domain: sde sdv

text:
Rebecca K French
email: rebecca.french@sydney.edu.au

Sandra H Anderson

Kristal E Cain

Terry C Greene

Maria Minor

Colin M Miskelly

Jose M Montoya

Chris G Muller

Michael W Taylor

Kākāpō Recovery Team

Edward C Holmes
email: edward.holmes@sydney.edu.au

Host Phylogeny Shapes Viral Transmission Networks in an Island Ecosystem

Viral transmission between host species underpins disease emergence. Both host phylogenetic relatedness and aspects of their ecology, such as species interactions and predator-prey relationships, may govern cross-species virus transmission and zoonotic risk, although their relative impact is unknown. By characterising the virome of a relatively isolated island ecological community linked through a food web we show that phylogenetic barriers result in distantly related host species sharing fewer viruses. Host ecology had a much smaller influence on overall virome composition. Network analysis revealed that hosts with a high diversity of viruses were more likely to gain new viruses, and that generalist viruses were more likely to infect new hosts. Such a highly connected ecological community heightens the risk of disease emergence, particularly among closely related species.

Determining how viruses move through ecosystems is central to understanding their occasional emergence as pathogens. Cross-species virus transmission is a near universal feature of viruses [START_REF] Geoghegan | Comparative analysis estimates the relative frequencies of co-divergence and cross-species transmission within viral families[END_REF]. However, most studies of virus host jumping have focused on single virus pathogens and/or single host species [START_REF] Wille | Strong phylogenetic and ecological effects on host competency for avian influenza in Australian wild birds[END_REF][START_REF] Fusaro | The introduction of fox rabies into Italy (2008-2011) was due to two viral genetic groups with distinct phylogeographic patterns[END_REF], limiting our capacity to reveal broad-scale host-virus interactions [START_REF] Wille | Virus-virus interactions and host ecology are associated with RNA virome structure in wild birds[END_REF][START_REF] Zhao | Viral metagenomics unveiled extensive communications of viruses within giant pandas and their associated organisms in the same ecosystem[END_REF]. In addition, recent metagenomic studies suggest that only a tiny proportion of viruses cause disease, with apparently healthy wildlife species commonly infected with multiple viruses [START_REF] Zhang | Expanding the RNA virosphere by unbiased metagenomics[END_REF]. In reality, viruses are a key component of global ecosystems, regularly moving between species in the absence of overt disease [START_REF] Holmes | The Ecology of Viral Emergence[END_REF]. A full understanding of infectious disease emergence therefore requires an ecosystem-level approach [START_REF] Holmes | The Ecology of Viral Emergence[END_REF][START_REF] French | An ecosystems perspective on virus evolution and emergence[END_REF], in which the viromes of entire interacting communities are investigated.

Two factors have been proposed to govern virus movement from one host species to another, shaping similarities and differences in virus composition among taxa, and hence determining virome structure at the ecosystem scale. First, it is possible that host phylogenetic relatedness directly impacts the frequency and pattern of cross-species virus transmission. Rates of cross-species virus transmission are expected to be higher among closely related host species, reflecting a greater similarity in the virus and host proteins that underpin successful virus-cell relationships such as host cell binding [START_REF]Host and viral traits predict zoonotic spillover from mammals[END_REF]. A fundamental difference in virus-host cell relationships explains why viruses from vertebrates and invertebrates are usually phylogenetically distinct [START_REF] Harvey | Diversity and evolution of the animal virome[END_REF], even though the latter are often dietary components of the former. A second proposed factor is that ecological properties of the host play a major role in virus movement between host taxa by determining the probability of virus exposure [START_REF] Luis | Network analysis of host-virus communities in bats and rodents reveals determinants of cross-species transmission[END_REF]. Each time species interact they provide opportunities for cross-species virus transmission. Hence, the more interactions, the greater the probability of host jumping. For example, changes in land use have increased human-animal interactions, driving disease emergence events in humans [START_REF] Holmes | COVID-19-lessons for zoonotic disease[END_REF]. Predator-prey interactions are common exposure events, with the consumption of prey providing direct exposure to viruses during digestion. Consequently, the structure of an ecosystem food web may have a large impact on the flow of viruses through communities. With the exception of marine microbial food webs in which viral lysis of microbial hosts impacts food web structure [START_REF] Weitz | A multitrophic model to quantify the effects of marine viruses on microbial food webs and ecosystem processes[END_REF], this prediction is yet to be tested.

Metagenomic sequencing allows virus diversity to be explored with precision, by -in theory -sequencing the entire virome of samples without bias [START_REF] Zhang | Expanding the RNA virosphere by unbiased metagenomics[END_REF] and leading to the increasingly rapid discovery of novel viruses [START_REF] Zayed | Cryptic and abundant marine viruses at the evolutionary origins of Earth's RNA virome[END_REF]. Far less attention has been directed toward understanding how viruses are embedded within ecosystems, and the relative role of host phylogeny and ecology in structuring virus diversity. Research on food webs including viruses has commonly either focused on single pathogens of a species of interest [START_REF] Haynes | Spatial synchrony propagates through a forest food web via consumer-resource interactions[END_REF], or on the microbial subset of the food web [START_REF] Weitz | A multitrophic model to quantify the effects of marine viruses on microbial food webs and ecosystem processes[END_REF]. The virome of an entire food web is yet to be characterised, in part because the large number of species in most ecological communities makes this impractical. However, in small island forest ecosystems, such as Pukenui/Anchor Island in the Fiordland region of southwestern New Zealand (fig. S1), the entire forest community is small enough that most broad taxonomic groups can be sampled, providing a snapshot of a food web virome.

The unique evolution of New Zealand wildlife (i.e., in the almost complete absence of native terrestrial mammals) means that unlike other forest ecosystems, the Pukenui/Anchor Island ecological community is dominated by birds, with no terrestrial mammals, and only one species of reptile. However, despite being small, isolated and having a low diversity (relative to forest ecosystems outside New Zealand), this community contains a breadth of ecological niches, including a variety of diets: carnivores, insectivores, plant-eaters (including herbivores, frugivores, nectarivores and granivores), piscivores and omnivores. This ecological community also contains numerous threatened species that may be vulnerable to disease emergence via cross-species transmission; for example, the critically endangered kākāpō (Strigops habroptila), endangered mohua (Mohoua ochrocephala), and critically endangered Te Kakahu/Chalky Island skink (Oligosoma tekakahu). The Pukenui/Anchor Island community also has a relative decoupling of phylogenetic relatedness and ecological niches. Some species that are distantly related (i.e., from different phylogenetic orders) possess a similar ecological niche and regularly interact. For example, yellow-crowned parakeets (Cyanoramphus auriceps, parrot) were frequently observed with passerines (brown creeper Mohoua novaeseelandiae, mohua, and the grey warbler Gerygone igata), travelling and foraging in multi-species flocks (pers. obs), which has been observed elsewhere in New Zealand [START_REF] Mclean | Mixed-species flocking of forest birds on Little Barrier Island[END_REF].

