# American Institute of Mathematical Sciences

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

 1 Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC 27412, USA 2 Department of Mathematics, Auburn University Montgomery, Montgomery, AL 36124, USA 3 Department of Mathematics and Statistics, University of Maine, Orono, ME 04469, USA

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

Received  July 2020 Revised  October 2020 Published  December 2020

Fund Project: 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.
Citation: Nalin Fonseka, Jerome Goddard II, Ratnasingham Shivaji, Byungjae Son. A diffusive weak Allee effect model with U-shaped emigration and matrix hostility. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020356
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##### References:
Illustration of $1-\alpha(s)$ and $f(s)$
Bifurcation diagram for the solution set of (3)
Bifurcation diagram for the solution set of (3) for $\gamma \gg 1$ and $\epsilon \approx 0$.
Graphs of $\kappa$ vs $B_1(\kappa)$ and $\frac{\kappa^2}{\gamma^2 (g(0))^2}$
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