Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Hopf bifurcation, stability switches and chaos in a prey-predator system with three stage structure and two time delays

1 School of Mathematics and Statistics, Qiannan Normal University for Nationalities, Guizhou, 558000, P.R. China
2 Key Laboratory of Complex Systems and Intelligent Computing,School of Mathematics and Statistics, Qiannan Normal University for Nationalities, Guizhou, 558000, P.R. China

A three stage-structured prey-predator model with digestion delay and density dependent delay for the predator is investigated. The stability of the equilibrium point and the Hopf bifurcation of the system by choosing time delay as a bifurcation parameter in five cases are considered, and the conditions for the positive equilibrium occurring local Hopf bifurcation are given in each case. Numerical results show that delayed system considered has not only periodic oscillation, stability switches but also chaotic oscillation, even unbounded oscillation. Finally, delays induced Hopf bifurcation, stability switches, complicated dynamic behaviors of the system are discussed in detail.
  Article Metrics

Keywords Prey-predator system; time delays; Hopf bifurcation; stability switches; chaos

Citation: Shunyi Li. Hopf bifurcation, stability switches and chaos in a prey-predator system with three stage structure and two time delays. Mathematical Biosciences and Engineering, 2019, 16(6): 6934-6961. doi: 10.3934/mbe.2019348


  • 1. W. Aiello and H. Freedman, A time-delay model of single-species growth with stage structure, Math. Biosci., 101 (1990), 139–153.
  • 2. W. Aiello, H. Freedman and J. Wu, Analysis of a model representing stage-structured populations growth with stage-dependent time delay, SIAM J. Appl. Math., 3 (1992), 855–869.
  • 3. W. Wang and L. Chen, A predator-prey system with stage-structure for predator, Comp. Math. Appl., 33 (1997), 83–91.
  • 4. S. Liu, L. Chen and R. Agarwal, Recent progress on stage-structured population dynamics, Math. Comput. Model., 36 (2002), 1319–1360.
  • 5. S. Gao, Models for single species with three life history stages and cannibalism, J. Biomath., 20 (2005), 385–391.
  • 6. S. Yang and B. Shi, Periodic solution for a three-stage-structured predator-prey system with time delay, J. Math. Anal. Appl., 341 (2008), 287–294.
  • 7. H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer Science+Business Media, LLC, 201l.
  • 8. E. Beretta and D. Breda, Discrete or distributed delay? Effects on stability of population growth, Math. Biosci. Eng., 13 (2016), 19–41.
  • 9. Z. Shen and J. Wei, Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect, Math. Biosci. Eng., 15 (2018), 693–715.
  • 10. S. Li and Z. Xiong, Bifurcation analysis of a predator-prey system with sex-structure and sexual favoritism, Adv. Differ. Equ., 219 (2013), 1–24.
  • 11. Z. Ma and S. Wang, A delay-induced predator Cprey model with Holling type functional response and habitat complexity, Nonl. Dyna., 93 (2018), 1519–1544.
  • 12. S. Kundu and S. Maitra Dynamical behaviour of a delayed three species predator Cprey model with cooperation among the prey species, Nonl. Dyna., 92 (2018), 627–643.
  • 13. L. Li and J. Shen, Bifurcations and Dynamics of a Predator CPrey Model with Double Allee Effects and Time Delays, Int. J. Bifurc. Chaos, 28 (2018), 1–14. (No. 1850135)
  • 14. T. Caraballo, R. Colucci and L. Guerrini, On a predator prey model with nonlinear harvesting and distributed delay, Comm. on Pure Appl. Anal., 17 (2018), 2703–2727.
  • 15. X. Xu, Y. Wang and Y. Wang, Local bifurcation of a Ronsenzwing-MacArthur predator prey model with two prey-taxis, Math. Biosci. Eng., 16 (2019), 1786-1797.
  • 16. S. Li, Y. Xue and W. Liu, Hopf bifurcation and global periodic solutions for a three-stage-structured prey-predator system with delays, Int. J. Info. Syst. Scie., 8 (2012), 142–156.
  • 17. S. Li and X. Xue, Hopf bifurcation in a three-stage-structured prey-predator system with predator density dependent, Comm. Comp. Info. Scie., 288 (2012), 740–747.
  • 18. S. Li and W. Liu, Global hopf bifurcation in a delayed three-stage-structured prey-predator system, Proceedings-5th Int. Conf. Info. Comp. Scie., (2012), 206–209.
  • 19. J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Am. Math. Soc., 350 (1998), 4799–4838.
  • 20. Z. Wang, A very simple criterion for characterizing the crossing direction of time-delay systems with delay-dependent parameters, Int. J. Bifu. Chaos, 22 (2012), 1–7.
  • 21. J. Hale, Theory of Functional Differential Equations, Springer, New York, 1977.
  • 22. D. Breda, S. Maset and R. Vermiglio, TRACE-DDE: a tool for robust analysis and characteristic equations for delay differential equations, Lect. Notes Cont. Info. Scie., 388 (2009), 145–155.
  • 23. D. Breda, S. Maset and R. Vermiglio, Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM J. Sci. Comput., 27 (2005), 482–495.
  • 24. D. Breda, S. Maset and R. Vermiglio, An adaptive algorithm for efficient computation of level curves of surfaces, Numer. Algorithms, 52 (2009), 605–628.
  • 25. Y. Zhao, X. Yu and L. Wang, Bifurcation and control in an inertial two-neuron system with time delays, Int. J. Bifurc. Chaos, 22 (2012), 1–15.
  • 26. S. Li, W. Liu and X. Xue, Hopf bifurcation, chaos and impulsive control in a sex-structured prey-predator system with time delay, J. Biomath., 30 (2015), 443–452.


This article has been cited by

  • 1. Firdos Karim, Sudipa Chauhan, Joydip Dhar, On the comparative analysis of linear and nonlinear business cycle model: Effect on system dynamics, economy and policy making in general, Quantitative Finance and Economics, 2020, 4, 1, 172, 10.3934/QFE.2020008

Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved