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+kρ(x)v)u]+u(a−u−c(x)v)=0inΩ,Δv+v(b+d(x)u−v)=0inΩ,∂νu=∂νv=0on∂Ω. Here ΩΩ is a bounded domain in RN(N≤3)RN(N≤3), aa and kk are positive constants, bb is a real constant, c(x)>0c(x)>0 and d(x)≥0d(x)≥0 are continuous functions and ρ(x)>0ρ(x)>0 is a smooth function with ∂νρ=0∂νρ=0 on ∂Ω∂Ω. From a viewpoint of the mathematical ecology, unknown functions uu and vv, respectively, represent stationary population densities of prey and predator which interact and migrate in ΩΩ. Hence, the set ΓpΓp of positive solutions (with bifurcation parameter bb) forms a bounded line in a spatially homogeneous case that ρρ, cc and dd are constant. This paper proves that if aa and |b||b| are small and kk is large, a spatial segregation   of ρ(x)ρ(x) and d(x)d(x) causes ΓpΓp to form a ⊂⊂-shaped curve with respect to bb. A crucial aspect of the proof involves the solving of a suitable limiting system   as a,|b|→0a,|b|→0 and k→∞k→∞ by using the bifurcation theory and the Lyapunov–Schmidt reduction.