Bifurcation curves of positive steady-state solutions for a reaction-diffusion problem of lake eutrophication
type: Item
instance of: scholarly article
title: Bifurcation curves of positive steady-state solutions for a reaction-diffusion problem of lake eutrophication
zbMATH Open document ID: 1365.34048
author: Tzung-Shin Yeh
published in: Journal of Mathematical Analysis and Applications
publication date: 2017-02-08T00:00:00Z
review text: The article contains the investigation of the exact multiplicity and bifurcation curves of positive solutions for a multiparameter Dirichlet problem \[ \begin{cases} u''(x)+\lambda \left [a-bu+\frac{u^{p}}{1+u^{p}} \right]=0, -1<x<1,\\ u(-1)=u(1)=0 ,\end{cases} \] where \(p\), \(a\), \(b\) are positive dimensionless parameters, \(\lambda>0\) is a bifurcation parameter, the function \(\frac{u^{p}}{1+u^{p}} \) with \(p>1\) is the Holling type III functional response function. For fixed \(p > 1\), it's assumed that either \(0 \leq a \leq a_{1,p}\), \(b > 0\) and \((a, b)\) lies below the curve \[ \Gamma_{1} =\left\{ (a,b): a(t)=\frac{t^{p}[p-1-t^{p}]}{(t^{p}+1)^{2}}, b(t)=\frac{pt^{p-1}}{(t^{p}+1)^{2}}, 0<t< \sqrt[p]{\frac{p-1}{p+1}}\right\} \] and the bifurcation curve is an \(S\)-shaped curve; at \(0 \leq a \leq a_{2,p}\) the point \((a, b)\) lies on or above the curve \(\Gamma_{1}\) for some positive constants \(a_{1,p}\) and \(a_{2,p}\). The main result of the article are two theorems, it includes a classification of three qualitatively different bifurcation curves on the \((\lambda, \| u\|_{\infty})\)-plane and corresponding exactly amount of positive solutions. The bifurcation curves have one of the shapes: an \(S\)-shaped curve, a broken \(S\)-shaped curve, and a monotone increasing curve. Hence, the article includes the method to determine the exact multiplicity of positive solutions by the values of \(a\), \(b\) and \(\lambda\).
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34B08
Mathematics Subject Classification ID: 37N25
zbMATH DE Number: 6682118
zbMATH Keywords: exact multiplicity
zbMATH Keywords: Holling type-III functional response
zbMATH Keywords: time map
reviewed by: Tatiana E. Badokina
MaRDI profile type: Publication
full work available at URL: j.jmaa.2016.12.063
OpenAlex ID: W2565228621
cites work: Q4816709
cites work: A theorem on S-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem
cites work: Q3646809
cites work: Q5617863
cites work: S-shaped bifurcation curves in ecosystems
cites work: Spatial patterning of the spruce budworm
cites work: Mathematical biology. Vol. 1: An introduction.
cites work: Mathematical biology. Vol. 2: Spatial models and biomedical applications.
cites work: Exact multiplicity of positive solutions for a class of semilinear problem. II
cites work: Persistence in reaction diffusion models with weak Allee effect
cites work: Q4022840
cites work: S-shaped and broken S-shaped bifurcation diagrams with hysteresis for a multiparameter spruce budworm population problem in one space dimension
cites work: S-shaped and broken S-shaped bifurcation curves for a multiparameter diffusive logistic problem with Holling type-III functional response
DOI: 10.1016/J.JMAA.2016.12.063
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description: scientific article; zbMATH DE number 6682118