Complex Eco-Evolutionary Dynamics Induced by the Coevolution of 2 Predator-Prey Movement Strategies

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bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

1 Complex eco-evolutionary dynamics induced by the coevolution of 2 predator-prey movement strategies

3 Christoph Netz, Hanno Hildenbrandt, Franz J. Weissing

4 Groningen Institute for Evolutionary Life Sciences, University of Groningen, Groningen 9747 5 AG, The Netherlands

6 Author for correspondence:

7 Franz J. Weissing

8 e-mail: [email protected]

9 Version: 7 December 2020

10 Abstract

11 The coevolution of predators and prey has been the subject of much empirical and theoretical 12 research, which has produced intriguing insights into the intricacies of eco-evolutionary dynamics. 13 Mechanistically detailed models are rare, however, because the simultaneous consideration of 14 individual-level behaviour (on which natural selection is acting) and the resulting ecological patterns is 15 challenging and typically prevents mathematical analysis. Here we present an individual-based 16 simulation model for the coevolution of predators and prey on a fine-grained resource landscape. 17 Throughout their lifetime, predators and prey make repeated movement decisions on the basis of 18 heritable and evolvable movement strategies. We show that these strategies evolve rapidly, inducing 19 diverse ecological patterns like spiral waves and static spots. Transitions between these patterns occur 20 frequently, induced by coevolution rather than by external events. Regularly, evolution leads to the 21 emergence and stable coexistence of qualitatively different movement strategies. Although the 22 strategy space considered is continuous, we often observe discrete variation. Accordingly, our model 23 includes features of both population genetic and quantitative genetic approaches to coevolution. The 24 model demonstrates that the inclusion of a richer ecological structure and higher number of 25 evolutionary degrees of freedom results in even richer eco-evolutionary dynamics than anticipated 26 previously.

28 Keywords: Individual variation, individual-based simulations, mechanistic models, polymorphism, 29 predator-prey interactions, spatial pattern formation, trait cycles 30 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

31 1. Introduction

32 Predator-prey coevolution has fascinated biologists for decades (Cott 1940, Pimentel 1961, Levin & 33 Udovic 1977, Dawkins & Krebs 1979). In the ecological arena, predator-prey interactions can lead to 34 complex non-equilibrium dynamics (Turchin 2003). On top of the ecological interactions, an 35 evolutionary arms race is taking place where adaptive changes in one species impose new selective 36 pressures on the other species. Experimental findings suggest that the ecological and the evolutionary 37 dynamics can be intertwined in an intricate manner (Yoshida et al. 2003, 2007, Becks et al. 2010). In 38 natural systems, it is a major challenge to unravel this complexity (Hendry 2019). In view of all this, it 39 is no surprise that theoretical models have played a crucial role for understanding the ecology and 40 evolution of predator-prey interactions (Fussmann et al. 2007, Govaert et al. 2019).

41 Predator-prey coevolution models have traditionally been based on the frameworks of population 42 genetics (Nuismer et al. 2005, Kopp & Gavrilets 2006, Cortez & Weitz 2014, Yamamichi & Ellner 2016), 43 quantitative genetics (Gavrilets 1997, Mougi & Iwasa 2010, Cortez 2018), and adaptive dynamics 44 (Dieckmann & Law 1996, Marrow et al. 1996, Flaxman & Lou 2009). Each approach encompasses a 45 diversity of models. The approaches differ in their assumptions on the nature of genetic variation 46 (discrete vs continuous), the occurrence and distribution of mutations, and the interaction of alleles 47 within and across loci, but they have in common that they strive for analytical tractability. To achieve 48 this, virtually all analytical models make highly simplifying assumptions on the traits that are the target 49 of selection. For example, predators are characterized by a one-dimensional attack strategy, prey by a 50 one-dimensional avoidance strategy, and prey capture rates are assumed to be maximal when the 51 predator attack strategy matches the prey avoidance strategy (Van der Laan & Hogeweg 1995, 52 Dieckmann & Law 1996, Marrow et al. 1996, Gavrilets 1997, Nuismer et al. 2005, Kopp & Gavrilets 53 2006, Yamamichi & Ellner 2016, see Abrams 2000 for a general overview). Such simplification allows 54 for an elegant and seemingly general characterisation of coevolutionary outcomes, but the question 55 arises whether it captures the essence of predator-prey interactions, which in natural systems are 56 mediated by complex behavioural action and reaction patterns.

57 Experiments have demonstrated that predators and prey show strong behavioural responses to each 58 other (Gilliam & Fraser 1987, Savino & Stein 1989, Ehlinger 1990, Hammond et al. 2007, Simon et al. 59 2019). These responses differ across species and spatial scales, and they are likely the product of 60 behavioural strategies that incorporate various information sources from the environment. In the 61 literature, behavioural interactions between predators and prey are often discussed in game- 62 theoretical terms as ‘behavioural response races’ (Sih 1984, 2005) or ‘predator-prey shell games’ 63 (Mitchell & Lima 2002). These games usually play out in space, where resources, prey and predators 64 are heterogeneously distributed among different patches. Prey have to balance foraging for a resource 65 with predator avoidance, while predators are faced with the task of predicting prey behaviour. A full 66 dynamical analysis of such interactions is a forbidding task (Flaxman & Lou 2009). Therefore, analytical 67 approaches typically use short-cuts, such as the assumption that at all times individual predators or bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

68 prey behave in such a way that they maximize their fitness under their given local circumstances 69 (Gilliam & Fraser 1987, Abrams 2007). It is often doubtful whether such short-cuts are realistic. For 70 example, it is unlikely that evolution will fine-tune behaviour to such an extent that it is locally optimal 71 under all circumstances and when conditions rapidly change (McNamara & Houston 2009, McNamara 72 & Weissing 2010). To make progress, we therefore have to consider the option of abandoning 73 analytical tractability and instead adopt a simulation-based approach that can make more realistic 74 assumptions on the ecological setting and the traits governing the behaviour of predators and prey.

