From theory to experiment and back again — Challenges in quantifying a trait-based theory of predator-prey dynamics

Food webs map feeding interactions among species, providing a valuable tool for understanding and predicting community dynamics. Trait-based approaches to food webs are increasingly popular, using e.g. species’ body sizes to parameterize dynamic models. Although partly successful, models based on body size often cannot fully recover observed dynamics, suggesting that size alone is not enough. For example, differences in species’ use of microhabitat or non-consumptive effects of other predators may affect dynamics in ways not captured by body size. Here, we report on the results of a pre-registered study (Laubmeier et al., 2018) where we developed a dynamic food-web model incorporating body size, microhabitat use, and non-consumptive predator effects and used simulations to optimize the experimental design. Now, after performing the mesocosm experiment to generate empirical time-series of insect herbivore and predator abundance dynamics, we use the inverse method to determine parameter values of the dynamic model. We compare four alternative models with and without microhabitat use and non-consumptive predator effects. The four models achieve similar fits to observed data on herbivore population dynamics, but build on different estimates for the same parameters. Thus, each model predicts substantially different effects of each predator on hypothetical new prey species. These findings highlight the imperative of understanding the mechanisms behind species interactions, and the relationships mediating the effects of traits on trophic interactions. In particular, we believe that increased understanding of the estimates of optimal predator-prey body-size ratios and maximum feeding rates will improve future predictions. In conclusion, our study demonstrates how iterative cycling between theory, data and experiment may be needed to hone current insights into how traits affect food-web dynamics.

more often in habitat h. Here, we measure p i,h empirically. 133 We also introduce a term that describes the decrease in predation by a predator due to non-134 consumptive effects of other predators. This may include fear of predation, leading to decreased 135 foraging, or physical interference (Preisser et al., 2007;Sih et al., 1998). We propose that the 136 magnitude of this effect depends on the likelihood of predator j being intraguild prey to predator 137 l, and therefore depends on the expected attack rate of l on j (a jl ). Microhabitat overlap will also 138 affect predator encounters and should therefore affect the magnitude of non-consumptive effects 139 (e.g. Knop et al., 2014). We account for the effects of microhabitat overlap on non-consumptive 140 predator-predator effects in the same way as described above for predator-prey interactions. We 141 sum over the potential attack rates of all species l on a single individual of species j to account for 142 time spent avoiding or evading species l while species j is attempting to capture its own prey. The 143 importance of non-consumptive predator-predator effects is described by the scaling constant t 0 , 144 where a large value indicates a high penalty to attack rates due to non-consumptive effects. Non-145 consumptive effects from a conspecific individual may not be distinguishable from non-consumptive 146 effects from another predator species, and so we remove the intraspecific competition term as used  In total, dynamics for the number of individuals N i of species i are therefore given by: where species i increases in proportion to its intrinsic growth rate r i and decreases due to 151 predation. We assume the intrinsic growth rate (r i ) for predators to be zero due to their much 152 longer generation time (a year) compared with the duration of our experiment. The realized per 153 capita attack rate of predator j on species i in a microhabitat h (α ijh ) increases with the intrinsic 154 attack rate determined by the predator-prey body-mass ratio (a ij , see below) and decreases with 155 the size of the microhabitat, (α ijh = a ij /A h ), because predator and prey encounter each other less 156 frequently in the larger area. Total predation in a microhabitat increases as the proportion of prey 157 species i (p i,h N i ) and predator species j (p j,h N j ), in habitat h increases, but decreases dependent 158 on the time predator j spends handling prey of the same or other species (h kj ), or spends avoiding 159 or interfering with other predators l. 160 As in Schneider et al. (2012), we assume that for species body masses W i and W j (corresponding 161 to prey i and predator j), the allometric parameters (i.e. those dependent on body mass) are given 162 by: The derivation of allometric parameters is described in Schneider et al. (2012). We note the 164 importance of scaling parameters a 0 , h 0 , and R opt,j . a 0 scales the frequency of attacks when species where R opt,j = 1 indicates that predator j is most successful when attacking prey as large as itself and R opt,j 1 indicates that predator j is most successful when attacking prey much smaller than 171 itself. Parameter φ (φ >= 0) tunes the width of this success curve, with φ = 0 indicating that 172 attack success is independent of prey size, while the greater the value of φ the more restricted the 173 attack success around R opt . In contrast to Schneider et al. (2012), we allow the value of R opt to 174 vary from predator to predator. This is to account for differences in traits not accounted for in the 175 model that may affect predator foraging behavior.

