Electron. J. Diff. Equ., Vol. 2012 (2012), No. 223, pp. 1-18.

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

Wan-Tong Li, Yu-Xia Wang, Jia-Fang Zhang

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.

Submitted October 10, 2012. Published December 4, 2012.
Math Subject Classifications: 35K57, 35R20, 92D25.
Key Words: Cross-diffusion; heterogeneous environment; stability; Hopf bifurcation; steady-state solution.

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Wan-Tong Li
School of Mathematics and Statistics
Lanzhou University
Lanzhou, Gansu 730000, China
email: wtli@lzu.edu.cn
Yu-Xia Wang
School of Mathematics and Statistics
Lanzhou University
Lanzhou, Gansu 730000, China
email: wangyux10@163.com
Jia-Fang Zhang
School of Mathematics and Information Sciences
Henan University
Kaifeng, Henan 475001, China
email: jfzhang@henu.edu.cn

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