75 Several such simulation models have been developed in the past (Huse et al. 1999, Kimbrell & Holt 76 2004, Flaxman et al. 2011, Patin et al. 2020). All these models are individual-based, meaning that both 77 the ecology and the evolution of predator-prey interactions reflect the fate of individual agents. Such 78 an approach is natural because selection acts on individual characteristics, while population-level 79 phenomena are aggregates and/or emerging features of these characteristics. Additionally, individual- 80 based models enable the implementation of considerable mechanistic complexity, both at the level of 81 individual behaviour and with regard to the environmental setting. The existing individual-based 82 models have shown that they can reproduce findings of analytical models (Huse et al. 1999, cf. Iwasa 83 1982) and theoretical expectations such as the ideal free distribution (Flaxman et al. 2011). As 84 illustrated by the recent study of Patin and colleagues (2020), such models can unravel the importance 85 of random movement, memory use, and other factors that are often neglected.

86 Still, current individual-based models for the coevolution of predators and prey are limited by the fact 87 that individual interactions are considered at a coarse-grained spatial scale (for an exception see 88 Kimbrell & Holt 2004). The environment is assumed to be structured in discrete patches and individuals 89 have the task of choosing a patch that provides an optimal balance between resource abundance and 90 safety. Within patches, predator-prey interactions are governed by patch-level population dynamics 91 and not based on single individuals moving and behaving in space. Accordingly, these models do not 92 capture interactions that take place at the individual level.

93 Here, we therefore consider a model where individuals move and interact in more fine-grained space, 94 where only few individuals co-occur at the same location. Predators and prey evolve situation- 95 dependent movement strategies that determine the likelihood of finding food resources and avoiding 96 predation. On an ecological (or behavioural) timescale, individual predators and prey repeatedly take 97 movement decisions based on an inherited strategy that evaluates the ‘suitability’ of environmental 98 situations. We assume that predator and prey individuals continually scan their environment in various 99 directions and sense the local density of resources, prey and predators in these directions. Based on 100 these inputs, each individual judges the suitability of the various movement directions (including its 101 current location), and makes a step in the direction of highest suitability. The movements individuals 102 make determine their survival and foraging success in case of prey, and their prey capture rate in case 103 of predators. These success rates in turn affect the number of offspring produced. On an evolutionary 104 timescale, successful individuals transmit their movement strategy to many offspring, subject to some bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

105 mutation. Over the generations, successful strategies will spread, thus potentially changing the nature 106 of ecological interactions and the spatial pattern of resource densities and abundances.

107 On purpose, we have kept our model as simple as possible. We are more interested in obtaining 108 conceptual insights than in accurately representing any specific real-world predator-prey system. To 109 this end, we will ask questions such as: What kinds of predator-prey movement strategies do evolve 110 and can they be classified in relation to each other? Which information sources are used by predators 111 and prey, respectively? How do predator-prey interactions reflect and shape the resource landscape? 112 Are the predictions of earlier models such as regular trait cycles recovered in our model? Are the 113 evolved populations monomorphic, or do different prey and predator strategies coexist in the same 114 population? How predictable is the outcome of coevolution; do replicate simulation runs produce 115 qualitatively similar outcomes? How fast is evolutionary change in relation to ecological change; does 116 the interplay of ecology and evolution lead to novel eco-evolutionary patterns and dynamics?

117

118 2. The Model

119 We consider an individual-based model for the antagonistic coevolution of movement strategies 120 between a population of herbivores (prey) and a population of predators in a spatially explicit 121 environment.

122 Ecological Interactions

123 To be concrete, we imagine the prey to be herbivores feeding on a resource, henceforth called grass, 124 and predators feeding on herbivores. All individuals live in an environment consisting of a grid of 125 512*512 cells with wrapped boundaries. A cell can host one or several herbivores and predators, but 126 this is unlikely since the number of grid cells is an order of magnitude larger than the number of 127 individuals. Ecological interactions occur in discrete time steps. A time step (we think of a day) contains 128 a grass growth phase, a movement phase and a foraging phase with predator-prey interactions. Grass 129 grows at a constant rate of 0.01 per time step up to a maximum level of 1. Next, herbivores and 130 predators move between cells based on their inherited movement strategy as described below. 131 Herbivores visiting a cell deplete the grass and gain the corresponding amount of energy. Herbivores 132 occupying the same cell share the amount of grass on that cell. If a predator encounters a herbivore 133 on the same cell, the predator succeeds to capture the herbivore with probability 0.5, in which case 134 the herbivore is killed and the predator gains one unit of energy. If several predators co-occur with 135 several herbivores (which only rarely ever happens), the successful predators kill and consume all the 136 herbivores present in the cell. The killed herbivores are equally distributed between successful 137 predators.

