The predator-dependent replicator dynamics

Abstract

Ecological interactions are central to understanding evolution. For example, Darwin noticed that the beautiful colours of the male peacock increase the chance of successful mating. However, the colours can be a threat because of the increased probability of being caught by predators. Eco-evolutionary dynamics takes into account environmental interactions to model the process of evolution. The selection of prey types in the presence of predators may be subjected to pressure on both reproduction and survival. Here, we analyze the evolutionary game dynamics of two types of prey in the presence of predators. We call this model the predator-dependent replicator dynamics. If the evolutionary time scales are different, the number of predators can be assumed constant, and the traditional replicator dynamics is recovered. However, if the time scales are the same, we end up with sixteen possible dynamics: the combinations of four reproduction's games with four predation's games. We analyze the dynamics and calculate conditions for the coexistence of prey and predator. The main result is that predators can change the equilibrium of the traditional replicator dynamics. For example, the presence of predators can induce polymorphism in prey if one type of prey is more attractive than the other, with the prey ending with a lower capture rate in this new equilibrium. Lastly, we provide two illustrations of the dynamics, which can be seen as rapid feedback responses in a predator-prey evolutionary arm's race.