Skip to main content
SpringerLink
Log in

Analysis of the Predator–Prey Interactions: A Stochastic Model Incorporating Disease Invasion

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

Environmental noise and infectious diseases are important factors affecting the development of the population. This paper develops a mathematical system to investigate the impacts of environmental noise and infectious diseases on predator–prey interactions. The globally unique positive solution is confirmed by using conventional methods. The stochastic uniform boundedness of the solution is obtained under certain conditions. Sufficient conditions for the persistence and extinction are given to measure the level of population size. Asymptotic dynamics of the solutions are carried out by two criteria parameters. The long-term dynamics of the solutions are demonstrated by numerical simulations. The results show that small-intensity environmental perturbations can cause population size to fluctuate around a certain level, while high-intensity environmental perturbations may lead to population extinction.

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. Aguirre, P., González-Olivares, E., Torres, S.: Stochastic predator–prey model with allee effect on prey. Nonlinear Anal. Real World Appl. 14(1), 768–779 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Bao, J., Mao, X., Yin, G., Yuan, C.: Competitive Lotka–Volterra population dynamics with jumps. Nonlinear Anal. Theory, Methods Appl. 74(17), 6601–6616 (2011)

    MathSciNet  MATH  Google Scholar 

  3. Baurmann, M., Gross, T., Feudel, U.: Instabilities in spatially extended predator–prey systems: spatio-temporal patterns in the neighborhood of turing-hopf bifurcations. J. Theor. Biol. 245(2), 220–229 (2007)

    MathSciNet  Google Scholar 

  4. Cai, Y., Kang, Y., Banerjee, M., Wang, W.: A stochastic sirs epidemic model with infectious force under intervention strategies. J. Differ. Equ. 259(12), 7463–7502 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Chakraborty, S., Kooi, B.W., Biswas, B., Chattopadhyay, J.: Revealing the role of predator interference in a predator-prey system with disease in prey population. Ecol. Complex. 21, 100–111 (2015)

    Google Scholar 

  6. Chang, Z., Meng, X., Zhang, T.: A new way of investigating the asymptotic behaviour of a stochastic sis system with multiplicative noise. Appl. Math. Lett. 87, 80–86 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Du, N.H., Nguyen, D.H., Yin, G.G.: Conditions for permanence and ergodicity of certain stochastic predator–prey models. J. Appl. Probab. 53(1), 187–202 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Feng, T., Qiu, Z.: Global dynamics of deterministic and stochastic epidemic systems with nonmonotone incidence rate. Int. J. Biomath. 11(7), 1850101 (2018)

    MathSciNet  MATH  Google Scholar 

  9. Feng, T., Qiu, Z., Meng, X.: Analysis of a stochastic recovery-relapse epidemic model with periodic parameters and media coverage. J. Appl. Anal. Comput. 9(3), 1022–1031 (2019)

    MathSciNet  Google Scholar 

  10. Feng, T., Qiu, Z., Meng, X.: Dynamics of a stochastic hepatitis c virus system with host immunity. Discrete Contin. Dyn. Syst. Ser. B 24(12), 6367–6385 (2019)

    MathSciNet  MATH  Google Scholar 

  11. Feng, T., Zhipeng, Q.: Global analysis of a stochastic TB model with vaccination and treatment. Discrete Contin. Dyn. Syst. Ser. B 24(6), 2923–2939 (2019)

    MathSciNet  MATH  Google Scholar 

  12. Fulton, E.A., Smith, A.D., Johnson, C.R.: Mortality and predation in ecosystem models: is it important how these are expressed? Ecol. Model. 169(1), 157–178 (2003)

    Google Scholar 

  13. Haque, M., Sarwardi, S., Preston, S., Venturino, E.: Effect of delay in a Lotka–Volterra type predator-prey model with a transmissible disease in the predator species. Math. Biosci. 234(1), 47–57 (2011)

    MathSciNet  MATH  Google Scholar 

  14. Hethcote, H.W., Wang, W., Han, L., Ma, Z.: A predator–prey model with infected prey. Theor. Popul. Biol. 66(3), 259–268 (2004)

    Google Scholar 

  15. Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525–546 (2001)

    MathSciNet  MATH  Google Scholar 

  16. Hilker, F.M., Schmitz, K.: Disease-induced stabilization of predator–prey oscillations. J. Theor. Biol. 255(3), 299–306 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Kooi, B.W., Venturino, E.: Ecoepidemic predator–prey model with feeding satiation, prey herd behavior and abandoned infected prey. Math. Biosci. 274, 58–72 (2016)

    MathSciNet  MATH  Google Scholar 

  18. Lipsitch, M., Cohen, T., Cooper, B., Robins, J.M., Ma, S., James, L., Gopalakrishna, G., Chew, S.K., Tan, C.C., Samore, M.H., Fisman, D., Murray, M.: Transmission dynamics and control of severe acute respiratory syndrome. Science 300(5627), 1966–1970 (2003)

    Google Scholar 

  19. Liu, M., Bai, C., Jin, Y.: Population dynamical behavior of a two-predator one-prey stochastic model with time delay. Discrete Contin. Dyn. Syst. A 37(5), 2513–2538 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Liu, M., He, X., Yu, J.: Dynamics of a stochastic regime-switching predator–prey model with harvesting and distributed delays. Nonlinear Anal. Hybrid Syst. 28, 87–104 (2018)

