Coexistence states for a prey-predator model with cross-diffusion

Kousuke Kuto; Yoshio Yamada

doi:10.3934/proc.2005.2005.536

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2005, Volume 2005: 536-545. Doi: 10.3934/proc.2005.2005.536 open access

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Coexistence states for a prey-predator model with cross-diffusion

Kousuke Kuto^1, and
Yoshio Yamada^2,

1.
Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555
2.
Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku 169-8555, Tokyo

Received: August 31, 2004

Revised: February 28, 2005

Published: August 2005

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Abstract

This paper discusses a prey-predator system with cross-diffusion. We can prove that the set of coexistence steady-states of this system contains an S or $\supset$-shaped branch with respect to a bifurcation parameter in a large cross-diffusion case. We give also 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.

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Mathematics Subject Classification: Primary: 35B32, 35J65; Secondary: 92D25.

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