Perturbation of A Nonlinear Elliptic Biological Interacting Model with Multiple Species

Abstract

In this paper, we investigate the effects of perturbations on the coexistence state of the general competition model for multiple species. Previous work by Kang, Lee and Oh (see [11]) established sufficient conditions for the uniqueness of the positive solution to the following general elliptic system for multiple competing species of animals: \[\Delta u_{i} +u_{i}g_{i}(u_{1},u_{2},\ldots ,u_{i},u_{i+1},\ldots ,u_{N}) =0 \ \mbox{in} \ \Omega, \ u_{i}|_{\partial\Omega} = 0\]for $i = 1,\ldots ,N$. That is, they proved that under certain conditions, the species can coexist and that the coexistence state is unique at fixed rates. In this paper, we extend their uniqueness results by perturbing functions $g_{i}$'s of the above model, and applying super-sub solutions, maximum principles and spectrum estimates. Our arguments also rely on some detailed properties for the solution of logistic equations. By applying these techniques, we obtain sufficient conditions for the existence and uniqueness of a time independent coexistence state for the perturbed general competition model.