Skip to main content
SpringerLink
Log in

Dynamics of a stochastic predator-prey model with two competitive preys and one predator in a polluted environment

  • Original Paper
  • Area 1
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we propose and study a stochastic predator-prey model with two competitive preys and one predator in a polluted environment. We first carry out the survival analysis and establish sufficient criteria for the extinction, non-persistence, weak persistence in the mean and strong persistence in the mean. The threshold between weak persistence in the mean and extinction is obtained for each population. Then, we derive sufficient conditions for global attractivity of the studied model. Numerical simulations are carried out to verify 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.

Institutional subscriptions

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

Similar content being viewed by others

References

  1. Mezquita, F., Hernndez, R., Rueda, J.: Ecology and distribution of ostracods in a polluted mediterranean river. Palaeogeogr. Palaeocl. 148(1–3), 87–103 (1999)

    Article  Google Scholar 

  2. Yang, S.Q., Liu, P.W.: Strategy of water pollution prevention in taihu lake and its effects analysis. J. Great Lakes Res. 36(1), 150–158 (2010)

    Article  Google Scholar 

  3. Hallam, T.G., Clark, C.E., Lassiter, R.R.: Effects of toxicants on populations: a qualitative approach. First order kinetics. J. Math. Biol. 18, 25–27 (1983)

    Article  MATH  Google Scholar 

  4. Freedman, H.I., Shukla, J.B.: Models for the effect of toxicant in single-species and predator-prey systems. J. Math. Biol. 30(1), 15–30 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gard, T.C.: Stochastic models for toxicant-stressed populations. Bull. Math. Biol. 54(5), 827–837 (1992)

    Article  MATH  Google Scholar 

  6. He, J., Wang, K.: The survival analysis for a single-species population model in a polluted environment. Appl. Math. Model. 31(10), 2227–2238 (2007)

    Article  MATH  Google Scholar 

  7. He, J., Wang, K.: The survival analysis for a population in a polluted environment. Nonlinear Anal-Real 10(3), 1555–1571 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Tao, F., Liu, B.: Dynamic behaviors of a single-species population model with birth pulses in a polluted environment. Rocky Mt. J. Math. 38(38), 1663–1684 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jiao, J., Chen, L.: Dynamical analysis of a chemostat model with delayed response in growth and pulse input in polluted environment. J. Math. Chem. 46(2), 502–513 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jiao, J., Long, W., Chen, L.: A single stage-structured population model with mature individuals in a polluted environment and pulse input of environmental toxin. J. Anshan Norm. Univ. 10(5), 3073–3081 (2009)

    MathSciNet  MATH  Google Scholar 

  11. Liu, B., Chen, L., Zhang, Y.: The effects of impulsive toxicant input on a population in a polluted environment. J. Biol. Syst. 11(03), 265–274 (2003)

    Article  MATH  Google Scholar 

  12. Liu, B., Zhang, L.: Dynamics of a two-species Lotka-Volterra competition system in a polluted environment with pulse toxicant input. Elsevier Science Inc. (2009)

  13. Liu, M., Wang, K.: Persistence and extinction of a stochastic single-species model under regime switching in a polluted environment ii. J. Theor. Biol. 264(3), 934–944 (2010)

    Article  MathSciNet  Google Scholar 

  14. Liu, H.P., Ma, Z.: The threshold of survival for system of two species in a polluted environment. J. Math. Biol. 30(1), 49–61 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, M., Wang, K., Wu, Q.: Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle. Bull. Math. Biol. 73(9), 1969–2012 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ma, Z., Luo, Z.: The threshold of survival for perdator-pery volterra system of three species in a polluted environment. Syst. Sci. Math. Sci. 8(4), 373–382 (1995)

    MATH  Google Scholar 

  17. Liu, M., Wang, K.: Dynamics of a two-prey one-predator system in random environments[j]. J. Nonlinear Sci. 23(5), 751–775 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, W., Ma, Z.: Permanence of populations in a polluted environment. Math. Biosci. 122(2), 235 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pan, J., Jin, Z., Ma, Z.: Thresholds of survival for an n-dimensional volterra mutualistic system in a polluted environment. J. Math. Anal. Appl. 252(2), 519–531 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhao, Z., Chen, L., Song, X.: Extinction and permanence of chemostat model with pulsed input in a polluted environment. Commun. Nonlinear Sci. Numer. Simul. 14(4), 1737–1745 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Duan, L., Lu, Q., Yang, Z., Chen, L.: Effects of diffusion on a stage-structured population in a polluted environment. Appl. Math. Comput. 154(2), 347–359 (2004)

    MathSciNet  MATH  Google Scholar 

  22. Dubeya, B., Hussainb, J.: Modelling the survival of species dependent on a resource in a polluted environment. Nonlinear Anal. Real World Appl. 7(2), 187–210 (2006)

    Article  MathSciNet  Google Scholar 

  23. Gard, T.C.: Persistence in stochastic food web models. Bull. Math. Biol. 46(3), 357–370 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gard, T.C.: Stability for multispecies population models in random environments. Nonlinear Anal. 10(12), 1411–1419 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gard, T.C.: Introductions to Stochastic Differential Equations. New York (1988)

  26. Bandyopadhyay, M., Chattopadhyay, J.: Ratio-dependent predator-prey model: effect of environmental fluctuation and stability. Nonlinearity 18(2), 913C936 (2005)

  27. May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (1973)

    Google Scholar 

  28. Qiu, H., Deng, W.: Stationary distribution and global asymptotic stability of a three-species stochastic food-chain system. Turk. J. Math. 41, 1292–1307 (2017)

    Article  MathSciNet  Google Scholar 

  29. Liu, M., Wang, K.: Persistence, extinction and global asymptotical stability of a non-autonomous predator-prey model with random perturbation. Appl. Math. Model. 36(11), 5344–5353 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mao, X.: Stochastic Differential Equations and Applications, 2nd edn. Woodhead Publishing (2008)

  31. Ikeda, N., Watanabe, S.: A comparison theorem for solutions of stochastic dfferential equations and its applications. Osaka J. Math. 14, 619–633 (1997)

    MATH  Google Scholar 

  32. Jiang, D., Shi, N.: A note on non-autonomous logistic equation with random perturbation. J. Math. Anal. Appl. 303, 164–172 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hung, L.C.: Stochastic delay population systems. Appl. Anal. 88(9), 1303–1320 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Cheng, S.R.: Stochastic population systems. Stoch. Anal. Appl. 27(4), 854–874 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus. Springer, Berlin (1991)

    MATH  Google Scholar 

  36. Mao, X., Yuan, C.: Stochastic differential equations with Markovian switching. Stochastic Differential Equations with Markovian Switching. Imperial College Press 77–110 (2006)

  37. Barbalat, I.: Systemes dequations differentielles doscillations non lineaires. Rev. Math. Pures Appl. 4, 267–270 (1959)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Civil Aviation University of China, Tianjin, 300300, China

    Yongxin Gao & Shiquan Tian

Authors
  1. Yongxin Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shiquan Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shiquan Tian.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, Y., Tian, S. Dynamics of a stochastic predator-prey model with two competitive preys and one predator in a polluted environment. Japan J. Indust. Appl. Math. 35, 861–889 (2018). https://doi.org/10.1007/s13160-018-0314-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-018-0314-z

Keywords

Mathematics Subject Classification

Search

Navigation