Weak Convergence of a Mass-Structured Individual-Based Model

Abstract

International audienceWe propose a model of chemostat where the bacterial population is individually-based, each bacterium is explicitly represented and has a mass evolving continuously over time. The substrate concentration is represented as a conventional ordinary differential equation. These two components are coupled with the bacterial consumption. Mechanisms acting on the bacteria are explicitly described (growth, division and washout). Bacteria interact via consumption. We set the exact Monte Carlo simulation algorithm of this model and its mathematical representation as a stochastic process. We prove the convergence of this process to the solution of an integro-differential equation when the population size tends to infinity. Finally, we propose several numerical simulation

Gaussian approximations for chemostat models in finite and infinite dimensions

“…As pointed out by (Campillo and Fritsch, 2014, Section 5.4), one can prove that these two quantities can be described as a solution to the classic chemostat equations. Moreover d dt…”

“…it is however proved in (Campillo and Fritsch, 2014, Proof of Theorem 5.2). Therefore, from Assumptions 1.1 and the Sobolev-type inequalities,…”

“…Following, for example, the approach of (Campillo and Fritsch, 2014, Lemma 5.8), we can prove that ζ Þ Ñ Ψ f t pζq is continuous from Dpr0, T s, W3 q to R in any point ζ P Cpr0, T s, W3 q. Indeed, using some rough bounds, we have the existence of a constant C ą 0, such that…”

“…This representation combines discrete mechanisms (birth and death events) as well as continuous mechanisms (mass growth and substrate dynamics). Campillo and Fritsch (2014) set the exact Monte Carlo simulation algorithm of this model and its mathematical representation as a stochastic process. They prove the convergence of this process to the solution of an integro-differential equation when the population size tends to infinity.…”

“…With these assumptions, Campillo and Fritsch (2014) show that if the sequence ps ν n 0 q n converges in distribution towards a deterministic, finite and positive measure ξ 0 then, under Assumptions 1.1, the following limit holds in distribution (see Section 2.1 for details about the topology),…”