138 In the graphs below, the ‘performance’ of the predator and herbivore population is quantified as 139 follows: Predator performance is equal to the average lifetime consumption of prey items per bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

140 predator. Herbivore performance corresponds to the average lifetime resource intake per herbivore, 141 where lifetime intake was set to zero for those herbivores that did not survive predation.

142 Movement strategies 143 Movement strategies are based on the evaluation of nearby cells (see fig. 1). For each evaluated grid

144 cell, a ‘suitability score’ S is calculated. S is the weighted sum SwGwHwP=++ghp , where G, H and P 145 are the grass density, herbivore density, and predator density in the cell, respectively. The weighing

146 factors wg, wh, and wp are genetically encoded and transmitted from parent to offspring.

A) *

Figure 1: Movement and decision-making. A) Herbivores and predators move on the same rectangular grid, but their movement range per time step can be different. The plot illustrates the movement range of herbivores (blue, radius = 1) and predators (red, radius = 2) for our standard configurationC) . B) Individuals evaluate all cells in their movement range as to their ‘suitability’ and move to the cell of highest suitability. The suitability S of each cell is the weighted sum of the local grass density G, the local herbivore density H, and the predator density P, where the weighing factors wg, wh, and wp are genetically determined and hence evolvable. 147

148 Grass density represents the total amount of grass in a given cell, while for herbivore and predator 149 densities we convoluted the otherwise discrete presence-absence values of agents via a Gaussian 150 filter (Lindeberg 1994). This yields continuous values of herbivore and predator densities that are bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

151 diffused around the actual positions of the agents, much like we would expect from olfactory or similar 152 cues. Individuals evaluate all cells in their movement range, and move to the cell with the highest 153 suitability. In the simulations shown, the movement range of herbivores had a radius of one, and the 154 predators a radius of two (9 cells and 25 cells, fig. 1a). If predators have the same movement radius as 155 their prey, herbivores can reliably escape predation by moving away from high predator densities. Only 156 when predators can move further than herbivores, both parties need to predict the behaviour of the 157 other party, and interactions become more intricate.

158 Evolution of the evaluation mechanism

159 We consider haploid parthenogenetic populations with discrete, non-overlapping generations. The 160 population sizes are constant at the start of each generation. The herbivore population is initiated each 161 generation to 10,000 individuals and is successively diminished by predation. The predator population 162 size remains constant (either 10,000 or 1,000 individuals). At the start of a new generation, individuals 163 are placed at random in the ‘dispersal range’ around the position of their parent. In the simulations 164 shown below we used a dispersal radius of 1 or 10 (for both species).

165 Each individual has three gene loci with alleles wg, wh, and wp that correspond to the weights encoding 166 the evaluation of environmental suitability and, hence, determine the individual’s movement strategy. 167 The movement strategy determines the types of habitat most likely visited and therefore the 168 individual’s intake rate and, in the case of the herbivore, the individual’s probability of escaping 169 predation. At the end of a generation (after 100 time steps), surviving individuals produce offspring 170 where the number of offspring is proportional to their total intake. Each offspring inherits the genetic 171 parameters of its parent, subject to rare mutations. A mutation occurs with probability µ=0.001 per 172 locus, in which case the original value is changed by an amount drawn from a Cauchy distribution with 173 location 0 and scale parameter 0.001.

174

175 3. Results

176 Transitions between spatial patterns

177 Figure 2 shows a representative simulation run of our model for the situation that offspring dispersal 178 is local (dispersal radius 1) and predators and herbivores occur in equal proportions. Within 179 generations, the movement strategies of predators and herbivores induce ecological patterns that are 180 characteristic for these strategies (fig. 2B). Over generations, the movement strategies evolve; the 181 evolutionary dynamics of the weights are shown in Supplementary figure S1. The change in movement 182 strategies alters the ecological pattern. Even when averaged over the landscape, the performance of 183 predators and herbivores neither reaches an equilibrium nor stable oscillations (fig. 2A). Instead, 184 evolution is stochastic, highly dynamic and continually produces new strategies and ecological bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

Figure 2: Eco-evolutionary pattern formation. (A) Change in the performance of predators (red, average number of consumed prey per predator) and herbivores (black, average lifetime consumption of resources per herbivore, set to zero for individuals captured by predators) over 50,000 generations. (B) Landscape snapshots at six time points of the evolutionary trajectory (indicated by arrows in panel A). Grass density is shown in green, herbivore density in blue and predator density in red. Other colors emerge from additive color mixing, yellow for example signifies areas of high grass and predator density, purple that herbivores and predators occupy the same area. Predator-prey ratio 1:1; offspring dispersal radius 1.

185 patterns. The performance of predators and herbivores undergoes distinct phases. In the first half of 186 the simulation (fig. 2A), we see about 8 phases with a relatively stable configuration, between which 187 the system rapidly shifts. Each of these configurations corresponds to a specific combination of 188 movement strategies (Supplementary fig. S1) and an associated ecological pattern (fig. 2B.1-4). 189 Predators and herbivores can, for example, be distributed relatively homogeneously (fig. 2B.2) or in bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

190 static spatial clusters forming spot patterns (fig. 2B.3); they can also produce dynamic spatial patterns 191 such as rotating spirals (fig. 2B.4).

