Interactions between Human Populations and Related Problems of Optimal Transport

Human populations are rarely homogeneous, and often one would like to understand interactions between different subpopulations within a given population. The global model we propose is built on a concatenation of a so-called Resource Dependent Branching Process. A distinguished characteristic of such branching processes, designed to mimic the “human” aspect, is that individuals live in a society and typically have to work in order to survive. The objective is to establish, under natural hypotheses, criteria for survival of subpopulations in a society, and also for reaching an equilibrium between them. Apart from natality rates, mean productivity of resources, and consumption of resources, the “society” must examine policies to distribute resources among individuals. This is one concern of the present article. Second, the search of good policies leads to problems of optimal control. As discussed with Marc Hallin in 2021, some of these problems can be seen as optimal transport problems, for which Marc has developed a strong interest in recent years. Third, society may want to find criteria for the convergence of its subpopulations to equilibria, which are often delicate. The main contribution of this paper is providing new insights and new criteria in form of closed equations.