<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>
In this article we prove a three solution type theorem for the following boundary value problem: \begin{equation*} \label{abs} \begin{cases} -\mathcal{M}_{\lambda,\Lambda}^+(D^2u) =f(u)& \text{in }\Omega,\\ u =0& \text{on }\partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ and $f\colon [0,\infty]\to[0,\infty]$ is a $C^{\alpha}$ function. This is motivated by the work of Amann \cite{aman} and Shivaji \cite{shivaji1987remark}, where a three solutions theorem has been established for the Laplace operator. Furthermore, using this result we show the existence of three positive solutions to above boundary value by explicitly constructing two ordered pairs of sub and supersolutions when $f$ has a sublinear growth and $f(0)=0.$
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