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Stay-eat or run-away: Two alternative escape behaviors
Physics Letters A 383 (2019) 593–599
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Physics Letters A www.elsevier.com/locate/pla
Stay-eat or run-away: Two alternative escape behaviors Shuai Zhang a,∗ , Mingyong Liu a , Xiaokang Lei b,c , Yunke Huang a a b c
School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an, Shaanxi, PR China School of Information and Control Engineering, Xi’an University of Architecture and Technology, Xi’an, Shaanxi, PR China KLINNS Lab, Xi’an Jiaotong University, Xi’an, Shaanxi, PR China
a r t i c l e
i n f o
Article history: Received 26 September 2018 Accepted 4 December 2018 Available online 16 January 2019 Communicated by F. Porcelli Keywords: Prey–predator Escape strategy Group chase and escape Energy management
a b s t r a c t Pursuit-and-evasion behavior in groups of animals is a phenomenon that can be easily observed in nature. Immediate flight upon detection is a common but not exclusive response for prey with both stay-eat behavior and run-away behavior occur during the predation process. It remains unclear why these two contrasting survival tactics appear, what the triggering conditions are and what internal mechanisms are at play. Here we investigate the effect of energy level on the behavior of prey during predation. We find that (1) the optimal escape speed of prey is context-dependent rather than the fastest always producing the best chance of survival. (2) The stay-eat or run-away decision depends mainly on the maximum speed and the energy dissipation ratio of prey to predator. (3) Stay-eat behavior is more effective when the prey have a higher escape speed and a higher energy dissipation rate, where aggregation can induce this stay-eat behavior. The reported findings are not only of relevance when considering survival tactics in nature, but such an understanding is useful in the design of swarm robotic systems where energy conservation and task optimization could be incorporated into any escape and hunting strategies. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Group chase and escape behaviors are widely observed in the animal kingdom. A number of good examples include coyotes, elk and wolves in Yellowstone National Park [1,2], wild chimpanzees in the Tai National Park [3], cooperative hunting in Harris’ hawks [4], subsocial spiders [5], and schools of fish [6]. A good approach to study escape tactics is an agent-based model, where each prey/predator is represented by a particle following several simple behavioral rules with the spatial-temporal variation performed by a numerical simulation. Different aspects of this prey– predator system have been studied using an agent-based model, such as mentality [7], cognition and social structure [8], prey’s field of view [9], different roles in a group [10] and a range of internal characteristics/properties [11]. This model helps researchers to understand how individuals determine their future actions based on their current local interaction or knowledge of the behavioral traits of either the predator or prey [12]. Many papers have considered how aggregation behavior can benefit animals that are prey. For example, aggregation improves group vigilance (many eyes effect) [13], makes the capture of an
*
Corresponding author. E-mail address: [email protected] (S. Zhang).
https://doi.org/10.1016/j.physleta.2018.12.046 0375-9601/© 2019 Elsevier B.V. All rights reserved.
individual prey more difficult by confusing the predators (confusion effect) and dilutes an individual’s risk of being caught (dilution effect) [14], and enables an effective defense against the predators (unity effect) [15]. Another fundamental issue that has been well documented is that of efficient chase and escape strategies. For example, spotted hyenas besiege and kill grazers whilst wolves threaten the prey group before striking [16,2]. These papers document a multitude of prey escape maneuvers, which include fleeing in a straight line, running in zigzags, jumping, burrowing, performing intermittent locomotion etc. In practice, the prey do not spend their entire lives fleeing from predators as they have to take opportunities to eat and rest. For example, a school of fish often delay their escape actions for a while, and then leave the predator behind with a rapid movement, such as the flash expansion and fountain effect [9,17]. The elks may keep feeding on grass if conspecifics nearby are caught by predators [2]. An explanation of this behavior based on economic models showed that animals should not flee directly when observing a predator, but rather flee when the costs of escape and remaining are equal [18]. Though it should be noted that it is not always clear where the equilibrium point is in a predator and prey system. Another study found that the prey should only flee when it assesses that the predator has detected it [19], but making such an assessment of detection is challenging when the prey is part of a larger escaping group.
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We argue that by considering energy management we can determine the optimal selection for a prey choosing between stay-eat and run-away behavior. There is evidence that energy management influences the survival behavior in juncos, where a junco with a lower energy reserve emerges as the pace-maker who determines when the group should forage or escape [20]. Prey can also avoid spending energy, and time, on false alarm escape attempts by only responding to the departures of others after completing a scan for potential attacks rather than immediately taking flight [21]. Most of the previous works on the group chase and escape problem have not considered energy management. Instead the agent is assumed to be a “perpetual motion agent” with the main influencing variables, affecting prey/predator behavior, as follows: position, velocity, current neighbors and spatial topology in Euclidean space [22–24]. Obviously, the energy level of an animal is an important factor when considering mobility, thus energy management will have a significant influence on the survivability of prey. This strong relationship explains the emergence of both stay-eat and run-away behavior. In this paper we investigate the survivability of prey, from the perspective of energy management, that have two possible escape strategies (stay-eat or run-away). We concentrate on the questions: why stay-eat behavior exists among prey groups and under what condition is it beneficial for the survival of prey choosing this behavior? We model each agent with an energy level that affects the maneuverability and determines the living status of an agent (dead or alive). The predator needs to stop for a period of time to handle its food (energy increase) and during this handling time, the prey may stay-eat to recover (energy increase) or run-away from the predator (energy decrease). We find that the selection of one of two behaviors depends on the maximum escape speed and the energy dissipation ratio of prey to predator. Our findings suggest that energy management could be relevance to many basic and applied disciplines, from understanding the mechanism of biological behaviors to designing diverse hunting/escape strategies in swarm robotics. 2. The model Each prey/predator has an energy level associated with it, which determines the movement speed and the living status (dead or alive). The energy of both prey and predator decline while the agents are in motion. The prey is able to recover its energy by stopping all motion referred to as handling time. For the predator, they can increase their energy levels by stopping for a limited time after capturing prey. This is intended to represent the period of time required to eat and digest the prey. Therefore, a prey/predator does not change its position during the handling time as it raises its energy level by consuming food. A predation event occurs when a prey enters the capture area (a circle of radius rdie ) of a predator and is eliminated from the model. We assume that a predator suffers from starvation when their energy level reaches zero. It should also be noted that there is no limit to the number of captured prey a predator can consume. As such, the state variables of each agent is described as si (t ) = (r i (t ), v i (t ), e i (t )), where r i (t ), v i (t ) and e i (t ) are the position, the velocity and the energy level, respectively. 2.1. Energy update Energy is the foundation of maintaining an animal’s movements. The role of internal energy reserves with regards to prey avoiding predators has been studied on multiple occasions [25,21, 26]. The mean increase in energy per unit time is typically described by [26] as
