Asymptotic behavior of a delayed stochastic logistic model with impulsive perturbations

Sanling Yuan; Xuehui Ji; Huaiping Zhu

doi:10.3934/mbe.2017077

Article Contents

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

* Corresponding author: Sanling Yuan
* Corresponding author: Sanling Yuan

Received: June 23, 2016

Accepted: September 27, 2016

Published: December 2017

The research is supported by the National Natural Science Foundation of China (11271260,11671260), Innovation program of Shanghai Municipal Education Committee (13ZZ116), Shanghai Leading Academic Discipline Project (No. XTKX2012), and the Hujiang Foundation of China (B14005).

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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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Mathematics Subject Classification: Primary: 65C20, 65C30; Secondary: 92D25.

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Figure 1. Trajectories of impulsive stochastic system (5) with different $\sigma$ and $b_k$. Here $(S(0), x(0))=(1, 1)$ and $E^*(x^*, y^*)=(1.0909, 1.8182)$. System (5) is asymptotically stable

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Figure 2. Probability densities of $x(t)$ for system (5) based on $10000$ sample pathes, computed with the noise intensity $\sigma=0.03$, and for (a) differential initial value and (b) different iterative times. There exists stationary distribution and the density is stable

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Figure 3. The dynamics of the system (5) with different $\sigma$. (a) The population $x(t)$ is extinct with $\sigma=0.8$. (c) The population $x(t)$ is persistent with $\sigma=0.1$

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