Multiple coexistence states for a prey-predator system with cross-diffusion. (English) Zbl 1205.35116

This paper concerns the nonnegative steady-states of the following parabolic system \[ \begin{cases} u_t= \Delta[(d_1+ \rho_{12}v)u]+ u(a_1- b_1u- c_1v)\quad &\text{in }\Omega\times (0,+\infty),\\ v_t= \Delta[(d_2+ \rho_{21}u)v]+ v(a_2+ b_2u- c_2v)\quad &\text{in }\Omega\times (0,+\infty),\\ u=v=0,\quad &\text{on }\partial\Omega\times (0,+\infty),\\ u(x,0)= u_0(x)\geq 0,\;v(x,0)= v_0(x)\geq 0,\quad & x\in\Omega,\end{cases} \] where \(\Omega\subset\mathbb{R}^N\) \((N\geq 1)\) is a bounded domain, \(\rho_{12}\), \(\rho_{21}\geq 0\), \(a_1,\,\alpha_i,\,b_i,\,c_i\in \mathbb{R}\) \((i= 1,2)\) are positive constants and \(a_2\in \mathbb{R}\).
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.


35J65 Nonlinear boundary value problems for linear elliptic equations
35B32 Bifurcations in context of PDEs
92D25 Population dynamics (general)
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