    MathSciNet  MATH  Google Scholar 

  21. Liu, M., Wang, K.: Stochastic Lotka–Volterra systems with Lévy noise. J. Math. Anal. Appl. 410(2), 750–763 (2014)

    MathSciNet  MATH  Google Scholar 

  22. Liu, Q., Jiang, D., Hayat, T., Ahmad, B.: Stationary distribution and extinction of a stochastic predator–prey model with additional food and nonlinear perturbation. Appl. Math. Comput. 320, 226–239 (2018)

    MathSciNet  MATH  Google Scholar 

  23. Liu, Q., Jiang, D., Hayat, T., Alsaedi, A.: Dynamics of a stochastic predator–prey model with stage structure for predator and holling type II functional response. J. Nonlinear Sci. 28(3), 1151–1187 (2018)

    MathSciNet  MATH  Google Scholar 

  24. May, R.M.: Stability and Complexity in Model Ecosystems, vol. 6. Princeton University Press, Princeton (2001)

    MATH  Google Scholar 

  25. Meng, X., Li, F., Gao, S.: Global analysis and numerical simulations of a novel stochastic eco-epidemiological model with time delay. Appl. Math. Comput. 339, 701–726 (2018)

    MathSciNet  MATH  Google Scholar 

  26. Numfor, E., Hilker, F.M., Lenhart, S.: Optimal culling and biocontrol in a predator–prey model. Bull. Math. Biol. 79(1), 88–116 (2017)

    MathSciNet  MATH  Google Scholar 

  27. Rao, F., Castillo-Chavez, C., Kang, Y.: Dynamics of a diffusion reaction prey–predator model with delay in prey: effects of delay and spatial components. J. Math. Anal. Appl. 461(2), 1177–1214 (2018)

    MathSciNet  MATH  Google Scholar 

  28. Sahoo, B., Poria, S.: Effects of additional food in a delayed predator–prey model. Math. Biosci. 261, 62–73 (2015)

    MathSciNet  MATH  Google Scholar 

  29. Wang, W., Ma, W.: Travelling wave solutions for a nonlocal dispersal HIV infection dynamical model. J. Math. Anal. Appl. 457(1), 868–889 (2018)

    MathSciNet  MATH  Google Scholar 

  30. Wang, W., Zhang, T.: Caspase-1-mediated pyroptosis of the predominance for driving CD4+ T cells death: a nonlocal spatial mathematical model. Bull. Math. Biol. 80(3), 540–582 (2018)

    MathSciNet  MATH  Google Scholar 

  31. Wu, S., Shi, J., Wu, B.: Global existence of solutions and uniform persistence of a diffusive predator–prey model with prey-taxis. J. Differ. Equ. 260(7), 5847–5874 (2016)

    MathSciNet  MATH  Google Scholar 

  32. Xiao, Y., Chen, L.: Analysis of a three species eco-epidemiological model. J. Math. Anal. Appl. 258(2), 733–754 (2001)

    MathSciNet  MATH  Google Scholar 

  33. Yang, Q., Mao, X.: Extinction and recurrence of multi-group seir epidemic models with stochastic perturbations. Nonlinear Anal. Real World Appl. 14(3), 1434–1456 (2013)

    MathSciNet  MATH  Google Scholar 

  34. Zhang, Q., Jiang, D., Liu, Z., O’Regan, D.: The long time behavior of a predator–prey model with disease in the prey by stochastic perturbation. Appl. Math. Comput. 245, 305–320 (2014)

    MathSciNet  MATH  Google Scholar 

  35. Zhang, S., Meng, X., Feng, T., Zhang, T.: Dynamics analysis and numerical simulations of a stochastic non-autonomous predator–prey system with impulsive effects. Nonlinear Anal. Hybrid Syst. 26, 19–37 (2017)

    MathSciNet  MATH  Google Scholar 

  36. Zhang, T., Xing, Y., Zang, H., Han, M.: Spatio-temporal dynamics of a reaction–diffusion system for a predator–prey model with hyperbolic mortality. Nonlinear Dyn. 78(1), 265–277 (2014)

    MathSciNet  Google Scholar 

  37. Zhang, T., Zang, H.: Delay-induced turing instability in reaction–diffusion equations. Phys. Rev. E 90, 052908 (2014)

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank the editor and the reviewers for their valuable suggestions and comments, which greatly improve the quality of the paper.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

    Tao Feng & Zhipeng Qiu

  2. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, People’s Republic of China

    Xinzhu Meng

  3. Department of Mathematics, Swinburne University of Technology, Hawthorn, VIC, 3122, Australia

    Tonghua Zhang

Authors
  1. Tao Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xinzhu Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tonghua Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhipeng Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinzhu Meng.

Ethics declarations

Conflict of interest

We declare that we have no conflict of interest.

Additional information

Publisher's Note

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

This work was supported by the National Natural Science Foundation of China (11671206, 11971232), the Scholarship Foundation of China Scholarship Council (201806840120), the Fundamental Research Funds for the Central Universities (30918011339) and the Research Fund for the Taishan Scholar Project of Shandong Province of China and the SDUST Research Fund (2014TDJH102)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feng, T., Meng, X., Zhang, T. et al. Analysis of the Predator–Prey Interactions: A Stochastic Model Incorporating Disease Invasion. Qual. Theory Dyn. Syst. 19, 55 (2020). https://doi.org/10.1007/s12346-020-00391-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-020-00391-4

Keywords

Mathematics Subject Classification

Search

Navigation