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
.
In the present paper,
we will show some criteria for the stability of positive stationary
solutions on
.
Our results assert that if
is
small enough, then unstable positive stationary solutions bifurcate
from semitrivial solutions, the stability changes only at every
turning point of
and no Hopf bifurcation occurs. While as
becomes large, the stability has a drastic change when
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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