[Paper Review] Gaussian approximations for chemostat models in finite and infinite dimensions

Bertrand Cloez, Coralie Fritsch|arXiv (Cornell University)|Sep 26, 2016

Stochastic processes and statistical mechanics51 references4 citations

TL;DR

This paper establishes central limit theorems for fluctuation processes in mass-structured individual-based chemostat models, proving convergence to an infinite-dimensional Gaussian process in Sobolev spaces. For the two-dimensional Crump-Young model, it derives a stochastic differential approximation with explicit invariant distribution and extinction time behavior, resolving long-term dynamics previously misunderstood.

ABSTRACT

In a chemostat, bacteria live in a growth container of constant volume in which liquid is injected continuously. Recently, Campillo and Fritsch introduced a mass-structured individual-based model to represent this dynamics and proved its convergence to a more classic partial differential equation. In this work, we are interested in the convergence of the fluctuation process. We consider this process in some Sobolev spaces and use central limit theorems on Hilbert space to prove its convergence in law to an infinite-dimensional Gaussian process.As a consequence, we obtain a two-dimensional Gaussian approximation of the Crump-Young model for which the long time behavior is relatively misunderstood. For this approximation, we derive the invariant distribution and the convergence to it. We also present numerical simulations illustrating our results.

Motivation & Objective

• Address the long-standing challenge of understanding the long-time behavior of the stochastic Crump-Young model, which lacks a complete characterization of extinction and quasi-stationary distributions.
• Extend the deterministic convergence of individual-based chemostat models to the fluctuation scale by proving weak convergence of the fluctuation process to a Gaussian process.
• Provide a rigorous infinite-dimensional central limit theorem for measure-valued processes in Sobolev spaces, overcoming the non-metrizability of the space of signed measures.
• Derive a stochastic differential equation approximation for the Crump-Young model that captures the invariant distribution and extinction time dynamics.
• Validate the approximation through numerical simulations and compare it with alternative diffusion processes exhibiting better mimicry of the original model in certain regimes.

Proposed method

• Model the chemostat as a mass-structured individual-based process with discrete birth/death events and continuous substrate dynamics, using a signed measure-valued process to represent population and substrate levels.
• Apply a tightness-uniqueness argument in a Sobolev space framework to prove weak convergence of the normalized fluctuation process to an infinite-dimensional degenerate Gaussian process.
• Use martingale techniques on Hilbert spaces (following Métivier, 1984) to establish tightness, leveraging the Hilbertian structure of Sobolev spaces to handle the signed measure nature of the process.
• Reduce the infinite-dimensional model to finite dimensions by focusing on the two-dimensional Crump-Young model, where the population and substrate are aggregated into a single population mass and substrate concentration.
• Derive a stochastic differential equation approximation for the Crump-Young model by applying the central limit theorem in finite dimensions, resulting in a diffusion process with drift and diffusion coefficients derived from the original model.
• Apply Itô’s formula and Feller’s test for explosions to analyze the extinction time of the diffusion approximation, proving positive probability of extinction in finite time under general conditions on the growth function.

Experimental results

Research questions

• RQ1 Can the fluctuation process of a mass-structured individual-based chemostat model converge weakly to a Gaussian process in infinite dimensions?
• RQ2 What is the long-time behavior of the stochastic Crump-Young model, particularly regarding the existence and properties of its invariant distribution and extinction time?
• RQ3 How does the Gaussian approximation derived from the central limit theorem compare to the original Crump-Young model in terms of numerical behavior and extinction dynamics?
• RQ4 Under what conditions does the diffusion approximation of the Crump-Young model exhibit finite-time extinction with positive probability?
• RQ5 Can the invariant distribution of the stochastic approximation be explicitly characterized, and does it reflect the true long-term behavior of the original model?

Key findings

• The normalized fluctuation process of the individual-based chemostat model converges in law to an infinite-dimensional degenerate Gaussian process in a suitable Sobolev space.
• For the two-dimensional Crump-Young model, the central limit theorem yields a stochastic differential equation approximation whose invariant distribution is explicitly characterized.
• The extinction time of the diffusion approximation is finite almost surely, and the probability of extinction within any finite time interval is positive for all positive initial conditions.
• The extinction time distribution satisfies a Lyapunov-type condition, and the process exhibits a quasi-stationary distribution that satisfies a necessary condition derived from the Fokker-Planck equation.
• Numerical simulations confirm that the derived diffusion process mimics the original Crump-Young model better than alternative approximations in certain parameter regimes, particularly in capturing extinction dynamics.
• The result provides a new proof of extinction in finite time for the Crump-Young model without requiring monotonicity assumptions on the growth function, extending previous results by Collet et al. (2013a).