Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 223, pp. 1-18.
Title: Stability of positive stationary solutions to a spatially
heterogeneous cooperative system with cross-diffusion
Authors: Wan-Tong Li (Lanzhou Univ., Gansu, China)
Yu-Xia Wang (Lanzhou Univ., Gansu, China)
Jia-Fang Zhang (Henan Univ., Kaifeng, China)
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 04, 2012.
Math Subject Classifications: 35K57, 35R20, 92D25.
Key Words: Cross-diffusion; heterogeneous environment;
stability; Hopf bifurcation; steady-state solution.