Youssef Yacine
email: youssef.yacine@sorbonne-universite.fr

Nicolas Loeuille

Stable coexistence in plant-pollinator-herbivore communities requires balanced mutualistic vs antagonistic interactions Authors

Keywords: pollination vs. herbivory Community ecology, Herbivory, Lotka-Volterra, Network motif, Plant-Animal interactions, Pollination

Ecological communities consist of multiple species interacting in diverse ways.

Understanding the mechanisms supporting coexistence requires accounting for such a diversity.

Because most works focus either on mutualism or predation, how pollination and herbivory interactively determine the stable coexistence in plant-pollinator-herbivore communities is still poorly understood. Studying the typical three-species module of such communities, we determine the conditions allowing stable coexistence then investigate how its maintenance constrains the relative interaction strengths. Our results show that coexistence is possible if pollination is sufficiently strong relative to herbivory, while its stability is possible if herbivory is sufficiently strong relative to pollination. A balance between pollination and herbivory is therefore required. Interestingly, shared preferences for plant phenotypes, that would favor such balance, have been frequently reported in the empirical literature. The identified ecological trade-off between attracting pollinators and deterring herbivores therefore also appears as an emergent property of stable plant-pollinator-herbivore communities.

Introduction

Multiple species interacting closely together form an ecological community. A topic of long-standing interest in community ecology is to understand what mechanisms drive the coexistence of species and its maintenance over time. It is now well established that the way ecological interactions connect species -the type of interactions, the network topology as well as the distribution of interaction strengths -plays a decisive role. Combining modelling approaches with empirical data, several works indicate for instance that weak trophic interactions are crucial to maintain the stability of complex food webs [START_REF] Mccann | Weak trophic interactions and the balance of nature[END_REF][START_REF] Neutel | Stability in real food webs: Weak links in long loops[END_REF]. Deriving general laws is, however, difficult. The network properties and topologies favoring the maintenance of coexistence indeed vary with the type of interaction characterizing the community, mutualism or antagonism in particular [START_REF] Thébault | Stability of ecological communities and the architecture of mutualistic and trophic networks[END_REF]. The ecological processes and structural patterns supporting the maintenance of coexistence within single-interaction-type communities can, moreover, considerably differ from the ones at play within communities with several interaction kinds (e.g. [START_REF] Mougi | Diversity of interaction types and ecological community stability[END_REF][START_REF] Sauve | Structure-stability relationships in networks combining mutualistic and antagonistic interactions[END_REF]. Studies of such communities should therefore significantly improve our understanding of ecological communities, especially given that most species get simultaneously involved in a diversity of interaction networks [START_REF] Fontaine | The ecological and evolutionary implications of merging different types of networks[END_REF][START_REF] Kéfi | More than a meal... integrating non-feeding interactions into food webs[END_REF]. Most terrestrial plant species (≈ 90% of flowering plants, [START_REF] Ollerton | How many flowering plants are pollinated by animals?[END_REF], for instance, are involved in a mutualistic interaction with their animal pollinators, while suffering from herbivorous predation (antagonism). Plant-pollinator-herbivore communities are, in addition, of particular interest due to their critical role in agricultural production [START_REF] Klein | Importance of pollinators in changing landscapes for world crops[END_REF][START_REF] Oerke | Crop losses to pests[END_REF], as well as the serious threats global change poses to them [START_REF] Atwood | Herbivores at the highest risk of extinction among mammals, birds, and reptiles[END_REF][START_REF] Potts | Global pollinator declines: Trends, impacts and drivers[END_REF]. The study of stable coexistence within these communities is thus of high applied relevance while offering the opportunity to gain new conceptual insights into the functioning of mutualistic-antagonistic communities.

Understanding stable coexistence within plant-pollinator-herbivore communities requires explicitly accounting for both the mutualistic (i.e. plant-pollinator) and the antagonistic (i.e. plant-herbivore) interaction. A large body of empirical evidence indeed documents nonadditive effects of pollination and herbivory on plant densities, in both uncultivated [START_REF] Gómez | Non-additive effects of herbivores and pollinators on Erysimum mediohispanicum (Cruciferae) fitness[END_REF][START_REF] Herrera | Measuring the Effects of Pollinators and Herbivores: Evidence for Non-Additivity in a Perennial Herb[END_REF][START_REF] Herrera | Interaction of pollinators and herbivores on plant fitness suggests a pathway for correlated evolution of mutualism-and antagonism-related traits[END_REF][START_REF] Pohl | Nonadditive effects of flower damage and hummingbird pollination on the fecundity of Mimulus luteus[END_REF] and cultivated [START_REF] Lundin | When ecosystem services interact: Crop pollination benefits depend on the level of pest control[END_REF][START_REF] Strauss | Towards an understanding of the mechanisms of tolerance: Compensating for herbivore damage by enhancing a mutualism[END_REF][START_REF] Sutter | Synergistic interactions of ecosystem services: Florivorous pest control boosts crop yield increase through insect pollination[END_REF] plant species. The strength of the mutualistic interaction is affected by the antagonistic interaction and vice versa, explaining such an interactive effect. Herbivores may, for instance, preferentially consume plant species bearing abundant flowers or developing fruits as a result of strong pollination [START_REF] Herrera | Measuring the Effects of Pollinators and Herbivores: Evidence for Non-Additivity in a Perennial Herb[END_REF][START_REF] Herrera | Interaction of pollinators and herbivores on plant fitness suggests a pathway for correlated evolution of mutualism-and antagonism-related traits[END_REF]. By decreasing floral display, herbivore damages can reduce pollination [START_REF] Adler | Direct and indirect effects of alkaloids on plant fitness via herbivory and pollination[END_REF][START_REF] Cardel | Effects of florivory on the pollination of flowers: An experimental field study with a perennial plant[END_REF][START_REF] Pohl | Nonadditive effects of flower damage and hummingbird pollination on the fecundity of Mimulus luteus[END_REF]. In addition to floral display, herbivoryinduced changes in plant chemistry can also deter pollinators [START_REF] Kessler | Herbivory-mediated pollinator limitation: Negative impacts of induced volatiles on plant-pollinator interactions[END_REF].

Indirect interactions between two species within a community can also be mediated by their effect on the density of a third species [START_REF] Wootton | Indirect effects in complex ecosystems: Recent progress and future challenges[END_REF]. Ubiquitous in natural communities, such indirect effects play a key part in the maintenance of coexistence [START_REF] Burns | Quantifying direct and indirect effects of perturbations using model ecosystems[END_REF][START_REF] Menge | Indirect effects in marine rocky intertidal interaction webs: Patterns and importance[END_REF]. By isolating the structural building blocks of complex ecological networks -modules or motifs -it becomes easier to unravel such indirect effects and their implications for community maintenance [START_REF] Milo | Network motifs: Simple building blocks of complex networks[END_REF]. Modules are therefore small sets of interacting species characteristic of the studied community, whose study enables deeper insights into the mechanisms at play at the broader scale [START_REF] Milo | Network motifs: Simple building blocks of complex networks[END_REF][START_REF] Stouffer | Understanding food-web persistence from local to global scales[END_REF].

In plant-pollinator-herbivore communities, the typical module consists of two animal species -a pollinator and a herbivore -sharing a common resource plant species (Fig. 1A.a).

The resulting indirect interaction between pollinators and herbivores is an antagonism (Fig. 1A.a, [START_REF] Holland | Consumer-resource dynamics of indirect interactions in a mutualism-parasitism food web module[END_REF]. Pollinators allow the community to sustain a higher herbivore density by increasing plant productivity while herbivores, on the other hand, decrease pollinator density by reducing resource availability. Theoretical works indicate that the presence of pollinators can even make the herbivore population viable [START_REF] Georgelin | Dynamics of coupled mutualistic and antagonistic interactions, and their implications for ecosystem management[END_REF]Mougi and Kondoh, 2014a). As illustrated by [START_REF] Georgelin | Dynamics of coupled mutualistic and antagonistic interactions, and their implications for ecosystem management[END_REF], direct vs. indirect ecological effects can be of similar magnitude. Their study indeed reports a constant herbivore density despite increasing herbivore mortality. In their model, over a wide range of herbivore mortalities, the direct mortality-induced losses on herbivores are totally offset by the indirect gain resulting from a higher pollinator density consecutive to the herbivorous predation release on plants. Further increasing mortality, however, leads to the abrupt collapse of the herbivore population, which illustrates that combining different interactions also has important implications in terms of community stability (Mougi and Kondoh, 2014b).

Mutualisms, such as pollination, intrinsically entail positive feedback loops (Fig. 1A.b).

Positive feedbacks are destabilizing [START_REF] Levins | Discussion paper: the qualitative analysis of partially specified systems[END_REF][START_REF] Neutel | Interaction strengths in balanced carbon cycles and the absence of a relation between ecosystem complexity and stability[END_REF] as they tend to amplify the direct effect of a perturbation. As a result, unstable behaviors have been identified in theoretical models of mutualism, including tipping points [START_REF] Lever | The sudden collapse of pollinator communities[END_REF] or unbounded population growths driven by an "orgy of mutual benefaction" [START_REF] May | Models for two interacting populations[END_REF]. The latter is, however, seldom observed in nature. One possible explanation is that antagonistic interactions, such as predation, could prevent this behavior in real systems. Negative feedback loops born from antagonistic interactions [START_REF] Neutel | Interaction strengths in balanced carbon cycles and the absence of a relation between ecosystem complexity and stability[END_REF] could restore stability by counterbalancing the positive loops arising from mutualisms (Fig. 1A.b). This hypothesis implies that the relative magnitude of pollination vs. herbivory plays a critical role, which is in line with the findings of several theoretical investigations on mutualistic-antagonistic modules [START_REF] Georgelin | Dynamics of coupled mutualistic and antagonistic interactions, and their implications for ecosystem management[END_REF][START_REF] Holland | Consumer-resource dynamics of indirect interactions in a mutualism-parasitism food web module[END_REF]Mougi and Kondoh, 2014b;Sauve et al., 2016a).

The goal of the present paper is to understand how stable coexistence within plantpollinator-herbivore communities constrains the relative strengths of pollination and herbivory, i.e. the relative per capita effects of each interacting animal species on plant population growth.

In contrast with most previous theoretical works on mutualistic-antagonistic modules, the relationships governing stable coexistence we obtain are analytical. Such relationships between pollination and herbivory are derived from the population dynamics of the characteristic threespecies module (Fig. 1A.a), in which both animal intake rates (i.e. functional responses) are assumed linear to achieve analytical tractability. We discuss such an assumption at the end of the present work (section 4). Finally, the per-capita effect of plant-animal interactions on community dynamics is mediated by animal densities, which in turn depend on other ecological parameters such as animal mortalities or intraspecific competition rates. We therefore extend our analysis by studying their influence, which confirms the robustness of our results. In what follows, we show that stable coexistence within plant-pollinator-herbivore communities requires a balance between the strengths of pollination and herbivory. Such a pattern ensues from the opposite effect each interaction has on coexistence and stability (Fig. 1A). Coexistence is favored by pollination and disfavored by herbivory, as a result of both direct and indirect ecological interactions (Fig. 1A.a). Stability is enhanced by herbivory and reduced by pollination, as a result of the respective feedback loops (Fig. 1A.b). It is interesting to note that a large body of empirical literature (e.g. [START_REF] Hutson | The Role of Herbivores in the Maintenance of a Flower Color Polymorphism in Wild Radish[END_REF] reports shared preferences for plant phenotypes between pollinators and herbivores that would favor balanced interactions, which appear here as an emergent property of stable plant-pollinator-herbivore communities.

Model presentation 2.1 Ecological dynamics

We formulate the dynamics of the biomass densities of three interacting species -a plant P, a pollinator M, and a herbivore H -using ordinary differential equations:

= ( - + - ) (1)
= ( - + ) = ( - + )
Plants have a positive intrinsic growth rate ( > 0, autotrophs), while both pollinators ( < 0) and herbivores ( < 0) have a negative one (heterotrophs). As in previous models (e.g. [START_REF] Sauve | Structure-stability relationships in networks combining mutualistic and antagonistic interactions[END_REF], we thus assume the plant-animal interaction to be obligate for animals and facultative for the plant. Intraspecific competition is accounted for. The animal competition rates ( , ) correspond to interference while, for the plant species ( ), this rate essentially captures the competition for resources such as light, water, and nutrients [START_REF] Craine | Mechanisms of plant competition for nutrients, water and light[END_REF]. Interspecific interactions, whose strength is for pollination and for herbivory, affect population growths proportionally to biomass densities. The use of a linear functional response for mutualism exposes the model to unbounded population growths [START_REF] May | Models for two interacting populations[END_REF]. It, however, enables testing whether this behavior could be top-down controlled by herbivory, placing our work in the line of research tackling how the community context could explain the stability of mutualisms in nature (e.g. [START_REF] Ringel | The stability and persistence of mutualisms embedded in community interactions[END_REF]. Finally, and are the conversion efficiencies from plants to animals. Parameter details are given in table 1.

Table 1: List of all model parameters and variables with their biological significance, value, and dimension ( for mass, for length, and for time).

Variables and parameters

Biological meaning Value Dimension

Ecological equilibria

When the three population growth rates vanish (equations (1) are null), we reach an ecological equilibrium ( * , * , * ). At equilibrium, each population can either be present or absent, leading to 8 potential equilibria whose expressions are given in table 2. The present work focuses on the "coexistence equilibrium", i.e. the equilibrium in which the three species are present. We study under which conditions of interaction strengths -( , ) -the coexistence equilibrium corresponds to positive biomass densities (i.e. is feasible), and is stable (i.e. is resilient to small perturbations, i.e. local stability as in [START_REF]Will a large complex system be stable ?[END_REF]. Detailed definitions are provided in Appendix B.I.

Table 2: Expressions of the eight potential ecological equilibria within our plantpollinator-herbivore community. Reminder of our assumptions: > 0, < 0, < 0. For two-species equilibria, an index indicates which two species coexist at equilibrium (e.g. *+ * ).

Equilibrium Expression

All

Three-species coexistence

* = + - - ! + ! * = + ( + ! ) - - ! + ! * = + + ( - ! ) - ! + ! 10

Two-population subcommunities

The plant-pollinator-herbivore community is constituted of two subcommunitiesplant-pollinator and plant-herbivore -sharing the same plant species. Such subcommunities have extensively been studied in the literature (e.g. [START_REF] Goh | Global Stability in Two Species Interactions[END_REF][START_REF] Vandermeer | Varieties of mutualistic interaction in population models[END_REF].

We briefly report here their dynamics (see Appendix B.II for details).

The plant-herbivore subcommunity is characterized by one feasible and globally stable equilibrium. Either the plant population at carrying capacity (/ ≝ -) can support the herbivore population and both coexist, or the herbivores go extinct while plants persist.

Two distinct regimes are possible for the plant-pollinator subcommunity, depending on the comparison between pollination strength and self-limiting competitions (Appendix B.II.2).

(1) If the pollination positive feedback loop (destabilizing, Fig. 1A.b) is smaller than that from competition, there is one feasible and globally stable equilibrium. This equilibrium corresponds to plant-pollinator coexistence when the carrying capacity of plants is large enough to support the pollinator population. Otherwise, plants persist while pollinators go extinct. ( 2) If the pollination positive feedback loop is stronger than that from competition, unbounded population densities are possible. In this case, when the carrying capacity of plants is sufficient to make pollinators viable, populations unboundedly grow irrespective of initial densities.

Otherwise, unbounded growth is observed if initial densities are large enough while only plants persist if it is not the case.

