Bifurcation branch of stationary solutions for a Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment

This paper is concerned with the following Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment (SP) {(Δ [(1 + k ρ (x) v) u] + u (a - u - c (x) v) = 0, in Ω,; Δ v + v (b + d (x) u - v) = 0, in Ω,; ∂_ν u = ∂_ν v = 0, on ∂ Ω .) Here Ω is a bounded domain in R^N (N ≤ 3), a and k are positive constants, b is a real constant, c (x) > 0 and d (x) ≥ 0 are continuous functions and ρ (x) > 0 is a smooth function with ∂_ν ρ = 0 on ∂ Ω. From a viewpoint of the mathematical ecology, unknown functions u and v, respectively, represent stationary population densities of prey and predator which interact and migrate in Ω. Hence, the set Γ_p of positive solutions (with bifurcation parameter b) forms a bounded line in a spatially homogeneous case that ρ, c and d are constant. This paper proves that if a and | b | are small and k is large, a spatial segregation of ρ (x) and d (x) causes Γ_p to form a ⊂-shaped curve with respect to b. A crucial aspect of the proof involves the solving of a suitable limiting system as a, | b | → 0 and k → ∞ by using the bifurcation theory and the Lyapunov-Schmidt reduction.