On the effects of the exterior matrix hostility and a U-shaped density dependent dispersal on a diffusive logistic growth model
On the effects of the exterior matrix hostility and a U-shaped density dependent dispersal on a diffusive logistic growth model
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外部矩阵敌意和 U 形密度依赖扩散对扩散逻辑增长模型的影响
DOI：
10.3934/dcdss.2020245
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发表时间：
2018
影响因子：
3.5
通讯作者：
B. Son
中科院分区：
文献类型：
--
作者：
N. Fonseka;R. Shivaji;J. Goddard;Q. Morris;B. Son
我们研究了种群动力学中出现的一个稳态反应扩散方程的正解，即：Begin{Document}$\Begin{等式*}\Label{abs}\Left\lbrace\Begin{Matrix}-\Delta u=\lambda u(1-u)；\；x\In\Omega\\Frc{\Partial\Eta}+\Gamma\Sqrt{\lambda}[(A-u)^2+\epsilon]u=0；\；x\in\Partial Omega\end{Matrix}\。其中，\Begin{Document}$\Omega$\end{Document}是\Begin{Document}$\mathbb{R}^N$\end{Document}；\Begin{Document}$N>；1$\end{Document}边界光滑\Begin{Document}$\Partial\Omega$\End{Document}或\Begin{Document}$\Omega=(0，1)$\end{Document}，\Begin{Document}$\frac{Partial\Eta}$\end{Document}是\Begin{Document}$\Partial\Omega$\End{Document}上的向外法导数，\Begin{Document}$\lambda$\end{Document}是域伸缩参数，\Begin{Document}$\Gamma$\end{Document}是外部矩阵(\Begin{Document}$\Omega^c$\end{Document})敌意的度量，而(0，1)$\end{Document}$\end{Document}和\Begin{Document}$\epsilon>；0$\end{文档}为常量。这里的边界条件代表了边界上的扩散是U形的情况。特别是，对于Begin{Document}$u，分散度在减小，而对于Begin{Document}$u>；A$\end{Document}，分散度在增加。我们将建立不存在、存在、多样性和唯一性的结果。特别地，我们将讨论在特定范围内发生的Allee效应。当Begin{Document}$\Omega=(0，1)$\end{Document}时，我们将提供更详细的正解分支图及其随敌意参数\Begin{Document}$\Gamma$\end{Document}变化时的演化。我们的结果表明，当Begin{Document}$\Gamma$\end{Document}较大时，对任何Begin{Document}$\lambda$\end{Document}都不存在Allee效应。当Begin{Document}$N>；1$\end{Document}时，我们使用子上解法得到存在性和多解性结果，并利用求积方法研究情形\Begin{Document}$N=1$\end{Document}。
We study positive solutions to a steady state reaction diffusion equation arising in population dynamics, namely, \begin{document}$ \begin{equation*} \label{abs} \left\lbrace \begin{matrix}-\Delta u = \lambda u(1-u) ;\; x\in\Omega\\ \frac{\partial u}{\partial \eta}+\gamma\sqrt{\lambda}[(A-u)^2+\epsilon]u = 0; \; x\in\partial \Omega \end{matrix} \right. \end{equation*} $\end{document} where \begin{document}$ \Omega $\end{document} is a bounded domain in \begin{document}$ \mathbb{R}^N $\end{document} ; \begin{document}$ N > 1 $\end{document} with smooth boundary \begin{document}$ \partial \Omega $\end{document} or \begin{document}$ \Omega = (0,1) $\end{document} , \begin{document}$ \frac{\partial u}{\partial \eta} $\end{document} is the outward normal derivative of \begin{document}$ u $\end{document} on \begin{document}$ \partial \Omega $\end{document} , \begin{document}$ \lambda $\end{document} is a domain scaling parameter, \begin{document}$ \gamma $\end{document} is a measure of the exterior matrix ( \begin{document}$ \Omega^c $\end{document} ) hostility, and \begin{document}$ A\in (0,1) $\end{document} and \begin{document}$ \epsilon>0 $\end{document} are constants. The boundary condition here represents a case when the dispersal at the boundary is U-shaped. In particular, the dispersal is decreasing for \begin{document}$ u and increasing for \begin{document}$ u>A $\end{document} . We will establish non-existence, existence, multiplicity and uniqueness results. In particular, we will discuss the occurrence of an Allee effect for certain range of \begin{document}$ \lambda $\end{document} . When \begin{document}$ \Omega = (0,1) $\end{document} we will provide more detailed bifurcation diagrams for positive solutions and their evolution as the hostility parameter \begin{document}$ \gamma $\end{document} varies. Our results indicate that when \begin{document}$ \gamma $\end{document} is large there is no Allee effect for any \begin{document}$ \lambda $\end{document} . We employ a method of sub-supersolutions to obtain existence and multiplicity results when \begin{document}$ N>1 $\end{document} , and the quadrature method to study the case \begin{document}$ N = 1 $\end{document} .