Towards understanding factors in ﬂ uencing the bene ﬁ t of diversity in predator communities for prey suppression

. It is generally assumed that high biodiversity is key to sustaining critical ecosystem services, including prey suppression by natural predator guilds. Prey suppression is driven by complex interactions between members of predator and prey communities, as well as their shared environment. Because of this, empirical studies have found both positive and negative effects of high predator diversity on prey suppression. However, we lack an understanding of when these different prey suppression outcomes will occur. In this work, we use a mechanistic, trait-based model to unravel how intraguild interactions, species body mass, predator foraging area, and ambient temperature can combine to produce different levels of prey suppression. Surprisingly, we ﬁ nd that prey suppression is only improved by high biodiversity under a limited set of conditions. The most important factor in determining whether diversity improves prey suppression is the amount of overlap between predators ’ foraging areas. The degree of overlap in foraging areas shapes species interactions, and as the overlap between species increases, we see decreasing bene ﬁ ts from species-rich communities. In contrast, diversity in body mass only improves prey suppression when there is signi ﬁ cant variation in temperature.


INTRODUCTION
There is a long-standing debate on the value of biodiversity in agricultural ecosystems, often in the context of prey suppression as an ecosystem service (Bianchi et al. 2006, Geiger et al. 2010, Winqvist et al. 2011. In many cases, herbivore prey populations can be effectively suppressed by a small group of specialist predators, such as ladybird beetles feeding on aphids or predatory mites feeding on spider mites. When these specialists are not abundant, such as early in a growing season, herbivore populations may be suppressed by a community of generalist predators (Pekár et al. 2015, Athey et al. 2016. Since generalist predators also consume other members of the predator community (intraguild predation), the benefit of a high number of predator species (predator diversity) is unclear. The outcome of prey suppression is determined by a range of possible intraguild (predator-predator) interactions. For example, two generalist predator species might engage in intraguild predation or interference competition. As predators spend more time attacking or competing with one another, the consumption of prey decreases (Sitvarin and Rypstra 2014). The number of possible interactions, and outcomes of those interactions, dramatically increases with the number of predator species (for an overview, see Tscharntke et al. Because of this, we hypothesize that diversity in body mass will permit consistent and effective prey suppression in environments that may experience al large range of temperatures. In order to predict the effect of increasing temperatures and variability due to climate change on the benefits of biodiversity for prey suppression, we must understand the effect of body mass on predatorprey interactions. Mathematical models are a powerful tool for predicting predator-prey interaction strengths resulting from the complicated relationships between body mass and other species characteristics (Brose et al. 2006, Berlow et al. 2009, Boit et al. 2012. Along with body mass, encounter rates between individuals are affected by species' foraging areas. We consider a foraging area to be the physical region a species traverses in search of resources. For example, an herbivorous prey species' foraging area might consist of its host plant. In order to consume the herbivore, a predator species' foraging area must overlap with the location of the prey species (e.g., plant leaves). Other species in the predator community may forage over different portions of their prey's habitat (e.g., plant stem). The degree to which two predator species' foraging areas overlap will determine the likelihood that predators encounter one another while foraging. A high degree of overlap permits increased levels of intraguild predation, while a low degree of overlap reduces competition for prey items. Foraging area therefore affects interactions between species (Schmitz 2007, Straub and, and empirical results indicate that this can affect intraguild interactions and prey suppression (Woodcock and Heard 2011). We expect that diversity in foraging area and a low degree of overlap will improve prey suppression.
In this manuscript, we present a generalizable, allometric model to investigate the benefit of high generalist predator diversity on suppressing herbivorous pest species in the absence of specialist predators. Populations of small herbivores, such as aphids or spider mites, increase rapidly during a growing season. Generalist predators often have a much longer generation time and consume a range of different prey species during their lifetime. This means that predator reproduction is decoupled from the consumption of a single prey species. In contrast, predator v www.esajournals.org 2 October 2020 v Volume 11(10) v Article e03271 mortality and the degree of herbivore suppression are affected by intraguild interactions. We utilize a model for these interactions which incorporates temperature-dependent trade-offs between mobility and metabolic rates and scales encounter rates according to overlap in predator foraging areas. We apply our model to a specific community of terrestrial arthropods in an agricultural field and simulate the prey suppression outcomes caused by a range of distributions in body size and foraging area in the predator community. Using an optimization approach, we identify predator communities which minimize the predicted average pest population size. The utility of this approach is that we can disentangle the interactions and traits which cause similar outcomes in true ecosystems. Using our theoretical model, we identify the conditions under which intraguild predation and interference reduce the efficiency of diverse predator communities as well as the types of diversity which improve prey suppression. We predicted that diversity in predator body mass and foraging area would both improve prey suppression. We also predicted that increased temperatures would lead to optimal communities comprised of smaller predators. Although diversity in foraging area had a strong effect on optimal communities, we find that diversity in body mass was less important than expected. We discuss these results and identify areas where additional empirical information will improve theoretical predictions of prey suppression.

