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

“…Introduction. Recently, there has been growing interest on the study of the reaction diffusion models where a parameter influences the equation as well as the boundary condition (see [1], [2], [6], [8], [7], and [9]). In this paper, we study reaction diffusion systems of the form:…”

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$ \Sigma $-shaped bifurcation curves for classes of elliptic systems

Acharya

Shivaji

Fonseka³

2022

DCDS-S

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We study positive solutions to classes of steady state reaction diffusion systems of the form:<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\lbrace \begin{matrix}-\Delta u = \lambda f(v) ;\; \Omega\\ -\Delta v = \lambda g(u) ;\; \Omega\\ \frac{\partial u}{\partial \eta}+\sqrt{\lambda} u = 0; \; \partial \Omega\\ \frac{\partial v}{\partial \eta}+\sqrt{\lambda}v = 0; \; \partial \Omega\ \end{matrix} \right. \end{equation*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M2">\begin{document}$ \lambda>0 $\end{document}</tex-math></inline-formula> is a positive parameter, <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded domain in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>; <inline-formula><tex-math id="M5">\begin{document}$ N > 1 $\end{document}</tex-math></inline-formula> with smooth boundary <inline-formula><tex-math id="M6">\begin{document}$ \partial \Omega $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M7">\begin{document}$ \Omega = (0, 1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \frac{\partial z}{\partial \eta} $\end{document}</tex-math></inline-formula> is the outward normal derivative of <inline-formula><tex-math id="M9">\begin{document}$ z $\end{document}</tex-math></inline-formula>. Here <inline-formula><tex-math id="M10">\begin{document}$ f, g \in C^2[0, r) \cap C[0, \infty) $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M11">\begin{document}$ r>0 $\end{document}</tex-math></inline-formula>. Further, we assume that <inline-formula><tex-math id="M12">\begin{document}$ f $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ g $\end{document}</tex-math></inline-formula> are increasing functions such that <inline-formula><tex-math id="M14">\begin{document}$ f(0) = 0 = g(0) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M15">\begin{document}$ f'(0) = g'(0) = 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M16">\begin{document}$ f''(0)>0, g''(0)>0 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M17">\begin{document}$ \lim\limits_{s \rightarrow \infty} \frac{f(Mg(s))}{s} = 0 $\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id="M18">\begin{document}$ M>0 $\end{document}</tex-math></inline-formula>. Under certain additional assumptions on <inline-formula><tex-math id="M19">\begin{document}$ f $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M20">\begin{document}$ g $\end{document}</tex-math></inline-formula> we prove that the bifurcation diagram for positive solutions of this system is at least <inline-formula><tex-math id="M21">\begin{document}$ \Sigma- $\end{document}</tex-math></inline-formula>shaped. We also discuss an example where <inline-formula><tex-math id="M22">\begin{document}$ f $\end{document}</tex-math></inline-formula> is sublinear at <inline-formula><tex-math id="M23">\begin{document}$ \infty $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M24">\begin{document}$ g $\end{document}</tex-math></inline-formula> is superlinear at <inline-formula><tex-math id="M25">\begin{document}$ \infty $\end{document}</tex-math></inline-formula> which satisfy our hypotheses.

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Analysis of reaction–diffusion systems where a parameter influences both the reaction terms as well as the boundary

Acharya

Fonseka²,

Shivaji

2021

Bound Value Probl

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We study positive solutions to steady-state reaction–diffusion models of the form $$ \textstyle\begin{cases} -\Delta u=\lambda f(v);\quad\Omega, \\ -\Delta v=\lambda g(u);\quad\Omega, \\ \frac{\partial u}{\partial \eta }+\sqrt{\lambda } u=0;\quad \partial \Omega, \\ \frac{\partial v}{\partial \eta }+\sqrt{\lambda }v=0; \quad\partial \Omega, \end{cases} $$ { − Δ u = λ f ( v ) ; Ω , − Δ v = λ g ( u ) ; Ω , ∂ u ∂ η + λ u = 0 ; ∂ Ω , ∂ v ∂ η + λ v = 0 ; ∂ Ω , where $\lambda >0$ λ > 0 is a positive parameter, Ω is a bounded domain in $\mathbb{R}^{N}$ R N $(N > 1)$ ( N > 1 ) with smooth boundary ∂Ω, or $\Omega =(0,1)$ Ω = ( 0 , 1 ) , $\frac{\partial z}{\partial \eta }$ ∂ z ∂ η is the outward normal derivative of z. We assume that f and g are continuous increasing functions such that $f(0) = 0 = g(0)$ f ( 0 ) = 0 = g ( 0 ) and $\lim_{s \rightarrow \infty } \frac{f(Mg(s))}{s} = 0$ lim s → ∞ f ( M g ( s ) ) s = 0 for all $M>0$ M > 0 . In particular, we extend the results for the single equation case discussed in (Fonseka et al. in J. Math. Anal. Appl. 476(2):480-494, 2019) to the above system.

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

Cited by 4 publications

References 11 publications

On a fractional reaction-diffusion models arising in population dynamics

On a fractional reaction-diffusion models arising in population dynamics

$ \Sigma $-shaped bifurcation curves for classes of elliptic systems

Analysis of reaction–diffusion systems where a parameter influences both the reaction terms as well as the boundary

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