Abstract
In this article, we give an in-depth analysis of the problem of optimising the total population size for a standard logistic-diffusive model. This optimisation problem stems from the study of spatial ecology and amounts to the following question: assuming a species evolves in a domain, what is the best way to spread resources in order to ensure a maximal population size at equilibrium? In recent years, many authors contributed to this topic. We settle here the proof of two fundamental properties of optimisers: the bang-bang one, which had so far only been proved under several strong assumptions, and the other one is the fragmentation of maximisers. We prove the bang-bang property in all generality using a new spectral method. The technique introduced to demonstrate the bang-bang character of optimisers can be adapted and generalised to many optimisation problems with other classes of bilinear optimal control problems where the state equation is semilinear and elliptic. We comment on it in a conclusion section. Regarding the geometry of maximisers, we exhibit a blow-up rate for the BV-norm of maximisers as the diffusivity gets smaller: if is an orthotope and if is an optimal control, then The proof of this results relies on a very fine energy argument.
Highlights
This article is devoted to the study of a problem of calculus of variations motivated by questions of spatial ecology
We present here a new method that we believe to be flexible and versatile enough to be applied to a wide class of bilinear optimal control problem, and that provides a positive answer to the question of knowing whether optimal resource distributions are bang-bang
We prove a fragmentation phenomenon, with explicit blow-up rates: as has been noticed [24, 21, 22], for the optimisation of the total population size, the characteristic dispersal rate of the population has a drastic influence on the geometry of optimal resource distributions
Summary
This article is devoted to the study of a problem of calculus of variations motivated by questions of spatial ecology. This problem is related to the ubiquitous question of optimal location of resources. While we further specify what we mean by “optimal” in what follows, let us note that optimisation problems related to the location of resources are a possible way to tackle the question of spatial. In this context, spatial heterogeneity is interpreted as heterogeneity of the resources available to a given population.
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