We used metatranscriptomic (i.e., total RNA) sequencing to document the virome of each host in the community, including those that directly infect the host, bacteriophages, and viruses present in the host diet. Hence, we use the total viral diversity of each host, rather than only those viruses that have established a true infection, to provide a broader view of virus transmission. If host phylogeny were the key driver of viral diversity we would expect viromes to cluster according to the major host phyla (e.g., Chordata, Arthropoda and Streptophyta), as well as grouping at lower phylogenetic levels, with more closely related hosts having more similar viromes than hosts that are more distantly related (Fig. 1A). In contrast, if host ecology were the main driver of virus diversity, we would expect viromes to cluster according to the major dietary associations, with hosts that have similar diets possessing similar viromes, and predators and prey clustering together (Fig. 1B). Our sampling of the Pukenui/Anchor Island forest community included all key vertebrate species in addition to representative sampling of invertebrates and plants (table S1). The sampled community comprised five host phyla, 13 classes and 37 orders. We conducted diversity and network analyses to determine the importance of host phylogeny (at the phylum, class and order level) and host ecology on viral diversity at the virus family level. A detailed account of the methods used in this study is provided in the supplementary materials.

Results

Viruses predominantly cluster according to host phylogeny

We identified 16,633 viral sequences (assembled contigs) with an abundance of 360 million reads from a total of 112 different viral families. Non-metric multidimensional scaling plots revealed that, overall, viral communities clustered to the level of host phyla (Fig. 2), which was confirmed using pairwise adonis tests (table S2). All comparisons between Chordata, Arthropoda and Streptophyta were significant (p<0.05, n=49, table S2). In addition, host phylogenetic order explained the most variation between viral communities (R 2 =0.8), compared to host phylum (R 2 =0.13) and class (R 2 =0.30), when used as the dependant variable in the adonis test (n=49).

Within Chordata, we used host order and host diet to compare how much variation in the viromes is explained by each factor. Host diet was described using two separate binary factors based on the primary diet types (table S1) -'insectivore' (yes/no) and 'plant-eater' (yes/no). Nectar, fruit and seed eaters were assigned as plant-eaters. Omnivores that feed on both invertebrates and plants were assigned as both an insectivore and a plant-eater.

Carnivores and piscivores were assigned as neither an insectivore nor a plant-eater. When controlling for host order, permutational multivariate analysis of variance (PERMANOVA) tests revealed that insectivores had significantly different viromes than non-insectivores, and plant-eaters had significantly different viromes from non-plant-eaters (p<0.05, n=19, table S2). However, host order explained much more of the variation (R 2 =0.47) than diet (insectivore R 2 =0.06, plant-eater R 2 =0.07). A similar result was obtained when including only viruses likely to infect chordates, with all comparisons significant (p<0.05, n=19, table S2), and host order explaining more variation (R 2 = 0.48) than host diet (insectivore R 2 =0.06, plant-eater R 2 =0.06). Module one contained only invertebrates (arthropods, annelids and a platyhelminth), module two predominately comprised plants (75% plants and 25% invertebrates), while modules three and four largely comprised chordates (86 and 81% chordates, 14 and 19% invertebrates, respectively) (Fig. 3). All modules contained both DNA and RNA viruses. Module three comprised mostly bacteriophage, while the other modules were dominated by viruses that infect the hosts in those modules.

Modules also had differing levels of host and virus richness, with module one containing the highest number of hosts [START_REF] Streicker | Host phylogeny constrains cross-species emergence and establishment of rabies virus in bats[END_REF], module two containing the highest number of virus families [START_REF] Camacho | BLAST+: architecture and applications[END_REF], and module three containing the lowest number of both hosts [START_REF] Holmes | The Ecology of Viral Emergence[END_REF] and virus families [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF]. The communities within each module were significantly different from one another (p<0.05, n=49, pairwise adonis test; table S2), which was robust to rarefication (table S3). Interestingly, the two modules containing chordates (modules three and four) had the smallest difference in communities (table S2), although still statistically significant.

Network structure

We next analysed the structure of the host-virome network using the degree distribution (i.e., the distribution of the number of links between nodes) in comparison to a network generated with a null model (a bipartite network with the same number of nodes and links, randomly assigned). The cumulative degree distributions for the host and virus nodes followed a truncated power-law distribution (Kolmogorov-Smirnov test p>0.05) of for the hosts and for the virus

Potential for virus spread within the network

To examine the strength of connections between hosts that shared viruses, we created a unipartite network (i.e., links between hosts with shared virus families, where a link is a connection between two hosts) based on the Bray-Curtis dissimilarity matrix, which we refer to here as the 'host community network' (Fig. 5). This network had a high level of connectivity between hosts and within host phyla, with the connections across host phyla generally weaker (i.e., a higher Bray-Curtis value). The maximum shortest path was eight, such that the most distantly connected nodes were still only eight links (connections between hosts) away from one another. The mean shortest path distance was only 3.19, such that on average each node is 3-4 links away from every other node. Also of note is a key cluster containing predominately chordates with a high number of strong connections.

In comparison to a null network, the transitivity ratio (i.e., the probability that the adjacent nodes are connected, expressed as a ratio comparing the null and host community network values) was high (3.1), while the path length ratio (the smallest number of links between each node) was similar (1.2). This signifies a small world network: both highly clustered and highly connected. Most nodes are a small number of links away from every other node, enabling viruses to easily move between species [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF]. Our analysis also revealed that if a virus were to move through the network, it would first move within a host phylum, then most likely from plants to arthropods, and to/from arthropods and chordates. Strikingly, direct links between plants and chordates were far less common, which may reflect the relatively small number of chordates on the island that eat plants, compared to those that eat invertebrates (table S1).

Host ecology affects viral diversity at smaller scales

To examine the diversity of virus species within and across host phyla we conducted phylogenetic analysis for one virus family per module: Parvoviridae (module 1), Caulimoviridae (module 2), Fiersviridae (module 3) and Caliciviridae (module 4) (Fig. 6, fig. S2-S5, table S6), chosen for their high viral diversity across multiple host species. In general, there was a high degree of virus clustering by host phyla, with each module containing a high richness and abundance of virus species from their predominant host.

However, there was also evidence of host ecology (i.e., predator-prey interactions)

influencing virus diversity, with closely related/identical viruses found in distantly related hosts. For example, we identified three viruses in the kākā (parrot) library from the plant virus family Caulimoviridae that clustered close to members of the viral genus Badnavirus found in the plants sampled here (Fig. 6). This pattern is indicative of a food web interaction in which the parrot consumed the plants. Similarly, we found near identical (>99% at the amino acid level) members of the Caliciviridae in a miromiro/tomtit (passerine, Petroica macrocephala) and slater/woodlouse (arthropod, Isopoda sp.) library, strongly indicating a food web interaction (fig. S5). As the arthropod is a detritivore and the bird predominately insectivorous, the interaction could have occurred in either or both directions.

Discussion

We used a metatranscriptomic approach to reveal the drivers of virome structure in a relatively isolated island community. We achieved this by creating the host-virome network (bipartite) based on the presence/absence of each virus family in each host library, and the host community network (unipartite), based on the level of virome similarity between hosts.

