Bifurcation Analysis of a Delayed Modified Holling-Tanner Predator-Prey Model with Refuge

  • Charu AroraEmail author
  • Vivek Kumar
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 655)


This paper deals with a delayed modified Holling-Tanner predator-prey model with refuge. The proposed model highlights the impact of delay and refuge on the dynamics of the system wherein analysis of the model in terms of local stability is performed. Both theoretical and experimental works point out that delay and refuge play an important role in the stability of the model and also it has been observed that due to delay, bifurcation occurred which results in considering delay as a bifurcation parameter. For some specific values of delay, Hopf bifurcation is investigated for the proposed model and direction of Hopf bifurcation with the stability of bifurcated periodic solutions by using normal form theory and central manifold reduction is also included in the domain of this study. At the end, few numerical simulations based on hypothetical set of parameters for the support of theoretical formulation are also carried out.


Predator-prey model Time delay Hopf bifurcation Stability Periodic solution 


  1. 1.
    Leslie, P.H., Gower, J.C.: The properties of a stochastic model for the predator-prey type of interaction between two species. Biometrika 47, 219–234 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Lu, Z., Liu, X.: Analysis of a predator prey model with modified Holling-Tanner functional response and time delay. Nonlinear Anal. Real World Appl. 9, 641–650 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Zhang, J.-F.: Bifurcation analysis of a modified Holling-Tanner predator-prey model with time delay. Appl. Math. Model. 36, 1219–1231 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kant, S., Kumar, V.: Delayed prey-predator system with habitat complexity and refuge. In: Proceedings of International Conference on Mathematical Sciences (ICMS 2014), pp. 584–591 (2014).
  5. 5.
    Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)zbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Delhi Technological UniversityDelhiIndia

Personalised recommendations