Turing Instability of Periodic Solutions and Stationary Patterns Induced by Cross-Diffusion in a Holling Type-II Prey–Predator Model

In this paper, we consider a Holling type-II prey–predator model with cross-diffusion under homogeneous Neumann boundary conditions. The existing literature shows that the corresponding ODE model admits a unique globally asymptotically stable periodic solutions and these periodic solutions become locally asymptotically stable homogeneous periodic solutions in the diffusive model, which means that passive diffusion does not create Turing instability of periodic solutions. When cross-diffusions are introduced in the model, the Turing instability of periodic solutions and the existence of nonconstant positive stationary solutions are studied. We derive exact conditions on the cross-diffusion coefficients so that under these conditions, the periodic solutions can undergo Turing instability. Once Turing instability of periodic solutions occurs, numerical simulations show that new irregular patterns emerge. In addition, by using the fixed point index theory and analytical techniques, we obtain sufficient conditions for the existence of nonconstant positive stationary solutions. Our results show that cross-diffusion plays a crucial role in the formation of spatiotemporal patterns, that is, it can create new irregular patterns, which is a sharp contrast to the case without cross-diffusion.