192 In the second half of the simulation, the stable configurations disappear and the system starts to 193 exhibit rapid large-scale fluctuations. This phase is characterized by large herbivore aggregations 194 enclosed by predators that remain stable for multiple generations (fig. 2B.5). Repeatedly, the 195 emergence of new movement strategies allows the herbivores to break out of these aggregations and 196 spread rapidly across the landscape (fig. 2B.6), until a new stable spatial formation between predators 197 and herbivores is reached. The pattern in figure 2B.5 corresponds to a period of low herbivore 198 performance, while herbivore performance is high in a pattern like figure 2B.6.

199 The performance phases and the corresponding ecological patterns reflect the evolved movement 200 strategies of herbivores and predators. For example, dynamic spatial patterns such as spirals tend to 201 be produced by herbivores that move into areas of high grass density, while predators chase after 202 them. In contrast, spatial aggregations as in figure 2B.5 reflect a ‘sit-and-wait’ strategy of the predator. 203 Individual predators prefer high grass densities and tend to remain there, while herbivores avoid 204 predators; the predators can thus monopolize high grass patches and consume those herbivores that 205 are eventually lured in by the high grass density. Although simulation replicates can vary a lot in their 206 behaviour (Supplementary material videos 1-3), the simulation run featured above is characteristic for 207 simulations with local offspring dispersal and a 1:1 predator-prey ratio. Under global offspring 208 dispersal, a landscape structure cannot emerge in the timeframe of our simulations (100 movement 209 steps per generation); under these conditions, the herbivore population often went practically extinct.

210 Polymorphism and trait cycles

211 We now turn to the case of a 1:10 predator-prey ratio and intermediate dispersal radius (r=10). Figure 212 3 shows the change of herbivore and predator performance over time and the evolution of weighing

213 factor wp in the herbivore population (for the evolution of all six weights see Supplementary fig. S2). 214 Under these conditions, spatial patterns are less pronounced and spatial differentiation is more limited 215 (see Supplementary fig. S3, Supplementary Material video 4). As shown in the top panel of figure 3, 216 the overall performance level of both populations is subject to relatively regular oscillations. In all but 217 one of the simulation replicates we investigated (n=30), a pronounced polymorphism in the herbivore

218 strategy evolved (fig. 3B): the evolved weight wp is strongly negative in some herbivores (purple) and

219 positive in all others (green). Since wp determines how the density of predators affects the suitability 220 assessment of a location by a herbivore individual, the two morphs respond very differently to the

221 presence of predators. The purple morph (with wp < 0) avoids locations with a high density of 222 predators, as one would expect. In contrast, the green morph is attracted by locations with a high 223 predator density. However, the avoidance of the purple morph is much stronger than the green 224 morph’s preference, and the two morphs differ more generally in their movement strategy: The green 225 morph moves primarily according to grass densities, while the purple morph increasingly moves to 226 avoid predator densities (see information usage, Supplementary fig. S4). bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

Figure 3: Polymorphism and trait cycles. A) Change in the performance of predators (red, average number of consumed prey per predator) and herbivores (black, average lifetime consumption of resources per herbivore, set to zero for individuals captured by predators) over 10,000 generations. B) Evolution of the distribution of the weighing factor wp (= the weight given to predator density) in the herbivore population. In all simulations, a polymorphism emerged, consisting of a predator-avoiding morph (purple) and a predator- prone morph (green). To enhance the visibility of rare morphs, weight values were multiplied with 100 and tanh-transformed, while weight frequencies were square-root transformed. C) Persistent oscillations in the relative frequency of the predator-avoiding (purple) and the predator-prone (green) morph. Predator-prey ratio 1:10; dispersal radius r=10.

227 At first sight, the behaviour of the green morph seems counterintuitive, but can be explained by the 228 fact that the predators evolved an aversion against locations with a high predator density (see 229 Supplementary fig. S2), presumably in order to avoid competition with other predators. This means 230 that predators tend to move away from locations with high predator density, leaving these locations bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

231 predator-free in the next time step. As long as predators are self-avoiding, the green morph is 232 therefore a viable strategy. Indeed, the green morph suffers less from predation than the purple 233 morph. In contrast, the purple morph is more efficient in exploiting resources. Hence the 234 polymorphism reflects two contrasting strategies when facing a trade-off between predator avoidance 235 and resource attraction.

236 As shown in figure 3C, the two morphs exhibit stochastic but regular oscillations in relative frequency. 237 Overall, the predator-avoiding purple morph is more frequent, but there are periods where the 238 predator-prone green morph has the highest frequency. The comparison of figures 3A and 3C reveals 239 that the fluctuations in herbivore and predator performance are driven by the oscillation in morph 240 frequency. In particular, predator performance (= overall predation success) is strongly correlated with 241 the frequency of the purple morph, which is less able to avoid predation than the green morph.

242 The described polymorphism is a characteristic feature of all simulations with local offspring dispersal 243 (dispersal radius 1 and 10); it did not occur in simulations with global offspring dispersal.