Mathematical model for predator-prey dynamics
We expand the model from Schneider et al. (2012), which describes intraguild interactions (intraspecific competition and intraguild predation) such that body mass determines encounter rates, feeding preferences, and metabolic constraints. We scale encounter rates according to overlap in foraging area and incorporate intraguild interference, as described in Laubmeier et al. (2018). The resulting model can be applied to communities of varying size or trophic complexity, but must be restricted to generalist, ectothermic predators which reproduce on a longer timescale than their herbivore prey. We model the population density N(t) of the herbivore and M i (t) of predator species i at time t in the range 0 < t < t f , where t f is necessarily less than the reproductive time of the predator species. We consider the case with s species such that i = 1, 2,. . ., s and denote the prey with index i = 1 and predators with indices i = 2, 3,. . ., s. The dynamics for population densities are given by.
N for the herbivore prey species; where j, l, and m are species indices with the same meaning as i. We describe the terms in the differential equations below and summarize the biological meaning of model parameters in Table 1. The growth rate for the herbivore prey population is a constant, positive value (r) that accounts for intrinsic birth and death processes. The herbivore prey population also decreases due to predation, which occurs at the rate a 1j v 1j M j /F j for every potential predator species j. Predator populations decline due to metabolic death rates (x i ) and suffer losses due to intraguild predation, following the same relationship as in the herbivore prey population.
Individuals of species i utilize the same foraging area of individuals of species j with probability ν ij . When the foraging areas of two species overlap, they can encounter one another, and an attack on species i by species j occurs at rate a ij . However, the amount of time that predators of species j can spend hunting is limited by the time spent on other activities and is quantified by the "functional response" (Skalski and Gilliam 2001) for species j (F j ). These other activities include the time a predator spends engaging in intraspecific competition (c j M j ), hunting and digesting herbivore prey (a 1j ν 1j h 1j N), hunting and digesting alternative prey items from other predators of species l(a lj ν lj h lj M l ), and evading intraguild interference from predators of species m (b 0 a jm ν jm M m ). Intraguild interference may overlap with intraspecific competition in cannibalistic species, and so we require that . Foraging area overlap, quantified by ν ij , impacts the rate at which intraguild predation and interference occur. We replace model parameters with descriptions of how body mass affects the mechanisms driving predator interactions. Attack rates are given by.
where W i is the average mass of an individual of species i and a 0 , R j , and ϕ are constants. We assume that a species' speed increases with its body mass. The speed of a species is proportional to W 1=4 i and the speed of two individuals determines the rate at which they encounter and attack one another. The attack rate between species i and species j is therefore.
where a 0 is a normalizing constant which scales the relationship between species speeds and attempted attacks. However, not every encounter between a predator and potential prey results in a predation event. The probability of a successful attack by species j on species i depends on the relative sizes of both species (W j =W i ). There is an ideal predator-prey body mass ratio (R j ) at which species j is always successful in attacking its prey. If the true predator-prey body mass ratio (W j =W i ) is close to this value, there is a high probability for species j to successfully attack species i. The probability of a successful attack by species j on species i is therefore given by.
where ϕ is a tuning parameter. When ϕ ¼ 0, attack rates are not affected by predator-prey body mass ratios. Encounters between individuals of the same species might result in intraspecific competition for resources. We therefore specify the rate of intraspecific competition as.
where c 0 is a scaling constant for the rate at which competition occurs. We assume that the amount of time a predator of species j handles a prey item of species i depends on the relative sizes of both species (W i =W j ) because predators require more time to consume relatively large prey. The time required for an individual of species j to handle a prey item of species i is.
where h 0 quantifies the proportional increase in handling time with increasing body mass ratio. Metabolic rates, and hence predator penalties, increase with temperature and body mass. The metabolic death rate for the predator population of species i is.
where x 0 is a scaling constant for the penalty, E is activation energy, k is the Boltzmann constant, and T is the ambient temperature. We summarize the biological meaning of model parameters in Table 2 and refer to Schneider et al. (2012) and references therein for a more detailed description of the meaning and units of measurement for body mass-dependent parameters (a ij , c j , h ij , x i ).

