On the effects of the exterior matrix hostility and a U-shaped density dependent dispersal on a diffusive logistic growth model

Fonseka, Nalin; Shivaji, R.; Goddard, Jerome; Morris, Quinn; Son, Byungjae

doi:10.3934/dcdss.2020245

Cited by 5 publications

(2 citation statements)

References 11 publications

(16 reference statements)

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“…Cantrell and Cosner have studied the case of density dependent dispersal in the logistic growth case (see [3,2,4]). See also [13,17,18,16,14,10] for additional results in the case of the logistic growth and [10,9] for the case of weak Allee growth. Our results for (1.1) will enrich this literature and also include the case when f (u) = f 3 (u).…”

Section: Introductionmentioning

confidence: 99%

Bifurcation, uniqueness and multiplicity results for classes of reaction diffusion equations arising in ecology with nonlinear boundary conditions

Mallick¹,

Sasi²,

Shivaji³

et al. 2022

CPAA

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We study the structure of positive solutions to steady state ecological models of the form:<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{array}{l} \left\{ \begin{split} -\Delta u& = \lambda uf(u)\; \; && {\rm{in}}\; \; \Omega,\\ \alpha(u)&\frac{\partial u}{\partial \eta}+[1-\alpha(u)]u = 0 &&\;\;\;{\rm{on}}\; \; \partial\Omega, \end{split} \right. \end{array} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded domain in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^n; $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M3">\begin{document}$ n>1 $\end{document}</tex-math></inline-formula> with smooth boundary <inline-formula><tex-math id="M4">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M5">\begin{document}$ \Omega = (0,1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ \frac{\partial}{\partial\eta} $\end{document}</tex-math></inline-formula> represents the outward normal derivative on the boundary, <inline-formula><tex-math id="M7">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> is a positive parameter, <inline-formula><tex-math id="M8">\begin{document}$ f:[0,\infty)\to \mathbb{R} $\end{document}</tex-math></inline-formula> is a <inline-formula><tex-math id="M9">\begin{document}$ C^2 $\end{document}</tex-math></inline-formula> function such that <inline-formula><tex-math id="M10">\begin{document}$ \tfrac{f(s)}{k-s}>0 $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M11">\begin{document}$ k>0 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M12">\begin{document}$ \alpha:[0,k]\to[0,1] $\end{document}</tex-math></inline-formula> is also a <inline-formula><tex-math id="M13">\begin{document}$ C^2 $\end{document}</tex-math></inline-formula> function. Here <inline-formula><tex-math id="M14">\begin{document}$ f(u) $\end{document}</tex-math></inline-formula> represents the per capita growth rate, <inline-formula><tex-math id="M15">\begin{document}$ \alpha(u) $\end{document}</tex-math></inline-formula> represents the fraction of the population that stays on the patch upon reaching the boundary, and <inline-formula><tex-math id="M16">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> relates to the patch size and the diffusion rate. In particular, we will discuss models in which the per capita growth rate is increasing for small <inline-formula><tex-math id="M17">\begin{document}$ u $\end{document}</tex-math></inline-formula>, and models where grazing is involved. We will focus on the cases when <inline-formula><tex-math id="M18">\begin{document}$ \alpha'(s)\geq 0 $\end{document}</tex-math></inline-formula>; <inline-formula><tex-math id="M19">\begin{document}$ [0,k] $\end{document}</tex-math></inline-formula>, which represents negative density dependent dispersal on the boundary. We employ the method of sub-super solutions, bifurcation theory, and stability analysis to obtain our results. We provide detailed bifurcation diagrams via a quadrature method for the case <inline-formula><tex-math id="M20">\begin{document}$ \Omega = (0,1) $\end{document}</tex-math></inline-formula>.

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Section: Introductionmentioning

confidence: 99%

Bifurcation, uniqueness and multiplicity results for classes of reaction diffusion equations arising in ecology with nonlinear boundary conditions

Mallick¹,

Sasi²,

Shivaji³

et al. 2022

CPAA

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“…Even though a recent literature review ( [10]) found that U-shaped density dependent emigration was ecologically important, little is known about the population dynamical effects of such an emigration relationship. Notwithstanding, the authors have studied the effects of UDDE on populations governed by a logistic growth in [7] and [5]. In order to state our main results, we first consider the following eigenvalue problem.…”

mentioning

confidence: 99%

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

Fonseka¹,

Goddard²,

Shivaji³

et al. 2021

DCDS-B

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A study of logistic growth models influenced by the exterior matrix hostility and grazing in an interior patch

Fonseka¹,

Machado²,

Shivaji³

2020

Electron. J. Qual. Theory Differ. Equ.

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On the effects of the exterior matrix hostility and a U-shaped density dependent dispersal on a diffusive logistic growth model

Cited by 5 publications

References 11 publications

Bifurcation, uniqueness and multiplicity results for classes of reaction diffusion equations arising in ecology with nonlinear boundary conditions

Bifurcation, uniqueness and multiplicity results for classes of reaction diffusion equations arising in ecology with nonlinear boundary conditions

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

A study of logistic growth models influenced by the exterior matrix hostility and grazing in an interior patch

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