Dynamics of a stochastic turbidostat model with sampled and delayed measurements

Tingting Yu; Sanling Yuan; Tingting Yu; Sanling Yuan

doi:10.3934/mbe.2023268

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 4: 6215-6236. doi: 10.3934/mbe.2023268

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Dynamics of a stochastic turbidostat model with sampled and delayed measurements

Tingting Yu ,
Sanling Yuan ^,

University of Shanghai for Science and Technology, Shanghai 200093, China

Received: 07 November 2022 Revised: 19 December 2022 Accepted: 21 December 2022 Published: 31 January 2023

In this paper, a stochastic turbidostat model with controllable output is established by using piecewise constant delayed measurements of the substrate concentration. We commence by proving the existence and uniqueness of the global positive solution of the stochastic delayed model. Then, sufficient conditions of extinction and stochastic strong permanence of the biomass are acquired. In quick succession, we investigate the stochastic asymptotical stability of the washout equilibrium as well as the asymptotic behavior of the random paths approaching the interior equilibrium of its corresponding deterministic model by employing the method of Lyapunov functionals. Numerical and theoretical findings show that the influence of environmental random fluctuations on the dynamics of the model may be more pronounced than that of time delay.
- turbidostat,
- delay,
- feedback control,
- strongly stochastically permanent,
- asymptotic behavior
Citation: Tingting Yu, Sanling Yuan. Dynamics of a stochastic turbidostat model with sampled and delayed measurements[J]. Mathematical Biosciences and Engineering, 2023, 20(4): 6215-6236. doi: 10.3934/mbe.2023268

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Abstract

In this paper, a stochastic turbidostat model with controllable output is established by using piecewise constant delayed measurements of the substrate concentration. We commence by proving the existence and uniqueness of the global positive solution of the stochastic delayed model. Then, sufficient conditions of extinction and stochastic strong permanence of the biomass are acquired. In quick succession, we investigate the stochastic asymptotical stability of the washout equilibrium as well as the asymptotic behavior of the random paths approaching the interior equilibrium of its corresponding deterministic model by employing the method of Lyapunov functionals. Numerical and theoretical findings show that the influence of environmental random fluctuations on the dynamics of the model may be more pronounced than that of time delay.

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