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A diffusive weak Allee effect model with U-shaped emigration and matrix hostility

  • * Corresponding author: Jerome Goddard II, jgoddard@aum.edu

    * Corresponding author: Jerome Goddard II, jgoddard@aum.edu 
The second author is supported by NSF grant DMS-1853372 and the third author is supported by NSF grant DMS-1853352
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  • We study positive solutions to steady state reaction diffusion equations of the form:

    $ \begin{equation*} \; \; \begin{matrix} -\Delta u = \lambda f(u);\; \Omega \\ \; \; \alpha(u)\frac{\partial u}{\partial \eta}+\gamma\sqrt{\lambda}[1-\alpha(u)]u = 0; \; \partial \Omega\end{matrix} \end{equation*} $

    where $ u $ is the population density, $ f(u) = \frac{1}{a}u(u+a)(1-u) $ represents a weak Allee effect type growth of the population with $ a\in (0,1) $, $ \alpha(u) $ is the probability of the population staying in the habitat $ \Omega $ when it reaches the boundary, and positive parameters $ \lambda $ and $ \gamma $ represent the domain scaling and effective exterior matrix hostility, respectively. In particular, we analyze the case when $ \alpha(s) = \frac{1}{[1+(A - s)^2 + \epsilon]} $ for all $ s \in [0,1] $, where $ A\in (0,1) $ and $ \epsilon\geq 0 $. In this case $ 1-\alpha(s) $ represents a U-shaped relationship between density and emigration. Existence, nonexistence, and multiplicity results for this model are established via the method of sub-super solutions.

    Mathematics Subject Classification: Primary: 35J25, 35J60; Secondary: 35J66.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  Illustration of $ 1-\alpha(s) $ and $ f(s) $

    Figure 2.  Bifurcation diagram for the solution set of (3)

    Figure 3.  Bifurcation diagram for the solution set of (3) for $ \gamma \gg 1 $ and $ \epsilon \approx 0 $.

    Figure 4.  Graphs of $ \kappa $ vs $ B_1(\kappa) $ and $ \frac{\kappa^2}{\gamma^2 (g(0))^2} $

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