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.