Coexistence states for a cross-diffusion predator–prey model with spatial heterogeneity

This paper investigates a quasilinear elliptic system modeling a prey–predator model incorporating cross-diffusion, herd behavior and spatial heterogeneity. By taking the parameter k as a bifurcation parameter, we establish the global bifurcation of coexistence states from the semi-trivial solution branch, employing bifurcation theory combined with the method of upper and lower solutions. We prove that when the prey birth rate r is sufficiently high, specifically, r>λ1[−D1(0)Δ; m(x)] and k lies in an intermediate range satisfying k∈(min{H(r),λ1[−D2(0)Δ]}, max{H(r),λ1[−D2(0)Δ]}), coexistence of prey and predator emerges; that is, the system admits positive solutions. Moreover, we derive explicit conditions that determine the direction of the bifurcation. Our results highlight the essential role played by the spatial distribution of prey resources m(x) in governing the existence of coexistence states. In addition, we find that the region of coexistence shrinks as the herd behavior coefficient increases.