Results

At the coexistence equilibrium when feasible and stable, all biomass densities increase with the strength of pollination (Fig. 1B). On the contrary, both plant and pollinator densities decrease as herbivory gets stronger, while herbivore density can either increase or decrease (Fig. 1B). Matching our predictions (Fig. 1A.a), these dynamics are illustrated in Fig. 2, which shows how densities depend on herbivory for three pollination levels (Fig. 2a-b-c), and on pollination for three herbivory levels (Fig. 2d-e-f). Fig. 2 especially demonstrates that population dynamics are determined by both pollination and herbivory interactively. For instance, the decline of herbivore density with the strength of herbivory is observed when pollination is strong (Fig. 1B &2c), which we interpret as a consequence of the strong indirect antagonism with pollinators (Fig. 1A.a). Another example is that the strength of one interaction affects the level the other interaction has to reach in order for the focal animal to persist in the community. As pollination increases, the minimal level of herbivory allowing herbivores to persist gets lower (Fig. 2a vs 2b). Pollination favors the feasibility of coexistence. On the contrary, herbivory disfavors the feasibility of coexistence. As herbivory gets stronger, the minimal level of pollination allowing pollinators to persist gets higher (Fig. 2d&e vs 2f).

In line with our predictions (Fig. 1A.b), stability displays opposite patterns: it is favored by herbivory and disfavored by pollination. For a given level of herbivory, populations display unstable dynamics for high pollination strengths (Fig. 2 d-e-f, blue background). This instability captures the unbounded growth of biomass densities driven by the mutualism. As herbivory gets stronger, higher pollination levels are needed for unbounded growth to happen (Fig. 2d vs 2e vs 2f). Herbivory can indeed restore stability (Fig. 2c): starting from an initially unbounded situation, increasing herbivory restores finite densities.

The strength of pollination contributes positively to the feasibility of coexistence and negatively to its stability. It is the opposite for herbivory. Although presented for a given parameter set (Fig. 2), these two main results are general as they derive from the analytical relationships governing stable coexistence (table 3). They imply that stable coexistence requires a balance between the strengths of pollination and herbivory to achieve both feasibility and stability. Such a balance can be observed in Fig. 2 that are necessary and sufficient for the stable coexistence of plants, pollinators and herbivores (Appendix C.II). These relationships, as well as their biological interpretations, are presented in Table 3. We illustrate the biological implications underlying them using Fig. 3, which

indicates the community composition depending on the strengths of pollination and herbivory. 3), indicated by the dotted red, orange, and blue curve, respectively. These three relationships are sufficient to achieve stable coexistence given the parameter set (as in Fig. 2), indicating that relationship (24) is less constraining here. The orange (resp. red) star indicates the level of pollination (resp.

herbivory) that makes pollinators (resp. herbivores) viable when only plants are present (hence at carrying capacity). Unbounded growth is possible in the plant-pollinator subcommunity when the strength of pollination is higher than the level figured by an infinity symbol. Note that stable coexistence (yellow area) requires the two interactions to be of similar magnitude. The interpretation of the second relationship (22) depends on whether unbounded population growth is possible or not within the plant-pollinator community, i.e. on the competition loop ( ) being weaker or stronger than the pollination feedback loop

( ! ).
If unbounded growth is not possible, relationship (22) indicates that the per capita growth rate of herbivores, when low in density and within a plant-pollinator community at ecological equilibrium, is positive. In this case, the relationship governs the transition (orange dotted curve, arrow ( 2)) between the plant-pollinator equilibrium (orange) and the coexistence equilibrium (yellow) in Fig. 3. We mathematically demonstrate that in such a case, the feasibility of coexistence implies its global stability (Appendix B.IV). Relationships (21) and

(22) are thus necessary and sufficient for stable coexistence (Fig. 3, left side of ∞).

Furthermore, stronger pollination ( ) makes herbivores viable at lower predation intensities ( ) (inequality (22 E . 1)) due to a higher plant density within the plant-pollinator community

( *+ * , expression in table 2) (Appendix C.II.2).
If unbounded growth is possible, relationship (22) sets an upper limit to the strength of herbivory (inequality (22 E . 2)), which we interpret as a condition for herbivores to not exclude pollinators by reducing plant biomass too strongly. In fact, inequality (22 E . 2) (feasibility of * , see equations ( 2)) is critical for a parameter configuration over which the persistence of herbivores is due to the presence of pollinators (Appendix C.II.2, Fig. S3). In such parameter instances, *+ * loses its biological meaning (as the plant-pollinator equilibrium is unstable) and decreases with pollination ( ), and alternative stable states are possible (Fig. S3). Note that no transition corresponds to inequality (22 E . 2) in Fig. 3 Pollinators can invade the plantherbivore community

+ (direct) -( *. * ↘) (1)
(2)

Other ecological parameters also affect stable coexistence

In addition to the per capita effect of plant-animal interactions (i.e. interaction strength), stable coexistence depends on the densities of animal species, which in turn depend on their intrinsic growth and competition rates, as well as their conversion efficiencies. We consequently study the effect of animal growth rates ( vs. , Stable coexistence is facilitated when the pollinator intrinsic growth rate is higher than the herbivore one. The range of pollination and herbivory strengths allowing stable coexistence indeed gets wider on the upper right of Fig. 4A. The explanation relies on two points: the feasibility of coexistence is favored by pollination and disfavored by herbivory; intrinsic growth rates play a major part in the feasibility of coexistence (relationships (21) & ( 22)), but only a minor part in its stability (no effect on relationship (23)). Analytical details are available in Appendix D. I. An increase in the pollinator growth rate makes it easier for pollinators to invade the plant-herbivore community (Fig. 4c vs 4d, point A). Due to a higher plant density, herbivores can also invade the plant-pollinator community more easily (Fig. 4a vs 4b, point B).

Likewise, a higher herbivore growth rate enables an easier invasion of the plant-pollinator community by herbivores (Fig. 4a vs 4c, point B). It, however, makes the invasion of the plantherbivore community by pollinators harder due to the reduction of plant density (Fig. 4a vs 4c, point A). Note finally that despite similar growth rates, the community is endangered when these growth rates are too low (Fig. S4, Appendix D).

