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

Nalin Fonseka; Jerome Goddard II; Ratnasingham Shivaji; Byungjae Son

doi:10.3934/dcdsb.2020356

Article Contents

2021, Volume 26, Issue 10: 5509-5517. Doi: 10.3934/dcdsb.2020356

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

Received: June 30, 2020

Revised: September 30, 2020

Early access: November 2020

Published: October 2021

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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Abstract

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.

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

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Figure 1. Illustration of $ 1-\alpha(s) $ and $ f(s) $

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

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

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

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