Fish–hook shaped global bifurcation branch of a spatially heterogeneous cooperative system with cross-diffusion

Abstract

In this paper, we will consider the following strongly coupled cooperative system in a spatially heterogeneous environment with Neumann boundary conditionwhere Ω is a bounded domain in R N (N 1) with smooth boundary ∂Ω; k is a positive constant, λ and μ are real constants which may be non-positive; b(x) ≡ 0 and d(x) ≡ 0 are continuous functions inΩ; ρ(x) is a smooth positive function inΩ with ∂ ν ρ(x)| ∂Ω = 0; ν is the outward unit normal vector on ∂Ω and ∂ ν = ∂/∂ν. For the case μ > 0, we show that if |μ| is small and k is large, a spatial segregation of ρ(x) and b(x) can cause the positive solution curve to form an unbounded fish-hook (⊂) shaped curve with parameter λ. For the case μ < 0, if |μ| is small and k is large, and the cooperative effect is strong for species u and very weak for species v, then the positive solution set also forms an unbounded fish-hook shaped continuum. These results are quite different from those of predator-prey systems and the cooperative system under Dirichlet boundary condition, both of which can form a bounded continuum. Our results deduce that the spatial heterogeneity of environments can produce multiple coexistence states. 1671 Our method of analysis is based on the bifurcation theory and the Lyapunov-Schmidt procedure.

The stability of bifurcating steady states for a spatially heterogeneous cooperative system with cross-diffusion

“…In this paper, we have studied the local stability of bifurcating solutions obtained in [14] of a spatially heterogeneous cooperative system with cross-diffusion. First, we give the bifurcating direction near bifurcation point.…”

“…Hutson et al [10][11][12][13] studied spatial effects of birth rates in some diffusive competitive models. Wang et al [14] investigated a Lotka-Volterra cooperative system with cross-diffusion in a spatially heterogeneous environment. Applying the bifurcation theory and the Lyapunov-Schmidt reduction, they obtained the global bifurcation branch of positive steady states.…”

“…Applying the bifurcation theory and the Lyapunov-Schmidt reduction, they obtained the global bifurcation branch of positive steady states. They also proved that the spatial segregation of ρ(x) and b(x) could cause the positive solution curve to form an unbounded fish-hook shaped curve with parameter λ. Li et al [15] showed some criteria for the stability of positive stationary solutions obtained in [14] and the Hopf bifurcation in certain circumstances. In this paper, we study the stability of the bifurcating steady states obtained in [14].…”

“…They also proved that the spatial segregation of ρ(x) and b(x) could cause the positive solution curve to form an unbounded fish-hook shaped curve with parameter λ. Li et al [15] showed some criteria for the stability of positive stationary solutions obtained in [14] and the Hopf bifurcation in certain circumstances. In this paper, we study the stability of the bifurcating steady states obtained in [14]. Here we list the local bifurcation result and some preliminary results in [14], which will be used in this paper.…”

“…In this paper, we study the stability of the bifurcating steady states obtained in [14]. Here we list the local bifurcation result and some preliminary results in [14], which will be used in this paper.…”