Coexistence states in a cross-diffusion system of a prey-predator model with an anti-predation mechanism

This paper is concerned with the coexistence states for a cross-diffusion system arising from a prey-predator model with an anti-predation mechanism. The necessary and sufficient conditions for the coexistence of two species are derived. Our mathematical analysis is based on a priori bounds, eigenvalue theory, bifurcation method and homogenization principle. Particularly, the introduction of the anti-predation mechanism causes some difficulties in establishing a priori bounds. We numerically explore how the dispersal rate and anti-predation strength of the species affected the coexistence of two species as well as the spatial distribution of two species.