Optimizing predator communities
In order to identify how diversity in predator body mass or foraging area impacts prey suppression, we seek predator communities which minimize the average prey population density over n time steps. The average population density is given by v www.esajournals.org where N t d ð Þ denotes the prey population density at time t d prior to the reproductive time of the predator species, for d ¼ 1, 2, ...,n and 0<t 1 <t 2 <...<t f . For a given prey population with initial density N 0 ð Þ, we begin by specifying a set of predator species j typically associated with the prey species. Each predator species j has average body mass W j , a specific foraging area, and feeding interactions with other species. For each set of initial predator densities M 0 2 , M 0 3 ,..., M 0 s , which we denote by M j 0 ð Þ È É , we simulate the population dynamics from (1) to obtain the prey population N t ð Þ for n time steps. The average prey population for these initial predator densities is.
We obtain the optimal predator community for prey suppression by minimizing C M j 0 ð Þ È É À Á over all possible initial densities of all predator species. We assume that there is not an unlimited supply of predators and require that.
for a constant number M * determined by the biology of the system. In the absence of this type of constraint, we would have the unrealistic outcome that predator communities could always include more predators, which would lower the number of prey indefinitely without ever attaining a minimum solution. We note that depending on the study system, an inequality constraint ∑M j ð0Þ≤ M * À Á or different constraining values, such as the biomass or management cost of a predator community, could be substituted here.
Our optimization problem is therefore to Á , subject to model dynamics (1) and the constraint (3). We refer to the Appendix S1 for a mathematical proof regarding the existence of solutions to this optimization problem. In practice, we solve the problem with the MATLAB function fmincon, which numerically minimizes a cost function according to specified constraints (for more details, see Appendix S2 or code at DOI:10.5281/ zenodo.4093174). However, this minimization might be sensitive to where it begins searching for a solution. We therefore employ the multistart function, which repeats the minimization over different starting points. This guards against the possibility of non-unique (locally minimizing) solutions, where the solution found by fmincon depends on the starting point of the search.

A case study: terrestrial arthropods
We illustrate the effect of body mass and foraging area on optimal predator communities for prey suppression through a specific example. Because a previous study provided us with empirical estimates for many model quantities, we consider the suppression of the bird cherryoat aphid, Rhopalosiphum padi, feeding on barley host plants in a spring field season. Aphids are widespread pests and frequently the subject of biological control studies (Brodeur andRosenheim 2000, Snyder andIves 2003). Additionally, R. padi are parthenogenetic and capable of telescoping generations, resulting in rapid reproduction (Villanuevab and Strong 1964) which occurs over shorter timescales than the generalist predators in this system. For details on the lifecycles and biology of R. padi, we refer to Dixon (1971) and Leather and Dixon (1981). This aphid species is prey to a generalist community of spiders and groundbeetles, which also engage in complex intraguild interactions (Hodge 1999, Lang 2003). Averaging over population density (M j ) and body mass (W j ) data from Curtsdotter et al. (2019), we consider the community of predators depicted in Fig. 1. This predator community is comprised of five differently sized groundbeetle species and four spider species with less variation in body mass than in the groundbeetle species. We use the population density data to specify the constraint (3) for the optimization problem. That is, letting M j È É be the densities of the predator community reported in Fig. 1, we have M * = ΣM j the total initial predator population density. We use the average start-of-season aphid populations in Curtsdotter et al. (2019) as the initial aphid population density. In addition to the aphid prey, predators can potentially consume every other predator species in the community, including members of their own species. We note that groundbeetles generally forage on the ground, possibly burrowing into the dirt or reaching up the base of a plant. Some species of spider actively forage on the ground or climb up plants (Tetragnathidae, Lycosidae), while others form webs to catch prey (Linyphiidae). However, we did not have specific measurements or observations of predator foraging area in this community. We therefore investigate the effect of different assumed overlaps in foraging area (ν ij ) on optimal predator communities, since foraging area is an important aspect of our model.
We obtain estimates of a 0 , ϕ, h 0 , and r 1 from Wootton et al. (2020) E and x 0 from Schneider et al. (2012). We consider temperatures in the range of T ¼ 15 ∘ C to T ¼ 45 ∘ C, which is consistent with temperatures observed in the field (Curtsdotter et al. 2019). The predators' preferred predator-prey body mass ratios (R j ) are difficult to measure and have minor effects on optimization outcomes. We therefore set these values such that every predator has a 100% success rate when consuming aphids. In doing so, we focus our investigation on the effect of predator body mass diversity on prey suppression, instead of on optimal predator-prey body mass ratios. Based on aphid population dynamics described in Curtsdotter et al. (2019), we restrict our simulations and optimization to a 30-d period. Over this period of time, aphid populations colonized barley fields and increased until reaching a peak density. After this, populations declined rapidly. The declines did not always line up with a decline in crop quality, and possible explanations include microclimate changes or avian predators.
We found a range of estimates in the literature for c 0 (between 0.12 and 1.09, Schneider et al. 2012, which is related to intraspecific competition between predators, and we only had one estimate of b 0 (Wootton et al. 2020), which is related to intraguild interference between predators. We explored c 0 values between 0 (no intraspecific competition) and 1