This analysis revealed a strong effect of host phylogeny on the host-virome network, with more viral families shared within than between host phyla. Such a clear effect provides strong evidence that host phylogenetic distance is a key constraint on the cross-species transmission of viruses, and hence on disease emergence. Similar trends have been observed at smaller scales (i.e., with specific viral families and hosts), with cross-species transmission decreasing in frequency with increasing host phylogenetic distance [START_REF] Streicker | Host phylogeny constrains cross-species emergence and establishment of rabies virus in bats[END_REF][START_REF] Longdon | Host phylogeny determines viral persistence and replication in novel hosts[END_REF]. Host phylogeny has also been shown to be an important factor in explaining the diversity of bacteria and eukaryotic parasites [START_REF] Gupta | Host phylogeny matters: Examining sources of variation in infection risk by blood parasites across a tropical montane bird community in India[END_REF][START_REF] Souza | Analysis of bacterial composition in marine sponges reveals the influence of host phylogeny and environment[END_REF], indicating that it has wide ranging impacts on all infectious agents within an organism. The strength of this host phylogenetic trend is surprising given that the viromes characterised included all viruses, not just those infecting the host in question. There were necessarily differences in sampling method between chordates (cloacal swab), invertebrates (tissue) and plants (tissue) which, while unavoidable, could have resulted in artificial similarities between similar sample types. However, cloacal swabs should have biased the results toward food web interactions since they represent a sample of the host and the digestive tract, which often contain diet-associated viruses [START_REF] Li | Bat guano virome: predominance of dietary viruses from insects and plants plus novel mammalian viruses[END_REF].

That viruses generally cluster by host phylogeny means that virus-host co-divergence will shape virome structure on deep evolutionary scales. Phylogenetic barriers may prevent frequent cross-species virus transmission between phyla because this can only occur between species with similar virus-cell receptor interactions. This results in a mixture of host-virus codivergence at deeper taxonomic levels and cross-species transmission at shallower taxonomic boundaries, with the combination of both processes meaning that more closely related species have generally similar viromes. This is in accord with most comparative studies that have revealed a relatively weak impact of host-virus codivergence [START_REF] Forni | Ancient evolution of mammarenaviruses: adaptation via changes in the L protein and no evidence for host-virus codivergence[END_REF][START_REF] Ramsden | Hantavirus evolution in relation to its rodent and insectivore hosts: no evidence for codivergence[END_REF].

Indeed, our phylogenetic analysis at the level of virus species provided direct evidence for cross-species transmission with, for example, a cluster of closely related caliciviruses in both the fantail (Rhipidura fuliginosa, passerine) and the tawaki (Eudyptes pachyrhynchus, penguin) (fig. S5).

A striking result was the limited impact of host diet on the host-virome network within Chordata, explaining only a small amount of variation. This suggests that predator-prey interactions are not associated with high levels of cross-species viral transmission, likely due to the constraints associated with phylogenetic barriers. Aside from the human gut [START_REF] Minot | The human gut virome: inter-individual variation and dynamic response to diet[END_REF], there is limited evidence for host diet impacting host viromes. However, other aspects of host ecology such as location, age and behaviour (within a narrower host range than our study) all influence host viromes, although to a lesser extent than phylogeny [START_REF] Wille | Virus-virus interactions and host ecology are associated with RNA virome structure in wild birds[END_REF][START_REF] Geoghegan | Virome composition in marine fish revealed by metatranscriptomics[END_REF]. Despite the overall weak influence of host ecology on viral diversity, we found clear examples of host predatorprey interactions at the virus species level, showing that viruses are moved between species via these processes. Overall, our results suggest that virus traffic from predator-prey interactions is only transient (i.e., the virus is only present for a short time in the predator's digestive system) and unlikely to result in productive infections.

Both the host-virome network (bipartite) and host community network (unipartite) had similar structures to other ecological networks, suggesting they have similar constraints. Our host-virome network followed a truncated power-law distribution that had lower cut-off values than expected by chance. This suggests that the structure is influenced by assembly mechanisms (i.e., ecological or phylogenetic traits of the host or virus that alter how the network is constructed). Power-law distributions occur when nodes are likely to get more links the more they have already, such that the 'rich get richer' [START_REF] Montoya | Ecological networks and their fragility[END_REF]. Accordingly, hosts that already have a high diversity of viruses are more likely to gain new viruses than hosts with a low diversity of viruses. Similarly, viruses with many hosts are more likely to gain more hosts, which may inform zoonotic risk assessments [START_REF] Wille | How accurately can we assess zoonotic risk?[END_REF]: rather than assigning the highest risk score to viruses that are similar to those that have already emerged, it may be of greater utility to give preference to the most generalist viruses. This power-law distribution is truncated when the underlying model no longer accurately predicts the distribution beyond a certain cut-off, effectively preventing the rich from getting richer beyond that point. The higher this cut-off, the greater the number of highly connected hosts and viruses [START_REF] Valverde | The architecture of mutualistic networks as an evolutionary spandrel[END_REF]. In our models, this cut-off value was higher than for the null network, indicating that the hostvirome network contains more highly connected hosts and viruses than expected by chance.

Truncated power-law distributions are often found in mutualistic networks (e.g. plantpollinators), with the truncation thought to be due to 'forbidden' -physically impossiblelinks [START_REF] Jordano | Invariant properties in coevolutionary networks of plant-animal interactions[END_REF]. In our network, forbidden links could be due to phylogenetic barriers preventing cross-species transmission.

The host community network (unipartite) was a 'small world' network. Although clear clusters emerge, most nodes (hosts) can be reached via a small number of connections to other nodes [START_REF] Montoya | Small world patterns in food webs[END_REF]. This is common in ecological networks, with most species only two links apart on average in complex food webs [START_REF] Williams | Two degrees of separation in complex food webs[END_REF]. The combination of strong modularity in our host virome network and high connectivity in our host community network implies that a pathogenic virus could rapidly move through the network, particularly once the virus crossed from one module to another. The host community network showed an especially strong clustering within chordates, with a high degree of similarity between hosts. This cluster included endangered species such as the mohua and Te Kakahu skink, suggesting that on Pukenui/Anchor Island these species are vulnerable to disease emergence from other chordates within that cluster. In contrast, the kākāpō was less closely connected to this cluster, suggesting it may be less vulnerable.

Our study shows that the phylogenetic relatedness of hosts is a strong driver of viral diversity in ecological communities. Phylogenetic barriers between distantly related hosts likely prevent frequent virus movement despite exposure events via predator-prey interactions. The ecological community is highly connected, which carries risks for disease emergence in vulnerable species. Our study sheds light on the processes that dictate viral movement through ecosystems. Further research is needed on the implications of disruption to these networks on disease emergence.

Fig. 1 .

 1 Fig. 1. Expected impact of the two drivers of viral diversity on the structure of virus

Fig. 2 .

 2 Fig. 2. The similarity/dissimilarity of viral communities among host species. The first two

Fig. 3 .

 3 Fig. 3. Host-virome network (bipartite) displayed using the Fruchterman-Reingold

Fig. 4 .

