Global analysis and optimal harvesting for a hybrid stochastic phytoplankton-zooplankton-fish model with distributed delays

Yuanpei Xia; Weisong Zhou; Zhichun Yang; Yuanpei Xia; Weisong Zhou; Zhichun Yang

doi:10.3934/mbe.2020326

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 6149-6180. doi: 10.3934/mbe.2020326

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Global analysis and optimal harvesting for a hybrid stochastic phytoplankton-zooplankton-fish model with distributed delays

1.
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401131, China
2.
School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Received: 15 June 2020 Accepted: 24 August 2020 Published: 14 September 2020

In this paper, we formulate a phytoplankton-zooplankton-fish model with distributed delays and hybrid stochastic noises involving Brownian motion and Markov chain, and propose an optimal harvesting problem pursuing the maximum of total economic income. By global analysis in terms of some system parameters, we investigate the dynamical behaviors on the well-posedness, bounded- ness, persistence, extinction, stability and attractiveness of the solutions for the stochastic delayed system. Moreover, we provide sufficient and necessary condition ensuring the existence of the optimization solution for the optimization problem and obtain the optimal harvesting effect and the maximum of sustainable yield. Lastly, two numerical examples and their simulations are given to illustrate the effectiveness of our results.
- phytoplankton-zooplankton-fish model,
- white noise,
- Markov switching,
- distributed delays,
- optimal harvesting
Citation: Yuanpei Xia, Weisong Zhou, Zhichun Yang. Global analysis and optimal harvesting for a hybrid stochastic phytoplankton-zooplankton-fish model with distributed delays[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 6149-6180. doi: 10.3934/mbe.2020326

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Abstract

In this paper, we formulate a phytoplankton-zooplankton-fish model with distributed delays and hybrid stochastic noises involving Brownian motion and Markov chain, and propose an optimal harvesting problem pursuing the maximum of total economic income. By global analysis in terms of some system parameters, we investigate the dynamical behaviors on the well-posedness, bounded- ness, persistence, extinction, stability and attractiveness of the solutions for the stochastic delayed system. Moreover, we provide sufficient and necessary condition ensuring the existence of the optimization solution for the optimization problem and obtain the optimal harvesting effect and the maximum of sustainable yield. Lastly, two numerical examples and their simulations are given to illustrate the effectiveness of our results.

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