Stationary patterns of the Holling–Tanner prey–predator model with diffusion and cross-diffusion

This paper is concerned with the Holling–Tanner prey–predator model subject to the homogeneous Neumann boundary condition. We will show that under certain hypotheses, even though the unique positive constant steady state is uniformly asymptotically stable for the dynamics with diffusion, non-constant positive steady states can exist due to the emergence of cross-diffusion. The biological implication of cross-diffusion means that the prey species exercise a self-defense mechanism to protect themselves from the attack of the predator. Our result also demonstrates that cross-diffusion can create Stationary Patterns.