Some challenging optimization problems for logistic diffusive equations and their numerical modeling

Abstract

This chapter is dedicated to the study of several shape optimization problems arising in population dynamics. We provide some answers to the generic question: what is the optimal way to arrange resources inside a domain? Of course, the wording “optimal” refers to different criteria, which may be the survival ability of a given species or the total population size. We consider here a very simple spatial ecology model in which the evolution of the population is governed by the logistic diffusive equation parametrized by the resources distribution inside the domain, denoted m ( ⋅ ) , which is the main optimization variable and models the favorable and unfavorable parts of the habitat. We investigate here two optimal design problems: the first one is related to the species persistence for large times. It boils down to the optimization of the principal eigenvalue associated with an elliptic operator with respect to the resources distributions m ( ⋅ ) . The second one deals with steady-states of the aforementioned reaction-diffusion equation and aims at maximizing the total size of the population with respect to resources distributions. In our analysis, we mainly focus on qualitative properties of maximizers, and illustrate it with the help of numerical illustrations. We also highlight related open problems and interesting numerical issues that remain to be investigated.