Stability and hopf bifurcation of coexistence steady-states to an skt model in spatially heterogeneous environment

This paper is concerned with the following Lotka-Volterra cross-diffusion system ? in a bounded domain ω ⊂ R^N with Neumann boundary conditions ∂_vu = ∂_vv = 0 on ∂ω. In the previous paper [18], the author has proved that the set of positive stationary solutions forms a fishhook shaped branch ⌈ under a segregation of p(x) and d(x). In the present paper, we give some criteria on the stability of solutions on ⌈. We prove that the stability of solutions changes only at every turning point of ⌈ if ? is large enough. In a different case that c(x) > 0 is large enough, we find a parameter range such that multiple Hopf bifurcation points appear on ⌈.