Asymptotic behavior of a delayed stochastic logistic model with impulsive perturbations

Sanling Yuan; Xuehui Ji; Huaiping Zhu; Sanling Yuan; Xuehui Ji; Huaiping Zhu

doi:10.3934/mbe.2017077

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 5&6: 1477-1498. doi: 10.3934/mbe.2017077

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Asymptotic behavior of a delayed stochastic logistic model with impulsive perturbations

1.
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
2.
College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China
3.
Lamps and Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada

Received: 23 June 2016 Accepted: 27 September 2016 Published: 01 October 2017
MSC : Primary: 65C20, 65C30; Secondary: 92D25

In this paper, we investigate the dynamics of a delayed logistic model with both impulsive and stochastic perturbations. The impulse is introduced at fixed moments and the stochastic perturbation is of white noise type which is assumed to be proportional to the population density. We start with the existence and uniqueness of the positive solution of the model, then establish sufficient conditions ensuring its global attractivity. By using the theory of integral Markov semigroups, we further derive sufficient conditions for the existence of the stationary distribution of the system. Finally, we perform the extinction analysis of the model. Numerical simulations illustrate the obtained theoretical results.
- Delayed stochastic logistic model,
- impulsive perturbation,
- Itô's formula,
- stationary distribution,
- extinction
Citation: Sanling Yuan, Xuehui Ji, Huaiping Zhu. Asymptotic behavior of a delayed stochastic logistic model with impulsive perturbations[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1477-1498. doi: 10.3934/mbe.2017077

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Abstract

In this paper, we investigate the dynamics of a delayed logistic model with both impulsive and stochastic perturbations. The impulse is introduced at fixed moments and the stochastic perturbation is of white noise type which is assumed to be proportional to the population density. We start with the existence and uniqueness of the positive solution of the model, then establish sufficient conditions ensuring its global attractivity. By using the theory of integral Markov semigroups, we further derive sufficient conditions for the existence of the stationary distribution of the system. Finally, we perform the extinction analysis of the model. Numerical simulations illustrate the obtained theoretical results.

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