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To determine the importance of the terms we introduce -microhabitat overlap and non-trophic in the same microhabitat zone as prey x and never overlaps with prey y. In a model accounting 197 for habitat use (as in models 1 and 2), R opt,i , therefore, will likely have an estimated value near 198 10, since predator i is ten times larger than prey x. If, however, we do not account for habitat 199 use (as in models 3 and 4), the model needs some other way to capture that predator i does not 200 interact with prey y in order to optimize the fit to empirical data. In the parameter estimation this 201 could be achieved by a larger value of R opt,i (moving its optimal prey size further from prey y), 202 or a higher value of φ (narrowing the effective feeding range), thus absorbing differences based on 203 which terms are present in the model and producing good model fit without necessarily reflecting 204 the 'true value' of a parameter. This is important to remember when interpreting the results of the 205 model fitting below. 206 We do not explicitly include non-consumptive mortality in this model. For the aphid (or basal) 207 prey, mortality not due to predation is included in the growth rate term r i , while for predators the 208 experiment is not long enough that we expect mortality other than that due to intraguild predation.

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Furthermore, without single individual controls, it would be difficult to separate "natural" mortality 210 from that due to predation or cannibalism. To empirically test and parameterize this model required a study system with rapid growth of 214 the prey population, a range of body sizes of both predators and prey, and distinct habitat zones.  ; bird cherry-oat aphid (Rhopalosiphum padi ); pea aphid (Acyrthosiphon pisum); and ground beetle (Bembidion spp). Arrows indicate potential feeding interactions which we then parameterized using the inverse method. Arrows point from prey to predator. Double headed arrows indicate that species could potentially eat each other and arrows beginning and ending with the same species indicate cannibalism. We removed all interactions to and from C. septempunctata except for C. septempunctata preying on aphids, and assumed that the aphids did not consume any predators. This arthropod community was dependent on two species of plants; barley (Hordeum vulgare) and fava beans (Vicia faba). search was repeated the next day to catch any missed in the initial search.

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Over the duration of the experiment, we found that C. septempunctata could occasionally 267 escape through gaps in the mesh cages. We assume that any C. septempunctata missing from cages 268 escaped in this manner, as other predators were never observed consuming C. septempunctata.

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Because this change in the population is not described by our mathematical model, we added  all possible parameterizations, we found the best-fitting parameterization. In order to compare the 295 importance of habitat use and predator interference, we repeated this fitting for each of the four 296 models.

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To fit the models, we first established a common baseline for aphid growth. Using the data 298 from control treatments, we estimated the intrinsic growth rate (r i ) for A. pisum and R. padi.

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The rate was distinct for each aphid species, but we assumed the same rate for each aphid species 300 across all aphid treatments (single-species or combined) and replicates. After this, we estimated 301 the remaining model parameters using data from predator-treated mesocosms. There were 10 302 predator treatments, each with 18 aphid populations; 6 replicates of each of A. pisum and R. padi 303 in isolation, as well as 6 replicates of A. pisum and R. padi when in combination with each other.

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interference (t 0 ) alongside initial aphid abundances. Although model parameters must be the same 306 across all treatments and replicates (but vary among the four models), initial aphid abundances were 307 permitted to vary in every replicate mesocosm. This allowed for differences in population outcomes 308 due to external, potentially stochastic factors, such as variation in plant growth. We constrained 309 estimates of these initial abundances to a range determined by observations from control mesocosms    -will lead to a higher predation rate. At the same time, differences in habitat overlap between 340 species decrease encounters, as do non-consumptive predator effects, which -all else being equal 341 -would lead to a lower predation rate. Thus, in models from which the latter two terms are 342 absent, a certain (observed) predation rate can only be achieved by a lower likelihood of attack (i.e.

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lower a 0 ) than in models where these terms are present (because here a higher a 0 can be countered 344 by a decrease in attacks due to fewer encounters). In a similar vein, R opt,j describes the optimal 345 predator-prey body-size ratio for predator j. A larger value means predator j preferentially attacks 346 prey much smaller than itself, while an R opt of 1 means that predators preferentially attack prey 347 the same size as themselves. Because we allowed R opt to vary among predators, we found that a 348 particular predator's R opt value could vary substantially among models, thereby compensating for 349 differences in attack rate which were or were not accounted for by habitat use or non-consumptive 350 predator effects in that particular model.

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As a result of all this, we saw different parameter values in the different models (Table 1)

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As expected, the scaling parameter for attack rate, a 0 was highest in the models with habitat 360 overlap (where encounters were limited), especially the model which also included non-consumptive 361 predator-predator effects (Table 1). R opt,j values also varied significantly among models, suggesting 362 that this parameter is also absorbing differences based on which terms are present in the model.