Stable coexistence is facilitated when competition is stronger among pollinators than among herbivores. The range of pollination and herbivory strengths allowing stable coexistence indeed gets wider in the lower-left of Fig. 4B. Such a pattern is due to the effect of competition rates on stability (relationship (23)), which is much stronger than their effect on feasibility (relationships To summarize, the results obtained from studying the effect of these other parameters support our main results, i.e. pollination favors feasibility at the expense of stability while it is the opposite for herbivory. Indeed, any parameter variation that benefits pollinators (higher growth rate , weaker competition or higher conversion efficiency (Appendix D.III))

favors feasibility, disfavors stability or both. Likewise, any parameter variation that benefits herbivores (higher growth rate , weaker competition or higher conversion efficiency (Appendix D.III)) disfavors feasibility, favors stability or both.

Discussion

At the core of community ecology, understanding the mechanisms that support the maintenance of species coexistence is of primary importance in a time of major threats to biodiversity due to global changes [START_REF] Barnosky | Has the Earth's sixth mass extinction already arrived?[END_REF]. In food webs, it has been shown that the coupling of weak and strong trophic interactions was among such mechanisms [START_REF] Mccann | Weak trophic interactions and the balance of nature[END_REF][START_REF] Neutel | Stability in real food webs: Weak links in long loops[END_REF]. Because weak links can dampen the oscillatory dynamics ensuing from strong links, this unbalanced interaction pattern promotes stable coexistence. In contrast, we suggest that in mutualistic-antagonistic communities, a balance between the strengths of the two interaction types is required to achieve stable coexistence. This main result of our study is in agreement with the findings of several previous theoretical investigations on mutualisticantagonistic communities, both at the module [START_REF] Holland | Consumer-resource dynamics of indirect interactions in a mutualism-parasitism food web module[END_REF]Sauve et al., 2016a) and the network [START_REF] Mougi | Diversity of interaction types and ecological community stability[END_REF] scale.

The balance between pollination and herbivory is driven by the opposite effects each type of interaction has on coexistence (i.e. feasibility) and stability.

In line with theoretical findings [START_REF] Georgelin | Dynamics of coupled mutualistic and antagonistic interactions, and their implications for ecosystem management[END_REF]Mougi and Kondoh, 2014b;Sauve et al., 2016a), we show that pollination increases herbivore density by enhancing plant density, while the effect of herbivory on pollinators is utterly opposite. This remains true when mutualism is modeled as a modified consumer-resource interaction, thus accounting for exploitative competition between animal species [START_REF] Holland | Consumer-resource dynamics of indirect interactions in a mutualism-parasitism food web module[END_REF]. Congruent direct effects on plant densities have been confirmed by several field experiments [START_REF] Herrera | Measuring the Effects of Pollinators and Herbivores: Evidence for Non-Additivity in a Perennial Herb[END_REF][START_REF] Herrera | Interaction of pollinators and herbivores on plant fitness suggests a pathway for correlated evolution of mutualism-and antagonism-related traits[END_REF][START_REF] Sutter | Synergistic interactions of ecosystem services: Florivorous pest control boosts crop yield increase through insect pollination[END_REF], but empirical documentation of the consecutive indirect ecological effects between herbivore and pollinator species remains weak (e.g. [START_REF] Gómez | Non-additive effects of herbivores and pollinators on Erysimum mediohispanicum (Cruciferae) fitness[END_REF]).

In contrast with feasibility, we find stability to be favored by herbivory and disfavored by pollination, in line with the theory on feedback loops (relationship (R3), [START_REF] Levins | Discussion paper: the qualitative analysis of partially specified systems[END_REF].

Several studies have indeed shown that pollination networks are prone to display unstable dynamics, such as sudden collapses consecutive to the crossing of tipping points [START_REF] Dakos | Critical slowing down as early warning for the onset of collapse in mutualistic communities[END_REF][START_REF] Kaiser-Bunbury | The robustness of pollination networks to the loss of species and interactions: A quantitative approach incorporating pollinator behaviour[END_REF][START_REF] Lever | The sudden collapse of pollinator communities[END_REF], as positive feedbacks amplify and propagate disturbances. The important role of predation (herbivorous here) in stabilizing population dynamics, on the other hand, has early been identified [START_REF] Menge | Species Diversity Gradients: Synthesis of the Roles of Predation, Competition, and Temporal Heterogeneity[END_REF][START_REF] Nicholson | An outline of the dynamics of animal populations[END_REF][START_REF] Oksanen | Exploitation Ecosystems in Gradients of Primary Productivity[END_REF]. Our results confirm that the consecutive negative feedback can stabilize the dynamics of mutualistic-antagonistic communities. It is important to note, however, that the effects of each interaction type on the stability of such communities are inconsistent across models [START_REF] Georgelin | Dynamics of coupled mutualistic and antagonistic interactions, and their implications for ecosystem management[END_REF][START_REF] Holland | Consumer-resource dynamics of indirect interactions in a mutualism-parasitism food web module[END_REF]Sauve et al., 2016a). The different assumptions on the variation of the animal intake rates with plant density (i.e. functional responses) largely explain such contrasting results. It is nonetheless frequent to observe that the stability of the whole community is driven by the subcommunity displaying stable dynamics when considered in isolation. Yet, unstable dynamics are possible when merging two stable subcommunities as shown by Mougi & Kondoh (2014b). In their work, cycling densities are reported, driven by a delayed plant recovery after its exploitation by herbivores. The delay ensues from the fact that most of the productivity gain from pollination is captured by herbivores, which might be particularly problematic in an agricultural context, especially given that it has been reported in empirical studies several times [START_REF] Gómez | Non-additive effects of herbivores and pollinators on Erysimum mediohispanicum (Cruciferae) fitness[END_REF][START_REF] Herrera | Interaction of pollinators and herbivores on plant fitness suggests a pathway for correlated evolution of mutualism-and antagonism-related traits[END_REF]. An integrative management of pollination and biological control can, fortunately, enable synergetic interactions between ecosystem services [START_REF] Sutter | Synergistic interactions of ecosystem services: Florivorous pest control boosts crop yield increase through insect pollination[END_REF].