Pteros. Harpalus Poecilus
Other  Fig. 1. The average body mass (left axis, solid bars) and average population density (right axis, dotted bars) of predator species in our study system. The dashed line indicates the break between groundbeetle groups (Pterostichus, Harpalus, Poecilus, "Other Carabid," and Bembidion) and spider groups ("Other Spider," Lycosidae, Tetragnathidae, and Linyphiidae). The groups "Other Carabid" and "Other Spider" are made up of rare species not included in the previous groundbeetle and spider categories.
(high levels of intraspecific competition) and b 0 values from 0 (no intraguild interference) to 10 (high levels of intraguild interference). In the example predator community (Fig. 1), the average effect of intraguild interference on prey suppression when b 0 ¼ 10 matches the effect of intraspecific competition on prey suppression when c 0 ¼ 1. We list the values for model parameters and quantites from Fig. 1 in the Appendix S3.

Effects of foraging area on optimal community composition
We first investigate the effect of predator foraging area on prey suppression efficiency in predator communities. We consider three cases for how predator foraging area might overlap: (1) Every predator forages in the same area, (2) there are two foraging areas, which overlap completely for predators in the same group ("groundbeetle" and "spider"), but only slightly with predators from the other group, and (3) every predator species has its own foraging area, which overlaps moderately with predators in the same group ("groundbeetle" or "spider") and slightly with predators from the other group. Within each predator species, foraging areas overlap completely, and predators encounter members of other species according to the degree of overlap between foraging areas. See Fig. 2 for a diagram of the different overlap scenarios. In all three cases, we solve the optimization problem over a range of ambient temperatures, assuming that average temperatures vary between seasons but are constant within a single 30-d simulation period.
When all predators share a single foraging area (Fig. 3a), the optimal predator community for each temperature is always comprised of a single-predator species. Temperature affects metabolic penalties (x i ), which causes optimal predator species to change from large species at lower temperatures to small species at higher temperatures. When foraging area overlaps for predator species in the same group (Fig. 3b), spiders are more likely to encounter other spiders than they are to encounter groundbeetles, and vice versa. In this case, the optimal predator community for a single temperature is always comprised of one spider and one groundbeetle species, and the optimal species from either group changes with temperature. When each predator species has its own foraging area (Fig. 3  c), multiple spider and groundbeetle species form an optimal predator community for a single temperature. Again, the composition of predator species that is best in suppressing prey populations varies with temperature. This change can occur gradually with changing temperature or suddenly; for example, the densities of Pterostichus, Harpalus, and Poecilus change gradually in optimal predator communities for temperatures ranging between 15°C and 25°C, but Poecilus is almost half of the optimal predator community at 29°C before "Other Carabid" suddenly replaces Poecilus at 30°C. For most of the remaining results, we focus on scenarios where every predator species has its own foraging area (as in Fig. 3c) because this is a necessary condition for high biodiversity to improve prey suppression.

