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

Abstract

<p style='text-indent:20px;'>We study the structure of positive solutions to steady state ecological models of the form:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded domain in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^n; $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M3">\begin{document}$ n>1 $\end{document}</tex-math></inline-formula> with smooth boundary <inline-formula><tex-math id="M4">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M5">\begin{document}$ \Omega = (0,1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ \frac{\partial}{\partial\eta} $\end{document}</tex-math></inline-formula> represents the outward normal derivative on the boundary, <inline-formula><tex-math id="M7">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> is a positive parameter, <inline-formula><tex-math id="M8">\begin{document}$ f:[0,\infty)\to \mathbb{R} $\end{document}</tex-math></inline-formula> is a <inline-formula><tex-math id="M9">\begin{document}$ C^2 $\end{document}</tex-math></inline-formula> function such that <inline-formula><tex-math id="M10">\begin{document}$ \tfrac{f(s)}{k-s}>0 $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M11">\begin{document}$ k>0 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M12">\begin{document}$ \alpha:[0,k]\to[0,1] $\end{document}</tex-math></inline-formula> is also a <inline-formula><tex-math id="M13">\begin{document}$ C^2 $\end{document}</tex-math></inline-formula> function. Here <inline-formula><tex-math id="M14">\begin{document}$ f(u) $\end{document}</tex-math></inline-formula> represents the per capita growth rate, <inline-formula><tex-math id="M15">\begin{document}$ \alpha(u) $\end{document}</tex-math></inline-formula> represents the fraction of the population that stays on the patch upon reaching the boundary, and <inline-formula><tex-math id="M16">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> relates to the patch size and the diffusion rate. In particular, we will discuss models in which the per capita growth rate is increasing for small <inline-formula><tex-math id="M17">\begin{document}$ u $\end{document}</tex-math></inline-formula>, and models where grazing is involved. We will focus on the cases when <inline-formula><tex-math id="M18">\begin{document}$ \alpha'(s)\geq 0 $\end{document}</tex-math></inline-formula>; <inline-formula><tex-math id="M19">\begin{document}$ [0,k] $\end{document}</tex-math></inline-formula>, which represents negative density dependent dispersal on the boundary. We employ the method of sub-super solutions, bifurcation theory, and stability analysis to obtain our results. We provide detailed bifurcation diagrams via a quadrature method for the case <inline-formula><tex-math id="M20">\begin{document}$ \Omega = (0,1) $\end{document}</tex-math></inline-formula>.</p>