 4 Fig. 4. The degree distribution of the host-virome network (large circles) and the

Fig. 5 .

 5 Fig. 5. Host community network (unipartite) displayed using the Fruchterman-

Fig. 6 .

 6 Fig. 6. Virus diversity at the species level. Phylogeny of the Caulimoviridae (module 2).

  (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is

	bioRxiv preprint	doi:	; https://doi.org/10.1101/2022.10.04.510907	this version posted October 5, 2022.	The copyright holder for this preprint
			. CC-BY 4.0 International license made available under a

  (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is

	bioRxiv preprint	doi:	; https://doi.org/10.1101/2022.10.04.510907	this version posted October 5, 2022.	The copyright holder for this preprint
			. CC-BY 4.0 International license made available under a
			Submitted Manuscript: Confidential
			Template revised February 2021
				20

Acknowledgments: We thank Andrew Digby, Jodie Crane, Brodie Philp, Dave Johnson and Chloe Corne for logistical assistance, and Brodie Philp, Bex Jackson and Nigel French for assistance with sample collection. We thank the Molecular Epidemiology and Public Health Laboratory at Massey University New Zealand for use of their laboratory space, and Kristene

Funding:

Australian Research Council Australian Laureate Fellowship FL170100022 (ECH). FRAGCLIM ERC Consolidator Grant under the European Union's Horizon 2020 (Grant Agreement Number 726176), and "Laboratoires d'Excellences (LABEX)" TULIP (ANR-10-LABX-41) (JMM)

families, as shown by the fit lines in Fig. 4. The null networks also follow truncated power-law distributions of and , but with lower cut-off values than the host-virome network. Viruses with few connections had fewer than the random expectation, while viruses with more connections had more than expected by chance, shown by the null and virus distributions intersecting at approximately 10 links (Fig. 4). In contrast, all hosts had systematically more connections than the random expectation.

Gedye for advice and assistance. This research utilised the high-performance computing

Supplementary Materials

Materials and Methods

Figs. S1 to S5

Tables S1 to S6

Data S1 (separate file)

References


Alexis D Synodinos
email: a.d.synodinos@gmail.com

Arnaud Sentis

José M Montoya

Bart Haegeman

The effects of warming on the stability of consumer-resource interactions

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

Temperature regulates the physiology and behaviour of organisms. Thus, changing temperatures induce dynamics in 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 is lacking. To investigate these systematically, we propose a simplified theoretical framework that can incorporate empirical data in three steps. First, we constrain stability to intrinsic oscillations to avoid comparing disparate stability notions. Second, we reduce complexity by utilising a one-dimensional stability metric. Third, we enable the direct comparison of all data by converting all thermal dependence parameterisations into a single function, with two parameters in the exponent determining its shape. The empirical data generate four different warming-stability relationships: stability increases, decreases, is hump-shaped or U-shaped with temperature. The diversity of warming-stability relationships, though partly attributable to context-dependence, is fundamentally caused by sensitivity to two factors: how the processes within the functional response are defined and the thermal dependence of carrying capacity. Consistency across studies regarding the former and acquiring more data on the latter should help uncover systematic patterns in the thermal dependence of stability in consumer-resource interactions.

.

Introduction

Temperature affects the biology of organisms in many ways; it can cause changes in metabolic demands, foraging behaviour, ingestion rates and growth rates among other things [START_REF] Pawar | Real versus artificial variation in the thermal sensitivity of biological traits[END_REF][START_REF] Savage | Effects of body size and temperature on population growth[END_REF][START_REF] Gillooly | Effects of size and temperature on metabolic rate[END_REF]. Such organism-and population-level responses scale up to ecological communities and ecosystems via interspecific interactions [START_REF] Montoya | Climate change, biotic interactions and ecosystem services[END_REF][START_REF] O'connor | Theoretical predictions for how temperature affects the dynamics of interacting herbivores and plants[END_REF][START_REF] Thomas | Extinction risk from climate change[END_REF]. Therefore, to predict how increasing ambient temperatures (IPCC, 2021) will alter communities, we first need to understand how interspecific interactions will respond to warming [START_REF] Alvarez-Codesal | Thermal mismatches explain consumer-resource dynamics in response to environmental warming[END_REF][START_REF] Bideault | Thermal mismatches in biological rates determine trophic control and biomass distribution under warming[END_REF].

The dynamics of consumer-resource interactions are particularly important in this regard, as they regulate energy and material flows from primary producers towards higher trophic levels [START_REF] Rip | Cross-ecosystem differences in stability and the principle of energy flux[END_REF]. Therefore, the impact of temperature change on stabilitya composite aspect of dynamics [START_REF] Donohue | On the dimensionality of ecological stability[END_REF] -in consumer-resource dynamics has been extensively studied (e.g., [START_REF] Rall | Temperature, predator-prey interaction strength and population stability[END_REF][START_REF] Amarasekare | Effects of temperature on consumer-resource interactions[END_REF][START_REF] O'connor | Theoretical predictions for how temperature affects the dynamics of interacting herbivores and plants[END_REF][START_REF] Fussmann | Ecological stability in response to warming[END_REF][START_REF] Amarasekare | Effects of climate warming on consumer-resource interactions: a latitudinal perspective[END_REF]; however, the evidence appears inconclusive [START_REF] Synodinos | Theory of temperaturedependent consumer-resource interactions[END_REF]. In this study we seek to resolve this issue by producing a comprehensive overview of the warming-stability relationships which emerge from empirical data.

Theory postulates that stability properties can be deduced from aggregates of empirically measurable quantities [START_REF] Yodzis | Body size and consumer-resource dynamics[END_REF][START_REF] Synodinos | Theory of temperaturedependent consumer-resource interactions[END_REF]. One such aggregate is consumer energetic efficiency, which quantifies the energy gained through consumption relative to that lost through metabolic demands [START_REF] Yodzis | Body size and consumer-resource dynamics[END_REF]. A second aggregate, interaction strength, quantifies the impact of the consumer on the resource, either in the short-term through the functional response [START_REF] Rall | Temperature, predator-prey interaction strength and population stability[END_REF] or in the long-term via the change in resource equilibrium due to the consumer [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]. Both aggregates are closely linked to various aspects of stability: intrinsic oscillations, consumer persistence and asymptotic resilience.

A reduction with warming in either energetic efficiency or interaction strength has been associated with increased stability via the dampening or elimination of oscillations [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]. Lower energetic efficiency or weaker interaction strength due to warming have also been argued to increase the risk of consumer extinction due to starvation [START_REF] Sentis | Using functional response modeling to investigate the effect of temperature on predator feeding rate and energetic efficiency[END_REF][START_REF] Rall | Universal temperature and body-mass scaling of feeding rates[END_REF][START_REF] Alvarez-Codesal | Thermal mismatches explain consumer-resource dynamics in response to environmental warming[END_REF]. 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).

