Optimal Random Avoidance Strategy in Prey-Predator Interactions

It has recently been reported that individual animals, ranging from insects to birds and mammals, exhibit a special class of random walks, known as Lévy walks, which can lead to higher search efficiency than normal random walks. However, the role of randomness or unpredictability in animal movements is not very well understood. In the present study, we used a theoretical framework to explore the advantage of Lévy walks in terms of avoidance behaviour in prey-predator interactions and analysed the conditions for maximising the prey’s survival rate. We showed that there is a trade-off relationship between the predictability of the prey’s movement and the length of time of exposure to predation risk, suggesting that it is difficult for prey to decrease both parameters in order to survive. Then, we demonstrated that the optimal degree of randomness in avoidance behaviour could change depending on the predator’s ability. In particular, Lévy walks resulted in higher survival rates than normal random walks and straight movements when the physical ability of the predators was high. This indicates that the advantage of Lévy walks may also be present in random avoidance behaviour and provides new insights into why Lévy walks can evolve in terms of randomness.


Introduction
To an observer, animal behaviour often seems random and unpredictable. Even when individual 38 animals are subjected to the same experimental apparatus or external stimuli, behavioural 39 a power law distribution, P(l)-l −µ where µ (1,3] is a power law exponent characterising the 74 movements. Lévy walks are characterised by rare ballistic movements among short steps ( figure  75 1b), resulting in anomalous diffusions. Theoretical studies reported that Lévy walks have higher 76 search efficiency than Brownian walks where the targets are distributed sparsely and patchily 77 [26,27]. Therefore, it has been suggested that the prevalence of Lévy walks in animal movements 78 is a consequence of natural selection that facilitates search efficiency. However, recent findings 79 suggest that the various mechanisms that can produce Lévy walk patterns depend on several 80 conditions [28,29]. Thus, it is imperative to explore why and when animals exhibit Lévy walks 81 depending on various ecological conditions. Nevertheless, the impact of Lévy walks, other than 82 search efficiency, on the fitness of individual animals remains poorly understood. 83 In the present study, we hypothesised that individual animals performing Lévy walks 84 would benefit in terms of avoiding a predator's attacks. The questions examined included what 85 movement strategy is the most efficient in surviving in prey-predator interactions and how is the 86 optimal behaviour affected by the ecological conditions and the predator's cognitive and physical 87 abilities [30]. To answer these questions, we constructed a general framework of the predator 88 avoidance behaviour based on a random walk paradigm (figure 1a, b) and analysed the theoretical 89 model to determine the efficiency of avoidance.

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To explore the most efficient avoidance behaviour, we constructed a framework in which a prey 93 individual avoided a predator that overlooked an open arena (figure 1a, b) and calculated the 94 survival rate of the prey by using simulations and analytical solutions. 95 First, we considered a prey individual to be at the centre of the 2D circular field with 96 radius R and a predator that overlooked the entire field from above (figure 1a, b). Then, the prey 97 started moving and performed a class of random walks (i.e. Lévy walks with a specified µ) at a 98 constant velocity (one space unit per unit time) without stopping. The step lengths l were drawn 99 from a truncated power law distribution which is often best fitted to empirical data [19,20,23,24] 100 and the turning angles between consecutive steps were drawn from a uniform distribution (see 101 electronic supplementary material for details). In our analysis, µ ranged from 1.1 to 3.0 by 0. 1. 102 Lévy walks with µ = 1.1 and 3.0 correspond approximately to straight-line movements and 103 Brownian movements, respectively. 104 We assumed that predators were rational and had relatively high cognitive ability. The 105 predator could perceive the present position of the prey, denoted by Xt = (xt, yt), and could also 106 memorise its past positions. Then, the predator would have to predict the future position Xt+τ = 107 (xt+τ, yt+τ) of the prey after a τ time step, based only on the information it had about the present (t) 108 ∈ and past positions of the prey. The predicted position was denoted by . While 109 the time lag τ could be considered as a predator attack characteristic, it could also be a 110 characteristic of the prey individual as its position after τ could be scaled by its velocity. Therefore, 111 if the prey's velocity was fast, τ could be relatively large. Moreover, if the predator attacked slowly 112 after making a prediction, τ could also become large. Intuitively, the larger the τ, the higher the 113 prey's advantage in escaping the attacks. 114 To model how the predator would predict the prey's position in a simple manner, we 115 assumed that the probability distributions of step lengths and turning angles of the prey did not 116 depend on its spatial positions. Therefore, we could straightforwardly derive the optimal 117 prediction rule adopted by predators by using a straight line (figure 1c). This is similar to 118 interception strategies which have been observed in birds, bats, fishes, and insects [31][32][33][34][35]. The 119 predicted position can be described as follows: 120 where (xt−1, yt−1) is the past position of the prey used to obtain the direction of the prediction 122 (figure 1c). When the distance between the predicted position and the actual position was less 123 than or equal to an attack radius r, the predator could successfully catch the prey. If the prediction 124 failed, the predator would predict the new position again after the waiting time (W) had elapsed. 125 W represents the time interval from the present attack to the next attack and is indicative of a 126 characteristic of the predator that cannot attempt the next attack immediately. In contrast, once 127 the prey reached the safety area (i.e. the edge of the circular field) (figure 1) without having been 128 predicted by the predator, it would never leave it and would be regarded as having survived. To 129 obtain the survival rate of the prey, this trial (from the beginning to the point of being caught or 130 having survived) was independently iterated 10 6 times for a specified parameter set in our 131

