Abstract
We revisit the chemostat model with Haldane growth function, here subject to bounded random disturbances on the input flow rate, as often met in biotechnological or waste-water industry. We prove existence and uniqueness of global positive solution of the random dynamics and existence of absorbing and attracting sets that are independent of the realizations of the noise. We study the long-time behavior of the random dynamics in terms of attracting sets, and provide first conditions under which biomass extinction cannot be avoided. We prove conditions for weak and strong persistence of the microbial species and provide lower bounds for the biomass concentration, as a relevant information for practitioners. The theoretical results are illustrated with numerical simulations.
Highlights
The chemostat refers to a laboratory device used for the growth of micro-organisms in a culture environment [1,2], that has been regarded as an idealization of the nature to study microbial ecosystems in stationary stage [3]
Let us underline that we do not impose the upper bound Dr of the variations of the removal rate to fulfill this inequality. This means that one could have realizations of the disturbances such that the effective value of the removal rate is above μ(sin) on large periods of time making the species to be arbitrary closed to the extinction but it will always persist
We have considered the chemostat model (1.1)–(1.2) with Haldane consumption kinetics, under bounded perturbations on the input flow rate, motivated by real cases in industrial setup and biotechnology
Summary
The chemostat refers to a laboratory device used for the growth of micro-organisms in a culture environment [1,2], that has been regarded as an idealization of the nature to study microbial ecosystems in stationary stage [3]. The approach proposed in this paper allows to prove the persistence of the bacterial species (under some conditions on the growth function), as it is observed by practitioners on very long time periods despite variations of the input flow rate This is not the case when considering the Wiener process where persistence cannot be ensured (see [16] and [17] where the Wiener process is used to model disturbances on the input flow and environmental perturbations in the classical deterministic chemostat).
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