Bifurcation, uniqueness and multiplicity results for classes of reaction diffusion equations arising in ecology with nonlinear boundary conditions

We study the structure of positive solutions to steady state ecological models of the form:\begin{document}$ \begin{array}{l} \left\{ \begin{split} -\Delta u& = \lambda uf(u)\; \; && {\rm{in}}\; \; \Omega,\\ \alpha(u)&\frac{\partial u}{\partial \eta}+[1-\alpha(u)]u = 0 &&\;\;\;{\rm{on}}\; \; \partial\Omega, \end{split} \right. \end{array} $\end{document}where \begin{document}$ \Omega $\end{document} is a bounded domain in \begin{document}$ \mathbb{R}^n; $\end{document}\begin{document}$ n>1 $\end{document} with smooth boundary \begin{document}$ \partial\Omega $\end{document} or \begin{document}$ \Omega = (0,1) $\end{document} , \begin{document}$ \frac{\partial}{\partial\eta} $\end{document} represents the outward normal derivative on the boundary, \begin{document}$ \lambda $\end{document} is a positive parameter, \begin{document}$ f:[0,\infty)\to \mathbb{R} $\end{document} is a \begin{document}$ C^2 $\end{document} function such that \begin{document}$ \tfrac{f(s)}{k-s}>0 $\end{document} for some \begin{document}$ k>0 $\end{document} , and \begin{document}$ \alpha:[0,k]\to[0,1] $\end{document} is also a \begin{document}$ C^2 $\end{document} function. Here \begin{document}$ f(u) $\end{document} represents the per capita growth rate, \begin{document}$ \alpha(u) $\end{document} represents the fraction of the population that stays on the patch upon reaching the boundary, and \begin{document}$ \lambda $\end{document} relates to the patch size and the diffusion rate. In particular, we will discuss models in which the per capita growth rate is increasing for small \begin{document}$ u $\end{document} , and models where grazing is involved. We will focus on the cases when \begin{document}$ \alpha'(s)\geq 0 $\end{document} ; \begin{document}$ [0,k] $\end{document} , which represents negative density dependent dispersal on the boundary. We employ the method of sub-super solutions, bifurcation theory, and stability analysis to obtain our results. We provide detailed bifurcation diagrams via a quadrature method for the case \begin{document}$ \Omega = (0,1) $\end{document} .