On the steady-state problem for the Volterra-Lotka competition model with diffusion. (English) Zbl 0644.92016
The paper concerns the system
\[
(*)\quad -\Delta u=u[a-u-cv],\quad - \Delta v=v[d-eu-v]
\]
on a smooth bounded domain \(\Omega \subseteq {\mathbb{R}}^ N\) subject to zero boundary data. A componentwise positive solution to (*) represents a steady-state to a corresponding diffusive Volterra-Lotka model for two competing species in which both species survive. Consequently, questions of existence, uniqueness, dependence on the parameters a,c,d,e, and stability (when viewed as a steady-state to a parabolic problem) of such solutions are of considerable interest.
Reviewer: J.Schmeelk
MSC:
92D25 | Population dynamics (general) |
35J65 | Nonlinear boundary value problems for linear elliptic equations |
35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
58E07 | Variational problems in abstract bifurcation theory in infinite-dimensional spaces |