| Abstract: |
| \noindent We study the positive solutions to the reaction diffusion model
\begin{equation*}
%\label{1.11}
(*) \left\lbrace \begin{matrix}
-\Delta u = \lambda u(1-u - b_1v);~\Omega\
-\Delta v = \lambda rv(1-v - b_2u) ;~\Omega\
\frac{\partial u}{\partial \eta}+\gamma_1 \sqrt{\lambda}u=0;~\partial\Omega\
\frac{\partial v}{\partial \eta}+ \gamma_2 \sqrt{\lambda}v=0;~\partial\Omega
\end{matrix} \right.
\end{equation*}
\noindent which describes the steady states of two species $u$ and $v$ competing in a habitat $\Omega$. Here $b_1, b_2$ represent the strengths of competition, $\lambda$ represents a patch size measure, and $\gamma_1, \gamma_2$ are related to the hostility of the exterior domain. We analyze the positive solutions of (*) as the parameters $b_1, b_2$ and $\gamma_1, \gamma_2$ vary. |
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