Asymptotic behavior of a delayed stochastic logistic model with impulsive perturbations

  • 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.

    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

    Related Papers:

    [1] Chentong Li, Jinyan Wang, Jinhu Xu, Yao Rong . The Global dynamics of a SIR model considering competitions among multiple strains in patchy environments. Mathematical Biosciences and Engineering, 2022, 19(5): 4690-4702. doi: 10.3934/mbe.2022218
    [2] Xue-Zhi Li, Ji-Xuan Liu, Maia Martcheva . An age-structured two-strain epidemic model with super-infection. Mathematical Biosciences and Engineering, 2010, 7(1): 123-147. doi: 10.3934/mbe.2010.7.123
    [3] Matthew D. Johnston, Bruce Pell, David A. Rubel . A two-strain model of infectious disease spread with asymmetric temporary immunity periods and partial cross-immunity. Mathematical Biosciences and Engineering, 2023, 20(9): 16083-16113. doi: 10.3934/mbe.2023718
    [4] Ali Mai, Guowei Sun, Lin Wang . The impacts of dispersal on the competition outcome of multi-patch competition models. Mathematical Biosciences and Engineering, 2019, 16(4): 2697-2716. doi: 10.3934/mbe.2019134
    [5] Abdelrazig K. Tarboush, Jing Ge, Zhigui Lin . Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment. Mathematical Biosciences and Engineering, 2018, 15(6): 1479-1494. doi: 10.3934/mbe.2018068
    [6] Azmy S. Ackleh, Mark L. Delcambre, Karyn L. Sutton, Don G. Ennis . A structured model for the spread of Mycobacterium marinum: Foundations for a numerical approximation scheme. Mathematical Biosciences and Engineering, 2014, 11(4): 679-721. doi: 10.3934/mbe.2014.11.679
    [7] Nancy Azer, P. van den Driessche . Competition and Dispersal Delays in Patchy Environments. Mathematical Biosciences and Engineering, 2006, 3(2): 283-296. doi: 10.3934/mbe.2006.3.283
    [8] Junjing Xiong, Xiong Li, Hao Wang . The survival analysis of a stochastic Lotka-Volterra competition model with a coexistence equilibrium. Mathematical Biosciences and Engineering, 2019, 16(4): 2717-2737. doi: 10.3934/mbe.2019135
    [9] Yanxia Dang, Zhipeng Qiu, Xuezhi Li . Competitive exclusion in an infection-age structured vector-host epidemic model. Mathematical Biosciences and Engineering, 2017, 14(4): 901-931. doi: 10.3934/mbe.2017048
    [10] Azmy S. Ackleh, Shuhua Hu . Comparison between stochastic and deterministic selection-mutation models. Mathematical Biosciences and Engineering, 2007, 4(2): 133-157. doi: 10.3934/mbe.2007.4.133
  • 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.


    [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.
  • This article has been cited by:

    1. Yixiang Wu, Necibe Tuncer, Maia Martcheva, Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion, 2017, 22, 1553-524X, 1167, 10.3934/dcdsb.2017057
    2. Junping Shi, Yixiang Wu, Xingfu Zou, Coexistence of Competing Species for Intermediate Dispersal Rates in a Reaction–Diffusion Chemostat Model, 2020, 32, 1040-7294, 1085, 10.1007/s10884-019-09763-0
    3. Yixiang Wu, Xingfu Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, 2016, 261, 00220396, 4424, 10.1016/j.jde.2016.06.028
    4. Lin Zhao, Zhi-Cheng Wang, Shigui Ruan, Dynamics of a time-periodic two-strain SIS epidemic model with diffusion and latent period, 2020, 51, 14681218, 102966, 10.1016/j.nonrwa.2019.102966
    5. Jing Ge, Ling Lin, Lai Zhang, A diffusive SIS epidemic model incorporating the media coverage impact in the heterogeneous environment, 2017, 22, 1553-524X, 2763, 10.3934/dcdsb.2017134
    6. Yuan Lou, Rachidi B. Salako, Control Strategies for a Multi-strain Epidemic Model, 2022, 84, 0092-8240, 10.1007/s11538-021-00957-6
    7. Jinsheng Guo, Shuang-Ming Wang, Threshold dynamics of a time-periodic two-strain SIRS epidemic model with distributed delay, 2022, 7, 2473-6988, 6331, 10.3934/math.2022352
    8. Rachidi B. Salako, Impact of population size and movement on the persistence of a two-strain infectious disease, 2023, 86, 0303-6812, 10.1007/s00285-022-01842-z
    9. Yuan Lou, Rachidi B. Salako, Mathematical analysis of the dynamics of some reaction-diffusion models for infectious diseases, 2023, 370, 00220396, 424, 10.1016/j.jde.2023.06.018
    10. Jonas T. Doumatè, Tahir B. Issa, Rachidi B. Salako, Competition-exclusion and coexistence in a two-strain SIS epidemic model in patchy environments, 2023, 0, 1531-3492, 0, 10.3934/dcdsb.2023213
    11. Azmy S. Ackleh, Nicolas Saintier, Aijun Zhang, A multiple-strain pathogen model with diffusion on the space of Radon measures, 2025, 140, 10075704, 108402, 10.1016/j.cnsns.2024.108402
    12. Jamal Adetola, Keoni G. Castellano, Rachidi B. Salako, Dynamics of classical solutions of a multi-strain diffusive epidemic model with mass-action transmission mechanism, 2025, 90, 0303-6812, 10.1007/s00285-024-02167-9
  • Reader Comments
  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3529) PDF downloads(501) Cited by(9)

Article outline

Figures and Tables

Figures(3)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog