Maximization of the total population in a reaction-diffusion model with logistic growth
type: Item
instance of: scholarly article
title: Maximization of the total population in a reaction-diffusion model with logistic growth
zbMATH Open document ID: 1398.35091
author: Kentaro Nagahara
author: Eiji Yanagida
published in: Calculus of Variations and Partial Differential Equations
publication date: 2018-07-16T00:00:00Z
review text: The paper deals with a nonlinear optimization problem arising in population biology. Precisely, the authors study the Neumann problem \[ \begin{cases} \Delta u+m(x)u-u^2=0 & \text{in}\;\Omega,\\ \dfrac{\partial u}{\partial n}=0 & \text{on}\;\partial \Omega, \end{cases} \] where \(\Omega\subset\mathbb{R}^N\) is a bounded and smooth domain, \(n\) is the outward normal to \(\partial\Omega,\) and \(m\) is a bounded Lebesgue measurable function. That problem appears as a stationary problem for a model of biological population growth introduced by [\textit{J.G. Skellam}, Biometrika, 38, 196--218, (1951; Zbl 0043.14401)]. Defining the total population by \[ J(m)=\int_\Omega u\;dx, \] the authors maximize \(J(m)\) under suitable constraints on \(m,\) corresponding to the real situation when resources are distributed in the habitat to control the intrinsic growth rate, but the total amount of resources is limited. They show that any local maximizer must be of ``bang-bang'' type, and this is done by computing the first and second variations of the total population. When the growth rate is not of bang-bang type, it is shown in some cases that the first variation becomes nonzero and hence the resource distribution is not a local maximizer. In the case of vanishing first variation, the authors prove that the second variation is positive.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35J91
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 6903811
zbMATH Keywords: Laplacian
zbMATH Keywords: semilinear equation
zbMATH Keywords: Neumann problem
zbMATH Keywords: population biology
MaRDI profile type: Publication
full work available at URL: s00526-018-1353-7
OpenAlex ID: W2800280465
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Wikidata QID: Q129886329
DOI: 10.1007/S00526-018-1353-7
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description: scientific article; zbMATH DE number 6903811