Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models

TL;DR

This paper compares stochastic and deterministic models for invasion in growth-fragmentation-death systems, establishing their equivalence and linking their mathematical frameworks to better understand population invasion dynamics.

Contribution

It introduces a unified analysis connecting stochastic and integro-differential approaches, proving their equivalence and providing conditions for their joint applicability.

Findings

Both approaches yield the same invasion criterion.

Existence of solutions to the eigenproblem is established.

Conditions for applying both models are identified.

Abstract

We present two approaches to study invasion in growth-fragmentation-death mod- els. The first one is based on a stochastic individual based model, which is a piecewise deterministic branching process with a continuum of types, and the second one is based on an integro-differential model. The invasion of the population is described by the survival probability for the former model and by an eigenproblem for the latter one. We study these two notions of invasion fitness, giving different characterizations of the growth of the population, and we make links between these two complementary points of view. In particular we prove that the two approaches lead to the same crite- rion of possible invasion. Based on Krein-Rutman theory, we also give a proof of the existence of a solution to the eigenproblem, which satisfies the conditions needed for our study of the stochastic model, hence providing a set of assumptions under which both approaches can be carried out. Finally, we motivate our work in the context of adaptive dynamics in a chemostat model.