It is important to highlight that instability, in our model, encompasses two behaviors whose biological implications are utterly different: (1) the loss of one or several species (Fig. 3, red-brown-green areas) vs. ( 2) the unbounded growth of population densities (Fig. 3, blue area) driven by an "orgy of mutual benefaction" [START_REF] May | Models for two interacting populations[END_REF]. While coexistence is not maintained in the first case, it is in the second case. Another notion of stability -permanence [START_REF] Hutson | The Role of Herbivores in the Maintenance of a Flower Color Polymorphism in Wild Radish[END_REF] -enables to distinguish between these two cases: a biological community is said to be permanent if the densities of all species are always above a minimal threshold. Unbounded population growth is thus a case of "permanent coexistence" [START_REF] Hutson | Permanent coexistence in general models of three interacting species[END_REF], a concept that captures the diversity of population dynamics that permit the coexistence of species in real biological communities. The orgy of mutual benefaction is, however, seldom observed in nature in spite of mutualisms being widespread [START_REF] Bronstein | Our current understanding of mutualism[END_REF].

This indicates that the assumptions of simple models of mutualism are likely violated in real biological systems. The functional response, which we assume linear for both interactions in order to gain analytical tractability, could saturate at high pollination levels when the handling time becomes limiting (e.g. [START_REF] Soberon | The dynamics of a plant-pollinator interaction[END_REF]. The community context can also impede unrealistic population growth [START_REF] Freedman | Obligate mutualism with a predator: Stability and persistence of three-species models[END_REF][START_REF] Heithaus | Models of Some Ant-Plant Mutualisms[END_REF][START_REF] Ringel | The stability and persistence of mutualisms embedded in community interactions[END_REF]. While intraspecific competitions prevent this behavior up to a given level of pollination [START_REF] Holland | Population dynamics and mutualism: Functional responses of benefits and costs[END_REF], we show here that the presence of a third species -the herbivore -allows for even stronger pollination levels to be compatible with biologically relevant finite population densities (relationship (23)). It is thus not surprising that orgies of mutual benefaction are not observed in nature as any two-species mutualism displaying such dynamics would accumulate enemies until restoring the balance required for stable coexistence.

Several mechanisms could underlie this community assembly process. Firstly, as the plant biomass is booming, more and more herbivore species are becoming viable in the focal patch (e.g. inequality (22 E . 1)). Because the plant population defines the threshold beyond which herbivore species can invade, as the plant density grows, the filter existing on the possible herbivore community weakens, and more herbivores species are susceptible to come and control the dynamics. Secondly, existing trophic links would likely strengthen as a result of adaptive foraging on the booming plant species in response to its abundance increase relative to other available resources. Adaptive foraging has notably been proposed as an important stabilizing process within complex trophic networks [START_REF] Kondoh | Foraging Adaptation and the Relationship Between Food-Web Complexity and Stability[END_REF]. In particular, Mougi & Kondoh (2014a) show how the interplay between adaptive foraging, pollination, and herbivory can support the maintenance of stable coexistence in plant-pollinator-herbivore communities.

Empirical evidence suggesting a balance between pollination and herbivory in natural communities does exist. At the module scale, several experimental studies manipulating the presence of animal species find the effects of pollination and herbivory on plant fecundity to be roughly of the same magnitude, approximately canceling each other [START_REF] Gómez | Non-additive effects of herbivores and pollinators on Erysimum mediohispanicum (Cruciferae) fitness[END_REF][START_REF] Herrera | Measuring the Effects of Pollinators and Herbivores: Evidence for Non-Additivity in a Perennial Herb[END_REF][START_REF] Herrera | Interaction of pollinators and herbivores on plant fitness suggests a pathway for correlated evolution of mutualism-and antagonism-related traits[END_REF][START_REF] Sutter | Synergistic interactions of ecosystem services: Florivorous pest control boosts crop yield increase through insect pollination[END_REF]. At the network scale, [START_REF] Melián | Diversity in a complex ecological network with two interaction types[END_REF] show that most strong interactions, mutualistic and antagonistic, are concentrated in the same few plant species of the Doñana Biological Reserve (Spain). Sauve et al. (2016b) exhibit a positive correlation between the number of pollinators and herbivores that interact with a given plant of the Norwood farm (UK). In line with our results, this correlation contributes positively to the stability of the community. Our results also imply that cascades of extinctions may be expected within plant-pollinator-herbivore networks as a result of the current global pollinator decline [START_REF] Potts | Global pollinator declines: Trends, impacts and drivers[END_REF], given the weakening of pollination relative to herbivory.

