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Zbl 1089.34041
Imhof, L.; Walcher, S.
Exclusion and persistence in deterministic and stochastic chemostat models. (English)
[J] J. Differ. Equations 217, No. 1, 26-53 (2005). ISSN 0022-0396

The authors first introduce and discuss a variant of the standard deterministic chemostat model. The principal difference is that microbe removal and growth rates depend on the biomass concentration with removal terms increasing faster than growth terms. In the single species scenario, they turn out that the qualitative behavior is virtually indistinguishable from the standard chemostat model. In the multiple species scenario, by using a comparison principle, they prove that persistence of all species is possible. Then, the authors turn to modelling the influence of random fluctuations by setting up and analyzing a stochastic differential equation. In the single species case, they show that if the deterministic counterpart admits persistence, and the stochastic effects are not too strong, then there are a recurrent system and, with some additional assumptions, the stochastic solution can be expected to remain close to the interior deterministic stationary point. In the two species case, they prove a transient result and show that under certain conditions, the stochastic model leads to extinction even though the deterministic counterpart predicts persistence.
[Wan-Tong Li (Lanzhou)]

MSC 2000:: *34D05 Asymptotic stability of ODE
34F05 ODE with randomness
34C60 Applications of qualitative theory of ODE
92D25 Population dynamics

Keywords: deterministic chemostat model; stochastic differential equation; exclusion; persistence

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