244 Evolutionary oscillations

245 When considering other simulation settings, we observed a variety of additional phenomena. For 246 example, we considered the implications of limiting the available information to one or both parties. 247 To illustrate this, we here show a simulation, in which all three information channels are available to 248 herbivores, while predators can only sense the density of other predators (Figure 4). In this scenario,

249 the weighing factor wp does not only exhibit a pronounced polymorphism in in the herbivores but also 250 in the predator population (Fig. 4B and C; the evolution of the other weights is shown in Supplementary 251 Fig. S5). At the start of the trajectory shown (at generation 9,000), both predator and herbivore 252 populations have a morph with a positive preference for predator density and a morph with a negative 253 preference for predator density. Each of these morphs exploits a strategic weakness in one of the other 254 morphs. The predator-avoiding (purple) herbivore morph is adapted to the predator-prone (red) 255 predator morph and will almost never be caught by such predators. The red predator morph exploits 256 the predator-attracted (green) herbivore morph, this morph in turn profits from the predator-avoiding 257 (blue) predator morph by walking in its steps. The blue predator morph in turn is the most efficient 258 predator of the equally predator-avoiding purple herbivore morph. As long as all four morphs are 259 present (between generations 9,000 and 10,500), they oscillate rapidly and in a regular fashion.

260 Around generation 10,500, the system enters a new phase, when the red predator morph goes extinct 261 (at t1, see fig. 4). The purple herbivore morph, being specialized on the red predator morph, follows 262 suit and also goes extinct. For about 400 generations, the performance of the predator population (= 263 the overall catching rate of prey) fluctuates at a relatively low level. This changes with the arrival of a 264 beneficial mutation in the predator population (at t2, fig. 4): a variant of the red (predator-prone) 265 morph suddenly reappears and becomes very frequent, exploiting the abundant green herbivore 266 morph. Here, the predator performance has a first peak. At this point the purple herbivore morph is bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

Figure 4: Predator extinction and two types of trait cycles. Cut-out from a simulation in which predators could only sense the density of other predators. A) Change in the performance of predators (red, average number of consumed prey per predator) and herbivores (black, average lifetime consumption of resources per herbivore, set to zero for individuals captured by predators). B) Evolution of the distribution of the weighing factor wp (= the weight given to predator density) in the predator population. Two distinct predator morphs emerge, a predator-prone morph (red) and a predator-avoiding morph (blue). C) Evolution of the distribution of the weighing factor wp (= the weight given to predator density) in the herbivore population. Two distinct herbivore morphs emerge, a predator-prone morph (green) and a predator-avoiding morph (purple). To enhance the visibility of rare morphs, weight values in B) and C) were multiplied with 100 and tanh-transformed, while weight frequencies were square-root transformed. Predator-prey ratio 1:10; dispersal radius r=10.

267 reintroduced via mutation and becomes very frequent. As the purple morph increases, the red 268 predator morph becomes rare and finally goes extinct again. The blue predator morph, which 269 remained present in the population, takes over again and benefits from the common purple herbivore bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

270 morph. Here, the predator performance peaks a second time. As the predator population is again 271 monomorphic and blue, the green herbivore morph becomes more common and the purple herbivore 272 morph goes extinct; the system returns to its original state. This cycle happens twice-over (t2, t3, fig. 4), 273 but the third time the red morph reappears (at t4, fig. 4), the blue predator morph goes extinct. As the 274 purple herbivore morph rapidly takes over the population, the now monomorphic predators do not 275 catch prey anymore and their predation rate goes to practically zero. The population also cannot shift 276 back to a negative preference, because this morph has gone extinct, and a new adaptive mutation 277 does not arise in time. Hence the predators practically go extinct.

278 279 4. Discussion

280 We here presented a model considering the antagonistic coevolution of movement strategies. 281 Predators and prey move around in a fine-grained resource landscape where they encounter 282 antagonists and conspecifics. Movement strategies are based on three heritable parameters 283 determining how individuals assess the densities of grass, herbivores and predators in their movement 284 decisions. As movement affects foraging success and predator-prey encounter rates, it is an important 285 determinant of lifetime reproductive success and, hence, subject to natural selection. Selection 286 pressures are strongly varying in space and time, leading to rapid evolution and rich eco-evolutionary 287 dynamics. Although our model is still very simple, we observe a range of phenomena that do not occur 288 in models with coarser spatial scales and with fewer evolutionary degrees of freedom. The populations 289 in our model do not evolve stable movement patterns but instead exhibit intricate evolutionary 290 dynamics, including regular cycles between movement strategies and the spontaneous advent of novel 291 strategies and counter-strategies. Regularly, stable polymorphisms emerge, where qualitatively 292 different movement strategies coexist for extended periods of time. For example, we observed the 293 evolutionary emergence of sit-and-wait predators, while other predators were chasing their prey. The 294 movement strategies in herbivores and predators determine the spatial pattern of resource depletion 295 and predator-prey encounters, which together induce the emergence of intriguing ecological patterns. 296 Over the generations, these patterns are fluent, as they change with the evolution of the underlying 297 movement strategies. Rapid transitions between patterns (e.g. between spiral waves and static spots) 298 can occur; these reflect coevolutionary changes rather than changes in external conditions. We will 299 now discuss some of these findings in more detail and in the context of other models of predator-prey 300 coevolution.

