Interactions between resource dependent branching processes and equilibria

TL;DR

This supplement to Bruss (2024) provides rigorous proofs and practical commentary on resource dependent branching processes (RDBP) with a focus on equilibria between sub-populations. It derives an asymptotic two-type equilibrium characterized by the pair $(\alpha,\tau)$ satisfying $m_h \int_0^{\tau} x\, dF_h(x) + \alpha m_i \int_0^{\tau} x\, dF_i(x) = r_h + \alpha r_i$ and the constraint $m_h F_h(\tau) = m_i F_i(\tau) \ge 1$, and establishes necessity and sufficiency via a four-part argument. It extends the analysis to multiple sub-populations through the BRS-inequality and connects the equilibrium problem to optimal transport under distribution-control constraints, outlining Monge-type solutions and computational considerations. It also discusses control and stopping perspectives, proposing a framework that blends measure transport with statistical decision theory to guide population dynamics.

Abstract

This paper is a supplement to the paper "Interactions between Human Populations and Related Problems of Optimal Transport" written by the same author in honour of Marc Hallin, Université Libre de Bruxelles, at the occasion of Hallin's $75$th birthday. It was announced in the main paper (Bruss (2024)) published in the Springer Festschrift entitled {\it Recent Advances in Econometrics and Statistics}. It contains the proofs which, given the space constraints required for the Festschrift, could not appear in the main paper. Moreover, we complement in the present supplement the main paper by brief comments on related problems which are likely to turn up in practice for problems of guiding human populations, namely problems of control and problems of optimal stopping.