Evolutionary Models

Before reviewing the evolutionary models of schooling fish, it should be clarified that although the terms ‘predator-prey’, ‘pursuit-evasion’ and ‘coevolution’ have been widely used in Evolutionary Computation since Benda (1986), their focus is different from the topic here. In Evolutionary Computation, ‘predator-prey’ models are built to solve complicated tasks or practise artificial intelligence (Hillis, 1990; Ficici & Pollack,

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1998; Goh & Tan, 2009). In these cases, a direct fitness function for the aimed goal is difficult to be explicitly defined. By cooperation or competition among agents, optimisation can be reached through a simpler fitness function (Haupt & Haupt, 2004; Bouffanais, 2016). The common focus of these techniques is put on optimisation of solutions rather than on evolutionary dynamics of the natural predator-prey interaction (Nowak & Sigmund, 2004).

With the concern of natural evolution in animal behaviours, some evolutionary simulations had focused on the competition between a predator and a prey. For example, Cliff & Miller (1995) and Nolfi & Floreano (1998) demonstrated how the coevolution between a prey agent’s evasive strategy and a predator’s chasing strategy leads to a balance of both fitness values. By these simulations, the ‘red queen hypothesis’ (Van Valen, 1973), which assumed a predator-prey coevolution is an arms race without endpoints, was highlighted and supported. However, the findings from this kind of one-on-one interactions are rarely considered to be an analogy to the evolution of gregarious animals and their collective behaviours because the intraspecies competition in a species, which influences the evolution much more than the interspecies competition (Connell, 1983), was overlooked.

The first evolutionary model focusing on animals’ collective behaviour can be traced back twenty years (Reynolds, 1993). However, the link between a computational simulation and the natural evolution of collective motion has only been established in recent years (Wood & Ackland, 2007). These evolutionary models are typically a combination of a self-organising model and a genetic algorithm. The self-organising model is to simulate agents’ spatiotemporal interaction as well as the emergent patterns.

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The genetic algorithm is employed to simulate the evolutionary mechanism of natural selection and adaptation.

Genetic algorithms are computer programmes that inspired from natural evolution (Holland, 1975). In a genetic algorithm, there is a population of ‘chromosomes’, each of which represents a strategy, a solution or an agent, depending on the design. A typical evolutionary process is as follows (Haupt & Haupt, 2004). First, each chromosome is assigned a score, referred to as a fitness value, based on its performance in a given task. Then, according to the principle of ‘survival of the fittest’, a chromosome’s reproduction probability and elimination probability are given based on its relative performance in the population. Finally, before entering the next generation, offspring are reproduced by the operation of crossover and mutation on the selected chromosomes. This process is repeated so that the dynamic of natural evolution is mimicked.

Earlier works, like Reynolds (1993) and Werner & Dyer (1993), were more similar to games of artificial agents. That is to say, given rewards and dangers in an arena, agents can evolve to develop an effective movement to gain better fitness. These preliminary works, however, drew a framework to simulate collective behaviour in evolution, that is, a spatial-explicit agent-based model combining with a genetic algorithm. Moreover, agents in these models were designed with sensory perceptions so that decision making can be adaptive to the surroundings.

Ward et al. (2001) simulated the behavioural coevolution of prey fish and their predators. In this model, a fish’s eyes and lateral lines were simulated as two sensory perceptions of an agent, and a neural network was employed to connect the movement

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decision with local information from these sensors. Although the output may not be as appealing, this is the first work to consider a real fish’s perceptions and responses in evolution.

Since Oboshi et al. (2003), Aoki’s framework (1982) of the design of a fish agent has been introduced to evolutionary models. Based on the predefined attraction, repulsion and paralleling behaviours in Aoki (1982), the simulation can be more robust and the output can be more similar to the appearance of real fish. In this kind of evolutionary models, the adaptation of a fish individual is simulated by evolving certain parameters of a chosen self-organising model. For example, the agent design in Oboshi et al. (2003) was based on Inada & Kawachi (2002). Through a genetic algorithm, the adaptive weight of escaping behaviour was studied.

Wood & Ackland (2007) designed agents based on the model in Couzin et al. (2002), with an additional escaping behaviour. The evolution of prey agents was simulated by evolving the movement speed and the orientation range, that is, neighbours in which can trigger a prey’s paralleling behaviour. This work exhibited that two Nash equilibria (should also be evolutionarily stable states according to its description) of collective patterns can be reached. One is a compact milling group of low-speed agents and the other is a polarised moving school of high-speed agents. This work stated that the milling aggregation is invasion-free although individuals in this pattern incur higher predation risk. Hence, the findings supported the selfish herd hypothesis (Hamilton, 1971). The significance of this work is that it is the first work to introduce game- theoretic concepts to validate the simulation (Sumpter, 2009).

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Similar to Wood & Ackland (2007), Ioannou et al. (2012) introduced the agent design in Couzin et al. (2002) and evolved the agents by the orientation range, where the designs different from Wood & Ackland (2007) were the fixed speed and the lack of escaping behaviour. Instead of setting an artificial predator to hunt the prey agents, Ioannou et al. (2012) used a real predatory fish, bluegill sunfish, to hunt the virtual prey agents and drive the evolutionary simulation. It showed that the bluegill sunfish preferred the isolated prey to prey aggregation, preferred the marginal prey to the central prey, and preferred the swarming prey to the schooling prey. It was demonstrated that these feeding preferences always drive the evolution of the virtual prey agents into the schooling pattern.

A series of latest works (Olson et al., 2013; Olson et al., 2016a; Olson et al., 2016b) simulated the evolution of prey aggregation as well as the coevolution between prey and predators with minimal predetermined rules. These works abandoned Aoki’s framework (1982) and designed an agent’s movement decision at a rather basic level, as turning right or turning left, based on the information from the visual perception. Olson et al. (2013) demonstrated that schooling is a transitional state and can be replaced by disordered swarms and milling groups in evolution. Olson et al. (2016a) experimented whether the cohesive swarming pattern can evolve given different hunting strategies. Olson et al. (2016b) demonstrated that prey’s swarming behaviour and the predator’s hunting strategy can form an evolutionary cycle in coevolution.