301 Previous work on the coevolution of predator-prey movement has also produced interesting patterns, 302 such as evolutionarily optimal strategies leading to ‘predator-prey chases’ across habitats (Abrams 303 2007) and the coupling of ecological dynamics across habitats via evolving conditional strategies 304 (Flaxman et al. 2011). However, in these studies ‘movement’ corresponds to a choice between a small 305 number of densely populated habitats. In the habitat patches, the interactions of predators and their 306 prey is not modelled at the individual level but by patch-level dynamic equations (but see Patin et al. bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

307 2020 for an exception). In contrast to these habitat choice models, we consider movement in a fine- 308 grained spatial environment. From ecological models, it is known that in such fine-grained 309 environments the interplay of predator and prey movement can induce a diversity of spatial ecological 310 patterns, including rotating spirals and static stripes or spots (Hassell et al. 1991, Comins & Hassell 311 1996, Alonso et al. 2002, Banerjee 2015). Our model shows that, by adding the evolution of movement 312 strategies to the ecological models, various of these patterns emerge in a single simulation run, and 313 that repeated switches between patterns can occur even though the ecological parameters have not 314 changed. .

315 Behavioural polymorphisms commonly occur in our simulations. They can either be fleeting 316 (Supplementary fig. S1) or persistent (fig. 3); they can be relatively stationary (fig. 3B) or strongly 317 fluctuating (fig. 2C); and they can either involve a small number of discrete behavioural types (fig. 3B) 318 or a continuous spectrum of coexisting strategies (fig. 3C). Polymorphisms are also predicted by 319 analytical models of coevolution (Senthilnathan & Gavrilets 2020), but they occur much more 320 frequently in more mechanistic individual-based models (Botero et al. 2010, Long & Weissing 2020). 321 The behavioural dimorphism emerging in the herbivore population in most of our simulations (figs 3 322 and 4) can be interpreted as the coexistence of two behavioural types that balance predator avoidance 323 and foraging in alternative ways. Behavioural biologists might classify the two types of individuals as 324 bold and shy personalities. From a single-species perspective, it is well known that resource 325 competition and predator-prey interactions can induce behavioural polymorphisms (Quinn & 326 Cresswell 2005, Bell & Sih 2007, Wolf et al. 2007, 2008, Wolf & Weissing 2010). Our simulations show 327 that systematic behavioural variation can also emerge in two species due to coevolution (Fig 4). The 328 existence of behavioural polymorphism can have important ecological and evolutionary implications 329 (Sih et al. 2012, Wolf & Weissing 2012). In predator-prey systems, intraspecific variation can, for 330 example, be crucial for the sustained persistence of both species (Senthilnathan & Gavrilets 2020). This 331 is illustrated by the simulation in Fig. 4 where the breakdown of behavioural polymorphism led to the 332 extinction of the predator species.

333 Our model produces evolutionary oscillations that are reminiscent of the trait cycles classically found 334 in models of unidimensional traits. These models either consider Mendelian traits, where trait cycles 335 consist in oscillating allele frequencies (population genetics models, Nuismer et al. 2005, Kopp & 336 Gavrilets 2006, Cortez & Weitz 2014), or quantitative traits, where trait cycles are produced by 337 oscillations in the trait value of a monomorphic population (quantitative genetics, Gavrilets 1997, 338 Mougi & Iwasa 2010, Cortez 2018, adaptive dynamics, Dieckmann et al. 1995, Dieckmann & Law 1996, 339 Marrow et al. 1996). A mixture of both approaches, where predators carry a quantitative trait and prey 340 a Mendelian trait, has been shown to produce complex population dynamics such as anti-phase cycles, 341 in-phase cycles and chaotic dynamics (Yamamichi & Ellner 2016). Both frequency-dependent and 342 mutation-limited oscillations naturally occur in our model. In polymorphic populations with a small 343 number of coexisting behavioural morphs (fig. 3), we observed regular oscillations in the frequencies 344 of these morphs. In the absence of behavioural types, other oscillations occurred, which are based on bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

345 the repeated substitution of traits. These oscillations are mutation-limited in that they depend on the 346 occurrence of specific mutations that can replace the current trait. Due to the stochasticity of the 347 mutation process, the latter trait cycles are much less regular than the oscillations between discrete 348 behavioural types.

349 We kept our model as simple as possible, in order to demonstrate that not much structure is required 350 for obtaining the eco-evolutionary patterns described above. It would be interesting to study the 351 implications of features such as sexual reproduction and inheritance, variable population sizes at birth, 352 and a spatially heterogeneous resource distribution, but this is beyond the scope of our study. Here

353 we only discuss the implementation of movement strategies by three weighing factors (wg, wh, and wp) 354 in our model. This way, there are three ‘evolutionary degrees of freedom’ in our model, which is 355 considerably more than in conventional predator-prey coevolution models that typically consider a 356 one-dimensional trait space in both the predator and the prey population. Enhancing the degrees of 357 freedom is important, because it allows for much more rapid and unpredictable evolution due to the 358 emergence of ‘surprising’ (and sometimes counterintuitive) strategies and counter-strategies, the 359 advent of novel forms of behaviour, and by allowing for behaviour that is (at least partly) stochastic 360 and unpredictable. We anticipate that, quite generally, models with more evolutionary degrees have 361 much richer eco-evolutionary dynamics than most conventional models.

362 Having said this, we are fully aware that our behavioural model is still unrealistically simple and well- 363 behaved. Our model could be extended quite naturally by basing the calculation of environmental 364 suitability on a more complex algorithm, such as an evolving artificial neural network (ANN) (Huse et 365 al. 1999, Enquist & Ghirlanda 2005, Morales et al. 2005). Evolving regulatory networks (including ANNs) 366 have a number of important features (Wagner 1979, Van den Berg & Weissing 2015, Van Gestel & 367 Weissing 2016), such as the emergence of cryptic variation (since the same phenotypic strategy can be 368 encoded by very different networks), which allows for much faster evolvability in the face of 369 environmental change. Perhaps most importantly, network models tend to have an intricate genotype- 370 phenotype map, implying that small mutations can have large and unexpected implications at the 371 phenotypic level. It is therefore not surprising that ANN-based pilot simulations on predator-prey 372 coevolution exhibit even richer eco-evolutionary dynamics than occurring in the present model.