Empirical evidence linked to species traits also supports the idea of a balanced interaction pattern. Indeed, a substantial amount of studies documents shared preferences for plant phenotypes between pollinators and herbivores. Favoring balanced pollination vs.

herbivory, shared preferences have been reported for a large number of plant traits, including flower color [START_REF] Hutson | The Role of Herbivores in the Maintenance of a Flower Color Polymorphism in Wild Radish[END_REF], floral display [START_REF] Cariveau | Direct and indirect effects of pollinators and seed predators to selection on plant and floral traits[END_REF][START_REF] Gómez | Herbivory reduces the strength of pollinator-mediated selection in the Mediterranean herb Erysimum mediohispanicum: Consequences for plant specialization[END_REF], chemical volatiles [START_REF] Andrews | Pollinator and herbivore attraction to Cucurbita floral volatiles[END_REF][START_REF] Theis | Attracting mutualists and antagonists: Plant trait variation explains the distribution of specialist floral herbivores and pollinators on crops and wild gourds[END_REF], nectar quantity [START_REF] Adler | Attracting antagonists: Does floral nectar increase leaf herbivory?[END_REF] or reproductive system [START_REF] Asikainen | Preferences of pollinators and herbivores in gynodioecious Geranium sylvaticum[END_REF]. This preference pattern implies that plants are subject to an ecological trade-off between attracting pollinators and deterring herbivores [START_REF] Strauss | Ecological Costs of Plant Resistance To Herbivores in the Currency of Pollination[END_REF]. Such a trade-off has been reported across a wide diversity of plant taxa [START_REF] Strauss | Direct and ecological costs of resistance to herbivory[END_REF]. Our work indicates that it might be ubiquitous as it fosters a balance between the strengths of pollination and herbivory, and hence stable plant-pollinatorherbivore coexistence. When sampling natural ecosystems, at any given time, stable communities should indeed be more frequently encountered than unstable ones owing to a longer 'lifespan'. Communities displaying such an ecological trade-off should accordingly be more frequently encountered. This trade-off might thus be construed as an emergent property of plant-pollinator-herbivore communities, i.e. a property that emerges when considering a large number of such communities rather than one that apply for each considered community.

Traits of plant species might be subject to conflicting selection arising from such a tradeoff [START_REF] Strauss | Non-pollinator agents of selection on floral traits[END_REF], with potentially important implications in terms of diversity maintenance. In the case of the wild radish Raphanus sativus, for instance, it has been shown that the maintenance of a flower color dimorphism (white vs. pink) was due to both the pollinators and the herbivores interacting preferentially with white morphs [START_REF] Hutson | The Role of Herbivores in the Maintenance of a Flower Color Polymorphism in Wild Radish[END_REF][START_REF] Mccall | Florivores prefer white versus pink petal color morphs in wild radish, Raphanus sativus[END_REF][START_REF] Stanton | Reproductive biology of petal color variants in wild populations of Raphanus sativus. I. Pollinator response to colour morphs[END_REF]. The question of whether such dimorphism emerged, in the first place, because of the interplay between pollination and herbivory, remains open. The study of mutualistic-antagonistic communities, plant-pollinator-herbivore in particular [START_REF] Strauss | Ecological and Evolutionary Consequences of Multispecies Plant-Animal Interactions[END_REF], thus offers opportunities to significantly improve our understanding of the ecological processes supporting the coexistence of species in natural systems, but also of the complex eco-evolutionary dynamics driving the maintenance of biodiversity.

Figure 1 :

 1 Figure 1: A. Predicted effects of interactions on stable coexistence. a. Predicted effects on densities. Solid (resp. dashed) arrows for direct (resp. indirect) interactions. b. Predicted effects on stability. B. Variation of biomass densities at stable coexistence with the strength of interactions. Biomass densities at equilibrium are denoted * , * , * for plants, pollinators and herbivores. The strengths of pollination and herbivory are respectively and . Other model parameters introduced here: intraspecific competition for plants and pollinators ( , ), conversion efficiency from plants to pollinators . See table 1 for a detailed presentation of all model parameters. These first analytical results are in line with predictions (see A.a). Analytical proofs in appendix B.IV.

  : as one interaction gets stronger, the range of the other interaction intensities allowing stable coexistence shifts toward larger values (yellow background, Fig. 2b vs 2c, Fig. 2e vs 2f).

Fig. 2 :

 2 Fig. 2: Biomass densities at equilibrium depend interactively on pollination and herbivory strengths. The curves are determined analytically (see appendix B.III). a-b-c Dependence of densities on herbivory for three pollination levels. In a, pollination intensity is too low for pollinators to persist. d-e-f Dependence of densities on pollination for three herbivory levels. Parameter set: = 10, = = -2.5, = 0.6, = = 0.4, = = 0.2.

Fig. 3 :

 3 Fig. 3: Stable coexistence requires balanced pollination vs. herbivory strengths. Given the strength of pollination ( ) and herbivory ( ), the stable equilibria are determined and the point of the graph is colored accordingly. In blue, no equilibrium is stable so densities grow unboundedly. Arrows (1), (2), and (3) indicate the transitions enabling the satisfaction of relationships (21), (22), and (23) (Table

  Fig. 1b)

  Fig. 4A & Fig. S4, appendix D), animal competition rates ( vs. , Fig. 4B & Fig. S5, appendix D) and conversion efficiencies ( vs. , Fig. S6, appendix D) on community composition. This investigation also constitutes a robustness check as we vary the parameters that were fixed hitherto. First of all, when stable coexistence is possible, it happens when the strengths of pollination and herbivory are approximately of the same magnitude (yellow area, Fig. 4 and S4 & S5 & S6 in Appendix D), as analyzed above (inequality (2)).

