shaped warming-stability relationship emerged (Fig. 3, Usz1). Introducing a monotonically decreasing $K$, $\varphi$ shifts to monotonically decreasing, simply moving along the $\alpha$ axis in the negative direction (Fig. 3, $Usz1 \to Usz2$). In contrast, introducing a hump-shaped $K$, shifts the point towards negative $\beta$, combined with a small decrease to more negative $\alpha$ (Fig. 3, $Usz1 \to Usz3$).

The lack of data on the carrying capacity is worth mentioning. The literature contains a vast number of studies measuring the temperature dependence of the functional response parameters. However, the five studies included here, were the only ones we found that reported the thermal dependence of carrying capacity from the same study system as that of the functional response measurements. From these, only one made this measurement within the scope of the functional response experiment (West and Post 2016). The others used observations of resource density from the field (Archer et al. 2019), derived self-regulation from the density-dependence of population growth from a different experiment on the same organisms (Amarasekare 2015) or calculated $K(T)$ based on the thermal sensitivity of growth and mortality rates (Uszko et al. 2017).

## Discussion

The impacts of changing mean temperatures on consumer-resource interactions – and food webs by extension – have been mostly studied through the thermal characterisation of specific biological rates (e.g. ingestion and respiration, Rall et al. 2010, Binzer et al. 2016). These rates are used to parameterise consumer-resource models and, thus, derive the predicted dependence of trophic community dynamics on temperature (Vasseur and McCann 2005, Rall et al. 2012, Kratina et al. 2022). Our analysis uses the Rosenzweig–MacArthur model with a saturating consumer growth (i.e. type II functional response) (Rosenzweig and MacArthur 1963). This represents pair-wise interactions in a rather simplified, yet useful, way. Thus, it has been used extensively both in theoretical analyses (Rosenzweig 1971, Yodzis and Innes 1992) and to evaluate the results of empirical work relating to temperature change (Fussmann et al. 2014, Amarasekare 2015, Uszko et al. 2017). In this context, our study unifies existing theoretical insights with empirical data on temperature-dependent trophic interactions.

Clearly, the Rosenzweig–MacArthur model is one of many consumer-resource models. For instance, changes in this model can improve its accuracy with respect to specific study systems (Harrison 1995, DeLong and Lyon 2020). Our approach benefited from the complexity reduction achievable in the Rosenzweig–MacArthur model, ultimately describing stability in one dimension. It may not be possible to achieve this level of simplification with all consumer-resource models; however, those of the Lotka–Volterra family could simplify significantly (Barbier and Loreau 2019, Galiana et al. 2021) and, thus, provide possible extensions to our approach.

## A simple framework for pair-wise trophic interactions

Within the pair-wise consumer-resource analyses, a key aspect of the dynamics which has been long studied is stability (Rosenzweig and MacArthur 1963, Yodzis and Innes 1992, Gilbert et al. 2014). However, we still lack a general expectation of how warming will alter the stability of consumer-resource interactions. Based on the Rosenzweig–MacArthur model, we developed an approach to facilitate progress in this respect. Our framework simplifies the inclusion and comparison of empirical data, produces a visual overview of empirically derived warming-stability relationship predictions and enables the identification of the origins of divergence in predictions.

