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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

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  • 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:

    \begin{equation} \\ \end{equation}
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  • 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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