Antibifurcation and the $n$-species Lotka-Volterra competition model with diffusion

Abstract

The system $$ \begin{align} -\Delta u_{k} &= ( a_{k} -u_{k} - \sum ^{n}_{ j\neq k,\, j=1} \gamma _{kj} u_{j} ) u_{k} \quad \text{in} \; \Omega \\ u_{k} & =0 \quad \text{on} \; \partial \Omega ,\end{align} \tag {*} $$ $k=1,\ldots, n$, where $\Omega $ is a bounded domain in $\Bbb R^{n}$ and $a_{k} , \gamma _{kj}$ are positive parameters, determines the possible equilibrium configurations for a diffusive Lotka-Volterra competition model and is of interest in the study of the role of competition in structuring communities where space or resources are limited. The componentwise nonnegative solutions to $(*)$ can perhaps best be understood for fixed $\gamma _{kj}$ and varying $a_{k}$ as a subset of the Banach space $\Bbb R^{n} \times [C^{1}_{0} (\overline{\Omega })]^{n}$. The aims of this article are to enhance understanding of the structure of this set and to provide a firmer foundation for future analysis. We accomplish these aims through some new observations regarding the set of componentwise nonnegative solutions to $(*)$ which enable us to unify some preceding analyses.