To resolve this issue, we focused exclusively on oscillations as the property of stability for which we sought general warming-stability relationships. Intrinsic (i.e., not induced by demographic or environmental perturbations) oscillations of coupled consumer and resource populations manifest when self-regulation (i.e., intraspecific density dependence) in the consumer is small compared to the strength of interspecific interactions. In the absence of self-regulation and under certain conditions, such as enrichment (the paradox of enrichment [START_REF] Rosenzweig | Paradox of enrichment: destabilization of exploitation ecosystems in ecological time[END_REF], consumers temporarily overshoot their equilibrium causing resource depletion, which is followed by a decline in consumers. These fluctuations in the consumer and resource populations can become self-sustained, leading to periodic oscillations. Such intrinsic oscillations are linked to the temporal variability in population densities, a property of the dynamics which has been often studied in relation to stability to perturbations (i.e., coefficient of variation, [START_REF] Donohue | On the dimensionality of ecological stability[END_REF]. [START_REF] Johnson | A metric for quantifying the oscillatory tendency of consumer-resource interactions[END_REF] demonstrated that the product of attack rate, handling time and carrying capacity quantifies the tendency of dynamics to oscillate periodically.

The product of attack rate and handling time determines the saturation of the functional response. Hence, it could be viewed as a measure of predation efficiency [START_REF] Yodzis | Body size and consumer-resource dynamics[END_REF]; more efficient consumer species achieve functional response saturation at lower resource abundances. Carrying capacity, on the other hand, has long been viewed as a proxy for environmental conditions, e.g., enrichment [START_REF] Rosenzweig | Paradox of enrichment: destabilization of exploitation ecosystems in ecological time[END_REF]. Thus, the stability metric aggregates information relating to the species' interaction and the environmental conditions. In fact, it is one of two components of interaction strength [START_REF] Synodinos | Theory of temperaturedependent consumer-resource interactions[END_REF]. Therefore, one could consider this stability metric as a proxy for predation efficiency, enrichment or interaction strength. Significantly, for the purpose of our study, this is a one-dimensional metric which significantly reduces the complexity of the study of stability.

The systematic understanding of warming-stability relationships has also been hindered by an apparent lack of agreement in the empirically determined thermal dependencies of the relevant biological processes; attack rate, handling time and carrying capacity (Table 1). Certain studies report that vital rates follow a monotonic exponential relationship across a sufficiently wide temperature range [START_REF] Rall | Temperature, predator-prey interaction strength and population stability[END_REF][START_REF] Fussmann | Ecological stability in response to warming[END_REF], while others have shown that a non-monotonic pattern describes this temperature-dependence more accurately [START_REF] Uiterwaal | Functional responses are maximized at intermediate temperatures[END_REF][START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF]. In this case, after an exponential increase, the function peaks at a temperature referred to as optimal, before rapidly declining with further warming.

U-shaped, Intraspecific competition peaks around the optimal temperature for growth, thus reducing the carrying capacity to a minimum around that temperature.

Amarasekare (2015)

General remark: There is a very limited number of empirical studies to have measured the temperature-dependence of carrying capacity. To overcome this obstacle, we converted all empirically-determined parameterisations from the relevant literature into a single exponential form that can accommodate both monotonic and unimodal functions. We were, thus, able to directly compare empirical findings which were parameterised differently across study systems.

Thus, our approach was based on three steps: defining stability solely in terms of the tendency for intrinsic oscillations, using a one-dimensional stability metric and converting all thermal dependence parameterisations to a single function. Using this approach, we created a comprehensive overview of how the empirical evidence suggests warming will alter stability. Our results revealed four distinct warming-stability relationships. The difference in relationships was determined by three factors: the dichotomy between monotonic and non-monotonic thermal dependencies, the relative thermal sensitivities of attack rate and handling time and the -most often assumed rather than measured -thermal dependence of carrying capacity.

Methods

We use the classical Rosenzweig-MacArthur model with a Holling type II functional response (eqs. 1 & 2), which will apply to a wide range of consumer-resource interactions [START_REF] Jeschke | Consumer-food systems: why type I functional responses are exclusive to filter feeders[END_REF]. For the temperature dependence of organisms, we follow past studies in assuming a direct mapping of environmental temperatures to body temperature [START_REF] Amarasekare | Effects of temperature on consumer-resource interactions[END_REF][START_REF] Vasseur | A mechanistic approach for modeling temperaturedependent consumer-resource dynamics[END_REF] 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.

The model

The Rosenzweig-MacArthur model:

dR dt = rR 1 - R K - aRC 1 + ahR (1) dC dt = e aRC 1 + ahR -mC (2)
The equations describe the rate of change of resource, R, and consumer, C, biomass density

([R] = [C] = mass/area or [R] = [C] = mass/volume,

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 (App. S1, Fig. S1a,b). In a stable limit cycle, populations oscillate periodically around an unstable equilibrium (App. S1, Fig. S1c).

A Hopf bifurcation occurs at the point where dynamics switch from stable equilibrium to stable limit cycle, though this transition is smooth rather than abrupt (App. S1, Fig. S2).

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 (App. S1).

We focus on one aspect of stability: the tendency for intrinsic oscillations in population abundances. [START_REF] Johnson | A metric for quantifying the oscillatory tendency of consumer-resource interactions[END_REF] proposed a single metric to quantify the tendency of dynamics to oscillate periodically in the Rosenzweig-MacArthur model (eq. 3).

Tendency for oscillations ∝ positive feedback negative feedback = ah

1 K = ahK (3)
This metric revolves around two processes impacting per capita resource growth: a positive and a negative feedback. The positive feedback manifests via the functional response product ah. Once the functional response saturates, 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 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).

φ = ahK (4)
As has been previously shown, the Rosenzweig-MacArthur model's dynamics can be simplified by grouping certain parameters together and deriving unitless aggregate parameters [START_REF] Yodzis | Body size and consumer-resource dynamics[END_REF][START_REF] Synodinos | Theory of temperaturedependent consumer-resource interactions[END_REF]. One such aggregate, φ, is precisely the stability metric defined above (eq. 4, see also App. S1 for its derivation). As described in the introduction, φ encapsulates information on the trophic interaction (product ah) and the environmental conditions (K) such that it can be considered a proxy for interaction strength, enrichment or predation efficiency.

Temperature-dependencies of parameters

We analyse the impacts of temperature on stability via the thermal dependence of φ, which is the product of three temperature-dependent parameters (eqn 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 [START_REF] Amarasekare | Effects of temperature on consumer-resource interactions[END_REF][START_REF] Vucic-Pestic | Warming up the system: higher predator feeding rates but lower energetic efficiencies[END_REF][START_REF] West | Impacts of warming revealed by linking resource growth rates with consumer functional responses[END_REF][START_REF] Archer | Consistent temperature dependence of functional response parameters and their use in predicting population abundance[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][START_REF] Binzer | Interactive effects of warming, eutrophication and size structure: impacts on biodiversity and food-web structure[END_REF]. Thus, we illustrate the different potential ways in which φ -and hence stability -depends on temperature.