373 Acknowledgements 374 We thank P.R. Gupte, R. Bilderbeek and the MARM group at the University of Groningen for valuable 375 discussion, comments, and suggestions. F.J.W. and C.N. acknowledge funding from the European 376 Research Council (ERC Advanced Grant No. 789240).

377 Data, code and materials 378 All data are available on https://dataverse.nl/privateurl.xhtml?token=4cb06f00-15f5-4949-9de5- 379 e0b87f74d1a5. The source code is available under https://github.com/christophnetz/Cinema_git 380 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

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1 Supplementary Material

2 In this supplement, we give additional information to figures 2,3 and 4. Simulation videos can be 3 accessed in the digital resources. The source code and an executable of the model can be downloaded 4 from GitHub.

5 Contents

6 Figure S1: Evolution of all three weights of prey and predators, compare figure 2

7 Figure S2: Evolution of all three weights of prey and predators, compare figure 3

8 Figure S3: Landscape snapshots, compare figure 3

9 Figure S4: Information usage, compare figure 3

10 Figure S5: Weight evolution of all weights of prey and predators, compare figure 4

11 Digital resources:

12 https://dataverse.nl/privateurl.xhtml?token=4cb06f00-15f5-4949-9de5-e0b87f74d1a5

13 Simulation videos 1-3: Landscape dynamics in replicate simulations of figure 2. Predator-prey ratio 14 1:1; offspring dispersal radius 1.

15 Simulation video 4: Landscape dynamics in replicate simulations of figure 3. Predator-prey ratio 1:10; 16 offspring dispersal radius 10.

17 Source Code:

18 https://github.com/christophnetz/Cinema_git

20 Supplementary information to figure 2

21 This simulation was conducted with a predator-prey ratio of 1:1 under local offspring dispersal with a 22 radius of 1 (for other default parameters see model description). Replicate simulation movies can be 23 viewed [here]. The landscape dynamics reflect the ecological interactions which in turn are shaped by

24 the evolution of movement decision mechanisms, encoded by the weights wg, wh and wp (see equation 25 1 in the main text). The evolution of all six weights, three per herbivore and predator population each, 26 is depicted in Supplementary figure S1. Notice that there are distinct periods with similar weight 27 configurations, between which rapid shifts occur. For example, from generation 10,000 to 18,000, the

28 herbivore population contains a dimorphism with regard to the predator density weight wp, similar to 29 the more stable polymorphism we observe in figure 3. The occurrence and vanishing of the positive- 30 valued morph is associated with a distinct phase of performance and ecological pattern shown in fig. 31 2B.2. From generation 6,000 to 10,000, the predator population even contains three morphs with

32 respect to the weights for grass and herbivore density, wg and wh. One of these morphs is positive for 33 both weights, the second is effectively neutral, whilst the third and least-frequent morph is negative 34 for both. Again this population composition is tightly associated with a performance phase and the 35 ecological pattern in fig. 2B.1. As the third morph goes extinct around generation 10,000, performance 36 and ecological pattern changes. Towards the end of the simulation, herbivores evolve a rather constant 37 strategy consisting in a strongly positive grass density weight wg, a strongly negative herbivore density

38 weight wh, and a strongly positive predator density weight wp. However, this state, corresponding to

39 the ecological pattern shown in fig. 2B.5, is punctuated by the repeated occurrence of herbivores with 40 a negative predator-density weight that escape the surrounding predators and spread across the 41 landscape (fig.2B.6). Whenever this morph arises, we observe a peak in herbivore performance. This 42 morph cannot establish itself permanently, though: Its agglomerations collapse under the pressure of 43 advancing predators as individuals move away from predators. The morph with a positive herbivore 44 density weight on the other hand does not move away from predators and can therefore form stable 45 agglomerations. It is worth noting that these results do not necessarily transfer to empirical systems, 46 as individuals in reality of course rarely move towards predators. Yet, this result demonstrates the 47 creativity, with which antagonists exploit each other’s strategic weaknesses created by phenotype and 48 external constraints (in this case given by our model, otherwise given by the laws of nature), and the 49 far-reaching ecological consequences this can have.

50 Supplementary information to figure 3

51 We next considered a predator-prey ratio of 1:10 and local offspring dispersal with a radius of 10. 52 Under these conditions the landscape is much less structured than in the previous simulation (see 53 Supplementary fig. S3, movie) and the predator-prey ratio is closer to general empirical expectations.

54 The herbivore population evolves a dimorphism for the predator density weight wp (see figure 3).

55 While one morph exhibits a negative-valued weight wp and therefore avoids predators, the other is 56 weakly positive-valued and predator-prone. This counter-intuitive strategy can be understood when 57 looking at the evolution of all six weights that is depicted in Supplementary figure S2: The predator 58 evolved a negative predator-density weight, and a (slightly) positive predator-density weight therefore 59 makes the movements of that herbivore morph less predictable to the predator. The other two weights 60 of the predator show a positive preference for grass and herbivore density. Thus the predators chase 61 prey, while at the same time expecting to find herbivores more often in areas of high grass. The other 62 two weights of the herbivore population show a preference for grass, but an avoidance of herbivore 63 densities. This could be an adaptation to avoid intraspecific competition for resources as well as 64 heightened predation risk in the presence of conspecifics.

