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2019 A diffusive logistic equation with U-shaped density dependent dispersal on the boundary
Jerome Goddard II, Quinn Morris, Catherine Payne, Ratnasingham Shivaji
Topol. Methods Nonlinear Anal. 53(1): 335-349 (2019). DOI: 10.12775/TMNA.2018.047

Abstract

We study positive solutions to the steady state reaction diffusion equation: \begin{equation*} \begin{cases} - \Delta v = \lambda v(1-v), & x \in \Omega_0, \\ \frac{\partial v}{\partial \eta} + \gamma \sqrt{\lambda} ( v-A)^2 v =0 , & x \in \partial \Omega_0, \end{cases} \end{equation*} where $\Omega_0$ is a bounded domain in $\mathbb{R}^n$; $n \ge 1$ with smooth boundary $\partial \Omega_0$, ${\partial }/{\partial \eta}$ is the outward normal derivative, $A \in (0,1)$ is a constant, and $\lambda$, $\gamma$ are positive parameters. Such models arise in the study of population dynamics when the population exhibits a U-shaped density dependent dispersal on the boundary of the habitat. We establish existence, multiplicity, and uniqueness results for certain ranges of the parameters $\lambda$ and $\gamma$. We obtain our existence and mulitplicity results via the method of sub-super solutions.

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Jerome Goddard II. Quinn Morris. Catherine Payne. Ratnasingham Shivaji. "A diffusive logistic equation with U-shaped density dependent dispersal on the boundary." Topol. Methods Nonlinear Anal. 53 (1) 335 - 349, 2019. https://doi.org/10.12775/TMNA.2018.047

Information

Published: 2019
First available in Project Euclid: 14 January 2019

zbMATH: 07068340
MathSciNet: MR3939159
Digital Object Identifier: 10.12775/TMNA.2018.047

Rights: Copyright © 2019 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.53 • No. 1 • 2019
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