In this talk, we present and study a model for two populations with a predator-prey interaction, where each population is composed of two types of individuals, denoted by 0 and 1, such that predators of a given type thrive in the presence of similar prey of type, while preys of a given type are more likely to survive in the presence of predators of the different type.
We consider a limit in large population and rare mutations at an intermediate scale, namely the individual mutation rate vanishes while the total mutation rate tends to infinity. We prove that depending on the parameters of the model different scenario can occur: successive prey and predator invasion leading to the coexistence of four types, or successive invasion of prey in a population of resident predator leading either to prey extinction or to the coexistence of all types, ...