simulation. 132
Although, based on our assumption, the prey could not obtain any information about 133 when and where the predator would attack, we considered two cases: the prey was either aware 134 (precaution case) or not aware (unwariness case) of the presence of the predator. The difference 135 of these cases would result in a different step length at an initial condition (t = 0). In the precaution 136 case, the prey could choose a new step length from the probability distribution at the initial 137 condition and then it could start moving. On the other hand, in the unwariness case, the prey was 138 on the move with step length l. Therefore, the initial step length is likely to be longer than that of 139 the precaution case. Under these assumptions, we conducted simulations to estimate the survival 140 rate of the prey. 141 To obtain an analytical solution of the survival rate based on the aforementioned 142 assumptions, we split it into two components; the prediction rate and the time length to reach the 143 safety area. In this case, the survival rate of a prey can be described as follows: 144 where Q is the predictability of the prey's movements by the predators and T/W is the average 146 number of attacks until the prey reaches the safety area. Q and T are functions of the parameter 147 associated with the movements, namely µ, but the explicit derived expression can be obtained 148 only for Q with r → 0 (see electronic supplementary material for details). Therefore, we used the 149 simulations for T. 150 In our framework, the most unpredictable strategy included movements with many 151 reorientations. In contrast, the straight-line movement was the easiest to predict. However,

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First, we demonstrated the relationship between the predators' prediction rate and the prey's time 158 length to reach the safety area for specified movement patterns, which were obtained from 159 simulations and analytical solutions (see figure 2 for the precaution case and figure S1 for the 160 unwariness case). The smaller the µ exponents, the shorter the time it would take prey to reach 161 the safety area. This roughly corresponds to the mean squared displacement, which characterises 162 the diffusion property and increases as the exponents decrease [19,26]. As for the prediction rate, 163 when the exponents µ were small, it was easier for the predators to predict the prey's position. 164 This is because Lévy walks with large µ are characterised by frequent reorientations. Note that 165 the prediction rate of Lévy walks with very small µ was close to one. This result indicates that 166 there is a trade-off relationship between the predictability and the time to reach the safety area for 167 any τ and R, that is, the shorter the time it takes to reach the safety area, the higher the 168 predictability and vice versa. An intuitive explanation for these results is shown in figure 1b. The 169 trade-off relationship also implies that the prey cannot decrease both parameters simultaneously, 170 although both the lower predictability and the shorter time length to reach the safety area can lead 171 to high survival rates for the prey. Therefore, we expect that Lévy walks with intermediate µ will  survival rates than Lévy walks with µ < 3 did in any scenario. When τ was small, the difference 180 in the prediction rates for large and small µ was relatively small (figure 2). Therefore, the 181 advantage of Brownian walks was negligible in terms of decreasing the prediction rate. 182 The results also showed that Brownian walks were favoured when the radius R of the 183 area was small (R = 10). This could be because of the short time length to reach the safety area 184 (e.g. T 40), even for Brownian walks, compared with the time length to reach the area for R = 185 50 and 100 (figure 2). Therefore, for small R, Brownian walks resulted in few attacks with a low 186 prediction rate. In contrast, for large R, Brownian walks were more likely to be attacked, owing 187 to the longer time it took to reach the safety area. 188 The waiting time W was also a key factor in determining the best avoidance strategy,

