Maximization of the total population in a reaction-diffusion model with logistic growth
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.
scientific article; zbMATH DE number 6903811
scientific article