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Zbl 1205.35116
Kuto, Kousuke; Yamada, Yoshio
Multiple coexistence states for a prey-predator system with cross-diffusion. (English)
[J] J. Differ. Equations 197, No. 2, 315-348 (2004). ISSN 0022-0396

This paper concerns the nonnegative steady-states of the following parabolic system $$\cases u_t= \Delta[(d_1+ \rho_{12}v)u]+ u(a_1- b_1u- c_1v)\quad &\text{in }\Omega\times (0,+\infty),\\ v_t= \Delta[(d_2+ \rho_{21}u)v]+ v(a_2+ b_2u- c_2v)\quad &\text{in }\Omega\times (0,+\infty),\\ u=v=0,\quad &\text{on }\partial\Omega\times (0,+\infty),\\ u(x,0)= u_0(x)\ge 0,\ v(x,0)= v_0(x)\ge 0,\quad & x\in\Omega,\endcases$$ where $\Omega\subset\bbfR^N$ $(N\ge 1)$ is a bounded domain, $\rho_{12}$, $\rho_{21}\ge 0$, $a_1,\,\alpha_i,\,b_i,\,c_i\in \bbfR$ $(i= 1,2)$ are positive constants and $a_2\in \bbfR$.\par This is a Lotka-Volterra prey-predator model with cross-diffusion effects. It is shown tha under certain assumptions (on the parameters) the system admits a branch of positive steady-states, which is $S$ or I shaped with respect to a bifurcation parameter. The analysis is based on the bifurcation theory and the Lyapunov-Schmidt procedure.
[Sebastian Anita (Iaşi)]

MSC 2000:: *35J65 (Nonlinear) BVP for (non)linear elliptic equations
35B32 Bifurcation (PDE)
92D25 Population dynamics

Keywords: cross-diffusion; steady state; bifurcation; Lyapunov-Schmidt reduction

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