In developing this framework, we made certain assumptions. First, we investigated stability solely with respect to the tendency for intrinsic oscillations, which result from the consumer-resource dynamics. For this, we used a one-dimensional stability metric, the product of attack rate, handling time and carrying capacity (Eq. 3, Johnson and Amarasekare 2015), which we termed $\varphi$ (Eq. 4). Clearly, $\varphi$ does not cover other aspects of stability, such as the return time to equilibrium. This is partly driven by the resource growth rate, which determines the speed at which dynamics unfold and can, thus, quickly stabilise dynamics or make them more excitable and less stable (McCann 2011). Nor is our analysis directly transferable to invariability with respect to external perturbations (i.e. coefficient of variation Donohue et al. 2013), since the two types of fluctuation have different mechanisms. Nevertheless, both types of fluctuation relate to the temporal variability of populations, which is the aspect of the dynamics most commonly impacted by temperature change (Jaureguiberry et al. 2022). Moreover, intrinsic oscillations in trophic interactions have long been studied in order to explain observed patterns (Stenseth et al. 1997, Ryabov et al. 2017, Blasius et al. 2020, DeLong and Lyon 2020), while they lend themselves to mathematical analyses (Rosenzweig 1971, Armstrong and McGehee 1980, Yodzis and Innes 1992). Due to the temperature dependence of the drivers of intrinsic oscillations, these have been extensively studied with respect to temperature change (Vasseur and McCann 2005, Rall et al. 2010, Amarasekare 2019).
As a final step we converted the different functions used to parameterise the empirically determined thermal dependencies of attack rate, handling time and carrying capacity into a single, canonical, form (Eq. 5). Thus, the shape of any empirically determined thermal dependence is dictated by two parameters of the canonical function, α and β, which determine the linear and quadratic sensitivity of the exponent, respectively (Fig. 2).

It is important to note that most empirical studies used here measured acute response to temperature change and did not consider acclimation or evolution, which would have accounted for phenotypic responses to temperature. We know that these adaptations can modulate the physiology (Sohlström et al. 2021), behaviour (Uszko et al. 2022) and morphology (Osmond et al. 2017, Sentis et al. 2017) of organisms and this can influence predator–prey dynamics (Sentis et al. 2015). Using our framework, it is possible to investigate the influence of thermal acclimation or evolution on stability by comparing parameter values from acclimated or non-acclimated organisms (Carbonell et al. 2022). Ultimately, our framework will only apply if the organismal responses to changing temperature manifest in the three parameters of φ.

An alternative would be the expansion of this framework from pair-wise interactions to whole food webs. For instance, we showed that increased foraging rates relative to handling times could destabilise dynamics. If this leads to an increase in direct interactions, it could help explain how warming will destabilise food webs due to the simplification of trophic interactions (O’Gorman et al. 2019). Clearly, a detailed and careful analysis would be required to obtain robust insights.

Another potential avenue of ‘theoretical unification’ would be to augment trophic with other interactions. Our simplified approach could increase in complexity through the addition of diversity within trophic levels, incorporating competitive or facilitative interactions. Even though complexity would increase, it could still be possible to reduce the analysis to few key drivers of the dynamics (Barbier et al. 2018, Hu et al. 2022). Yet, despite recent developments (García et al. 2023), thermal dependence data on other interactions remain sparse.

Reducing stability to a one dimensional metric, φ, and using the canonical temperature dependence function create a powerful tool to simplify and compare empirically determined impacts of warming on consumer-resource oscillations. In particular, our framework facilitates the direct comparison of the thermal dependence of stability and its constituent parameters. Thus, our approach does not only provide a conclusive overview of the reported warming-stability relationships (Fig. 3), but it can further elucidate why each study system yields a given relationship through the interaction of attack rate, handling time and the carrying capacity (Fig. 4).

## No strict boundaries between warming-stability relationships

Empirical evidence supported all four warming-stability relationships of our framework: stability increased monotonically,

decreased monotonically, was U-shaped or hump-shaped with increasing temperature (Fig. 3). Though these outcomes appear distinct, they are not separated by some strict boundary. The α – β plane describes a continuum along which the shape of the canonical function changes smoothly (Fig. 2). Therefore, the boundaries (coloured regions) are illustrative rather than definite. For instance, outcomes that are close to the boundaries become sensitive to small changes in the individual parameters (e.g. due to measurement errors). Alternatively, uncertainty in parameter estimates (Fig. 3, shaded ellipse) can shift the warming-stability relationship along the α – β plane.