Optimal predator communities under varied temperatures
First, we consider how sensitive the optimal predator community with diverse foraging area is to changes in temperature between years (Fig. 4). We focus our comparison on three single-species communities representing a range of body masses: Poecilus (a large predator), the  v www.esajournals.org group "Other Spider" (a medium-sized predator), and Tetragnathidae (a small predator). These predator species suppressed prey populations the most over the full temperature range, compared to species in the same size categories. In our comparison, we also include the predator composition found in the field from Fig. 1 ("baseline community") and a "diverse community" where the initial density of each predator species is the species' average initial density over all optimal predator communities in Fig. 3c. Compared to the "baseline community," the   (Fig. 1). Single-predator communities are indicated by thin, dashed lines, and a multi-predator community is indicated with a thick, solid line. Foraging area is as in Fig. 3c. v www.esajournals.org "diverse community" results in a 14.6% decrease in the prey population when averaged over temperatures between 15°C and 45°C (Fig. 4). No single-species community matches this performance; Poecilus decreases the average prey population by 7.6%, the group "Other Spider" decreases the average prey population by 10.4%, and Tetragnathidae increases the average prey population by 31.6% over the same range of temperatures. As we found previously, larger predators improve prey suppression at lower temperatures, while smaller predators improve prey suppression at high temperatures.
Next, we investigate the effect of temperature variation within a growing season. We use temperature recordings from 9 field sites in Curtsdotter et al. 2019, which had mean 20.2°C and median 17.7°C. We preserve autocorrelation in the data by taking every possible 3-day window of temperatures. We conduct 5000 simulations with 30-d temperature fluctuations randomly drawn from these windows and calculate the average daily prey population. We compare single-predator communities to the true community and two communities comprised of optimal predators from Fig. 3. The initial distribution of optimal predators is determined by relative abundances in the true community or by weighting predators according to the frequency of true temperatures at which they are optimal (see Fig. 5). We find that without diversity in foraging area (Fig. 6a), single-species communities comprised of the most efficient beetles (Harpalus and Poecilus) outperform diverse communities. When predators utilize different foraging areas (Fig. 6b), the diverse communities using optimal predators attain slightly lower levels of prey suppression and, more importantly, reduce variation in prey suppression across simulations.

Interplay between intraguild interactions and diversity
We next explore the effect of different combinations of intraspecific competition (scaled by the parameter c 0 ) and intraguild interference true relative densities   Fig. 6. Communities utilize optimal predators identified in Fig. 3a (left) or Fig. 3c (right). Initial densities are determined according to true relative densities from the field (top) or weighted by the frequency of corresponding optimal temperatures over one season in the field (bottom). v www.esajournals.org (scaled by the parameter b 0 ) on optimal predator communities. For each pair of b 0 and c 0 values, we find the optimal predator community for temperatures ranging between 15°C and 45°C and foraging area overlap as in Fig. 3c. We count the number of species in each optimal predator community and report the average number of predator species over all temperatures in Fig. 7. We find that the optimal number of predator species increases with increasing levels of intraspecific competition and decreases with increasing levels of intraguild interference. The value of c 0 determines how strongly predators' attack rates decrease due to competition within a predator species. As c 0 increases, the number of species in the optimal predator community increases (vertical slices of the contour diagram in Fig. 7). Regardless of how many species are present, the optimizing constraint (3) requires that every optimal predator community be Aphid density under single-predator and diverse communities with time-varying temperature. Daily temperatures were drawn from field data, and simulations were repeated over 5,000 random draws from daily temperatures. Foraging area is as in Fig. 3a (top) or Fig. 3c (bottom). v www.esajournals.org initially comprised of the same number of individuals; each population in a community of many species must have lower initial densities than each population in a community of fewer species. Communities with many predator species minimize the effect of intraspecific competition by reducing the number of individuals within each species. The value of b 0 determines how strongly predators' attack rates decrease due to interference from predators in other species. As b 0 increases, the number of species in the optimal predator community decreases (horizontal slices of the contour diagram in Fig. 7). In comparison with c 0 , changes in b 0 have a small effect on the average number of predators in the optimal community. The relative magnitude of these two parameters (and therefore the cost of intraspecific competition compared to intraguild interference) determines the number of species in an optimal community.

