Adaptation of Predators

The adaptation of predators’ feeding preference is simulated through the transition from a state to its adjacent state in the full-range map. In the previous sections, 𝑿 is

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considered as the risk distribution in the environment from the view at the prey side. Here, from the view at the predator side, 𝑿 = 𝒙 is treated as the feeding preference adopted by a homogeneous population of predators, which leads to its associated state,

𝑫. Therefore, the evolution of predators should convert a state to an adjacent one if the resident strategy of the nearby state is more adaptive than the original tactic.

The adaptive dynamic of predators is largely influenced by the energy cost, 𝑒𝑗. For example, assuming that the central position is always risk-free (Figure 5.6A), the evolutionarily stable states of the strongest predators (𝑒𝑗= 1) fall around the junction of the three emergent types. The junction is not directly attained because of the errors brought by a computational simulation. On average, the adaptive strategy is 𝑿 = (0.37,0.35,0.28,0), or 𝑿 = (1.32𝑋𝑀, 1.25𝑋𝑀, 𝑋𝑀, 0), which can be linked to the risk distribution of fast-predator situations in Chapter IV. In contrast, when predators are weaker (𝑒𝑗< 1), that is, their energy costs of chasing a moving school is higher than the costs of attacking a stationary herd, the evolutionarily stable state shifts along the boundary of school patterns and line abreast formations (Figure 5.6). For example, when 𝑒𝑗= 0.1, the average feeding preference is 𝑿 = (0.2,0.19,0.61,0) or equivalently,

(0.33𝑋𝑀, 0.31𝑋𝑀, 𝑋𝑀, 0), similar to the slow-predator situations in the previous chapter. This kind of adaptive dynamics can be observed in any 𝑋𝐶 = 𝑘𝑋𝑀situation, even if 𝑘 > 1. The adaptation of predators given 𝑋𝐶 _{= 𝑋}𝑀 _{is shown in Figure 5.6B as another} example.

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Figure 5.6: Adaptive Dynamic of Predators

The adaptation of predators causes the transition of states, as plotted by the grey paths and the black arrows. Given the environment where 𝑋𝐶 _{is fixed to zero, as in subfigure}

A, these paths are toward to the junction of the three emergent types when 𝑒𝑗= 1. It is an

evolutionarily stable state in the coevolution system. When 𝑒𝑗< 1, for example, 𝑒𝑗= 0.1,

the equilibrium shifts towards the zero point. Subfigure B shows the case of 𝑋𝐶_{= 𝑋}𝑀_.

When predators can evolve their feeding preferences freely, those weaker predators (𝑒𝑗< 0.8) still evolve the preference of marginal predation, that is, 𝑋𝐶 = 0 (Figure 5.7 & Figure 5.8). However, the extremely strong predators which 𝑒𝑗 > 0.8 lead to different evolutionarily stable states which resident strategies are 𝑿 = (0.23,0.19,0.16,0.42) on average (Figure 5.7 & Figure 5.8), or 𝑿 = (0.27,0.24,0.25,0.25) subject to 𝑋𝐶 ≤ 𝑋𝑀. Since these final states of predators’ adaptation have been evolutionarily stable at the prey side, they are the evolutionarily stable states in the coevolution system. It is observed that the coevolution of predators and prey can only drive extremely strong predators to hunt the central position of fish aggregation. In contrast, the weaker predators should evolve to feed on marginal prey of a stationary herd based on

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evolutionary selfishness. On the other hand, prey evolve into the schooling pattern, composed of pioneers and followers equally, in front of any predators (Figure 5.7).

Figure 5.7: Evolutionarily Stable States

Given a group of predators, which energy difference is ej, and a group of prey, the

evolutionarily stable state of their behavioural adaptations is plotted. In a stable state, prey generally form a schooling pattern with an equal amount of pioneers and followers, and the frequency of herding prey increases when ej becomes larger. On the other hand,

only those extremely strong predators (𝑒𝑗> 0.8) can adopt the preference to hunt the

central position of a prey group. The preference to hunt the margin of a stationary herd is enhanced when ej decreases.

For those extremely strong predators (𝑒𝑗> 0.8), the feeding preference in an evolutionarily stable state can lead to the maximal foraging benefit, which is defined as

(𝑥𝑗𝑃𝑃 + 𝑥𝑗𝐹𝐹)𝑒𝑗+ 𝑥𝑗𝑀𝑀 + 𝑥𝑗𝐶𝐶, among almost all strategies in their emergent states (Figure 5.8A). However, the adaptive strategies of weaker predators are not the optimal ones to the population (Figure 5.8B), which situation becomes worse with the decrease of 𝑒𝑗 (Figure 5.8C & Figure 5.8D). If cooperation is allowed in evolution, a species of group-hunting predators should add risk on the central cowards unselfishly so that they can gain better fitness from the emergent swarming pattern (Figure 5.8). Another possibility to reach these optimal states in evolution is when a species of predators have evolved to hunt alone. In this case, a predator’s feeding preference is directly equal to

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the risk distribution on prey aggregation, so that it can add risk on the central positions without the worry of selfish predators.

Figure 5.8: ESSs versus Optimal States

The azure paths draw the adaptive dynamic when predators evolve freely, which move towards the evolutionarily stable states pointed by the black arrows. These evolutionarily stable states can be different from the optimal states where the predator population receives the maximal foraging fitness. Those states which provide higher group benefit to the predator population are plotted in the figure, which colours are based on their types, as green for swarming patterns and orange for schooling patterns.

5.4 Analysis

The computational simulation has demonstrated all possibilities of adaptations between predators and prey in their coevolution. In this subchapter, theoretical analyses are provided to validate the experimental outputs. These derivations are partly based on the knowledge acquired from the simulation, so the analysis and the simulation are complements to each other.

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