Changes in the choice of temperature range over which the analysis or experiments are conducted can also cause shifts between the outcomes. In our framework, the temperature range of interest was included within the general thermal dependence function (T<sub>w</sub> in Eq. 5). Altering the width of the temperature range can shift the warming-stability relationship in the α – β plane. In a specific example, Amarasekare (2015) reported that resource self-regulation (inverse of carrying capacity) peaked at 32°C, while attack rate and handling time had optima at 24 and 22°C, respectively. Considering a temperature range wide enough to include the minimum in carrying capacity, results in a U-shaped φ (Fig. 3, Ama). However, if one constrains the temperature range to values below 32°C (e.g. due to experimental design limitations), carrying capacity would be an exponentially decreasing function. This would shift φ in the α – β plane into the region of monotonically decreasing thermal dependence and alter the warming-stability outcome (Supporting information).

These examples illustrate how the transition between different, opposite even, warming-stability relationships could occur smoothly, rather than as a result of a certain boundary in the system. We argue that such transitions are sensitive to subtle differences in the assumptions or measurements of the thermal dependencies of *a*, *b* or *K*. This is one reason that focusing on the individual empirically measured parameters has hindered generalisations. Instead, we need to consider their combined effects to be able to predict how temperature affects stability.

## Determinants of warming-stability relationships

Our analysis of the empirical evidence did not yield a single warming-stability outcome. Rather, studies were distributed in the two-dimensional plane, which represents the different shapes φ takes along the temperature gradient (Fig. 3). We identified three major contributing factors to this variation among findings.

First, there exists a dichotomy between the reported thermal dependence for both attack rate and handling time. Data with a monotonic dependence for these parameters, yielded monotonic warming-stability outcomes (Table 1). Most parameterisations with non-monotonic attack rates and handling times produced non-monotonic warming-stability outcomes (Table 1, Usz2, Usz3).
The evidence points to attack rate and handling time having a hump- and U-shaped thermal dependence, respectively (Englund et al. 2011, Uiterwaal and DeLong 2020) or, more generally, that the functional response has a unimodal thermal dependence (DeLong et al. 2023). Nevertheless, divergence from this pattern has been reported (Rall et al. 2010, Vucic-Pestic et al. 2011, Archer et al. 2019). This could be the result of a narrower studied temperature range, a less complete database in the case of meta-analyses or different resource levels (Uiterwaal and DeLong 2020). The first two causes reflect aspects of study design. The last one, however, hints at a multi-dimensionality of the thermal dependence curve (Litchman and Thomas 2023). Nevertheless, if one assumed that attack rate will tend to be hump-shaped and handling time U-shaped, would that yield fewer, a single even, warming-stability relationships? Probably not.

Given that attack rate and handling time almost always (Rall et al. 2010, Uiterwaal and DeLong 2020, but see Table 1, Vuc2) have opposite thermal dependencies (increasing versus decreasing or hump-shaped versus U-shaped, respectively), the process with the strongest thermal dependence would shift the thermal dependence of stability accordingly in the α – β plane (e.g. Fig. 4). In one meta-analysis of ectotherms, attack rate had a steeper slope than handling time (Englund et al. 2011), whereas a more recent meta-analysis found evidence of the opposite (Uiterwaal and DeLong 2020); handling time changed faster than attack rate with temperature. Thus, the second obstacle to simplifying generalisations arises.

The issue could originate in the different processes ascribed to handling time. In certain cases, handling time corresponds to the time required to ingest prey while in others it includes both the manipulation of prey and its ingestion (Jeschke et al. 2002). Interestingly, it has been shown that the former is more sensitive to temperature than the latter (Sentis et al. 2013). Thus, for predators limited only by ingestion, handling time could change faster than attack rate with temperature. On the other hand, predators limited by the manipulation and ingestion of the prey could have attack rates which change faster with temperature. A potential fix would be to break down the various processes represented within the functional response and to explicitly account for each one (Sentis et al. 2013, Wootton et al. 2021), or at

minimum to explicitly define within each experiment what is being measured.

