The Emergent Patterns

By computing prey’s adaptive dynamics in the full range of risk distributions, all possible emergent patterns can be plotted as a map in Figure 5.4. It is implemented by dividing each of the four positional risks into 100 segments, and then simulating prey’s adaptation given the predation risk 𝑿 + 𝝐𝐺 in each of the 1004 _{grids, where 𝑿 is the} average risk distribution in a grid and 𝝐𝐺 _{provides stochasticity to the distribution in} every generation. In general, the emergent pattern in each grid is evolutionarily stable at the prey side, since it is demonstrated that any small proportion of different positional changes cannot perturb the state.

Three primary types of emergent patterns in the map are as follows (Figure 5.4A). The first type is the line abreast formation, which only consists of pioneers. This state can emerge when the risk of pioneers is less than that of followers. The second type is the

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swarming pattern, which consists of cowards in a herd. This state can be observed when the risk of pioneers is much higher than the others. The third type is the schooling pattern, where a few pioneers, many followers and few cowards are present. It happens when the risk of pioneers is slightly higher than that of followers, at the same time lower or slightly higher than the risk of cowards.

Figure 5.4: Full-range Map of Emergent Patterns

The emergent patterns of prey aggregation under all possible risk distributions are displayed. To plot these states in a 3-D panel, the major position in a state is used to represent the general type, as in subfigure A. If the pioneers are in the majority, the state is a line abreast formation. If the followers are in the majority, it is a schooling pattern. Lastly, if central cowards are in the majority, it is a swarming pattern. Subfigure B provides complete information of the positional frequencies when XC_{=0, which is the} surface slice in subfigure A. Subfigure B shows the case when XC_=XM_{, which is the blue} slice in subfigure A. The white rectangle on the XC_{=0 slice marks the experimental range} in Chapter IV.

The boundary between the line abreast formation and the other types is generally fixed on the surface 𝑋𝑃 = 𝑋𝐹. In contrast, the boundary between the schooling pattern and the swarming pattern shifts when the risk of central cowards alters. When this risk equals that of marginal cowards (Figure 5.4C), the range of schooling patterns shrinks to the minimum. As noted in the previous section, the stable states in the area 𝑋𝐶 > 𝑋𝑀 can be

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less reasonable considering that the prey group cannot become compact through the selfish herd scenario.

Figure 5.5: Two Specific Risk Distributions

The risk distribution, 𝐗 + 𝛜𝐺_{, of those grids on the junction of the three types cannot lead}

to an evolutionarily stable state, as subfigure A. It is because the noise, 𝛜𝐺_{, randomly}

converts the expected stable state from one to another, which is an error due to the discretisation. Subfigure B shows a special type of the emergent pattern when XP_=XF_. Under this condition, a prey with even frequencies to be a pioneer and a follower can average the noise from 𝛜𝐺 _{and avoid being eliminated.}

The adaptive strategy, 𝑐𝑖, of all prey in a line abreast formation is {𝑝𝑖𝑀2𝑃, 𝑝𝑖𝑃2𝐹, 𝑝𝑖𝐹2𝑃} =

{1,0,1} due to the avoidance of being a follower. In a swarming pattern, since being a pioneer is highly in danger, the typical strategy of all prey is 𝑐𝑖 = {0,1,0}. The adaptive strategy to cause a schooling pattern contains two features. One is that the probability to turn from a marginal coward to a pioneer, 𝑝𝑖𝑀2𝑃, is small (around 0.13), even if the risk of marginal cowards is much higher than that of pioneers. As explained in the former section, it is because being brought as a follower is more beneficial than being a pioneer itself. The other feature is that the probability to turn from a follower to a pioneer, 𝑝𝑖𝐹2𝑃, is always zero and the opposite probability, 𝑝_𝑖𝑃2𝐹, is significantly greater than zero. This

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adaptation leads to a behavioural dynamic similar to the natural fish, which often form a temporary milling herd, that is, a stationary aggregate where individuals move around the centre, during their collective movement.

There is a group of grids which evolutionarily stable states cannot be computed, as those fall on the junction of the three types, for example, 𝑿 = (0.36,0.36,0.28,0) in Figure 5.5A. In these grids, prey’s adaptive strategies cannot be converged permanently because the noise to the positional risk, 𝝐𝐺_{, converts the evolutionarily stable state from} one type to another with generations. This is a phenomenon caused by the error of discretisation, and if further segments are applied, these grids are actually composed of three types of evolutionarily stable states. The expected states of these grids are estimated by averaging the positional distributions from the 10000th generation to the 15000th generation. Another special case happens on the boundary of the line abreast formation. The noise 𝝐𝐺 in this kind of grids creates a subtle type of stable states, as 𝑫 = (50,50,0,0) (Figure 5.5B). It is because a prey with even frequencies of being a pioneer and a follower can average the noises at the two positions, and hence is less likely to be the worst-performed ones (as well as the best-performed ones). In addition, on the boundary of the swarming pattern, there can be multiple evolutionarily stable states (Figure 5.2). These states are averaged based on their frequencies of being reached by simulations, which does not influence the adaptation at the predator side, as analysed in Chapter 5.4.