Persistence and Turing instability in a cross-diffusive predator-prey system with generalist predator

In this paper, we propose and investigate persistence and Turing instability of a cross-diffusion predator-prey system with generalist predator. First, by virtue of the comparison principle, we obtain sufficient conditions of persistence for a corresponding reaction-diffusion system without self-and cross-diffusion. Second, by using the linear stability analysis, we prove that under some conditions the unique positive equilibrium solution is locally asymptotically stable for the corresponding ODE system and the corresponding reaction-diffusion system without cross-diffusion and self-diffusion. Hence it does not belong to the classical Turing instability. Third, under some appropriate sufficient conditions, we obtain that the uniform positive equilibrium solution is linearly unstable for the cross-reaction-diffusion and partial self-diffusion system. The results indicate that cross-diffusion and partial self-diffusion play an important role in the study of Turing instability about reaction-diffusion systems with generalist predator. Fourth, we elaborate on the relations between the theoretical results and the cross-diffusion predator-prey system by relying on some examples. In the end, we conclude our findings and give a brief discussion.