Skip to main content
SpringerLink
Log in

Positive Periodic Solutions of an Eco-Epidemic Model with Crowley–Martin type Functional Response and Disease in the Prey

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

Abstract

In this paper, we consider an eco-epidemic predator–prey model with Crowley–Martin type functional response and disease in prey population, which is a branch of study in biomathematics which reflects both ecological and epidemiological cases simultaneously. By using Mawhin continuation theorem and constructing a suitable Lyapunov function, we obtained some sufficient conditions for the existence, uniqueness and global attractivity of a positive periodic solution for the predator–prey model.

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

Similar content being viewed by others

References

  1. Wang, M.: A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment. J. Funct. Anal. 270, 483–508 (2016)

    MathSciNet  MATH  Google Scholar 

  2. Du, Z., Feng, Z., Zhang, X.: Traveling wave phenomena of n-dimensional diffusive predator–prey systems. Nonlinear Anal. Real World Appl. 41, 288–312 (2018)

    MathSciNet  MATH  Google Scholar 

  3. KumarSasma, S., Takeuchi, Y.: Dynamics of a predator-prey system with fear and group defense. J. Math. Anal. Appl. 481, 123471 (2020)

    MathSciNet  MATH  Google Scholar 

  4. Wang, J., Wei, J., Shi, J.: Global bifurcation analysis and pattern formation in homogeneous diffusive predator–prey systems. J. Differ. Equ. 260, 3495–3523 (2016)

    MathSciNet  MATH  Google Scholar 

  5. Wang, C., Zhang, X.: Canards, heteroclinic and homoclinic orbits for a slow-fast predator–prey model of generalized Holling type III. J. Differ. Equ. 267, 3397–3441 (2019)

    MathSciNet  MATH  Google Scholar 

  6. Li, C., Zhu, H.: Canard cycles for predator–prey systems with Holling types of functional response. J. Differ. Equ. 254, 879–910 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Du, Z., Feng, Z.: Existence and asymptotic behaviors of travelling waves of a modified vector-disease model. Commun. Pure Appl. Anal. 17, 1899–1920 (2018)

    MathSciNet  MATH  Google Scholar 

  8. Li, S., Wu, J.: Asymptotic behavior and stability of positive solutions to a spatially heterogeneous predator–prey system. J. Differ. Equ. 265, 3754–3791 (2018)

    MathSciNet  MATH  Google Scholar 

  9. Chen, X., Du, Z.: Existence of positive periodic solutions for a neutral delay predator–prey model with Hassell–Varley type functional response and impulse. Qual. Theory Dyn. Syst. 17, 67–80 (2018)

    MathSciNet  MATH  Google Scholar 

  10. Upadhyay, R.K., Naji, R.K.: Dynamics of a three species food chain model with Crowley–Martin type function response. Chaos Solitons Fractals 42, 1337–1346 (2009)

    MathSciNet  MATH  Google Scholar 

  11. Gaines, R., Mawhin, J.: Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin (1977)

    MATH  Google Scholar 

  12. Li, X., Lin, X., Liu, J.: Existence and global attractivity of positive periodic solutions for a predator–prey model with Crowley–Martin functional response. Elec. J. Differ. Equ. 2018, 1–17 (2018)

    MathSciNet  MATH  Google Scholar 

  13. Mortoja, S.G., Panja, P., Mondal, S.K.: Dynamics of a predator–prey model with nonlinear incidence rate, Crowley–Martin type functional response and disease in prey population. Ecol. Genet. Genom. 10, 100035 (2019)

    Google Scholar 

  14. Bairagi, N., Roy, P.K., Chattopadhyay, J.: Role of infection on the stability of a predator–prey system with several response functions—a comparative study. J. Theor. Biol. 248, 10–25 (2007)

    MathSciNet  Google Scholar 

  15. Sari, N., Véron, E.A.: Periodic orbits of a seasonal SIS epidemic model with migration. J. Math. Anal. Appl. 423, 1849–1866 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Zheng, H., Guo, L., Bai, Y., Xia, Y.: Periodic solutions of a non-autonomous predator-prey system with migrating prey and disease infection: via Mawhin coincidence degree theory. J. Fixed Point Theory Appl. 260, 21–37 (2019)

    MathSciNet  MATH  Google Scholar 

  17. Silva, C.M.: Existence of periodic solutions for periodic eco-epidemic models with disease in the prey. J. Math. Anal. Appl. 453, 383–397 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Zhu, Y., Wang, K.: Existence and global attractivity of positive periodic solutions for a predator–prey model with modified Leslie–Gower Holling-type II schemes. J. Comput. Appl. Math. 384, 408–408 (2011)

    MathSciNet  MATH  Google Scholar 

  19. Chen, C., Chen, F.: Conditions for global attractivity of multispecies ecological competition-predator system with Holling-type III functional response. J. Biomath. 19, 136–140 (2004)

    MathSciNet  Google Scholar 

  20. Gopalasamy, K.: Stability and Oscillation in Delay Equation of Population Dynamics. Kluwer Academic Publishers, Dordrecht (1992)

    Google Scholar 

Download references

Acknowledgements

We express our sincere thanks to the anonymous reviewers for their valuable comments and suggestions for improving the quality of the paper.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, China

    Min Cai, Shuling Yan & Zengji Du

Authors
  1. Min Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shuling Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zengji Du
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zengji Du.

Additional information

Publisher's Note

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

This work is supported by the Natural Science Foundation of China (Grant No. 11871251, 11801231 and 11771185).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cai, M., Yan, S. & Du, Z. Positive Periodic Solutions of an Eco-Epidemic Model with Crowley–Martin type Functional Response and Disease in the Prey. Qual. Theory Dyn. Syst. 19, 56 (2020). https://doi.org/10.1007/s12346-020-00392-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-020-00392-3

Keywords

Mathematics Subject Classification

Search

Navigation