65 The weight values together form the movement strategy of an individual, but it is still difficult to assess 66 the overall importance of the three environmental cues: grass density, herbivore density and predator 67 density follow different distributions, and therefore influence the movement decisions to different 68 extends even if all three weights were the same. We could derive the distributions from the overall 69 landscape, but this would ignore the heterogeneous distribution of individuals and the bias introduced 70 by selective movement. Thus to correct for this, we produced summary statistics of the cues 71 experienced by all individuals of a population during a single generation. We can then multiply weight 72 values (here we can segregate by morph) with the mean absolute deviation of grass density, herbivore 73 density and predator density cues, giving us the information usage during each generation

74 (Supplementary fig. 4). In the beginning of the simulation, predators as well as wp-positive and wp- 75 negative herbivore morphs move primarily according to grass density. The wp-positive herbivore 76 morph considers herbivore densities second, while considering predator densities least, and maintains 77 this profile of information usage throughout the simulated time. The wp-negative herbivore morph on 78 the other hand undergoes a gradual shift between generations 3,000 and 5,000 towards considering 79 predator densities more strongly. After generation 5,000, this morph moves primarily according to 80 predator densities, with grass density and herbivore density ranking second and third. The predator 81 population follows this development in a rapid shift around generation 6,000, such that it then

82 matches the information usage profile of the wp-negative herbivore morph. These results demonstrate 83 how difficult it can be to determine the general importance of predator densities, herbivore densities

84 and resource densities. Movement strategies can be heterogeneous within populations and change 85 dynamically through time.

87 Supplementary information to figure 4

88 Figure 4 shows a special case in which predators were only allowed to sense their own presence, with 89 a predator-prey ratio of 1:10 and local offspring dispersal with a radius of 10. We chose to show this 90 special case, because it illustrates two types of trait cycles and evolutionary-induced extinction. For 91 completeness, figure S5 shows the all 4 weights. One can see that the grass and herbivore density 92 weights of the herbivore populations hitchhike on the oscillations induced by the interaction between 93 the predator density weights in predators and herbivores. However, the initial genetic variation is 94 depleted during the transitions indicated by t1, t2, t3 and t4.

95 Simulation Code and Availability

96 Our model program implements a visual representation of the simulated landscape, in which users can 97 visually inspect pattern formation and even the movements and interactions of single individuals. We 98 encourage the reader to watch the video clips of our simulations that we uploaded here 99 [https://dataverse.nl/privateurl.xhtml?token=4cb06f00-15f5-4949-9de5-e0b87f74d1a5], as they 100 confer a much more detailed impression of the eco-evolutionary dynamics than the landscape 101 snapshots (fig. 2, Supplementary fig. S) Everyone who would like to experiment themselves with our 102 model finds our model code on GitHub, [https://github.com/christophnetz/Cinema_git]. The program 103 produces a graphical output of the landscape each time step (as shown in videos 1-4), a heatmap of 104 the weight distributions as well as a performance plot. Users can also pause the simulation and zoom 105 in on single individuals. Please refer to the readme file for more detailed instructions, but note that 106 the program depends on C++ libraries that are only available for the Windows operating system.

Figure S1: Weight evolution of herbivores and predators, compare figure 2. Frequencies go from 0.0 (=white) to 0.3 (=red) and 1 (=blue). Weight values were multiplied with 100 and transformed by a hyperbolic tangent. Predator-prey ratio 1:1; offspring dispersal radius 1. 4 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

Figure S2: Weight evolution of herbivores and predators, compare figure 3. Frequencies go from 0.0 (=white) to 0.3 (=red) and 1 (=blue). Weight values were multiplied with 100 and transformed by a hyperbolic tangent. Predator-prey ratio 1:10; offspring dispersal radius 10. 5 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

Figure S3: Landscape snapshots of a replicate simulation of figure 3 at generations 1000, 5000 and 10000. Grass density is shown in green, herbivore density in blue and predator density in red. Other colors emerge from additive color mixing, yellow for example signifies areas of high grass and predator density, purple that herbivores and predators occupy the same area. Predator-prey ratio 1:10; offspring dispersal radius 10.

6 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

Figure S4: Average information usage of the predator-prone herbivore morph (wp-positive morph), the predator-avoiding herbivore morph (wp- negative morph), and the predator population, compare figure 3. Predator-prey ratio 1:10; offspring dispersal radius 10.

7 bioRxiv preprint doi: https://doi.org/10.1101/2020.12.14.422657; this version posted December 15, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

Figure S5: Cut-out from a simulation in which predators could only sense other predators. Evolution of all three weights of herbivores and the single predator weight, compare figure 4. Frequencies go from 0.0 (=white) to 0.3 (=red) and 1 (=blue). Weight values were multiplied with 100 and transformed by a hyperbolic tangent. Predator-prey ratio 1:10; offspring dispersal radius 10. t1 marks the extinction of the predator-prone predator morph at generation 10544, t2, t3 and t4 the reintroduction of this morph via mutation at generations 10953, 11249 and 11891. 8