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Erschienen in: Journal of Applied Mathematics and Computing 4/2022

07.10.2021 | Original Research

A stochastic turbidostat model coupled with distributed delay and degenerate diffusion: dynamics analysis

verfasst von: Xiaojie Mu, Daqing Jiang, Ahmed Alsaedi, Bashir Ahmad

Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 4/2022

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Abstract

Time delay, where it depends on the current state and on the past situation, is often occurred in biological activities, for example, the process by which microorganism consume nutrients into their available biomass is not instantaneous. This investigation inspects the dynamic behavior of stochastic turbidostat model coupled with distributed delay and degenerate diffusion, including sufficient conditions of the extinction and the existence of a unique stationary distribution. What’s more, the existence and uniqueness of globally positive equilibrium of the exploited model are studied. The findings manifest that the turbidostat system is ergodic only when the intensity of white noise is very small. Finally, some numerical examples are proposed to indicate the validity of the theoretical results.

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Literatur
1.
Zurück zum Zitat Guo, H., Chen, L.: Qualitative analysis of a variable yield turbidostat model with impulsive state feedback control. J. Appl. Math. Comput. 33(1–2), 193–208 (2010)MathSciNetCrossRef Guo, H., Chen, L.: Qualitative analysis of a variable yield turbidostat model with impulsive state feedback control. J. Appl. Math. Comput. 33(1–2), 193–208 (2010)MathSciNetCrossRef
2.
Zurück zum Zitat Hu, X., Li, Z., Xiang, X.: Feedback control for a turbidostat model with ratio-dependent growth rate. J. Appl. Math. Inform. 31(3–4), 385–398 (2013)MathSciNetCrossRef Hu, X., Li, Z., Xiang, X.: Feedback control for a turbidostat model with ratio-dependent growth rate. J. Appl. Math. Inform. 31(3–4), 385–398 (2013)MathSciNetCrossRef
3.
4.
Zurück zum Zitat Li, Z., Chen, L.: Periodic solution of a turbidostat model with impulsive state feedback control. Nonlinear Dyn. 58(3), 525–538 (2009)MathSciNetCrossRef Li, Z., Chen, L.: Periodic solution of a turbidostat model with impulsive state feedback control. Nonlinear Dyn. 58(3), 525–538 (2009)MathSciNetCrossRef
5.
Zurück zum Zitat Yao, Y.: Dynamics of a delay turbidostat system with contois growth rate. Math. Biosci. Eng. 16(1), 56–77 (2018)MathSciNetCrossRef Yao, Y.: Dynamics of a delay turbidostat system with contois growth rate. Math. Biosci. Eng. 16(1), 56–77 (2018)MathSciNetCrossRef
6.
Zurück zum Zitat Yao, Y., Li, Z., Xiang, H., et al.: Dynamic behaviors of a turbidostat model with Tissiet functional response and discrete delay. Adv. Differ. Equ. 2018(1), 106 (2018)MathSciNetCrossRef Yao, Y., Li, Z., Xiang, H., et al.: Dynamic behaviors of a turbidostat model with Tissiet functional response and discrete delay. Adv. Differ. Equ. 2018(1), 106 (2018)MathSciNetCrossRef
7.
Zurück zum Zitat Yao, Y., Li, Z., Liu, Z.: Hopf bifurcation analysis of a turbidostat model with discrete delay. Appl. Math. Comput. 262, 267–281 (2015)MathSciNetMATH Yao, Y., Li, Z., Liu, Z.: Hopf bifurcation analysis of a turbidostat model with discrete delay. Appl. Math. Comput. 262, 267–281 (2015)MathSciNetMATH
9.
Zurück zum Zitat Mu, Y., Li, Z., Xiang, H., et al.: Bifurcation analysis of a turbidostat model with distributed delay. Nonlinear Dyn. 90, 1315–1334 (2017)MathSciNetCrossRef Mu, Y., Li, Z., Xiang, H., et al.: Bifurcation analysis of a turbidostat model with distributed delay. Nonlinear Dyn. 90, 1315–1334 (2017)MathSciNetCrossRef
10.
Zurück zum Zitat Macdonald, N.: Time lags in biological models. In: Lecture notes in biomathematics. Springer, Heidelberg (1978) Macdonald, N.: Time lags in biological models. In: Lecture notes in biomathematics. Springer, Heidelberg (1978)
11.
Zurück zum Zitat Xu, C., Yuan, S., Zhang, T.: Stochastic sensitivity analysis for a competitive turbidostat model with inhibitory nutrients. Int. J. Bifurc. Chaos 26(10), 707–723 (2016)MathSciNetCrossRef Xu, C., Yuan, S., Zhang, T.: Stochastic sensitivity analysis for a competitive turbidostat model with inhibitory nutrients. Int. J. Bifurc. Chaos 26(10), 707–723 (2016)MathSciNetCrossRef
12.
Zurück zum Zitat Yu, M., Lo, W.: Dynamics of microorganism cultivation with delay and stochastic perturbation. Nonlinear Dyn. 101(6), 501–519 (2020)MathSciNet Yu, M., Lo, W.: Dynamics of microorganism cultivation with delay and stochastic perturbation. Nonlinear Dyn. 101(6), 501–519 (2020)MathSciNet
13.