γ (u , x) = a(x)u − b(x)
(1)
where 0 ≤ u ≤ 1 represents the proportion of time that an animal spends feeding, and x is the energy reserves. If expenditure is independent of u, b(x) is the mass-dependent metabolic rate and a(x) is the rate at which energy is gained per unit time spent feeding. The model also assumes that the cost of flight means that b(x) increases with increasing energy reserves, and a(x) either is constant or decreases with x. When x corresponds to the body mass of a growing organism, a(x) is likely to be an increasing function of x. In our model, the energy update function is modified from the eq. (1) as
e (u , v ) = a(·)u − b( v )(1 − u )
(2)
where u = [0, 1] is a switching function. 0 represents escape/chase behavior and 1 represents feed behavior. a(·) is the rate at which energy is gained per unit time spent feeding and b( v ) is the speeddependent metabolic rate. For prey, a(·) = m, where m > 0 is constant. For predators, a(·) = e tar , where e tar is the energy level of captured prey.1 It is assumed that the cost of movement, in the form of metabolic rate b(x), increases with speed. This increase may be linear or non-linear, thus b( v ) = kv ϕ is described as energy loss function for both prey and predators, where k > 0 is a coefficient and ϕ > 0 determines whether the rate changes linearly or non-linearly. The energy update functions for prey and predators are presented as follows
e tar (u , v ) = um − (1 − u )kv ϕ e cha (u , v ) = u etar − (1 − u )kv ϕ
(3) (4)
where it is worth noting that there are three situations the prey may encounter. 1) The free situation: there is no predator within sight of the prey. The prey may have rest and take food, thus u = 1. 2) The warning situation: there are moving predators within sight of the prey. The prey has to escape from the chasers rather than stay or eat, thus u = 0. 3) The relaxed situation: all detected predators are feeding on prey. This impermanent safe period allows the rest of prey to have two different behaviors: stay-eat behavior, u = 1, or run-away behavior until all of the predators are beyond the field of view of the prey, u = 0. For the duration of each simulation, each prey is labeled as one of two phenotypes, stay-eat or run-away. 2.2. Kinematic equation We expanded Angelani’s model [27] by considering the variable speed and the energy label. For a prey, its position and velocity are updated according to
ri (t + t ) = ri (t ) + vi (t + t )t di (t ) vi (t + t ) = v i (t )η [ ] di (t )
(5) (6)
where v i (t ) is the velocity modulus of prey i. η [·] is the effect of random noise, which is simulated by rotating the desired direction for a random angle with a Gaussian distribution in the interval [−ηπ , ηπ ], where η ∈ [0, 1] is the noise strength. di (t ) is the desired direction of prey i in the next step.
1 If etar is a constant, the predator obtains a constant energy over time regardless of any difference in the prey captured. Therefore etar is considered as a variable that is dependent on the size of the captured prey, which ensures that the transfer of energy is in accordance with the law of energy conservation and the inheritance of energy in the pursuit-evasion problem.
S. Zhang et al. / Physics Letters A 383 (2019) 593–599
Fig. 1. Sigmoid speed-energy function. For a = 5, the speed increases almost linearly with energy (c). For a bigger value of a, such as a = 10, a = 20, there is a nonlinear increase. The speed increases significantly during the medium energy level. The nonlinear increase (a = 10, c = 0.5) is used in the following simulations. The control parameters do not affect the final conclusions but change the threshold conditions.
Unlike the models of Angelani [27] and Yang [28], the speed of a prey/predator is energy-dependent and not constant. The speed increases with energy level but is limited to a maximum speed. A sigmoid speed-energy function is introduced to describe this monotonically increasing relation between the speed and the energy level as
v (t ) =
V max
(7)
1 + exp(−a(e (t ) − c ))
where V max is the maximum speed, and a, c > 0 are control parameters. c denotes the energy point where the speed increases fastest. a denotes the non-linear degree, where a smaller value means the speed increases with energy more linearly, while a higher value results in a sudden transition from low to the maximum value as the energy level increases, see Fig. 1. In eq. (6), the desired direction di (t ) describes interactions, including both the synergy among prey–prey and the pursuit-evasion relationship among predator–prey. For the prey there are three pos components, velocity matching fvel and i , position adjustment fi esp escape interaction fi .
fvel i
di (t ) = α fvel i =
fvel i
pos
+β
esp
fi
pos fi
+γ
fi
esp
fi
vj
(8)
pos
=
pos
j∈ S i esp
fi
=
(
1
ri − r j
−
r0
ri − r j 3
ri − r j − w rˆ i j
)ˆri j
(10)
(11)
α , β, γ denote the relative strength of three parts, and
r −r
interaction. Based on Angelani’s work [27], a weighted average of neighboring opponents with a power law is used to describe this prey–predator interaction, the exponent w describes the influence of distance between the prey and the predator on the escape direction of the prey as seen in eq. (11). We adopt w = 2, which was identified by [27] to be optimal for prey survivability. For a predator, the equations for updating position and velocity are similar to the prey. However, the desired direction di (t ) that captures the intra- and inter-group interacting force is limited to position adjustment and chase interaction. The position adjustment with short-range repulsion is needed because of individual space requirements and collision avoidance [23]. The cooperative hunting behavior of predators is not considered in our model. We assume that the predators try to chase the nearest prey (energy saving) and follow a group of prey (intrinsically greedy), therefore a power-law weighted average of near prey is also used for the predators (we adopt w = 2 again from [27]). The desired direction of a predator can be written as pos
di (t ) = β
fi
−γ
fchs i
pos fchs fi i = ri − r j − w rˆ i j fchs i
(12)
(13)
(9)
esp j∈ S i
where
Fig. 2. Snapshot of a prey–predator configuration from simulation run M4. The green circles are prey and the red triangles are predators. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
j ∈ S chs i
j ∈ S vel i
fi
595
rˆ i j = ri −r j is a unit vector from agent j to agent i. S vel and i i j pos Si define a set of prey performing velocity matching and position adjustment with the same aggregation range rp . S esp is a set of detected predators with a limited field of view rc for each prey. fvel is responsible for synchronizing the velocities as seen in i pos [29]. fi is responsible for aggregation and collision avoidance by adopting a long range attraction and short range repulsion function [30], where r0 is a predefined constant which regulates the exesp pected distance among the prey. fi is responsible for intergroup
where S chs is the set of detected prey within the same limited field of view rc for each predator. Note that during the predation event, a prey will be caught and eliminated as it has moved into the capture area (a circle of radius rdie ) of a predator. The predator then stops for a period of time τ to eat and digest their prey [31]. The proposed model is suitable for the study of the pursuitand-evasion problem, with a specific focus on two energydependent survival strategies: stay-eat or run-away. A snapshot from a simulation run, M4, is shown in Fig. 2 with aggregated prey (green circles) attempting to avoid the predators (red triangles). Video footage of the simulations can be found in the electronic supplementary materials. 3. Results At the beginning of every simulation, the predators and prey are placed in random locations inside a 40 × 40 unit plane with a periodic boundary. The initial, maximum and minimum energy
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S. Zhang et al. / Physics Letters A 383 (2019) 593–599
Table 1 Model parameters.