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The optimal predator-prey body-mass ratio for Pardosa was equal to one in three out of four 364 models, implying that Pardosa is capable of and willing to attack prey of its own size. Such feeding rates (lines in figure 5) were higher in the models including habitat overlap, but were (in 406 most cases) dramatically decreased when accounting for the actual habitat use of species. This 407 decrease is shown by the difference between the dots and the lines in figure 5. The few cases where 408 the realized feeding rate was larger than the potential feeding rate (e.g. the interaction strength 409 of Bembidion with itself) occurred when species had similar (or in this case, identical) habitat use, 410 especially in a relatively small habitat area. These potential feeding rates are primarily relevant 411 when extrapolating to other species combinations or habitat configurations; when fitting the data 412 from our experiment, it is the realized feeding rates, including the effects of habitat overlap and 413 non-consumptive predator-predator effects, that matter. This is why such different models can 414 produce such similar fits to observations.

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However, the fact that the different models all produced similar fits to observations, especially 416 of aphid dynamics, does not mean the models are equivalent. When we used the models to make 417 predictions for a new species, each model gave very different predictions (bottom row in figure   418 3). Using a hypothetical, entirely ground-dwelling prey species slightly larger than A. pisum, each 419 model predicted vastly different impacts of the predators on the prey population. Without habitat 420 use, the models predicted that Bembidion and Pardosa would have no impact on the prey, C.

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septempuntata would be a strong predator, and O. majusculus would fall somewhere in between.

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With habitat, C. septempuntata was predicted to have very little impact on the prey (since C.

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septempuntata was almost never on the ground). Pardosa and Bembidion were predicted to have 424 the strongest effect when habitat, but not non-consumptive effects, was included in the model.

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With respect to our model performance criteria, we found that all four models performed rel-

Full model Only habitat
Only pred avoidance Minimal model Figure 5: Predictions of each model for each predator population's feeding rate (y axis) on prey of different body sizes (x axis). Lines show the null expectation or potential feeding rate, i.e. predicted feeding rate as a function of prey size if predator and prey used all habitats in proportion to the area of the habitat (no difference in habitat use between predator and prey). This can also be understood as the likelihood that a predator will successfully attack a prey individual after they encounter each other, and is clearly much higher for the models that include habitat use. Line color corresponds to different models. Points show realized feeding rates of predators on each species in the experiment, based on the prey's body size (position along the x-axis, denoted by vertical dashed lines) and the actual amount of time predators and prey spend in different habitats. This can be understood as the likelihood that a predator first encounters and then successfully attacks a prey individual. Points are colored according to model, the same as lines. The two aphid species are identified by larger points and darker vertical lines, while predator species (which become intraguild prey to other predators) are shown by smaller points and lighter colored lines. Body sizes were R. padi = 0.155mg, A. pisum = 0.67mg, O. majusculus = 0.58mg, Bembidion = 2.15mg, Pardosa = 18mg, and C. septempuntata = 37mg. Relationships here are shown for predator populations of 20 Bembidion, 2 C. septempuntata, 20 O. majusculus or 10 Pardosa individuals with an aphid population of 100 individuals. Despite having a higher a 0 value, full model feeding-rate predictions are lower than the model with only habitat for all species except C. septempuntata due to predatorpredator non-consumptive effects. Note the varying scales of the y-axis. is therefore crucially important before we begin to make predictions. that it is important to understand model limitations before applying a model to an experimental 531 system, so that some aspect of the system is not more complex than that assumed by the model.

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In hindsight, we realise that some aspects of our model design might seem overambitious, if the 533 goal were to discriminate between alternative model formulations and estimate parameter values 534 for some universal relationship. More specifically, we explicitly aimed for predator diversity in 535 our experimental design, leading us to include predators like O. majusculus, Pardosa, and C. 536 septempuntata into the parameterization. As the foraging behavior of some of these predators 537 differs from more traditional 'grab-and-chew' predators like Bembidion, it is probably too simplistic 538 to describe trophic interaction strengths of all these predators using the same universal relationship 539 and based on one trait (body size) only. On the other hand, our attempt at doing so has revealed 540 the need to develop models that accommodate a diversity of foraging behaviour in predators.

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In the current case, we will stop short of these proposed added steps, as the current study was 542 explicitly designed to span the steps reported here (Laubmeier et al., 2018). Our intent is not to 543 arrive at the final solution, but to point to the next step in the iterative process between theory 544 and empirical insight.

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In this study, we tested an approach explicitly developed in Laubmeier et al. (2018). Our aim 547 was to arrive at an optimized design for generating empirical data to inform theoretical models. O. majusculus may be the more effective predators. Before we can convert the proposed models 558 to predictive tools, we thus need to do more ground-work and conduct smaller experiments to 559 estimate parameters such as optimal prey body size and attack rate scaling parameters -thereby 560 gaining the resolution to select among multiple models. In conclusion, our study demonstrates how 561 iterative cycling between theory, data and experiment may be needed to hone current insights into 562 how traits affect food-web dynamics.