Individual-based stochastic model with unbounded growth, birth and death rates: a tightness result

TL;DR

This paper develops a population dynamics model with unbounded growth and death rates based on individual energy traits, establishing tightness and convergence results for large populations and illustrating with allometric rate cases.

Contribution

It introduces a novel stochastic model with arbitrarily high jump rates depending on individual energies and proves tightness and convergence in large-population limits.

Findings

Established tightness of the process laws in large-population asymptotics.

Characterized accumulation points as solutions to integro-differential equations.

Presented numerical simulations for allometric rate cases.

Abstract

We study population dynamics through a general growth/degrowth-fragmentation process, with resource consumption and unbounded growth/degrowth, birth and death rates. Our model is structured in a positive trait called energy (which is a proxy for any biological parameter such as size, age, mass, protein quantity...), and the jump rates of the process can be arbitrarily high depending on individual energies, which has not been considered yet in the literature. After a preliminary study to construct well-defined objects (which is necessary contrary to similar works, because of the explosion of individual rates), we consider a classical sequence of renormalizations of the underlying process and obtain a tightness result for the associated laws in large-population asymptotics. We characterize the accumulation points of this sequence as solutions of an integro-differential system of equations, which proves the existence of measure solutions to this system. Furthermore, if such a measure solution is unique, then our tightness result becomes a convergence result towards this unique process. We illustrate our work with the case of allometric rates (i.e. they are assumed to be power functions) and eventually present numerical simulations in this allometric setting.