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Exclusion and persistence in deterministic and stochastic chemostat models. (English) Zbl 1089.34041
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.
MSC:
34D05Asymptotic stability of ODE
34F05ODE with randomness
34C60Qualitative investigation and simulation of models (ODE)
92D25Population dynamics (general)