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Dynamics of microorganism cultivation with delay and stochastic perturbation

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Abstract

In the microorganism cultivation process, delay and stochastic perturbations are inevitably accompanied, which results in complicated dynamical behaviors for microorganisms. In this paper, a mathematical model with discrete delay and random perturbation is constructed to understand how the dynamics of microorganisms in the turbidostat can be characterized. The existence, uniqueness and boundedness of the positive solution are first determined for the mathematical model. Furthermore, sufficient conditions for microorganism extinction and permanence in the turbidostat are obtained with the theory of stochastic differential equations. The system has the stationary distribution under a low-level intensity of stochastic perturbation from the environment; that is, microorganism in the turbidostat is persistent fluctuating around a positive value. On the contrary, microorganisms will be extinct with a strong enough intensity of noise. Several numerical simulations are applied to validate the theoretical results for the dynamics of the system.

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Acknowledgements

This work was partially supported by a ECS grant from the Research Grants Council of Hong Kong (Project No. CityU 21303615) and a CityU Strategic Research Grant (Project No. CityU 11300518).

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  1. Department of Mathematics, City University of Hong Kong, Hong Kong, People’s Republic of China

    Yu Mu & Wing-Cheong Lo

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  1. Yu Mu
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  2. Wing-Cheong Lo
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Correspondence to Wing-Cheong Lo.

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Mu, Y., Lo, WC. Dynamics of microorganism cultivation with delay and stochastic perturbation. Nonlinear Dyn 101, 501–519 (2020). https://doi.org/10.1007/s11071-020-05718-z

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  • DOI: https://doi.org/10.1007/s11071-020-05718-z

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