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In the present study, we created a theoretical framework to investigate random avoidance 200 behaviour in terms of Lévy walks, which have been observed widely in biological movements 201 [19-25]. We constructed a framework for prey-predator interactions based on the prey's 202 movements and the predator's prediction (figure 1) and calculated the predator's prediction rate, 203 the time length to reach the safety area and the prey's survival rate. We found that there is a trade-204 off relationship between the predator's prediction rate and the prey's time length to reach the 205 safety area (figure 2). We can interpret the latter as the degree of exposure to predation risk. Thus, 206 the trade-off corresponds to the relationship between the predictability and the degree of exposure 207 to predation risk, suggesting that prey individuals cannot decrease both of these factors, although 208 they need to. Consequently, they must identify a point of compromise in order to achieve higher 209 survival rates. 210 Then, we showed that Lévy walks could outperform Brownian walks in terms of the 211 survival rate when the time lag τ between the prediction and the attack was short and the area was 212 When the time lag was large, Brownian walks represented a better escape strategy 227 ( figure 3). If the movements were sufficiently random, they decreased the chances of the prey's 228 position being predicted by the predator. Furthermore, when W was small, the predator had a 229 higher chance of attacking; in this case, the random movements represented an advantage. In both 230 cases, eluding the attack was relatively more important for the prey's survival than reaching the 231 refuge rapidly was. This result suggests that Brownian walks still represent an effective strategy 232 of escaping from a predator, which aims for and catches prey. 233 So far, the main advantage of Lévy walks has been the high encounter rate with targets 234 (e.g. food, mates and habitats), which are distributed scarcely and patchily [26,27]. However, 235 recent findings suggest that the origins of Lévy walks may not be unique, but may depend on 236 ecological and physical conditions [28,29]. Therefore, it is crucial to explore when and why 237 animal individuals would exhibit Lévy walks and which ecological traits, physical constraints or 238 environmental factors are responsible for this phenomenon. In this study, we focused on the prey-239 predator interactions once they encounter each other. This scenario occurs in relatively shorter 240 time and smaller spatial scales than the scenario in which a predator moves in search for prey 241 moving in the environment [36,37]. Our results suggest a novel advantage of Lévy walks in the 242 context of avoiding a predator's attacks. Moreover, although the essential role of variability or 243 randomness in Lévy walks is not very well understood, our results indicate that the extent of 244 randomness plays a key role in determining the optimal avoidance strategy. Additionally, based 245 on our framework, Lévy walks should be expressed as spontaneous behaviour, which is generated 246 from the internal state of the prey, as the prey determines the reorientation and step length 247 independent of external factors such as environmental cues. Therefore, these results point to one of the evolutionary origins of spontaneously produced Lévy walks [38]. 249 A hunting strategy by predators with high cognitive abilities would not just be a 250 reflective response to their prey's movements. It has been shown that predators, including 251 vertebrates and invertebrates, can predict the future position of a prey by using internal models 252 [31,39,40]. In our model, we assumed that predators simply predicted the prey's future position 253 based on the prey's direction, which reflects the prediction that is based on internal models. The 254 evolution of the neural circuits associated with this prediction may be driven by the movements 255 of prey. Further studies could reveal the co-evolution of prey's movements and predators' 256 cognitive abilities. 257 In our framework, the environmental conditions were limited (e.g. circle). Indeed, in 258 natural conditions, complex obstacles may be present, however the trade-off between the 259 prediction rate and the time of exposure to the predation risk would still exist. It is important that 260 future research addresses how environmental heterogeneity and the strategy's dependency on 261 environmental influences, affect prey-predator interactions [41]. Furthermore, considering 262 situations in which a predator and a prey predict one another's movements, is important in helping 263 us gain a full understanding of prey-predator interactions.