This article discusses a prey-predator system with cross-diffusion. We obtain multiple positive steady-state solutions of this system. More precisely, we prove that the set of positive steady-states possibly contains an S or ⊃-shaped branch with respect to a bifurcation parameter in the large cross-diffusion case. Next we give some criteria on the stability of these positive steady-states. Furthermore, we find the Hopf bifurcation point on the steady-state solution branch in a certain case. Our method of analysis uses the idea developed by Du and Lou [6] and is based on the bifurcation theory and the Lyapunov-Schmidt reduction technique.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc66-0-13,
author = {Kousuke Kuto and Yoshio Yamada},
title = {Multiple existence and stability of steady-states for a prey-predator system with cross-diffusion},
journal = {Banach Center Publications},
volume = {65},
year = {2004},
pages = {199-210},
zbl = {1235.35158},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc66-0-13}
}
Kousuke Kuto; Yoshio Yamada. Multiple existence and stability of steady-states for a prey-predator system with cross-diffusion. Banach Center Publications, Tome 65 (2004) pp. 199-210. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc66-0-13/