| Abstract: |
| We study positive solutions to a steady state reaction diffusion equation arising in population dynamics, namely,
\begin{equation*}
\label{abs}
\left\lbrace \begin{matrix}-\Delta u=\lambda u(1-u) ;~x\in\Omega\
\frac{\partial u}{\partial \eta}+\gamma\sqrt{\lambda}[(A-u)^2+\epsilon]u=0; ~x\in\partial \Omega
\end{matrix} \right.
\end{equation*}
\noindent where $\Omega$ is a bounded domain in $\mathbb{R}^N$; $N > 1$ with smooth boundary $\partial \Omega$ or $\Omega=(0,1)$, $\frac{\partial u}{\partial \eta}$ is the outward normal derivative of $u$ on $\partial \Omega$, $\lambda$ is a domain scaling parameter, $\gamma$ is a measure of the exterior matrix ($\Omega^c$) hostility, and $A\in (0,1)$ and $\epsilon>0$ are constants. The boundary condition here represents a case when the dispersal at the boundary is U-shaped. In particular, the dispersal is decreasing for $uA$. We will establish non-existence, existence, multiplicity and uniqueness results. In particular, we will discuss the occurrence of an Allee effect for certain range of $\lambda$. When $\Omega=(0,1)$ we will provide more detailed bifurcation diagrams for positive solutions and their evolution as the hostility parameter $\gamma$ varies. Our results indicate that when $\gamma$ is large there is no Allee effect for any $\lambda$. We employ a method of sub-supersolutions to obtain existence and multiplicity results when $N>1$, and the quadrature method to study the case $N=1$. |
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