In the literature, different functions have been used to parameterise the data on the thermal dependencies of a, h and K (e.g., Arrhenius, exponential quadratic with optimal temperature, etc., see Sup. inf. S2 for a comprehensive list). To simplify the analysis, improve comparability between studies and facilitate generalisations, we converted these parameterisations to a canonical form (eq. 5). This function, the general Gaussian, can produce both unimodal and monotonic thermal dependencies and represents a simple choice used previously in the literature (e.g., [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]. The transforma-tions are based on a Taylor approximation (up to the second order) of the log-transformed temperature-dependent variable around the temperature T 0 ≈ T . We present these transformations in the supplementary material (App. S2) and contend that they can provide a useful tool for comparing empirical findings.

p(T ) = p 0 exp α p T -T 0 T w + β p T -T 0 T w 2 , (5)
where

T w = T max -T min (6)
T 0 = T min + T max 2 (7)
T w (eq. 6) is the width of the temperature range of interest with T 0 the mid-point of this temperature range (eq. 7). 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 α p and β p are dimensionless constant parameters of the Gaussian function specific to p(T ). We refer to these as linear and quadratic sensitivities, respectively. Depending on the relative magnitudes and the signs of α p and β p , p(T ) will attain different shapes (Fig. 1 and App. S2.1 for exact conditions).

Figure 1: 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).

Our convention of using a single, general temperature parameterisation (eq. 5) simplifies and clarifies the analysis of the thermal sensitivity of stability. First, the transformed thermal dependence of individual parameters works well for all the original parameteristions we identified in the literature (App. S2, Fig. S3). Second, comparing like-for-like parameterisations across studies allows for the extraction of broader conclusions. Third, the stability metric, φ, becomes the product of three identical exponential functions. Mathematically, this simplifies things drastically: multiplying exponential functions amounts to adding the exponents. Given that a, h and K all share the same form, the values of α φ and β φ will be the sum of the respective α's and β's of a, h and K.

α φ = α a + α h + α K (8a) βφ = β a + β h + β K (8b)
We note that 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 p 0 .

Results

Four warming-stability outcomes

Warming-stability relationships derived from the literature were distributed across the αβ plane, indicating that our approach provided a good tool to encapsulate the available information and compare studies (Fig. 2). Significantly, our analysis revealed that empirical evidence fell into four distinct groupings for the thermal dependence of φ(T ). Below we describe these patterns and look into the causes of this variation.

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, illustrate how the four outcomes emerge from the interaction between these parameters.

Monotonic and non-monotonic thermal dependencies

Seven parameterisations produced a monotonic thermal dependence of φ. In six of these, the temperature-dependent (i.e., not constant) constituent parameters had a monotonic thermal dependence (Table 2, Arc1, Arc2, Bin, Fus, Vuc1 Vuc2). In the remaining case, attack rate was hump-shaped, handling time monotonic and carrying capacity constant (Table 2, Sen).

Both exponents of handling time (α h , β h ) were relatively large, with |α h | > β h (see App. S1 for all values). Attack rate had a quadratic exponent of similar magnitude but opposite sign to handling time, β a < 0, and a relatively small linear exponent, α a > 0. Thus, the sum of the two βs led to a smaller value than the sum of the αs

S2.4, Table

(|β a + β h | < |α a + α h |).
Hence, |α φ | > |β φ | and φ(T ) was monotonic. The remaining four parameterisations yielded a non-monotonic φ(T ) (Table 2, Amr, Usz, Wes1, Wes2). In these cases attack rate and handling time were non-monotonic, though carrying capacity was monotonic in one study (Table 2,Wes1,Wes2).

The split between monotonic and non-monotonic constituent parameters can provide a first clue for the shape of φ. However, the case of 'Sen' (Fig. 2, Table 2) illustrates it is not always robust. Next we look into a subtler difference that may provide an additional criterion in determining the shape of φ 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 2). a increased monotonically or was hump-shaped, while h decreased monotonically or was Ushaped, 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. 3).

Figure 3: 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 α and β values, respectively. By consecutively adding the three vectors (one for each parameter), we end up with the position of φ 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 Usz, K has no thermal dependence (K = constant) and we included a loop-arrow to illustrate this. a is hump-shaped (β a < 0) and h U-shaped (β h > 0) with β h > |β a |. Thus, φ is U-shaped (i.e., β φ > 0). In Wes2, a and h are again hump-and U-shaped, respectively. However, in this case, β h < |β a |. Moreover, the thermal dependence of K is both weak and monotonic, so its contribution in the β direction is minimal. Thus, φ emerges as hump-shaped.

In both cases, attack rate was hump-shaped and handling time U-shaped (Fig. 3 black and grey vectors, respectively). However, in the first study (Fig. 3 Usz), a had a weaker thermal sensitivity than h, so the addition of the two vectors landed φ(T ) in the U-shaped region (green). Additionally, carrying capacity was assumed constant (Fig. 3, Usz white loop), thus not having any further impact on the thermal dependence of φ. In the second study (Fig. 3 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 φ remained in the blue region, where φ(T ) is hump-shaped.

Incomplete data on carrying capacity

Our approach shows that the contribution of each parameter to the sign and magnitude of α φ and β φ is equal (eq. 8a, 8b). Further, the thermal dependencies of a and h most often have opposite signs, and hence weaken each other. Therefore, the thermal dependence of carrying capacity could be significant in determining the thermal dependence of stability.

However, the thermal dependence of carrying capacity is rarely measured for the same study system as the functional response (a, h) parameters have been. From our data set, only one study did that (Wes1, Wes2 [START_REF] West | Impacts of warming revealed by linking resource growth rates with consumer functional responses[END_REF]). For one study we used resource density observations [START_REF] Archer | Consistent temperature dependence of functional response parameters and their use in predicting population abundance[END_REF], another used data from the literature sourced from different experiments on the same host-parasitoid pair [START_REF] Amarasekare | Effects of temperature on consumer-resource interactions[END_REF].

The remaining parameterisations of carrying capacity were derived from a general (i.e., not species-specific) estimate of primary production [START_REF] Vucic-Pestic | Warming up the system: higher predator feeding rates but lower energetic efficiencies[END_REF], came from metaanalyses of species with no strong links to the consumer-resource pairs under consideration [START_REF] Fussmann | Ecological stability in response to warming[END_REF][START_REF] Binzer | Interactive effects of warming, eutrophication and size structure: impacts on biodiversity and food-web structure[END_REF] or were assumed constant [START_REF] Sentis | Using functional response modeling to investigate the effect of temperature on predator feeding rate and energetic efficiency[END_REF][START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF].

Sensitivity of warming-stability outcomes

For consumer-resource pairs close to the origin (i.e., where α = 0 and β=0), the thermal dependence of φ (and stability) will be sensitive to small changes in the individual parameters (e.g., due to measurement errors). Our analysis yielded three such cases (Fig. 2, Sen, Vuc1, Wes2), whose sensitivity arose because a and h had opposite thermal dependencies of similar magnitude and K did not have a strong thermal dependence

(|α K |, |β K | ≲ 1).