Third, the lack of data on the thermal dependence of carrying capacity constitutes a significant obstacle to uncovering systematic warming-stability relationships (Table 2). The three different scenarios produced by Uszko et al. (2017) demonstrate as much: the warming-stability relationship shifts from hump-shaped (Fig. 3, Usz1), to monotonically decreasing (Fig. 3, Usz2), to almost U-shaped (Fig. 3, Usz3) with a constant, decreasing and hump-shaped carrying capacity, respectively. This highlights the significance of having measurements from the same study system for which the thermal dependence of the other parameters were established. Though there is no guarantee that a systematic relation between carrying capacity and temperature must emerge, we nevertheless highlight the need for more measurements.

Carrying capacity represents an aggregate of processes which ultimately constrain resource population growth, playing a significant role in the dynamics of populations and communities (Rosenzweig 1971, Yodzis and Innes 1992, Lemoine 2019). Because it is hard to define as a specific biological quantity, empirical evidence for its thermal dependence remains scarce (Bernhardt et al. 2018).

Measured as the resource equilibrium in the absence of consumers, carrying capacity has been shown to decrease with warming (Table 2, O'Connor et al. 2009, West and Post 2016, Bernhardt et al. 2018). An explanation has been attributed to the metabolic theory of ecology: given constant nutrient supply, per capita metabolic demands increase with temperature and cause the total population biomass to decline. Interestingly, if body-size decreases with temperature, then per capita metabolic demands increase less, dampening the effect of warming on the carrying capacity expressed in terms of individuals (Bernhardt et al. 2018).

Alternatively, the thermal dependence of carrying capacity could be hump-shaped (Table 2). Based on a mathematical approach to the logistic growth function, the carrying capacity can be a linear function of the resource growth rate (Uszko et al. 2017, Vasseur 2020). In this case, the carrying capacity would share the thermal dependence of the resource growth rate, which has been found to be hump-shaped (Amarasekare and Savage 2012, Uszko et al. 2017). This relationship can vary with nutrient availability (Bieg and Vasseur

<table><caption>Table 2. Assumed or measured thermal dependencies of carrying capacity with their sources.</caption><thead><tr><td>Reported thermal dependence</td><td>Comments</td><td>Sources</td></tr></thead><tbody><tr><td>Monotonically decreasing, \</td><td>Warming increases metabolic demands per unit of biomass. Therefore, assuming constant nutrient supply, warming leads to a reduction in carrying capacity with respect to biomass. However, if warming also causes a decrease in body-size, the negative thermal dependence of carrying capacity becomes weaker in terms of total individuals (rather than total biomass)</td><td>Uszko et al. 2017, Bernhardt et al. 2018</td></tr><tr><td>Constant, –</td><td>Assuming that stoichiometric constraints are not sensitive to temperature change and that the system is closed.</td><td>Uszko et al. 2017</td></tr><tr><td>Hump-shaped, ∩</td><td>Driven by the thermal dependence of resource growth rate, found to be hump-shaped in many cases</td><td>Uszko et al. 2017, Vasseur 2020</td></tr><tr><td>U-shaped, ∪</td><td>Intraspecific competition peaks around the optimal temperature for growth, thus reducing the carrying capacity to a minimum around that temperature</td><td>Amarasekare 2015</td></tr></tbody></table>
2024). With ample nutrients, resource growth and carrying capacity could both be hump-shaped, with very similar temperature optima. However, as nutrients become limiting, the thermal optimum of the carrying capacity can shift to lower temperatures, to the extent that its thermal dependence becomes practically monotonically decreasing.

More generally, temperature dependencies should change when accounting for other environmental factors (Litchman and Thomas 2023) and targeted approaches are required to deal with this nuance (van Moorsel et al. 2023). Accounting for such multi-dimensionality could be the missing piece required to unify apparently different thermal dependence shapes. Carrying capacity, specifically, is known to emerge as an aggregate of nutrient supply, stoichiometry and metabolism among others (Uszko et al. 2017, Lemoine 2019, Sentis et al. 2022). In this case, the hump-shaped thermal dependence remains the most likely outcome (Lemoine 2019); however, temperature invariant or exponentially decreasing relationships have also been reported (Uszko et al. 2017).

