Stochastic models for the chemostat at different scales

International audienceThe evolution of the state of a single species/single substrate chemostat is usually described by a set of ordinary differential equations (ODE) derived from a mass balance principle. In this case, the modeling process relies on the fact that the stochastic effects can be neglected thanks to the law of large numbers. This is possible only at macroscopic scale, for high population sizes, and under homogeneity conditions. At all other scales or when the homogeneity conditions are not met, random effects cannot be neglected. Our goal is to establish a set of stochastic models that are valid at different scales: from the small population scale to the scale immediately preceding the one corresponding to the deterministic model. At a microscopic scale we present a pure jump stochastic model that gives rise, at the macroscopic scale, to the ordinary differential equation model. At an intermediate scale, an approximation diffusion allows us to propose a model in the form of a system of stochastic differential equations. The convergence of the pure jump model or of the diffusion approximation to the deterministic model can be rigorously established. We expound the mechanism to switch from one model to another, together with the associated simulation procedures. Three associated simulation algorithms that will be valid at different scales are presented. We also describe the domain of validity of the different models. The pure jump model can be exactly simulated thanks to the Gillespie algorithm, also called stochastic simulation algorithm. In standard cases, that is for high population levels, this procedure is not feasible as it requires us to simulate too many events. In this case, we present the Poisson approximation and the normal approximation, both in discrete-time