Hopf bifurcation and stability analysis of a delay differential equation model for biodegradation of a class of microcystins

Luyao Zhao; Mou Li; Wanbiao Ma; Luyao Zhao; Mou Li; Wanbiao Ma

doi:10.3934/math.2024899

AIMS Mathematics

2024, Volume 9, Issue 7: 18440-18474. doi: 10.3934/math.2024899

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Research article

Hopf bifurcation and stability analysis of a delay differential equation model for biodegradation of a class of microcystins

School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

Received: 14 March 2024 Revised: 20 April 2024 Accepted: 28 April 2024 Published: 03 June 2024
MSC : 34K18, 34K20, 92B05

In this paper, a delay differential equation model is investigated, which describes the biodegradation of microcystins (MCs) by Sphingomonas sp. and its degrading enzymes. First, the local stability of the positive equilibrium and the existence of the Hopf bifurcation are obtained. Second, the global attractivity of the positive equilibrium is obtained by constructing suitable Lyapunov functionals, which implies that the biodegradation of microcystins is sustainable under appropriate conditions. In addition, some numerical simulations of the model are carried out to illustrate the theoretical results. Finally, the parameters of the model are determined from the experimental data and fitted to the data. The results show that the trajectories of the model fit well with the trend of the experimental data.
- microcystins,
- sphingomonas sp.,
- enzyme,
- biodegradation,
- delay differential equations,
- Hopf bifurcation,
- Lyapunov functional
Citation: Luyao Zhao, Mou Li, Wanbiao Ma. Hopf bifurcation and stability analysis of a delay differential equation model for biodegradation of a class of microcystins[J]. AIMS Mathematics, 2024, 9(7): 18440-18474. doi: 10.3934/math.2024899

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Abstract

In this paper, a delay differential equation model is investigated, which describes the biodegradation of microcystins (MCs) by Sphingomonas sp. and its degrading enzymes. First, the local stability of the positive equilibrium and the existence of the Hopf bifurcation are obtained. Second, the global attractivity of the positive equilibrium is obtained by constructing suitable Lyapunov functionals, which implies that the biodegradation of microcystins is sustainable under appropriate conditions. In addition, some numerical simulations of the model are carried out to illustrate the theoretical results. Finally, the parameters of the model are determined from the experimental data and fitted to the data. The results show that the trajectories of the model fit well with the trend of the experimental data.

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