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

This material is based upon work supported by the National Science Foundation under Grant Nos. DMS-1516519 & DMS-1516560

• 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 $N = 1$.

Mathematics Subject Classification: Primary: 35J25, 35J60, 35J66, 34B18.

 Citation: • • Figure 1.  Habitat $\Omega$ and the exterior matrix $\Omega^c$

Figure 2.  An example that illustrates U-shaped density dependent dispersal ($1-\alpha(u)$) on the boundary

Figure 3.  Eigencurve $B(\kappa)$ and principal eigenvalue of (1.5)

Figure 4.  Bifurcation diagrams for (1.4)

Figure 5.  Shape of a positive solution

Figure 6.  Plot that illustrates the existence of $\epsilon_{\lambda}$

Figure 7.  The graph of $H(q)$

Figure 8.  Evolution of bifurcation diagrams for (1.8) as $\gamma$ varies when $\epsilon = 0.1$ and $A = 0.5$

Figure 9.  Bifurcation diagrams for (1.8) for several values of $\gamma$, when $\epsilon = 0.01$ and $A = 0.8.$

Figure 10.  Picture that illustrates that if $\lambda < E_1(\gamma,D)$ then $\sigma_1(\lambda,\gamma,D)>0$

Figure 11.  The plot illustrates the existence of $\kappa_1(\lambda,\gamma,D)$

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