| Abstract: |
| We analyze the structure of positive steady states for a population model designed to explore the effects of habitat fragmentation, density dependent emigration, and Allee effect growth. The steady state reaction diffusion equation is:
$\begin{equation*}
\left\lbrace \begin{matrix} -\Delta u = \lambda f(u);~ \Omega \
\frac{\partial u}{\partial \eta}+ \gamma \sqrt{\lambda}g(u) u=0 ; ~\partial \Omega \end{matrix} \right.
\end{equation*}$
where $f(s) = \frac{1}{a}s(1-s)(a+s)$ can represent either logistic-type growth ($a \geq 1$) or weak Allee affect growth ($a \in (0,1)$), $\lambda, \gamma > 0$ are parameters, $\Omega$ 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 $u$, and $g(u)$ is related to the relationship between density and emigration. In particular, we consider three forms of emigration: density independent emigration ($g = 1$), a negative density dependent emigration of the form $g(s) = \frac{1}{1 +\beta s}$, and a positive density dependent emigration of the form $g(s) = 1 +\beta s $, where $\beta > 0$ is a parameter representing the interaction strength. We establish existence, nonexistence, and multiplicity results for ranges of $\lambda$ depending on the choice of the function $g$. Our existence and multiplicity results are proved via the method of sub-super-solutions and study of certain eigenvalue problems. For the case $\Omega = (0,1),$ we also provide exact bifurcation diagrams for positive solutions for certain values of the parameters $a, \beta$ and $\gamma$ via a quadrature method and Mathematica computations. Our results shed light on the complex interactions of density dependent mechanisms on population dynamics in the presence of habitat fragmentation. |
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