Effects of cross-diffusion and heterogeneous environment on positive steady states of a prey–predator system

In this paper, we consider the positive steady state problem of a spatially heterogeneous cross-diffusion prey–predator model with modified Leslie–Gower and Holling-Type II schemes. The heterogeneity here is created by a protection zone for the prey. By the bifurcation method and a priori estimates, we discuss the existence and non-existence of positive steady states. Moreover, uniqueness and stability of positive steady states for small birth rate of the predator are shown as well as the asymptotic behavior of positive steady states when some coefficients tend to infinity. Our result reveals that large cross-diffusion in a heterogeneous environment has a profound effect on the positive steady state set, and the bifurcation continuum of positive steady states changes from a bounded one to an unbounded one as the cross-diffusion varies from 0 to a large number for suitable ranges of the parameter. Whereas it has little effect in the homogeneous environment  . The impact of the protection zone is also quite important, and it deduces a critical number for the birth rate λλ of the prey which determines the bifurcation continuum of positive steady states to be bounded or unbounded. Furthermore, the modified Leslie–Gower term yields essentially different results of positive solutions from the Leslie–Gower term.