Stochastic modeling of algal bloom dynamics with delayed nutrient recycling

Xuehui Ji; Sanling Yuan; Tonghua Zhang; Huaiping Zhu; Xuehui Ji; Sanling Yuan; Tonghua Zhang; Huaiping Zhu

doi:10.3934/mbe.2019001

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 1: 1-24. doi: 10.3934/mbe.2019001

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Stochastic modeling of algal bloom dynamics with delayed nutrient recycling

1.
College of Science, University of Shanghai for Science and Technology,Shanghai, 200093, China
2.
College of Science, Zhongyuan University of Technology,Henan Zhengzhou, 450007, China
3.
Department of Mathematics, Swinburne University of Technology,Melbourne, VIC. 3122, Australia
4.
Department of Mathematics and Statistics, York University,Toronto, M3J 1P3, Canada

Received: 25 March 2018 Accepted: 22 June 2018 Published: 05 December 2018

Using the discrete Markov chain, in this paper we develop a stochastic model for algal bloom, in which white noise terms are introduced to describe the e ects of environmental random fluctuations and time delay to account for the time needed in the conversion of detritus into nutrient. For the proposed model, we firstly discuss the well-posedness, namely the existence and uniqueness of the global positive solution. Then, it is followed by seeking the sufficient conditions for the stochastic stability of its washout equilibrium. Then by using Fourier transform method, the spectral densities of the nutrient and the algae population are estimated. Finally, we show that larger noise can make the algae population extinct exponentially with probability one. Our theoretical and numerical results suggest that the environmental random fluctuations may have more significant influences on the dynamics of the model than the delay. These findings are helpful for a better understanding of the formation mechanism of algal blooms.
- delayed stochastic nutrient-algae model,
- discrete Markov chain,
- spectral density,
- Itô’s formula,
- lyapunov function
Citation: Xuehui Ji, Sanling Yuan, Tonghua Zhang, Huaiping Zhu. Stochastic modeling of algal bloom dynamics with delayed nutrient recycling[J]. Mathematical Biosciences and Engineering, 2019, 16(1): 1-24. doi: 10.3934/mbe.2019001

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Abstract

Using the discrete Markov chain, in this paper we develop a stochastic model for algal bloom, in which white noise terms are introduced to describe the e ects of environmental random fluctuations and time delay to account for the time needed in the conversion of detritus into nutrient. For the proposed model, we firstly discuss the well-posedness, namely the existence and uniqueness of the global positive solution. Then, it is followed by seeking the sufficient conditions for the stochastic stability of its washout equilibrium. Then by using Fourier transform method, the spectral densities of the nutrient and the algae population are estimated. Finally, we show that larger noise can make the algae population extinct exponentially with probability one. Our theoretical and numerical results suggest that the environmental random fluctuations may have more significant influences on the dynamics of the model than the delay. These findings are helpful for a better understanding of the formation mechanism of algal blooms.

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