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The existence of Hopf bifurcation for a delayed Holling–Tanner type predator–prey model

Part of: Functional-differential and differential-difference equations

Published online by Cambridge University Press:  05 January 2026

Canan Çelik Open the ORCID record for Canan Çelik [Opens in a new window]  and
Adalet Zeynep Esirgen Open the ORCID record for Adalet Zeynep Esirgen [Opens in a new window]
Canan Çelik
Affiliation:
Departments of Mathematics, Yildiz Technical University , Türkiye e-mail: celikcan@yildiz.edu.tr
Adalet Zeynep Esirgen*
Affiliation:
Departments of Mathematics, Yildiz Technical University , Türkiye e-mail: celikcan@yildiz.edu.tr
*
e-mail: zeynep.esirgen@std.yildiz.edu.tr
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Abstract

In this study, a Holling–Tanner type predator–prey model with a discrete time delay is investigated, where the functional response of the predator dynamics is ratio-dependent. We first analyze the local stability of the equilibrium point and examine the existence of Hopf bifurcations. The Hopf bifurcation, also known as the Poincaré–Andronov–Hopf bifurcation, is named after the French mathematician Jules Henri Poincaré, the Russian mathematician Alexander A. Andronov, and the German mathematician Heinz Hopf, whose fundamental contributions laid the foundation of this theory. By treating the delay parameter $\tau $ as the bifurcation parameter, we show that a Hopf bifurcation occurs when the delay crosses certain critical values. Finally, numerical simulations are carried out to support and illustrate our theoretical results.

Keywords

Predator–prey systemdiscrete delayHopf bifurcationstability

MSC classification

Primary: 34K18: Bifurcation theory 34K20: Stability theory

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Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 13
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Copyright
© The Author(s), 2026. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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References

Allen, L. J., An introduction to mathematical biology, Pearson/Prentice Hall, London, 2007.Google Scholar
Arditi, R., Abillon, J. M., and Silva, J. V., The Effect of a time-delay in a predator-prey model . Math. Biosci. 33(1977), nos. 1–2, 107–120.Google Scholar
Begon, M., Townsend, C. R., and Harper, J. L., Ecology: From individuals to ecosystems, Wiley, New York, 2009.Google Scholar
Celik, C., The stability and Hopf bifurcation for a predator–prey system with time delay . Chaos Solitons Fractals 37(2008), no. 1, 87–99.Google Scholar
Çelik, C., Hopf bifurcation of a ratio-dependent predator–prey system with time delay . Chaos Solitons Fractals 42(2009), no. 3, 1474–1484.Google Scholar
Celik, C. and Çekiç, G., Bifurcation analysis of a logistic predator–prey system with delay . ANZIAM J. 57(2016), no. 4, 445–460.Google Scholar
Çelik, C. and Degirmenci, E., Stability and Hopf bifurcation of a predator-prey model with discrete and distributed delays . J. Appl. Nonlinear Dyn. 5(2016), no. 1, 73–91.Google Scholar
Çelik, C. and Merdan, H., Hopf bifurcation analysis of a system of coupled delayed-differential equations . Appl. Math. Comput. 219(2013), no. 12, 6605–6617.Google Scholar
Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics. Vol. 74, Springer, 2013.Google Scholar
Hu, D. and Cao, H., Stability and bifurcation analysis in a predator–prey system with Michaelis–Menten type predator harvesting . Nonlinear Anal. Real World Appl. 33, Elsevier, Amsterdam, 2017, 58–82.Google Scholar
Karaoglu, E. and Merdan, H., Hopf bifurcation analysis for a ratio-dependent predator–prey system involving two delays . ANZIAM J. 55(2014), no. 3, 214–231.Google Scholar
Khabyah, A. A., Ahmed, R., Akram, M. S., and Akhtar, S., Stability, bifurcation, and chaos control in a discrete predator-prey model with strong Allee effect . AIMS Math. 8(2023), no. 4, 8060–8081.Google Scholar
Kuang, Y., Delay differential equations: With applications in population dynamics, Mathematics in Science and Engineering, 45, Academic Press, 1993.Google Scholar
Li, K. and Wei, J., Stability and Hopf bifurcation analysis of a prey–predator system with two delays. Chaos Solitons Fractals 42, Elsevier, Amsterdam, 2009, no. 5, 2606–2613.Google Scholar
Murray, J. D., Mathematical biology: I. An introduction. Vol. 17, Springer, New York, 2007.Google Scholar
Ruan, S., On nonlinear dynamics of predator-prey models with discrete delay. Math. Model. Nat. Phenom. Free Access, 4 EDP Sciences, Les Ulis, 2009, no. 2, 140–188.Google Scholar
Singh, H., Dhar, J., and Bhatti, H. S., Discrete-time bifurcation behavior of a prey-predator system with generalized predator . Adv. Differ. Equ. 1(2015), 206.Google Scholar
Smith, T. M. and Smith, R. L., Elements of ecology, Wiley, 2012.Google Scholar
Volterra, V., Variations and fluctuations of the number of individuals in animal species living together. ICES J. Mar. Sci. 3, Oxford University Press, Oxford, 1928, no. 1, 3–51.Google Scholar
Wang, J., Shi, H., Xu, L., and Zang, L., Hopf bifurcation and chaos of tumor-lymphatic model with two time delays . Chaos Solitons Fractals 157(2022), 111922.Google Scholar
Yafia, R., El Adnani, F., and Alaoui, H. T., Limit cycle and numerical similations for small and large delays in a predator–prey model with modified Leslie–Gower and Holling-type II schemes . Nonlinear Anal. Real World Appl. 9(2008), no. 5, 2055–2067.Google Scholar
Yuan, R., Jiang, W., and Wang, Y., Addle-node-Hopf bifurcation in a modified Leslie–Gower predator-prey model with time-delay and prey harvesting . J. Math. Anal. Appl. 422(2015), no. 2, 1072–1090.Google Scholar
Zhang, W., Jin, D., and Yang, R., Hopf bifurcation in a predator–prey model with memory effect in predator and anti-predator behaviour in prey . Mathematics 11(2023), no. 3, 556.Google Scholar