| Abstract: |
| \noindent We analyze positive solutions $(u,v)$ to the reaction-diffusion system:
\begin{equation*}
\label{system}
\left\lbrace \begin{matrix}-\Delta u=\lambda u (1-u) ;~\Omega\
-\Delta v =\lambda r v (1-v) ;~\Omega\
\frac{\partial u}{\partial \eta}+ \sqrt{\lambda} g(v)u = 0; ~\partial \Omega\
\frac{\partial v}{\partial \eta}+\sqrt{\lambda}h(u)v =0; ~\partial \Omega\
\end{matrix} \right.
\end{equation*}
\noindent
where $\Omega$ is a bounded domain (patch) in $\mathbb{R}^N$ of unit length, area, or volume; $N \in \{1, 2, 3\}$ with smooth boundary $\partial \Omega$, $ \lambda > 0$ is a parameter proportional to the square of patch size and $\frac{\partial z}{\partial \eta}$ is the outward normal derivative of $z$. Here $u$ and $v$ represent the normalized densities of two species which inhabit the patch, surrounded by a hostile matrix, where the level of hostility is determined by the functions $g, h \in C^1([0,1], (0, \infty))$. Finally, $r > 0$ compares the two species by the ratio of patch intrinsic growth to patch diffusion rate. We explore two cases: (1) $u$ and $v$ are competing species ($g$ and $h$ are increasing) and (2) $v$ competes with $u$ while $u$ cooperates with $v$ ($g$ increasing, $h$ decreasing). We establish coexistence and nonexistence results analytically for certain ranges of $\lambda$ depending on the characteristics of $g$ and $h$. We prove our coexistence results via the method of subsolutions and supersolutions. We discuss some interesting ecological phenomena observed in our numerical simulations. |
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