Table 2 The average results of optimal speed ratio and optimal survival possibility against
ϕ.
Parameter
Value
Description
ϕ
0.5
1
1.5
2
2.5
3
3.5
4
α
1 1 1 5 0.01 1 1 10 1 -
velocity matching strength position adjustment strength predation strength handling time strength interval of one time step distance of collision avoidance radius of dead area range of perception prey’s energy growth rate range of aggregation predators’ energy dissipation rate prey’s energy dissipation rate predators’ maximum speed prey’s maximum speed exponent of energy loss
Optimal survival probability
0.98
0.98
0.97
0.83
0.84
0.83
0.82
0.84
Optimal speed ratio
∞
∞
∞
1
0.71
0.69
0.68
0.67
β
γ τ t r0 rdie rc m rp kc kt V max,c V max,t
ϕ
levels are 0.8, 1 and 0.1 respectively for both prey and predators.2 The number of initial prey and predators are set as 100 and 5 respectively. Other parameters are presented in Table 1. Each simulation is terminated after 1500 time steps, which is long enough that it reduces the influence of the random initial agent positions and presents a divergence in survival prospects based on the tactic selected. For simplicity, the maneuverability of an agent is assumed to be influenced by two aspects, the maximum speed V max and the energy dissipation rate k. In order to quantify the survivability of prey in the two escape tactics, we define the survival rate P alv , which is counted at the end of each simulation, as
P alv =
N alv N ini
(14)
where N ini is the total number of prey, and N alv is the number of prey still alive at the end of each simulation. The effectiveness [24], or the typical life time [32] contains information on the lifetime of individual prey. But when evaluating two escape tactics at a group level, a statistical result for the survival rate of prey is a more appropriate metric as it captures the survivability of the whole group. 3.1. Effect of maximum speed ratio We first analyze the performances of the two escape behaviors influenced by the maximum speed of the agents (movies M1 and M2 in the electronic supplementary material). Define the maximum speed ration V max,t / V max,c , where V max,t is the maximum speed of prey and V max,c = 6 is a reference maximum speed of predators. For ϕ = 1 (ϕ is the exponent of energy loss function b( v ) = kv ϕ , see eq. (3) and (4)), the survival rate P alv increases with the maximum speed ratio, and there is no significant difference between the two tactics (see Fig. 3(a)). The results show that the maximum speed ratio is the dominant factor for the prey’s survivability and a prey running away at a higher speed is less likely to be caught. This suggests that evolution should favor infinite escape speed, which is obviously unrealistic. For ϕ = 2, a remarkable result occurs where there is an optimal value in the survival curve at V max,t / V max,c = 1 (see Fig. 3(b)). In this circumstance, the highest ratio of escape speed will not produce the best survival scenario for the prey. Further study of the optimal speed ratio and the optimal survival probability against ϕ is detailed in Table 2, where there is no optimal escape speed ratio when ϕ < 2. For ϕ = 2 the optimal speed ratio V max,t / V max,c = 1 means that
2 The energy level of a prey/predator never reduces to zero and, hence, they always have enough energy to move.
the prey and predator have the same maximum speed. The optimal survival probability is not significantly affected when ϕ ≥ 2. Note that ϕ cannot be infinity, a prey/predator can barely move within a very short time as the energy decreases significantly by b( v ) = kv ϕ , therefore the results of ϕ ≤ 4 are given. When the optimal speed ratio is less than one this can be explained as follows. A relatively slow speed saves energy, but the slower movement will place the prey in a dangerous situation that could be avoided by increasing the escape speed. However, a very fast speed consumes a lot of energy and results in the prey moving very slowly in the following predation round, having just escaped predation, thus increasing its chances of being caught when compared to those prey with a lower but sufficient maximum speed. This conclusion is also supported by Wilson et al. [33]. Their results showed that prey cannot accelerate forwards with or without turning, making its behavior highly predictable. A slower moving prey has a wider variety of escape options and is therefore less predictable. Another important result is that the stay-eat behavior produces a great advantage if the escape speed is higher than the optimal speed (Fig. 3(b)), which means the stay-eat tactic improves the survival possibility when the prey has to move at a relatively high speed to successfully escape. If the prey does not stop to eat then they will eventually get caught as they lose more energy than the predator because they move more quickly. The prey which employs a safe distance and handling time to recover (stay-eat) will have the better performance in the following predation, than a prey continuously running away (run-away) that will have a poorer performance after each predation event. 3.2. Effect of energy dissipation ratio We now investigate the role that different energy dissipation ratios have in the performances of the two tactics. As shown in Fig. 4, the survival rate P alv monotonically decreases with the energy dissipation ratio kt /kc . It indicates that the greater the energy dissipation ratio, the lower the survival possibility is for the prey. For stay-eat behavior and run-away behavior, there is no survival difference for the prey when kt /kc < 1. This means that as long as the energy dissipation rate of prey is low relative to its predator, the prey has a nearly optimal survival possibility ( P alv ≈ 1) no matter which behavior it performs. This is easy to understand given that a prey with the same maximum speed as its predator but a relatively lower energy dissipation rate could consistently escape predation. However, when kt /kc > 1, the survival curve of stay-eat is higher than that of run-away. This means when the energy dissipation rate of prey is high relative to its predator, stay-eat behavior is a better choice. The advantage of stay-eat behavior in this circumstance can be explained as follows. A prey choosing to stay and recover during the handling time is able to increase its energy, thus increasing its chances for survival in the following predation rounds. However, a prey that continues to move away during the handling time wastes energy, which results in slower movement and increased possibility of capture in the following predation rounds.
S. Zhang et al. / Physics Letters A 383 (2019) 593–599
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Fig. 3. Comparison of the survival rate of prey for the two escape tactics changing with maximum speed ratio V max,t / V max,c , where V max,t is the maximum speed of prey and V max,c = 6 is a reference maximum speed of predators. m = 1, kc = kt = −0.0001. The results are obtained by 1000 independent simulations.
3.4. Effect of aggregation preference
Fig. 4. Survival rate of prey under the two escape tactics changing with energy dissipation ratio kt /kc , where kt is the energy dissipation rate of prey and kc = −0.0001 is a reference energy dissipation rate of predators. m = 1, ϕ = 2, V max,t = V max,c = 6. The results are obtained by 1000 independent simulations.
3.3. Comprehensive effect of maximum speed ratio and energy dissipation ratio
Aggregation is very common in animal groups and many researchers argue that this phenomenon benefits the prey [13–15]. We study the influence of prey’s preference for aggregation on the two behaviors considered previously (see movies M3 and M4 in the electronic supplementary material). As shown in Fig. 6, the survival rate of stay-eat behavior is higher than that of run-away behavior when considering the aggregation preference of prey. It indicates that aggregation does not improve the survivability when employing either of the escape tactics. This is due to the additional movement that prey carry out during aggregation that leads to an increase in energy dissipation. Based on the previous result, that a prey with high energy dissipation is more likely to choose stay-eat behavior, we argue that aggregation stimulates the stay-eat behavior. This might be the reason why those animals living in a group are more likely to show stay-eat behavior. Surprisingly, our model shows that the aggregation preference decreases the survival probability of prey when compared with no aggregation groups. This is due to the fact that the predators capture the prey one by one if the prey move as a group, which reduces the predators’ hunting time and energy loss. The results also suggest that the prey need to take care of the “balance of aggregation” when the predators have a preference of hunting a group of prey. 4. Discussion and conclusion
We now examine an extensive range of values for different combinations of maximum speed ratio V max,t / V max,c and energy dissipation ratio kt /kc . The mean values are shown in Fig. 5. The results show that each color map includes three phases, phase 1, 2 and 3. The phase 1 is similar for both tactics, which indicates that too low an escape speed is deadly to the prey no matter which escape tactic is used. The minimum threshold of escape speed is V max,t / V max,c = 0.6. The phase 2 of both tactics are also similar, which presents that the prey survive well when it has a lower escape speed with a higher energy dissipation rate or a higher escape speed with a lower energy dissipation rate. This result proves the argument that the optimal escape speed depends on the situation rather than the highest always being best when taking into account energy loss when a prey is moving. The phase 3 of the two tactics is different. The phase 3 in stay-eat behavior is brighter than the phase 3 in run-away behavior, which indicates that the stay-eat behavior performs better than the run-away behavior when the maximum speed and the energy dissipation rate of prey is high relative to its predators.
The predation phenomenon has been extensively studied from the perspective of space-interaction (e.g., distance [19], direction [28], and velocity angle [34]). The spatial dynamics of predation is characterized by spatial interactions but this alone is not enough to explain all predation behaviors. The internal properties of animals, such as energy level, play a key role in avoiding predation [35] as they are intrinsically linked with an animal’s maneuverability. Our results shed some light on the seemingly irrational pursuit-andevasion behavior of staying in a predator’s vicinity to eat. Energy management was identified as a key aspect with prey deciding on a trade-off between current survival prospects and future energy requirements. The behavior of energy management is also observed in predators. For example, the wolf usually does not chase the nearest elk in a group. It prefers to run after a fleeing group, or lunge at a standing group, and make an assessment of the group members from this interaction. When an optimal potential target appears, usually a weak or young elk, it begins to run after this solitary
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Fig. 5. Survival rate of prey under the two escape tactics changing with both maximum speed ratio V max,t / V max,c and energy dissipation ratio kt /kc . m = 1, denotes the survival rate of prey as shown in color bar. The results are obtained by 1000 independent simulations.
ϕ = 2. The color
larly for those long-period or multi-round confrontations, such as cooperative hunting and group competition. Acknowledgements This work was partially supported by the National Natural Science Foundation of China (NSFC) under Grants 51879219, 51679201. Appendix A. Supplementary material Supplementary material related to this article can be found online at https://doi.org/10.1016/j.physleta.2018.12.046. References Fig. 6. Survival rate of prey by considering the aggregation preference. V max,c = 6, kc = −0.0001, V max,t = 9, kt = −0.0002, m = 1, ϕ = 2. Aggregation behavior is achieved by setting rp = 5, α = 1, β = 1, while rp = 1, α = 0, β = 1 could make the aggregation preference disappear and allow the collision avoidance only. The results are obtained by 1000 independent simulations.
individual while ignoring all the other group members [2]. This strategy could be explained by considering energy management for the predator, because a reserved chase process in the beginning could keep the prey moving and consuming large amounts of energy, therefore exposing the weakest prey that will be easily caught when the predator commits to the final chase. In addition, wolves could manage their own energy loss by taking turns at chasing the prey group before a weak member is exposed. In this paper, the agent-based model is used to analyze stay-eat and run-away behavior by considering the handling time, the internal energy level and the energy-speed relationship in a predator– prey scenario. The model reveals, to some extent, why and when a group of animals perform stay-eat behavior or run-away behavior. We find that (1) the optimal escape speed of prey is contextdependent rather than highest always being best. (2) The optimal decision between stay-eat or run-away depends primarily on the maximum speed and the energy dissipation ratio of prey to predator. (3) Stay-eat behavior is more effective when a prey has a higher escape speed and a higher energy dissipation rate, and their aggregation can induce stay-eat behavior. We suggest that these preliminary findings may be extended to diverse studies of the game dynamics of animal groups, particu-
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Stay-eat or run-away: Two alternative escape behaviors Shuai Zhang a,∗ , Mingyong Liu a , Xiaokang Lei b,c , Yunke Huang a a b c
School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an, Shaanxi, PR China School of Information and Control Engineering, Xi’an University of Architecture and Technology, Xi’an, Shaanxi, PR China KLINNS Lab, Xi’an Jiaotong University, Xi’an, Shaanxi, PR China
a r t i c l e
i n f o
Article history: Received 26 September 2018 Accepted 4 December 2018 Available online 16 January 2019 Communicated by F. Porcelli Keywords: Prey–predator Escape strategy Group chase and escape Energy management
a b s t r a c t Pursuit-and-evasion behavior in groups of animals is a phenomenon that can be easily observed in nature. Immediate flight upon detection is a common but not exclusive response for prey with both stay-eat behavior and run-away behavior occur during the predation process. It remains unclear why these two contrasting survival tactics appear, what the triggering conditions are and what internal mechanisms are at play. Here we investigate the effect of energy level on the behavior of prey during predation. We find that (1) the optimal escape speed of prey is context-dependent rather than the fastest always producing the best chance of survival. (2) The stay-eat or run-away decision depends mainly on the maximum speed and the energy dissipation ratio of prey to predator. (3) Stay-eat behavior is more effective when the prey have a higher escape speed and a higher energy dissipation rate, where aggregation can induce this stay-eat behavior. The reported findings are not only of relevance when considering survival tactics in nature, but such an understanding is useful in the design of swarm robotic systems where energy conservation and task optimization could be incorporated into any escape and hunting strategies. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Group chase and escape behaviors are widely observed in the animal kingdom. A number of good examples include coyotes, elk and wolves in Yellowstone National Park [1,2], wild chimpanzees in the Tai National Park [3], cooperative hunting in Harris’ hawks [4], subsocial spiders [5], and schools of fish [6]. A good approach to study escape tactics is an agent-based model, where each prey/predator is represented by a particle following several simple behavioral rules with the spatial-temporal variation performed by a numerical simulation. Different aspects of this prey– predator system have been studied using an agent-based model, such as mentality [7], cognition and social structure [8], prey’s field of view [9], different roles in a group [10] and a range of internal characteristics/properties [11]. This model helps researchers to understand how individuals determine their future actions based on their current local interaction or knowledge of the behavioral traits of either the predator or prey [12]. Many papers have considered how aggregation behavior can benefit animals that are prey. For example, aggregation improves group vigilance (many eyes effect) [13], makes the capture of an
*
Corresponding author. E-mail address: [email protected] (S. Zhang).
https://doi.org/10.1016/j.physleta.2018.12.046 0375-9601/© 2019 Elsevier B.V. All rights reserved.
individual prey more difficult by confusing the predators (confusion effect) and dilutes an individual’s risk of being caught (dilution effect) [14], and enables an effective defense against the predators (unity effect) [15]. Another fundamental issue that has been well documented is that of efficient chase and escape strategies. For example, spotted hyenas besiege and kill grazers whilst wolves threaten the prey group before striking [16,2]. These papers document a multitude of prey escape maneuvers, which include fleeing in a straight line, running in zigzags, jumping, burrowing, performing intermittent locomotion etc. In practice, the prey do not spend their entire lives fleeing from predators as they have to take opportunities to eat and rest. For example, a school of fish often delay their escape actions for a while, and then leave the predator behind with a rapid movement, such as the flash expansion and fountain effect [9,17]. The elks may keep feeding on grass if conspecifics nearby are caught by predators [2]. An explanation of this behavior based on economic models showed that animals should not flee directly when observing a predator, but rather flee when the costs of escape and remaining are equal [18]. Though it should be noted that it is not always clear where the equilibrium point is in a predator and prey system. Another study found that the prey should only flee when it assesses that the predator has detected it [19], but making such an assessment of detection is challenging when the prey is part of a larger escaping group.
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We argue that by considering energy management we can determine the optimal selection for a prey choosing between stay-eat and run-away behavior. There is evidence that energy management influences the survival behavior in juncos, where a junco with a lower energy reserve emerges as the pace-maker who determines when the group should forage or escape [20]. Prey can also avoid spending energy, and time, on false alarm escape attempts by only responding to the departures of others after completing a scan for potential attacks rather than immediately taking flight [21]. Most of the previous works on the group chase and escape problem have not considered energy management. Instead the agent is assumed to be a “perpetual motion agent” with the main influencing variables, affecting prey/predator behavior, as follows: position, velocity, current neighbors and spatial topology in Euclidean space [22–24]. Obviously, the energy level of an animal is an important factor when considering mobility, thus energy management will have a significant influence on the survivability of prey. This strong relationship explains the emergence of both stay-eat and run-away behavior. In this paper we investigate the survivability of prey, from the perspective of energy management, that have two possible escape strategies (stay-eat or run-away). We concentrate on the questions: why stay-eat behavior exists among prey groups and under what condition is it beneficial for the survival of prey choosing this behavior? We model each agent with an energy level that affects the maneuverability and determines the living status of an agent (dead or alive). The predator needs to stop for a period of time to handle its food (energy increase) and during this handling time, the prey may stay-eat to recover (energy increase) or run-away from the predator (energy decrease). We find that the selection of one of two behaviors depends on the maximum escape speed and the energy dissipation ratio of prey to predator. Our findings suggest that energy management could be relevance to many basic and applied disciplines, from understanding the mechanism of biological behaviors to designing diverse hunting/escape strategies in swarm robotics. 2. The model Each prey/predator has an energy level associated with it, which determines the movement speed and the living status (dead or alive). The energy of both prey and predator decline while the agents are in motion. The prey is able to recover its energy by stopping all motion referred to as handling time. For the predator, they can increase their energy levels by stopping for a limited time after capturing prey. This is intended to represent the period of time required to eat and digest the prey. Therefore, a prey/predator does not change its position during the handling time as it raises its energy level by consuming food. A predation event occurs when a prey enters the capture area (a circle of radius rdie ) of a predator and is eliminated from the model. We assume that a predator suffers from starvation when their energy level reaches zero. It should also be noted that there is no limit to the number of captured prey a predator can consume. As such, the state variables of each agent is described as si (t ) = (r i (t ), v i (t ), e i (t )), where r i (t ), v i (t ) and e i (t ) are the position, the velocity and the energy level, respectively. 2.1. Energy update Energy is the foundation of maintaining an animal’s movements. The role of internal energy reserves with regards to prey avoiding predators has been studied on multiple occasions [25,21, 26]. The mean increase in energy per unit time is typically described by [26] as
γ (u , x) = a(x)u − b(x)
(1)
where 0 ≤ u ≤ 1 represents the proportion of time that an animal spends feeding, and x is the energy reserves. If expenditure is independent of u, b(x) is the mass-dependent metabolic rate and a(x) is the rate at which energy is gained per unit time spent feeding. The model also assumes that the cost of flight means that b(x) increases with increasing energy reserves, and a(x) either is constant or decreases with x. When x corresponds to the body mass of a growing organism, a(x) is likely to be an increasing function of x. In our model, the energy update function is modified from the eq. (1) as
e (u , v ) = a(·)u − b( v )(1 − u )
(2)
where u = [0, 1] is a switching function. 0 represents escape/chase behavior and 1 represents feed behavior. a(·) is the rate at which energy is gained per unit time spent feeding and b( v ) is the speeddependent metabolic rate. For prey, a(·) = m, where m > 0 is constant. For predators, a(·) = e tar , where e tar is the energy level of captured prey.1 It is assumed that the cost of movement, in the form of metabolic rate b(x), increases with speed. This increase may be linear or non-linear, thus b( v ) = kv ϕ is described as energy loss function for both prey and predators, where k > 0 is a coefficient and ϕ > 0 determines whether the rate changes linearly or non-linearly. The energy update functions for prey and predators are presented as follows
e tar (u , v ) = um − (1 − u )kv ϕ e cha (u , v ) = u etar − (1 − u )kv ϕ
(3) (4)
where it is worth noting that there are three situations the prey may encounter. 1) The free situation: there is no predator within sight of the prey. The prey may have rest and take food, thus u = 1. 2) The warning situation: there are moving predators within sight of the prey. The prey has to escape from the chasers rather than stay or eat, thus u = 0. 3) The relaxed situation: all detected predators are feeding on prey. This impermanent safe period allows the rest of prey to have two different behaviors: stay-eat behavior, u = 1, or run-away behavior until all of the predators are beyond the field of view of the prey, u = 0. For the duration of each simulation, each prey is labeled as one of two phenotypes, stay-eat or run-away. 2.2. Kinematic equation We expanded Angelani’s model [27] by considering the variable speed and the energy label. For a prey, its position and velocity are updated according to
ri (t + t ) = ri (t ) + vi (t + t )t di (t ) vi (t + t ) = v i (t )η [ ] di (t )
(5) (6)
where v i (t ) is the velocity modulus of prey i. η [·] is the effect of random noise, which is simulated by rotating the desired direction for a random angle with a Gaussian distribution in the interval [−ηπ , ηπ ], where η ∈ [0, 1] is the noise strength. di (t ) is the desired direction of prey i in the next step.
1 If etar is a constant, the predator obtains a constant energy over time regardless of any difference in the prey captured. Therefore etar is considered as a variable that is dependent on the size of the captured prey, which ensures that the transfer of energy is in accordance with the law of energy conservation and the inheritance of energy in the pursuit-evasion problem.
S. Zhang et al. / Physics Letters A 383 (2019) 593–599
Fig. 1. Sigmoid speed-energy function. For a = 5, the speed increases almost linearly with energy (c). For a bigger value of a, such as a = 10, a = 20, there is a nonlinear increase. The speed increases significantly during the medium energy level. The nonlinear increase (a = 10, c = 0.5) is used in the following simulations. The control parameters do not affect the final conclusions but change the threshold conditions.
Unlike the models of Angelani [27] and Yang [28], the speed of a prey/predator is energy-dependent and not constant. The speed increases with energy level but is limited to a maximum speed. A sigmoid speed-energy function is introduced to describe this monotonically increasing relation between the speed and the energy level as
v (t ) =
V max
(7)
1 + exp(−a(e (t ) − c ))
where V max is the maximum speed, and a, c > 0 are control parameters. c denotes the energy point where the speed increases fastest. a denotes the non-linear degree, where a smaller value means the speed increases with energy more linearly, while a higher value results in a sudden transition from low to the maximum value as the energy level increases, see Fig. 1. In eq. (6), the desired direction di (t ) describes interactions, including both the synergy among prey–prey and the pursuit-evasion relationship among predator–prey. For the prey there are three pos components, velocity matching fvel and i , position adjustment fi esp escape interaction fi .
fvel i
di (t ) = α fvel i =
fvel i
pos
+β
esp
fi
pos fi
+γ
fi
esp
fi
vj
(8)
pos
=
pos
j∈ S i esp
fi
=
(
1
ri − r j
−
r0
ri − r j 3
ri − r j − w rˆ i j
)ˆri j
(10)
(11)
α , β, γ denote the relative strength of three parts, and
r −r
interaction. Based on Angelani’s work [27], a weighted average of neighboring opponents with a power law is used to describe this prey–predator interaction, the exponent w describes the influence of distance between the prey and the predator on the escape direction of the prey as seen in eq. (11). We adopt w = 2, which was identified by [27] to be optimal for prey survivability. For a predator, the equations for updating position and velocity are similar to the prey. However, the desired direction di (t ) that captures the intra- and inter-group interacting force is limited to position adjustment and chase interaction. The position adjustment with short-range repulsion is needed because of individual space requirements and collision avoidance [23]. The cooperative hunting behavior of predators is not considered in our model. We assume that the predators try to chase the nearest prey (energy saving) and follow a group of prey (intrinsically greedy), therefore a power-law weighted average of near prey is also used for the predators (we adopt w = 2 again from [27]). The desired direction of a predator can be written as pos
di (t ) = β
fi
−γ
fchs i
pos fchs fi i = ri − r j − w rˆ i j fchs i
(12)
(13)
(9)
esp j∈ S i
where
Fig. 2. Snapshot of a prey–predator configuration from simulation run M4. The green circles are prey and the red triangles are predators. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
j ∈ S chs i
j ∈ S vel i
fi
595
rˆ i j = ri −r j is a unit vector from agent j to agent i. S vel and i i j pos Si define a set of prey performing velocity matching and position adjustment with the same aggregation range rp . S esp is a set of detected predators with a limited field of view rc for each prey. fvel is responsible for synchronizing the velocities as seen in i pos [29]. fi is responsible for aggregation and collision avoidance by adopting a long range attraction and short range repulsion function [30], where r0 is a predefined constant which regulates the exesp pected distance among the prey. fi is responsible for intergroup
where S chs is the set of detected prey within the same limited field of view rc for each predator. Note that during the predation event, a prey will be caught and eliminated as it has moved into the capture area (a circle of radius rdie ) of a predator. The predator then stops for a period of time τ to eat and digest their prey [31]. The proposed model is suitable for the study of the pursuitand-evasion problem, with a specific focus on two energydependent survival strategies: stay-eat or run-away. A snapshot from a simulation run, M4, is shown in Fig. 2 with aggregated prey (green circles) attempting to avoid the predators (red triangles). Video footage of the simulations can be found in the electronic supplementary materials. 3. Results At the beginning of every simulation, the predators and prey are placed in random locations inside a 40 × 40 unit plane with a periodic boundary. The initial, maximum and minimum energy
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Table 1 Model parameters.
Table 2 The average results of optimal speed ratio and optimal survival possibility against
ϕ.
Parameter
Value
Description
ϕ
0.5
1
1.5
2
2.5
3
3.5
4
α
1 1 1 5 0.01 1 1 10 1 -
velocity matching strength position adjustment strength predation strength handling time strength interval of one time step distance of collision avoidance radius of dead area range of perception prey’s energy growth rate range of aggregation predators’ energy dissipation rate prey’s energy dissipation rate predators’ maximum speed prey’s maximum speed exponent of energy loss
Optimal survival probability
0.98
0.98
0.97
0.83
0.84
0.83
0.82
0.84
Optimal speed ratio
∞
∞
∞
1
0.71
0.69
0.68
0.67
β
γ τ t r0 rdie rc m rp kc kt V max,c V max,t
ϕ
levels are 0.8, 1 and 0.1 respectively for both prey and predators.2 The number of initial prey and predators are set as 100 and 5 respectively. Other parameters are presented in Table 1. Each simulation is terminated after 1500 time steps, which is long enough that it reduces the influence of the random initial agent positions and presents a divergence in survival prospects based on the tactic selected. For simplicity, the maneuverability of an agent is assumed to be influenced by two aspects, the maximum speed V max and the energy dissipation rate k. In order to quantify the survivability of prey in the two escape tactics, we define the survival rate P alv , which is counted at the end of each simulation, as
P alv =
N alv N ini
(14)
where N ini is the total number of prey, and N alv is the number of prey still alive at the end of each simulation. The effectiveness [24], or the typical life time [32] contains information on the lifetime of individual prey. But when evaluating two escape tactics at a group level, a statistical result for the survival rate of prey is a more appropriate metric as it captures the survivability of the whole group. 3.1. Effect of maximum speed ratio We first analyze the performances of the two escape behaviors influenced by the maximum speed of the agents (movies M1 and M2 in the electronic supplementary material). Define the maximum speed ration V max,t / V max,c , where V max,t is the maximum speed of prey and V max,c = 6 is a reference maximum speed of predators. For ϕ = 1 (ϕ is the exponent of energy loss function b( v ) = kv ϕ , see eq. (3) and (4)), the survival rate P alv increases with the maximum speed ratio, and there is no significant difference between the two tactics (see Fig. 3(a)). The results show that the maximum speed ratio is the dominant factor for the prey’s survivability and a prey running away at a higher speed is less likely to be caught. This suggests that evolution should favor infinite escape speed, which is obviously unrealistic. For ϕ = 2, a remarkable result occurs where there is an optimal value in the survival curve at V max,t / V max,c = 1 (see Fig. 3(b)). In this circumstance, the highest ratio of escape speed will not produce the best survival scenario for the prey. Further study of the optimal speed ratio and the optimal survival probability against ϕ is detailed in Table 2, where there is no optimal escape speed ratio when ϕ < 2. For ϕ = 2 the optimal speed ratio V max,t / V max,c = 1 means that
2 The energy level of a prey/predator never reduces to zero and, hence, they always have enough energy to move.
the prey and predator have the same maximum speed. The optimal survival probability is not significantly affected when ϕ ≥ 2. Note that ϕ cannot be infinity, a prey/predator can barely move within a very short time as the energy decreases significantly by b( v ) = kv ϕ , therefore the results of ϕ ≤ 4 are given. When the optimal speed ratio is less than one this can be explained as follows. A relatively slow speed saves energy, but the slower movement will place the prey in a dangerous situation that could be avoided by increasing the escape speed. However, a very fast speed consumes a lot of energy and results in the prey moving very slowly in the following predation round, having just escaped predation, thus increasing its chances of being caught when compared to those prey with a lower but sufficient maximum speed. This conclusion is also supported by Wilson et al. [33]. Their results showed that prey cannot accelerate forwards with or without turning, making its behavior highly predictable. A slower moving prey has a wider variety of escape options and is therefore less predictable. Another important result is that the stay-eat behavior produces a great advantage if the escape speed is higher than the optimal speed (Fig. 3(b)), which means the stay-eat tactic improves the survival possibility when the prey has to move at a relatively high speed to successfully escape. If the prey does not stop to eat then they will eventually get caught as they lose more energy than the predator because they move more quickly. The prey which employs a safe distance and handling time to recover (stay-eat) will have the better performance in the following predation, than a prey continuously running away (run-away) that will have a poorer performance after each predation event. 3.2. Effect of energy dissipation ratio We now investigate the role that different energy dissipation ratios have in the performances of the two tactics. As shown in Fig. 4, the survival rate P alv monotonically decreases with the energy dissipation ratio kt /kc . It indicates that the greater the energy dissipation ratio, the lower the survival possibility is for the prey. For stay-eat behavior and run-away behavior, there is no survival difference for the prey when kt /kc < 1. This means that as long as the energy dissipation rate of prey is low relative to its predator, the prey has a nearly optimal survival possibility ( P alv ≈ 1) no matter which behavior it performs. This is easy to understand given that a prey with the same maximum speed as its predator but a relatively lower energy dissipation rate could consistently escape predation. However, when kt /kc > 1, the survival curve of stay-eat is higher than that of run-away. This means when the energy dissipation rate of prey is high relative to its predator, stay-eat behavior is a better choice. The advantage of stay-eat behavior in this circumstance can be explained as follows. A prey choosing to stay and recover during the handling time is able to increase its energy, thus increasing its chances for survival in the following predation rounds. However, a prey that continues to move away during the handling time wastes energy, which results in slower movement and increased possibility of capture in the following predation rounds.
