Stability of positive stationary solutions to a spatially heterogeneous cooperative system with cross-diffusion

Abstract

In the previous article [Y.-X. Wang and W.-T. Li, J. Differential Equations, 251 (2011) 1670-1695], the authors have shown that the set of positive stationary solutions of a cross-diffusive Lotka-Volterra cooperative system can form an unbounded fish-hook shaped branch $Gamma_p$. In the present paper, we will show some criteria for the stability of positive stationary solutions on $Gamma_p$. Our results assert that if $d_1/d_2$ is small enough, then unstable positive stationary solutions bifurcate from semitrivial solutions, the stability changes only at every turning point of $Gamma_p$ and no Hopf bifurcation occurs. While as $d_1/d_2$ becomes large, the stability has a drastic change when $mu<0$ in the supercritical case. Original stable positive stationary solutions at certain point may lose their stability, and Hopf bifurcation can occur. These results are very different from those of the spatially homogeneous case.

Keywords

• Cross-diffusion
• heterogeneous environment
• stability
• Hopf bifurcation
• steady-state solution