Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

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

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.

  Article Metrics


[1] S. Aida,S. Kusuoka,D. Strook, On the support of Wiener functionals, Longman Scient. Tech., 284 (1993): 3-34.

[2] T. Alkurdi,S. Hille,O. Gaans, Ergodicity and stability of a dynamical system perturbed by impulsive random interventions, J. Math. Anal. Appl., 407 (2013): 480-494.

[3] L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York-London-Sydney, 1974.

[4] I. Barbalat, Systems dequations differentielles d'osci d'oscillations nonlineaires, Rev. Roumaine Math. Pures Appl., 4 (1959): 267-270.

[5] G. Ben Arous,R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale (Ⅱ), Probab. Theory Related Fields, 90 (1991): 377-402.

[6] A. Freedman, Stochastic differential equations and their applications, Stochastic Differential Equations, 77 (1976): 75-148.

[7] S. Foguel, Harris operators, Israel J. Math., 33 (1979): 281-309.

[8] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Springer-Verlag, New York, 1992.

[9] R. Z. Has'minskii, Stochastic Stability of Differential Equations, Sijthoof & Noordhoof, Alphen aan den Rijn, The Netherlands, 1980.

[10] D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001): 525-546.

[11] D. Jiang,N. Shi, A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303 (2005): 164-172.

[12] D. Jiang,N. Shi,X. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 340 (2008): 588-597.

[13] I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Springer Verlag, Berlin, 1991.

[14] Y. Kuang, Delay differential equations with applications in population dynamics, in Mathematics in Science and Engineering, Academic Press, New York, 1993.

[15] X. Li,X. Mao, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete Contin. Dyn. Syst., 24 (2009): 523-545.

[16] M. Liu,K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl., 375 (2011): 443-457.

[17] M. Liu,K. Wang, On a stochastic logistic equation with impulsive perturbations, Comput. Math. Appl., 63 (2012): 871-886.

[18] Z. Ma and Y. Zhou, Qualitative and Stability Method of Ordinary Differential Equation, Science Press, Beijing, 2001.

[19] M. Mackey,M. Kamińska,R. Yvinec, Molecular distributions in gene regulatory dynamics, J. Theoret. Biol., 247 (2011): 84-96.

[20] X. Mao, Stochastic Differential Equations and their Applications, Horwood publishing, Chichester, England, 1997.

[21] J. Norris, Simplified Malliavin calculus, in SLeminaire de probabilitiLes XX, Lecture Notes in Mathematics, Springer, New York, 1024 (1986), 101–130.

[22] K. Pichór,R. Rudnicki, Stability of Markov semigroups and applications to parabolic systems, J. Math. Anal. Appl., 215 (1997): 56-74.

[23] K. Pichór,R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., 249 (2000): 668-685.

[24] S. Ruan, Delay differential equations in single species dynamics, in Delay Differential Equations and Applications, Springer, Berlin, 205 (2006), 477–517.

[25] R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Math., 43 (1995): 245-262.

[26] J. Yan, On the oscillation of impulsive neutral delay differential equations, Chinese Ann. Math., 21A (2000): 755-762.

Copyright Info: © 2017, Sanling Yuan, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved