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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:251

Article

Wang, Yu-Xia¹  Li, Wan-Tong¹ 

[1] Lanzhou Univ, Sch Math & Stat, Lanzhou 730000, Gansu, Peoples R China

关键词: Cross-diffusion;    Heterogeneous environment;    Bifurcation;    Lyapunov-Schmidt reduction;    Stationary solution;   

DOI  :  10.1016/j.jde.2011.03.009

来源: Elsevier

【 摘 要 】

In this paper, we will consider the following strongly coupled cooperative system in a spatially heterogeneous environment with Neumann boundary condition {Delta u + u(lambda - u + b(x)v) = 0, x is an element of Omega, Delta[(1 + k rho(x)u)v] + v(mu - v + d(x)u) = 0, x is an element of Omega, partial derivative(nu)u = partial derivative(nu)v = 0, x is an element of partial derivative Omega, where Omega is a bounded domain in R-N (N >= 1) with smooth boundary partial derivative Omega; k is a positive constant, lambda and mu are real constants which may be non-positive; b(x) >= not equivalent to 0 and d(x) >= not equivalent to 0 are continuous functions in (Omega) over bar; rho(x) is a smooth positive function in (Omega) over bar with partial derivative(nu)rho(x)vertical bar partial derivative Omega = 0; nu is the outward unit normal vector on partial derivative Omega and partial derivative(nu) = partial derivative/partial derivative nu). For the case mu > 0, we show that if vertical bar mu vertical bar is small and k is large, a spatial segregation of rho(x) and b(x) can cause the positive solution curve to form an unbounded fish-hook (c) shaped curve with parameter lambda. For the case mu < 0, if vertical bar mu vertical bar 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. Our method of analysis is based on the bifurcation theory and the Lyapunov-Schmidt procedure. (C) 2011 Elsevier Inc. All rights reserved.

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