DISCUSSION
There is substantial ecological interest in understanding the relationship between the control of insect pests and biodiversity in natural enemy guilds. Reducing species richness (number of species) (Hooper et al. 2005, Cardinale et al. 2006) and evenness (skewed relative abundance distributions) (Hillebrand et al. 2008, Crowder et al. 2010 can weaken or improve biological control. However, our understanding of these outcomes' underlying mechanisms is limited (Crowder and Jabbour 2014). Our model suggests that overlap in foraging area within predator communities is key to predicting the effect of biodiversity on prey suppression (Fig. 3a-c). If predator species forage in overlapping areas, predators are more likely to encounter one another while hunting for prey. This results in frequent occurrences of intraguild predation and interference, which reduces prey suppression. In the case of largely v www.esajournals.org overlapping foraging areas, optimal communities for prey suppression are comprised of a singlepredator species, for a given temperature. These results align with multiple observations that functional diversity positively impacts prey suppression (Greenop et al. 2018). However, temperature determines which species is the most efficient predator, and so in environments with significant temperature fluctuation between years, predator biodiversity is beneficial regardless of foraging area overlap.

Diversity in foraging area improves prey suppression
In Fig. 3, we find that it is an advantage for optimal predator communities to be comprised of multiple predator species which forage in different areas. This is in line with empirical studies, which have shown that prey suppression is often improved in complex habitats where foraging behavior is likely more variable (Finke and Denno 2006) and when predators exhibit different patterns of habitat use or hunting behaviors (Woodcock and Heard 2011). The importance of foraging area overlap to prey suppression is also demonstrated by Northfield et al. (2017) spatial using a three species model (two predators, one prey). We expand this by increasing the number of possible predators, all of which are characterized by a combination of traits representative of a real community (Fig. 1), as well as considering non-fatal and non-consumptive effects of intraguild interference and interactions with temperature. In our model, predator interactions depend on foraging area overlap, but predator success is also determined by the fundamental trade-off between body size, prey size preferences, and temperature. Studying predator traits in isolation does not necessarily leads to meaningful predictions of prey suppression. Prey suppression by a predator community might be more or less effective than predicted when different traits interact in unexpected ways.
Empirical studies of intraguild predation and its effect on prey suppression (Rosenheim et al. 1993) suggest that the presence of very small predator species may impair prey suppression by very large predator species. Our model provides an explanation for this observation when comparing optimal predator communities at different temperatures (Fig. 3c). When one predator completely replaces another in the optimal community, it shows that neither predator can be effective alongside the other, due to intraguild interference or predation. For instance, at 29°C the optimal predator community includes the relatively large species Poecilus without the relatively small group "Other Carabid." However, a one degree increase in temperature causes Poecilus to be replaced by "Other Carabid" in the optimal predator community. Poecilus represents a highly mobile predator that consumes individuals much smaller than itself, which would impose a large predation pressure on "Other Carabid" if both species were present in the community. In contrast, other optimal predator communities included Poecilus alongside large groundbeetles (Pterostichus and Harpalus, similar foraging area) or small spiders ("Other Spider" and Lycosidae, dissimilar foraging area). Taken together, these modeling results illustrate how the interplay between diversity in body mass and diversity in foraging area can lead to positive or negative prey suppression outcomes. Specifically, we see that high diversity in body mass paired with overlapping foraging areas can impede prey suppression.