Discussion

The impacts of changing mean temperatures on consumer-resource interactions -and food webs by extension -have been studied through the thermal characterisation of specific biological rates (e.g., ingestion and respiration, [START_REF] Rall | Temperature, predator-prey interaction strength and population stability[END_REF], [START_REF] Binzer | Interactive effects of warming, eutrophication and size structure: impacts on biodiversity and food-web structure[END_REF]). These rates are used to parameterise consumer-resource models and, thus, derive the predicted dependence of trophic community dynamics on temperature [START_REF] Vasseur | A mechanistic approach for modeling temperaturedependent consumer-resource dynamics[END_REF][START_REF] Rall | Universal temperature and body-mass scaling of feeding rates[END_REF][START_REF] Kratina | The combined effects of warming and body size on the stability of predator-prey interactions[END_REF]. A key aspect of dynamics which has been long studied is stability [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] Gilbert | A bioenergetic framework for the temperature dependence of trophic interactions[END_REF]; however, we still lack a general expectation of how warming will alter the stability of consumer-resource interactions.

For one, the notion of stability has been applied to different properties of the dynamics [START_REF] Rall | Temperature, predator-prey interaction strength and population stability[END_REF]. A lack of stability has referred to consumer extinction [START_REF] Sohlström | Thermal acclimation increases the stability of a predator-prey interaction in warmer environments[END_REF][START_REF] Daugaard | Warming can destabilize predatorprey interactions by shifting the functional response from type III to type II[END_REF], weaker asymptotic resilience [START_REF] Gilbert | A bioenergetic framework for the temperature dependence of trophic interactions[END_REF] or increased tendency for intrinsic periodic oscillations [START_REF] Johnson | A metric for quantifying the oscillatory tendency of consumer-resource interactions[END_REF][START_REF] Kratina | The combined effects of warming and body size on the stability of predator-prey interactions[END_REF], though these three aspects of stability are not necessarily correlated [START_REF] Yodzis | Body size and consumer-resource dynamics[END_REF][START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF][START_REF] Synodinos | Theory of temperaturedependent consumer-resource interactions[END_REF]. Doubtless, this is not the first time that the complexity of the notion of stability has lead to confusion [START_REF] Donohue | On the dimensionality of ecological stability[END_REF], 2016[START_REF] Grimm | Babel, or the ecological stability discussions: an inventory and analysis of terminology and a guide for avoiding confusion[END_REF].

To avoid such complications, we defined stability with respect to the tendency for intrinsic oscillations. These oscillations result from the intrinsic consumer-resource dynamics and should not be confused with oscillations induced by external perturbations. Further, we used a one-dimensional stability metric, the product of attack rate, handling time and carrying capacity [START_REF] Johnson | A metric for quantifying the oscillatory tendency of consumer-resource interactions[END_REF] (eq. 3), which we termed φ (eq. 4). 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.

Defining stability in terms of φ and using the canonical temperature dependence function helps illuminate why each study system yields a specific outcome. On one hand, the thermal dependence of φ encapsulates the combined effect of the three constituent parameters determining stability. On the other hand, the canonical form of temperature dependence maps the individual contribution of each parameter and the resulting thermal dependence of φ onto the α-β plane. Thus, our approach does not only provide a conclusive overview of the reported warming-stability relationships (Fig. 2), 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. 3, Appendix S3.1, Fig. S5).

No strict boundaries between warming-stability relationships

Our approach illustrates how warming-stability relationships fall into four groupings: stability can be monotonically increasing, monotonically decreasing, U-shaped or hump-shaped.

Though these outcomes appear distinct, they are not separated by some strict boundary.

In particular, subtle differences in the measured or assumed thermal dependence of either attack rate, handling time or carrying capacity can cause a smooth transition into a different warming-stability outcome.

For instance, the parameter values reported in one study from our dataset were derived from a meta-analysis of global invertebrate data [START_REF] Fussmann | Ecological stability in response to warming[END_REF]. The original study reported a stabilising effect of warming (Fig. 2, Fus). However, carrying capacity could arguably have a weaker thermal dependence either due to a concomitant reduction in body-size with warming [START_REF] Bernhardt | Metabolic theory and the temperature-size rule explain the temperature dependence of population carrying capacity[END_REF][START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF]. or by accounting for the uncertainty (i.e., variance) in the derivation of the mean parameter values of the meta-analysis. This uncertainty could also yield a slightly stronger thermal sensitivity of attack rate. In this case, warming would destabilise dynamics, producing the opposite warming-stability relationship compared to the original findings (App. S3, Fig. S6).

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 w 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, [START_REF] Amarasekare | Effects of temperature on consumer-resource interactions[END_REF] reported that resource selfregulation (inverse of carrying capacity) peaked at 32 • C, while attack rate and handling time had optima at 24 • C and 22 • C, respectively. Considering a temperature range wide enough to include the minimum in carrying capacity, results in a U-shaped φ (Fig. 2 Amr). 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 (App. S3, Fig. S6).

These examples illustrate how the transition between different, opposite even, warmingstability 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, h 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 across the two-dimensional plane, which represents the different shapes φ takes along the temperature gradient (Fig. 2). 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. Studies reporting a monotonic dependence for these parameters, yielded monotonic warming-stability outcomes (Table 2). Studies with non-monotonic attack rates and handling times produced non-monotonic warming-stability outcomes, with one exception (Table 2, Sen).

Overwhelming evidence suggests that attack rate and handling time have a hump-and Ushaped thermal dependence, respectively [START_REF] Englund | Temperature dependence of the functional response[END_REF][START_REF] Uiterwaal | Functional responses are maximized at intermediate temperatures[END_REF].

Nevertheless, divergence from this pattern has been reported (e.g., [START_REF] Vucic-Pestic | Warming up the system: higher predator feeding rates but lower energetic efficiencies[END_REF], [START_REF] Rall | Temperature, predator-prey interaction strength and population stability[END_REF], [START_REF] Archer | Consistent temperature dependence of functional response parameters and their use in predicting population abundance[END_REF]) and ascribed to context-dependence [START_REF] Uiterwaal | Functional responses are maximized at intermediate temperatures[END_REF]. 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 have opposite thermal dependencies (increasing vs decreasing or hump-shaped vs 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. 3). In one meta-analysis of ectotherms, attack rate had a steeper slope than handling time [START_REF] Englund | Temperature dependence of the functional response[END_REF], whereas a more recent meta-analysis found evidence for the opposite [START_REF] Uiterwaal | Functional responses are maximized at intermediate temperatures[END_REF]; handling time changed faster with temperature than attack rate. 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 [START_REF] Jeschke | Predator Functional Responses: Discriminating between Handling and Digesting Prey[END_REF]. Interestingly, it has been shown that the former is more sensitive to temperature than the latter [START_REF] Sentis | Parsing handling time into its components: implications for responses to a temperature gradient[END_REF]. 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 [START_REF] Wootton | Towards a modular theory of trophic interactions[END_REF][START_REF] Sentis | Parsing handling time into its components: implications for responses to a temperature gradient[END_REF], 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. Given how predictions can shift continuously (i.e., no discrete boundary) across the α -β plane, this lack of data could have profound effects. Therefore, the potential thermal dependence of carrying capacity requires careful treatment and ideally more measurements from the same study system for which the thermal dependence of the other parameters were established. Carrying capacity represents an aggregate of processes which ultimately constrain resource population growth (i.e., self-regulation), playing a significant role in the dynamics of populations and communities [START_REF] Rosenzweig | Paradox of enrichment: destabilization of exploitation ecosystems in ecological time[END_REF][START_REF] Yodzis | Body size and consumer-resource dynamics[END_REF][START_REF] Lemoine | Considering the effects of temperature × nutrient interactions on the thermal response curve of carrying capacity[END_REF]. Because it is hard to define as a specific biological quantity, empirical evidence for its thermal dependence remains scarce (but see [START_REF] Bernhardt | Metabolic theory and the temperature-size rule explain the temperature dependence of population carrying capacity[END_REF]).