  (21) & (22)). Analytical details are available in Appendix D.II. As competition among herbivores gets stronger, the plant density within the plant-herbivore community increases as a result of predation release. It becomes easier for pollinators to invade (Fig. 4g vs 4e, point A'). Unbounded dynamics are, however, facilitated (Fig. 4h vs 4f, point C) because the positive destabilizing loop increases more than the negative stabilizing loops (relationship (23)). In the plant-pollinator community, a lower pollinator density ensuing from a stronger competition rate is responsible for a lower plant density. It thus becomes harder for herbivores to invade (Fig. 4h vs 4g, point B'). Stability is, however, enhanced due to the stronger control of the pollination positive feedback in both the plant-pollinator subcommunity (Fig. 4f vs 4e, infinity symbol) and the three-species community (Fig. 4f vs 4e, point C).

  3.1 Relationships governing stable coexistencePositive animal densities necessarily imply a positive plant density because animals are obligate plant-interactors (Appendix C.I.1). In other words, the coexistence equilibrium is feasible if, and only if, both animal species have positive densities, which leads to two inequalities. It is stable if, and only if, all three eigenvalues of the Jacobian matrix (Appendix B.III.1) calculated at the coexistence equilibrium have a negative real part, which is equivalent to the three Routh-Hurwitz inequalities (Appendix B.III.3). One of these inequalities is satisfied if feasibility is assumed. Therefore, there are four relationships (i.e.(21, 22, 23, 24) in table3)

  Assuming stability (relationship (23) actually), coexistence is feasible if and only if relationships (21) and (22) are satisfied. Relationship (21) indicates that the per capita growth rate of pollinators, when low in density and within a plant-herbivore community at ecological equilibrium, is positive

	-45657 *. * 9:;<=>?9:@ ≤ A9B > CD	(21 E )

(Appendix C.II.1)

. Pollinators are thus able to invade the plant-herbivore community so that this relationship governs the transition (red dotted curve, arrow

(1)

) between the plant-herbivore equilibrium (red) and the coexistence equilibrium (yellow) in Fig.

3

. Besides, the plant density within the plant-herbivore community ( *.

* , expression in table

2

) decreases when herbivory ( ) intensifies (Appendix C.II.1). In such a situation, pollination ( ) has to get stronger as well in order for pollinators to invade the plant-herbivore community (inequality (21'), the prime sign indicates that this inequality is obtained by rewriting relationship (21)).

  as relationship (22) is only constraining at the left of the infinity symbol (∞) for the given parameter set.

	G H I H J G H I H J Inequalities (21 E ) and (22 E ) (derived from relationships (21) & (22) respectively) ! ≤ -*+ * 467 K=;<=>?9:@ A9B > CL ≤ (22 E . 1) ≤ ! ≤ -*+ * 467 9:;<=>?9:@ A9B > CL (22 E . 2) clearly show that pollination favors the feasibility of coexistence while herbivory disfavors it. These inequalities indeed tend to be satisfied when pollination strengthens or herbivory weakens. Assuming feasibility, coexistence is stable if and only if relationships (23) and (24) are satisfied. Relationship (23) corresponds to the total feedback at level 3 (i.e. summation of the strengths of all three-element combinations of non-overlapping feedback loops, details in Appendix C.II.3) being negative. Pollination disfavors stability by contributing positively to this feedback, while it is the opposite for herbivory. Stability requires the competitive and the herbivory feedback loops to overcome the pollination feedback loop. Rewriting relationship + ! 455555565555557 9:;<=>?9:@ A9B > CD (23 E ) Relationship (24) is harder to interpret. Given that relationships (23) and (24) imply that the feedback at each level is negative, relationship (24) could be interpreted as proposed by Levins (1974): the negative feedback with long time lags (level 3) is weaker than the shorter-loop negative feedback (level 1 & 2) (details in Appendix C.II.4). Also, the constraints imposed by this relationship on interaction strengths are not analytically tractable, due to the effect of (23) into inequality (23 < N interactions on equilibrium densities.

E ) emphasizes the consecutive constraint limiting pollination. It governs the transition (blue dotted curve, arrow (3)) from unbounded growth (blue) to stable coexistence (yellow) in Fig. 3.

Table 3 : Analytical relationships governing stable coexistence. The

 3

	Relationship	Mathematical meaning	Biological interpretation	Effect of interaction strengths
	+	(21) *. * ≥ 0	Feasibility (PQR(

fourth column indicates how each relationship is affected by the strength of interactions (favored +, disfavored -). Note that the third and fourth columns present a simplified summary of our analysis (see subsequent text and Appendix C, especially tables S3 & S4). Notations: *+ * plant density at plant-pollinator equilibrium; *. * plant density at plant-herbivore equilibrium; "num" numerator; "den" denominator (see table 2 for expressions). Finally, the interplay between pollination and herbivory is difficult to disentangle in relationship (22), which led us to distinguish two cases (inequality (22 E . 1) for (1) and (22 E . 2) for (2) below). An increase in pollination ( ) makes the relationship shift from (1) to (2). In (2), *+ * loses its biological significance as the plant-pollinator subcommunity grows unboundedly. *+ * is, in this case, a mathematical function (expression in table 2), which explains its counterintuitive behavior with the variation of pollination strength *. * ) ≥ 0)

Acknowledgments

The authors would like to thank Prof. Sharon Y. Strauss, and Dr. François Duchenne, for their helpful feedback on the manuscript.