Body mass diversity is sometimes beneficial
When predators do not utilize different foraging areas (Fig. 3a), diversity in predator body mass is only beneficial if there is large annual variation in average temperature. At low temperatures, large predators (Pterostichus, Harpalus, and Poecilus) are most effective, since they are highly mobile and encounter prey more frequently than small predators. As temperatures increase, these same predators are ineffective because metabolic rates increase more quickly with temperature for large predators than small predators. If temperatures vary between years, the optimal predator community for prey suppression therefore might include small and large species which are highly efficient in controlling the prey population for any temperature in a given year. These expectations align with the insurance hypothesis that increasing biodiversity insures ecosystems against declines in their functioning caused by environmental fluctuations (Naeem andLi 1997, Yachi andLoreau 1999). In the context of this hypothesis, temperature is the environmental driver fluctuating between years and prey suppression is a type of ecosystem function which depends on how species in the predator community respond to temperature.
In Fig. 4, we explore the conditions under which the insurance against temperature variability between years can outweigh the penalties of a v www.esajournals.org suboptimal predator community, when combined with diversity in foraging area. We find that there is a small range of temperatures (~26-32°C) where the "diverse community" identified by our optimization procedure outperforms communities comprised of only small or large predators. Outside this range, the net benefit attained by the diverse community depends on the distribution of temperatures between years. However, averaging across the range of temperatures considered, the positive effects of diversity in body mass outweigh the negative effects of intraguild interference and predation associated with a larger number of species. For instance, large groundbeetles effectively suppress the prey population at low temperatures, but they are ineffective at higher temperatures. The difference in performance between large groundbeetles and the diverse community is larger at high temperatures (when the diverse community is more effective) than at lower temperatures (when large groundbeetles are more effective).
In Fig. 6, we investigate whether insurance against temperature variation within a single year can demonstrate a benefit to diversity in body mass. We find that without diversity in foraging area, single-predator communities consistently attain the lowest level or prey suppression. The optimal predators at the mean or median of the season's temperatures (large beetles Harpalus and Poecilus) outperform diverse communities. We again see that when predators utilize different foraging areas, the diverse communities outperform single-predator communities. Despite different distributions of initial predator abundances (comprised mainly of small spiders or large beetles), the two optimal communities we consider attain similar levels of prey suppression, and the inclusion of smaller predators in these communities results in smaller variation in prey suppression across different simulations (years). This provides some support for the insurance hypothesis, but at field-realistic temperatures, we cannot demonstrate a benefit to prey suppression which arises from only diversity in body mass (foraging areas must overlap).

Frequency of intraguild interactions drives diversity effects
Predator body mass and foraging area overlap are key characteristics of optimal predator communities largely because they determine the frequency of intraguild interactions, such as intraguild interference and intraspecific competition. However, quantifying the significance of these interactions and their effects on prey suppression in the field presents a challenge. In our model, intraguild interference and intraspecific competition are controlled by the parameters b 0 and c 0 , respectively. We find that when intraspecific competition dominates (large values of c 0 ), prey suppression improves with increasingly diverse predator communities (Fig. 7). This is a by-product of reducing the number of competing individuals within predator groups and is not directly related to variation in predator body mass or foraging area. Although larger groundbeetles are more mobile and engage in intraspecific competition more frequently, foraging area does not interact with intraspecific competition at all because intraspecific foraging area is identical. In contrast, when intraguild interference dominates (large values of b 0 ), smaller groundbeetles and spiders face higher intraguild interference from larger predators. Decreasing the overlap in predator foraging areas can reduce this effect. Our model indicates that the effect of predator diversity on prey suppression depends on the balance of intraspecific competition and intraguild interference within predator guilds. Hence, modeling intraspecific competition without considering general intraguild interference may inflate the predicted value of predator richness by underestimating negative intraguild interactions. Similarly, models which do not include intraspecific competition may underestimate the positive effects of diversity on prey suppression.

Key takeaways for prey suppression
In summary, our results further the understanding of when, and under which conditions, increased biodiversity improves prey suppression in real ecosystems. We find that the most important factor for determining prey suppression is overlap in predator foraging area. Overlap in foraging area is, to a large degree, influenced by the underlying landscape in which predators interact. For instance, complex environments are more likely to foster diversity in foraging areas. This suggests that farmers may improve pest control by maintaining weedy crop margins, growing two or more crops in proximity (intercropping) (Zhang et al. 2017), or planting cover crops (Bryant et al. 2013). When negative v www.esajournals.org predator-predator interactions are mediated by diversity in foraging area, body mass diversity (even within a community of "suboptimal" predators) can improve resilience and overall prey suppression of a community when temperatures vary between years. However, if an environment does not permit diversity in foraging area, then low-diversity communities will be optimal. In this case, managers should identify the most efficient predators for their field conditions and augment predator communities with these species to improve prey suppression. The predators we identify as optimal are specific to our study system, but similar methods could be employed to identify optimal predators in other systems.