Skip to main content
SpringerLink
Log in

Stability and Hopf bifurcation analysis of a delayed predator–prey model with age-structure and Holling III functional response

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

This study investigates a delayed predator–prey model with age-structure and a Holling III response function. The considered model is first presented as an abstract Cauchy problem, then the criteria of the presence for the positive equilibrium point are gained. The global asymptotic stability of the boundary equilibrium and the local asymptotic stability of the positive equilibrium are examined, respectively, through detailed investigation of the location of the solutions for the model’s characteristic equation. Then, using the Hopf bifurcation theorem, we show that Hopf bifurcation exists under certain criteria. At last, some numerical examples are performed to demonstrate the obtained 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

Similar content being viewed by others

References

  1. Beretta, E., Kuang, Y.: Global analyses in some delayed ratio-dependent predator-prey systems. Nonlinear Anal. 32, 381–408 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, J., De, Z.: The qualitative analysis of two species predator-prey model with Holling’s type III functional response. Appl. Math. Mech. 7, 77–86 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, S., Shi, J., Wei, J.: The effect of delay on a diffusive predator-prey systemwith Holling type-II predator functional response. Commun. Pure Appl. Anal. 12, 481–501 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chu, J., Ducrot, A., Magal, P., Ruan, S.: Hopf bifurcation in a size structured population dynamic model with random growth. J. Differ. Equ. 247, 956–1000 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cushing, J.M., Saleem, M.: A predator prey model with age structure. J. Math. Biol. 14, 231–250 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cushing, J.: Integrodifferential Equations and Delay Models in Population Dynamics. Springer, Heidelberg (1977)

    Book  MATH  Google Scholar 

  7. Ducrot, A., Magal, P., Ruan, S.: Projectors on the generalized eigenspaces for partial differential equations with time delay. Infin. Dimens. Dyn. Syst. 64, 353–390 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Iannelli, M.: Mathematical theory of age-structured population dynamics, Giardini editori e stampatori in Pisa (1995)

  9. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993)

    MATH  Google Scholar 

  10. Li, J.: Dynamics of age-structured predator-prey population models. J. Math. Anal. Appl. 152, 399–415 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu, Z., Li, N.: Stability and bifurcation in a predator-prey model with age structure and delays. J. Nonlinear Sci. 25(4), 937–957 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Liu, Z., Magal, P., Ruan, S.: Hopf bifurcation for non-densely defined Cauchy problems. Z. Angew. Math. Phys. 62, 191–222 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Liu, Z., Yuan, R.: Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response. J. Math. Anal. Appl. 296(2), 521–537 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lotka, A.J.: Elements of physical biology. Sci. Prog. Twent. Century 21, 341–343 (1926)

    Google Scholar 

  15. Magal, P.: Compact attractors for time-periodic age structured population models. Electron. J. Differ. Equ. 65, 1–35 (2001)

    MathSciNet  MATH  Google Scholar 

  16. Magal, P., Ruan, S.: Center manifolds for semilinear equations with non-dense domain and applications on Hopf bifurcation in age structured models. Mem. Am. Math. Soc. 202 (2009)

  17. Magal, P., Ruan, S.: Theory and Applications of Abstract Semilinear Cauchy Problems. Springer, New York (2018)

    Book  MATH  Google Scholar 

  18. Martcheva, M., Thieme, H.R.: Progression age enhanced backward bifurcation in an epidemic model with super-infection. J. Math. Biol. 46, 385–424 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  20. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    Book  MATH  Google Scholar 

  21. Song, Y., Yuan, S.: Bifurcation analysis in a predator-prey system with time delay. Nonlinear Anal. (RWA) 7, 265–284 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tang, H., Liu, Z.: Hopf bifurcation for a predator-prey model with age structure. Appl. Math. Model. 40, 726–737 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Thieme, H.R.: Convergence results and a Poincar\(\acute{e}\)-Bendixson trichotomy for asymptotically autonomous differential equations. J. Math. Biol. 30, 755–763 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. Volterra, V.: The role of simple mathematical models in malaria elimination strategy design, Roma. Acad. Naz. Lincei 2, 31–113 (1926)

    Google Scholar 

  25. Wang, L., Dai, C., Zhao, M.: Hopf bifurcation in an age-structured prey-predator model with Holling III response function. Math. Biosci. Eng. 18(4), 3144–3159 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang, W., Chen, L.: A predator-prey system with stage structure for predator. Comput. Math. Anal. Appl. 262, 499–528 (2001)

    Google Scholar 

  27. Wang, J.F.: Spatiotemporal patterns of a homogeneous diffusive predator-prey system with Holling type III functional response. J. Dyn. Differ. Equ. 29, 1383–1409 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Xiao, D., Ruan, S.: Stability and bifurcation in a delayed ratio-dependent predator-prey system. Proc. Edinb. Math. Soc. 45, 205–220 (2002)

    Article  MathSciNet  Google Scholar 

  29. Yan, D., Cao, H., Xu, X., et al.: Hopf bifurcation for a predator-prey model with age structure. Phys. A 526, 120953 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yang, R.Z., Wei, J.J.: Stability and bifurcation analysis of a diffusive prey-predator system in Holling type III with a prey refuge. Nonlinear Dyn. 79, 631–646 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhang, X., Liu, Z.: Periodic oscillations in age-structured ratio-dependent predator-prey model with Michaelis-Menten type functional response. Phys. D 389, 51–63 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhou, J.: Bifurcation analysis of a diffusive predator-prey model with ratio-dependent Holling type III functional response. Nonlinear Dyn. 81(3), 1535–1552 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

    Dongxue Yan

  2. School of International Economics and Business, Nanjing University of Finance and Economics, Nanjing, 210023, China

    Yu Cao

  3. School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241, China

    Yuan Yuan

Authors
  1. Dongxue Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuan Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

DY and YC wrote the main manuscript text and YY prepared all the figures in the manuscript. All authors reviewed the manuscript.

Corresponding author

Correspondence to Yu Cao.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

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 National Natural Science Foundation of China (Grant No. 12101323), Natural Science Foundation of Jiangsu Province of China (Grant No. BK20200749), The Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 21KJB630013), Nanjing University of Posts and Telecommunications Science Foundation (Grant No. NY220093).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yan, D., Cao, Y. & Yuan, Y. Stability and Hopf bifurcation analysis of a delayed predator–prey model with age-structure and Holling III functional response. Z. Angew. Math. Phys. 74, 148 (2023). https://doi.org/10.1007/s00033-023-02036-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-023-02036-3

Keywords

Mathematics Subject Classification

Search

Navigation