On the effects of the exterior matrix hostility and a U-shaped density dependent dispersal on a diffusive logistic growth model

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*} $

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 $ u<A $ and increasing for $ u>A $. 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 $.