Article Contents
Article Contents

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

• * 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
• 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.

 Citation:

• 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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