Quasi-stationary behavior for a piecewise deterministic Markov model of chemostat: the Crump–Young model
Annales Henri Lebesgue, Volume 6 (2023), pp. 1371-1427.


Keywords Quasi-stationary distribution, Chemostat model, Lyapunov function, Crump–Young model, Piecewise Deterministic Markov Process (PDMP)


The Crump–Young model consists of two fully coupled stochastic processes modeling the substrate and micro-organisms dynamics in a chemostat. Substrate evolves following an ordinary differential equation whose coefficients depend of micro-organisms number. Micro-organisms are modeled though a pure jump process whose jump rates depend on the substrate concentration.

It goes to extinction almost-surely in the sense that micro-organism 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 are highly degenerated. The proof is therefore original and consists of technically precise estimates and new approaches for quasi-stationary convergence.


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