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arXiv · 2205.05318

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

Bertrand Cloez·Coralie Fritsch
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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.

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Bertrand Cloez, Coralie Fritsch. 2022-05-11. Quasi-stationary behavior for an hybrid model of chemostat: the Crump-Young model. https://arxiv.org/abs/2205.05318

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