Zurück zum Zitat Shang, Y.: The limit behavior of a stochastic logistic model with individual time-dependent rates. J. Math. 2013, 1–7 (2013)MathSciNetCrossRef Shang, Y.: The limit behavior of a stochastic logistic model with individual time-dependent rates. J. Math. 2013, 1–7 (2013)MathSciNetCrossRef
14.
Zurück zum Zitat Li, Z., Mu, Y., Xiang, H., et al.: Mean persistence and extinction for a novel stochastic turbidostat model. Nonlinear Dyn. 97(1), 185–202 (2019)CrossRef Li, Z., Mu, Y., Xiang, H., et al.: Mean persistence and extinction for a novel stochastic turbidostat model. Nonlinear Dyn. 97(1), 185–202 (2019)CrossRef
16.
Zurück zum Zitat Xu, C., Yuan, S., Zhang, T.: Competitive exclusion in a general multi-species chemostat model with stochastic perturbations. Bull. Math. Biol. 83(1), 4 (2021)MathSciNetCrossRef Xu, C., Yuan, S., Zhang, T.: Competitive exclusion in a general multi-species chemostat model with stochastic perturbations. Bull. Math. Biol. 83(1), 4 (2021)MathSciNetCrossRef
17.
Zurück zum Zitat Yu, X., Yuan, S., Zhang, T.: Asymptotic properties of a stochastic chemostat model with two distributed delays and nonlinear perturbation. Discrete Contin. Dyn. Syst. Ser. B 25(7), 2273–2290 (2020)MathSciNet Yu, X., Yuan, S., Zhang, T.: Asymptotic properties of a stochastic chemostat model with two distributed delays and nonlinear perturbation. Discrete Contin. Dyn. Syst. Ser. B 25(7), 2273–2290 (2020)MathSciNet
18.
Zurück zum Zitat Rudnicki, R., Pichór, K., Tyran-Kamińska, M.: Markov Semigroups and their Applications. Dynamics of Dissipation. Springer, Berlin (2002)MATH Rudnicki, R., Pichór, K., Tyran-Kamińska, M.: Markov Semigroups and their Applications. Dynamics of Dissipation. Springer, Berlin (2002)MATH
19.
Zurück zum Zitat Rudnicki, R., Pichór, K.: Influence of stochastic perturbation on prey–predator systems. Math. Biosci. 206(1), 108–119 (2007)MathSciNetCrossRef Rudnicki, R., Pichór, K.: Influence of stochastic perturbation on prey–predator systems. Math. Biosci. 206(1), 108–119 (2007)MathSciNetCrossRef
20.
Zurück zum Zitat Rudnicki, R.: Asymptotic Properties of the Fokker–Planck Equation, vol. 457. Springer, Berlin, pp 517–521 (1995) Rudnicki, R.: Asymptotic Properties of the Fokker–Planck Equation, vol. 457. Springer, Berlin, pp 517–521 (1995)
21.
Zurück zum Zitat Mao, X.: Stochastic Differential Equations and Applications, 2nd edn. Horwood Publishing, Cambridge (1997)MATH Mao, X.: Stochastic Differential Equations and Applications, 2nd edn. Horwood Publishing, Cambridge (1997)MATH
22.
Zurück zum Zitat Bao, K., Rong, L., Zhang, Q.: Analysis of a stochastic SIRS model with interval parameters. Discrete Contin. Dyn. Syst. B 24(9), 4827–4849 (2019)MathSciNetMATH Bao, K., Rong, L., Zhang, Q.: Analysis of a stochastic SIRS model with interval parameters. Discrete Contin. Dyn. Syst. B 24(9), 4827–4849 (2019)MathSciNetMATH
23.
Zurück zum Zitat Ben Arous, G., Léandre, R.: Décroissance exponentielle du noyau de la chaleur sur la diagonale (II). Probab. Theory Relat. Fields 90, 377–402 (1991)CrossRef Ben Arous, G., Léandre, R.: Décroissance exponentielle du noyau de la chaleur sur la diagonale (II). Probab. Theory Relat. Fields 90, 377–402 (1991)CrossRef
24.
Zurück zum Zitat Pichór, K., Rudnicki, R.: Stability of Markov semigroups and applications to parabolic systems. J. Math. Anal. Appl. 215, 56–74 (1997)MathSciNetCrossRef Pichór, K., Rudnicki, R.: Stability of Markov semigroups and applications to parabolic systems. J. Math. Anal. Appl. 215, 56–74 (1997)MathSciNetCrossRef
25.
Zurück zum Zitat Higham, D.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 433, 525–546 (2001)MathSciNetCrossRef Higham, D.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 433, 525–546 (2001)MathSciNetCrossRef
26.
Zurück zum Zitat Gao, M., Jiang, D., Hayat, T.: The threshold of a chemostat model with single-species growth on two nutrients under telegraph noise. Commun. Nonlinear Sci. Numer. Simul. 75, 160–173 (2019)MathSciNetCrossRef Gao, M., Jiang, D., Hayat, T.: The threshold of a chemostat model with single-species growth on two nutrients under telegraph noise. Commun. Nonlinear Sci. Numer. Simul. 75, 160–173 (2019)MathSciNetCrossRef
Metadaten
Titel
A stochastic turbidostat model coupled with distributed delay and degenerate diffusion: dynamics analysis
verfasst von
Xiaojie Mu
Daqing Jiang
Ahmed Alsaedi
Bashir Ahmad
Publikationsdatum
07.10.2021
Verlag
Springer Berlin Heidelberg
Erschienen in
Journal of Applied Mathematics and Computing / Ausgabe 4/2022
Print ISSN: 1598-5865
Elektronische ISSN: 1865-2085
DOI
https://doi.org/10.1007/s12190-021-01639-1

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