Skip to main content
SpringerLink
Log in

Dynamics analysis of stochastic modified Leslie–Gower model with time-delay and Michaelis–Menten type prey harvest

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper, we study the dynamics of stochastic time-delayed predator–prey model with modified Leslie–Gower and ratio-dependent schemes including additional food for predator and prey with Michaelis–Menten type harvest. We introduce the effects of time delay, Michaelis–Menten type harvest and stochastic perturbation under the structure of the original model to make the model more consistent with the actual system. We first prove that the system has a globally unique positive solution. Secondly we obtain conditions for the persistence in mean and extinction of the system. Besides, we verify that the system is stochastic permanence under certain conditions. In addition to that, we prove that the system has an ergodic stationary distribution when the parameters satisfy certain conditions. Finally, some numerical simulations were performed to verify the correctness and validity of the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Lotka, A.J.: Elements of Physical Biology. Williams and Wilkins, Galtimore (1925)

    MATH  Google Scholar 

  2. Volterra, V.: Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, Mem.R.Accad.Naz.dei.Lincei, 6(2), 31–113 (1926)

  3. Leslie, P.H.: Some further notes on the use of matrices in population mathematic. Biometrica 35, 213–245 (1948)

    Article  MathSciNet  Google Scholar 

  4. Leslie, P.H., Gower, J.C.: The properties of a stochastic model for the predator-prey type of interaction between two species. Biometrica 47, 219–234 (1960)

    Article  MathSciNet  Google Scholar 

  5. Aziz-Alaoui, M.A., Okiye, M.D.: Boundedness and global stability for a predator-prey model with modified Leslie–Gower and Holling-type II schemes. Appl. Math. Lett. 16(7), 1069–1075 (2003)

    Article  MathSciNet  Google Scholar 

  6. Nindjin, A.F., Aziz-Alaoui, M.A., Cadivel, M.: Analysis of a predator-prey model with modified Leslie–Gower and Holling-type II schemes with time delay. Nonlinear Anal. RWA 7, 1104–1118 (2006)

    Article  MathSciNet  Google Scholar 

  7. Alves, M.T.: Des interactions indirectes entre les proies,modlisation et influence du comportement du pr-dateur commun. PhD thesis, Universit de Nice Sophia Antipolis (2013)

  8. Fan, M., Kuang, Y.: Dynamics of a nonautonomous predator-prey system with the Beddington–DeAngelis functional response. J. Math. Anal. Appl. 259(1), 15–39 (2004)

    Article  MathSciNet  Google Scholar 

  9. Xu, S.H.: Global stability of the virus dynamics model with Crowley–Martin functional response. Electron. J. Qual. Theory 9, 1–10 (2012)

    Google Scholar 

  10. Shi, H.B., Ruan, S.G., Su, Y., Zhang, J.F.: Spatiotemporal dynamics of a diffusive Leslie–Gower predator-prey model with ratio-dependent functional response. Int. J. Bifurcat. Chaos. 25(5), 1530014 (2015)

    Article  MathSciNet  Google Scholar 

  11. Flores, J.D., Gonzalez-Olivares, E.: A modified Leslie–Gower predator-prey model with ratio-dependent functional response and alternative food for the predator. Math. Method. Appl. Sci. 40(7), 2313–2328 (2017)

    Article  MathSciNet  Google Scholar 

  12. Xu, J., Tian, Y., Guo, H.J., Song, X.Y.: Dynamical analysis of a pest management Leslie–Gower model with ratio-dependent functional response. Nonlinear Dyn. 93(2), 705–720 (2018)

    Article  Google Scholar 

  13. Gupta, R.P., Chandra, P.: Bifurcation analysis of modified Leslie–Gower predator-prey model with Michaelis–Menten type prey harvesting. J. Math. Anal. Appl. 398(1), 278–295 (2013)

    Article  MathSciNet  Google Scholar 

  14. Yan, X.P., Zhang, C.H.: Global stability of a delayed diffusive predator-prey model with prey harvesting of Michaelis–Menten type. Appl. Math. Lett. (2021). https://doi.org/10.1016/j.aml.2020.106904

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang, S.L., Xie, Z., Zhong, R., Wu, Y.L.: Stochastic analysis of a predator-prey model with modified Leslie–Gower and Holling type II schemes. Nonlinear Dyn. 101(2), 1245–1262 (2020)

    Article  Google Scholar 

  16. Xu, C.H., Yu, Y.G., Ren, G.J.: Dynamic analysis of a stochastic predator–prey model with Crowley–Martin functional response, disease in predator, and saturation incidence. J. Comput. Nonlinear Dyn. 15(7), 071004 (2020)

    Article  Google Scholar 

  17. Zou, X.L., Lv, J.L., Wu, Y.P.: A note on a stochastic Holling-II predator–prey model with a prey refuge. J. Franklin I 357(7), 4486–4502 (2020)

    Article  MathSciNet  Google Scholar 

  18. Jiang, X.B., Zu, L., Jiang, D.Q., O’Regan, D.: Analysis of a stochastic Holling type II predator–prey model under regime switching. Bull. Malays. Math. Sci. Soc. 43(3), 2171–2197 (2020)

