Adaptation of prey and predators between patches

Mathematical models are proposed to simulate migrations of prey and predators between patches. In the absence of predators, it is shown that the adaptation of prey leads to an ideal spatial distribution in the sense that the maximal capacity of each patch is achieved. With the introduction of co-adaptation of predators, it is proved that both prey and predators achieve ideal spatial distributions when the adaptations are weak. Further, it is shown that the adaptation of prey and predators increases the survival probability of predators from the extinction in both patches to the persistence in one patch. It is also demonstrated that there exists a pattern that prey and predators cooperate well through adaptations such that predators are permanent in every patch in the case that predators become extinct in each patch in the absence of adaptations. For strong adaptations, it is proved that the model admits periodic cycles and multiple stability transitions.