Measured as the resource equilibrium in the absence of consumers, carrying capacity has been shown to decrease with warming [START_REF] O'connor | Warming and resource availability shift food web structure and metabolism[END_REF][START_REF] Bernhardt | Metabolic theory and the temperature-size rule explain the temperature dependence of population carrying capacity[END_REF][START_REF] West | Impacts of warming revealed by linking resource growth rates with consumer functional responses[END_REF]). 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 [START_REF] Bernhardt | Metabolic theory and the temperature-size rule explain the temperature dependence of population carrying capacity[END_REF].

Alternatively, the thermal dependence of carrying capacity could be hump-shaped. Based on a mathematical approach to the logistic growth function, the carrying capacity can be a linear function of the resource growth rate [START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF][START_REF] Vasseur | The impact of temperature on population and community dynamics[END_REF]. In this case, the carrying capacity would share the thermal dependence of the resource growth rate, which has been found to be hump-shaped [START_REF] Amarasekare | A framework for elucidating the temperature dependence of fitness[END_REF][START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF]. More realistically, carrying capacity emerges as an aggregate of nutrient supply, stoichiometry and metabolism among others [START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF][START_REF] Lemoine | Considering the effects of temperature × nutrient interactions on the thermal response curve of carrying capacity[END_REF][START_REF] Sentis | Stoichiometric constraints modulate temperature and nutrient effects on biomass distribution and community stability[END_REF]. In this case, the hump-shaped thermal dependence remains the most likely outcome [START_REF] Lemoine | Considering the effects of temperature × nutrient interactions on the thermal response curve of carrying capacity[END_REF]; however, temperature invariant or exponentially decreasing relationships have also been reported [START_REF] Uszko | Effects of warming on predator-prey interactions -a resource-based approach and a theoretical synthesis[END_REF].

Finally, carrying capacity could be U-shaped, as a result of stoichiometric constraints [START_REF] Sentis | Stoichiometric constraints modulate temperature and nutrient effects on biomass distribution and community stability[END_REF] or from intra-specific competition. The latter has been shown to be humpshaped, peaking at the same temperature as resource growth [START_REF] Amarasekare | Effects of temperature on consumer-resource interactions[END_REF]. In the formulation of the logistic growth with intra-specific competition as self-regulation, carrying capacity would be expressed as the inverse of this competition. Therefore, carrying capacity would have a U-shaped thermal dependence.

Based on our discussion of the above scenarios, certain warming-stability relationships would appear more likely. If attack rate has a stronger sensitivity than handling time (i.e., combined manipulation and ingestion), their product would be hump-shaped (e.g., Fig. 3 Wes2). In this case, a hump-shaped carrying capacity would ensure a hump-shaped φ, so the warming-stability relationship would be U-shaped. This outcome would be still likely in the event of a monotonically decreasing carrying capacity (e.g., Fig. 3 Wes2). If, however, handling time (ingestion only) changes faster than attack rate with warming, then their product would be U-shaped (e.g., Fig. 3 Usz). A hump-shaped carrying capacity would counter this and weaken the overall thermal sensitivity of stability. Whether this would end up being hump-shaped or U-shaped, would depend on the details of the three thermal dependencies.

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 for the existence of general warming-stability relationships. Though it appears obvious, the best way to seek systematic patterns is to measure the thermal dependence of φ in multiple and well-defined experiments representing different systems.

Caveats

Our approach of simplifying potential warming-stability outcomes invariably includes certain restrictive assumptions. Using φ reduces a complex problem to a one-dimensional metric. In certain cases, this could lead to a loss of accuracy as the consumer-resource pair approaches its thermal boundaries [START_REF] Synodinos | Theory of temperaturedependent consumer-resource interactions[END_REF]. Despite this, φ will give a good indication of the qualitative warming-stability relationship [START_REF] Johnson | A metric for quantifying the oscillatory tendency of consumer-resource interactions[END_REF].

Relative to the thermal boundaries of the consumer-resource pair, we considered resources with a wider thermal niche than consumers. If this assumption failed, resource growth would become zero at temperatures for which consumers would not be physiologically limited.

This would cause the abrupt loss of the resource (and consumer), however, our warmingstability predictions would not be affected if carrying capacity does not also become zero. If the carrying capacity also dropped to zero [START_REF] Vasseur | The impact of temperature on population and community dynamics[END_REF], then φ would rapidly decrease and stability -with respect to intrinsic oscillations -would increase prior to resource extinction.

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.

Though context-dependence did play a part, 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.

Figure 2 :

 2 Figure 2: Computed values for α φ and β φ pairs based on empirical studies. The exact values for α φ and β φ are provided in App. 3. These are Amarasekare (2015), Archer et al. (2019), Binzer et al. (2016), Fussmann et al. (2014), Sentis et al. (2012), Uszko et al. (2017), Vucic-Pestic et al. (2011), West & Post (2016). In Archer et al. (2019), Arc1 and Arc2 correspond to two different years of measurement; in Vucic-Pestic et al. (2011), Vuc1 and Vuc2 correspond to two different prey types; in West & Post (2016), Wes1 and Wes2 correspond to two different predators. Studies in the yellow region have a monotonically decreasing φ(T ), in the red region monotonically increasing φ(T ), in the blue region humpshaped φ(T ) and in the green region U-shaped φ(T ).

Table 1 :

 1 Evidence of the thermal dependencies of attack rate, handling time and carrying capacity from the literature.

	Even though such differences in reported thermal dependencies of the various biological
	processes complicate the search of general patterns, a further issue has hindered the direct

Table 2 :

 2 The thermal dependence shapes of attack rate, a, handling time, h and carrying capacity, K, and the resulting shaped in φ, in the studies used.

		study	a	h	K	φ
	Amr	Amarasekare (2015)
	Arc1 Archer et al. (2019)	⧸ const.	⧸	⧸
	Arc2		⧸ const.	⧸	⧸
	Bin	Binzer et al. (2016)	⧸	⧹	⧹	⧹
	Fus	Fussmann et al. (2014)	⧸	⧹	⧹	⧹
	Sen	Sentis et al. (2012)		⧹	const. ⧹
	Usz	Uszko et al. (2017)			const.
	Vuc1 Vucic-Pestic et al. (2011) ⧸	⧹	⧹	⧹
	Vuc2		⧹	⧹	⧹	⧹
	Wes1 West & Post (2016)			⧹
	Wes2				⧹