  19. Xu, D.S., Liu, M., Xu, X.F.: Analysis of a stochastic predator–prey system with modified Leslie–Gower and Holling-type IV schemes. Physica A 537, 122761 (2020)

    Article  MathSciNet  Google Scholar 

  20. Liu, C., Liu, M.: Stochastic dynamics in a nonautonomous prey-predator system with impulsive perturbations and Levy jumps. Commun. Nonlinear. Sci. 78 (2019)

  21. Li, H.H., Cong, F.Z.: Dynamics of a stochastic Holling–Tanner predator–prey model. Physica A 531, 121761 (2019)

    Article  MathSciNet  Google Scholar 

  22. Lv, J.L., Wang, K., Chen, D.D.: Analysis on a stochastic two-species ratio-dependent predator–prey model. Methodol. Comput. Appl. 17(2), 403–418 (2015)

    Article  MathSciNet  Google Scholar 

  23. Bai, L., Li, J.S., Zhang, K., Zhao, W.J.: Analysis of a stochastic ratio-dependent predator–prey model driven by Levy noise. Appl. Math. Comput. 233, 480–493 (2014)

    Article  MathSciNet  Google Scholar 

  24. Wang, Z.J., Deng, M.L., Liu, M.: Stationary distribution of a stochastic ratio-dependent predator–prey system with regime-switching. Chaos Solitons Fractals 142, 110462 (2021)

    Article  MathSciNet  Google Scholar 

  25. Zhou, D.X., Liu, M., Liu, Z.J.: Persistence and extinction of a stochastic predator–prey model with modified Leslie–Gower and Holling-type II schemes. Adv. Differ. Equ-ny. 2020(1), 179 (2020)

    Article  MathSciNet  Google Scholar 

  26. Khasminskii, R.Z., Klebaner, F.C.: Long term behavior of solutions of the Lotka–Volterra system under small random perturbations. Ann. Appl. Probab. 11, 952–963 (2001)

    Article  MathSciNet  Google Scholar 

  27. Ji, C.Y., Jiang, D.Q., Shi, N.Z.: Analysis of a predator-prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation. J. Math. Anal. Appl. 359(2), 482–498 (2009)

    Article  MathSciNet  Google Scholar 

  28. Liu, M., Wang, K.: Survival analysis of a stochastic cooperation system in a polluted environment. J. Biol. Syst. 19, 183C204 (2011)

    MathSciNet  Google Scholar 

  29. Cai, Y.L., Kang, Y., Wang, W.M.: A stochastic SIRS epidemic model with nonlinear incidence rate. Appl. Math. Comput. 305, 221–240 (2017)

    Article  MathSciNet  Google Scholar 

  30. Bellet, L.R.: Ergodic properties of Markov processes. Open Quantum Syst. II Markovian Approach 2006, 1–39 (1881)

    MATH  Google Scholar 

  31. Hasminskii, R.Z.: Stochastic Stability of Differential Equations. Sijthoff Noordhoff, Alphen aan den Rijn (1980)

    Book  Google Scholar 

  32. Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525–546 (2001)

    Article  MathSciNet  Google Scholar 

  33. Liu, M.: Dynamics of a stochastic regime-switching predator–prey model with modified Leslie–Gower Holling-type II schemes and prey harvesting. Nonlinear Dyn. 96(1), 417–442 (2019)

    Article  Google Scholar 

  34. Liu, Q., Jiang, D.Q., Shi, N.Z., Hayat, T., Alsaedi, A.: Stochastic mutualism model with Lvy jumps. Commun. Nonlinear Sci. 43, 78–90 (2017)

    Article  Google Scholar 

  35. Ji, W.M., Liu, M.: Optimal harvesting of a stochastic commensalism model with time delay. Physica A 527, 121284 (2019)

    Article  MathSciNet  Google Scholar 

  36. Liu, M.: Optimal harvesting policy of a stochastic predator–prey model with time delay. Appl. Math. Lett. 48, 102–108 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This study was funded by Fundamental Research Funds for the Central Universities (No.2572020BC09), Heilongjiang Provincial Natural Science Foundation of China(No.LH2019A001) and Fundamental Research Funds for the Universities of Heilongjiang Province (No.RCCXYJ201910).

Author information

Authors and Affiliations

  1. Department of Mathematics, Northeast Forestry University, Harbin, 150040, Heilongjiang, People’s Republic of China

    Yu Liu & Ming Liu

  2. School of Mathematical Sciences, Heilongjiang University, Harbin, 150080, Heilongjiang, People’s Republic of China

    Xiaofeng Xu

Authors
  1. Yu Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ming Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaofeng Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ming Liu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, Y., Liu, M. & Xu, X. Dynamics analysis of stochastic modified Leslie–Gower model with time-delay and Michaelis–Menten type prey harvest. J. Appl. Math. Comput. 68, 2097–2124 (2022). https://doi.org/10.1007/s12190-021-01612-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-021-01612-y

Keywords

Mathematics Subject Classification

Search

Navigation