Quasi-stationary behavior for an hybrid model of chemostat: the Crump-Young model

TL;DR

This paper studies the Crump-Young chemostat model, showing that conditioned on non-extinction, the microorganism population's distribution converges exponentially fast to a quasi-stationary distribution, despite the model's degenerated dynamics.

Contribution

It introduces new methods and estimates to prove exponential convergence to a quasi-stationary distribution for a degenerated hybrid stochastic model.

Findings

Conditional distribution converges exponentially fast to a quasi-stationary distribution.

The model's degenerated dynamics require novel analytical approaches.

Extinction occurs almost surely, but conditioned on survival, the system stabilizes.

Abstract

The Crump-Young model consists of two fully coupled stochastic processes modeling the substrate and microorganisms dynamics in a chemostat. Substrate evolves following an ordinary differential equation whose coefficients depend of microorganisms number. Microorganisms are modeled though a pure jump process whose the jump rates depend on the substrate concentration. It goes to extinction almost-surely in the sense that microorganism population vanishes. In this work, we show that, conditionally on the non-extinction, its distribution converges exponentially fast to a quasi-stationary distribution. Due to the deterministic part, the dynamics of the Crump-Young model is highly degenerated. The proof is then original and consists of technical sharp estimates and new approaches for the quasi-stationary convergence.