S. Zhang et al. / Physics Letters A 383 (2019) 593–599
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Fig. 3. Comparison of the survival rate of prey for the two escape tactics changing with maximum speed ratio V max,t / V max,c , where V max,t is the maximum speed of prey and V max,c = 6 is a reference maximum speed of predators. m = 1, kc = kt = −0.0001. The results are obtained by 1000 independent simulations.
3.4. Effect of aggregation preference
Fig. 4. Survival rate of prey under the two escape tactics changing with energy dissipation ratio kt /kc , where kt is the energy dissipation rate of prey and kc = −0.0001 is a reference energy dissipation rate of predators. m = 1, ϕ = 2, V max,t = V max,c = 6. The results are obtained by 1000 independent simulations.
3.3. Comprehensive effect of maximum speed ratio and energy dissipation ratio
Aggregation is very common in animal groups and many researchers argue that this phenomenon benefits the prey [13–15]. We study the influence of prey’s preference for aggregation on the two behaviors considered previously (see movies M3 and M4 in the electronic supplementary material). As shown in Fig. 6, the survival rate of stay-eat behavior is higher than that of run-away behavior when considering the aggregation preference of prey. It indicates that aggregation does not improve the survivability when employing either of the escape tactics. This is due to the additional movement that prey carry out during aggregation that leads to an increase in energy dissipation. Based on the previous result, that a prey with high energy dissipation is more likely to choose stay-eat behavior, we argue that aggregation stimulates the stay-eat behavior. This might be the reason why those animals living in a group are more likely to show stay-eat behavior. Surprisingly, our model shows that the aggregation preference decreases the survival probability of prey when compared with no aggregation groups. This is due to the fact that the predators capture the prey one by one if the prey move as a group, which reduces the predators’ hunting time and energy loss. The results also suggest that the prey need to take care of the “balance of aggregation” when the predators have a preference of hunting a group of prey. 4. Discussion and conclusion
We now examine an extensive range of values for different combinations of maximum speed ratio V max,t / V max,c and energy dissipation ratio kt /kc . The mean values are shown in Fig. 5. The results show that each color map includes three phases, phase 1, 2 and 3. The phase 1 is similar for both tactics, which indicates that too low an escape speed is deadly to the prey no matter which escape tactic is used. The minimum threshold of escape speed is V max,t / V max,c = 0.6. The phase 2 of both tactics are also similar, which presents that the prey survive well when it has a lower escape speed with a higher energy dissipation rate or a higher escape speed with a lower energy dissipation rate. This result proves the argument that the optimal escape speed depends on the situation rather than the highest always being best when taking into account energy loss when a prey is moving. The phase 3 of the two tactics is different. The phase 3 in stay-eat behavior is brighter than the phase 3 in run-away behavior, which indicates that the stay-eat behavior performs better than the run-away behavior when the maximum speed and the energy dissipation rate of prey is high relative to its predators.
The predation phenomenon has been extensively studied from the perspective of space-interaction (e.g., distance [19], direction [28], and velocity angle [34]). The spatial dynamics of predation is characterized by spatial interactions but this alone is not enough to explain all predation behaviors. The internal properties of animals, such as energy level, play a key role in avoiding predation [35] as they are intrinsically linked with an animal’s maneuverability. Our results shed some light on the seemingly irrational pursuit-andevasion behavior of staying in a predator’s vicinity to eat. Energy management was identified as a key aspect with prey deciding on a trade-off between current survival prospects and future energy requirements. The behavior of energy management is also observed in predators. For example, the wolf usually does not chase the nearest elk in a group. It prefers to run after a fleeing group, or lunge at a standing group, and make an assessment of the group members from this interaction. When an optimal potential target appears, usually a weak or young elk, it begins to run after this solitary
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Fig. 5. Survival rate of prey under the two escape tactics changing with both maximum speed ratio V max,t / V max,c and energy dissipation ratio kt /kc . m = 1, denotes the survival rate of prey as shown in color bar. The results are obtained by 1000 independent simulations.
ϕ = 2. The color
larly for those long-period or multi-round confrontations, such as cooperative hunting and group competition. Acknowledgements This work was partially supported by the National Natural Science Foundation of China (NSFC) under Grants 51879219, 51679201. Appendix A. Supplementary material Supplementary material related to this article can be found online at https://doi.org/10.1016/j.physleta.2018.12.046. References Fig. 6. Survival rate of prey by considering the aggregation preference. V max,c = 6, kc = −0.0001, V max,t = 9, kt = −0.0002, m = 1, ϕ = 2. Aggregation behavior is achieved by setting rp = 5, α = 1, β = 1, while rp = 1, α = 0, β = 1 could make the aggregation preference disappear and allow the collision avoidance only. The results are obtained by 1000 independent simulations.
individual while ignoring all the other group members [2]. This strategy could be explained by considering energy management for the predator, because a reserved chase process in the beginning could keep the prey moving and consuming large amounts of energy, therefore exposing the weakest prey that will be easily caught when the predator commits to the final chase. In addition, wolves could manage their own energy loss by taking turns at chasing the prey group before a weak member is exposed. In this paper, the agent-based model is used to analyze stay-eat and run-away behavior by considering the handling time, the internal energy level and the energy-speed relationship in a predator– prey scenario. The model reveals, to some extent, why and when a group of animals perform stay-eat behavior or run-away behavior. We find that (1) the optimal escape speed of prey is contextdependent rather than highest always being best. (2) The optimal decision between stay-eat or run-away depends primarily on the maximum speed and the energy dissipation ratio of prey to predator. (3) Stay-eat behavior is more effective when a prey has a higher escape speed and a higher energy dissipation rate, and their aggregation can induce stay-eat behavior. We suggest that these preliminary findings may be extended to diverse studies of the game dynamics of animal groups, particu-
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