Finally, carrying capacity could be U-shaped, as a result of stoichiometric constraints (Sentis et al. 2022) or from intra-specific competition (Table 2). Amarasekare (2015) defined density-dependent growth limitation as intra-specific competition and found it to be hump-shaped, peaking at the same temperature as resource growth. This competition term was included within the logistic growth function, so carrying capacity would be expressed as the inverse of this competition. Therefore, carrying capacity would have a U-shaped thermal dependence.

From this variety in reported thermal dependencies of the carrying capacity, one could question our ability to uncover systematic patterns. Based on the metabolic theory, a decreasing relationship would be expected (Bernhardt et al. 2018). However, this is based on the change in body-size. When nutrient concentrations and quality come into play, this relationship can shift (Uszko et al. 2017). We can only encourage the execution of more experiments which replicate the same conditions across organisms, or vary the conditions systematically for the same organisms. This could be a first step towards a more general answer, whether positive or negative.

As we noted above, slight changes in assumptions or measurements can shift the thermal dependence of φ. Therefore, in the absence of sufficient data, it is hard to argue in favour of one or other general warming-stability relationship. Our framework provides a simple tool for incorporating data from multiple sources and, thus, producing a broad overview of the empirical evidence. As we have stated, this simplicity which helps reveal qualitative patterns will come at a cost of accuracy. In particular, when consumer energetic efficiency (ρ) becomes small, the consumer becomes vulnerable to extinction. There, small temperature variations can induce disproportionate responses, making the dynamics unpredictable (Synodinos et al. 2021). In our data, a 'low' ρ (ρ < 2) was mostly confined to high temperatures (Supporting information), where metabolic costs increase faster than handling time decreases, when this is monotonically decreasing, or where handling time also increases, when it is U-shaped. Adding

experiments with well-defined measurements and temperature ranges representing different systems into this dataset would enhance the power of the tool and provide a basis for more definite insights and conclusions.

## Conclusions

Understanding the temperature-dependence of consumer-resource dynamics is one of the building blocks for predicting how increasing ambient temperatures will impact food webs and ecological communities. Therefore, the stability of consumer-resource interactions and its thermal dependence represent a topic of much interest, however, with little consensus. We isolated a specific aspect of stability – the tendency for intrinsic oscillations – and simplified the analysis of empirically determined warming-stability relationships. Our approach mapped existing findings onto the thermal dependence of a single aggregate parameter, φ, summarising all empirical results in a simple plot. We further illustrated the usefulness of φ as a tool to decompose the warming-stability outcome into the individual contributions of attack rate, handling time and carrying capacity. The absence of obvious systematic impacts of warming on stability should not preclude the existence of such systematic effects, as our analysis revealed how sensitive predictions can be to subtle assumptions or measurement errors. Well-defined experiments to characterise the thermal dependence of attack rate and handling time, and more measurements of the thermal sensitivity of carrying capacity should help reveal general warming-stability relationships. Our approach simplified a complex problem and provided a blueprint to improve our understanding. Ultimately, however, any resolution to the search for systematic warming-stability relationships rests on the existence and quality of sufficient empirical data.

**Acknowledgements** – We would like to thank Francois Massol for helpful comments. Open Access funding enabled and organized by Projekt DEAL.

**Funding** – The research was 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 no. 726176) and by the 'Laboratoires d'Excellences (LABEX)' TULIP (ANR-10-LABX-41).

## Author contributions

**Alexis D. Synodinos:** Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Visualization (lead); Writing – original draft (lead); Writing – review and editing (lead). **Arnaud Sentis:** Writing – original draft (equal); Writing – review and editing (equal). **Jose M. Montoya:** Funding acquisition (lead); Supervision (lead); Writing – original draft (equal); Writing – review and editing (equal). **Bart Haegeman:** Data curation (equal); Methodology (lead); Visualization (lead); Writing – original draft (equal); Writing – review and editing (equal).
## Data availability statement

There are no additional data for this paper.

## Supporting information

The Supporting information associated with this article is available with the online version.

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