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Nonautonomous Dynamical Systems

formerly Nonautonomous and Stochastic Dynamical Systems

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Mathematical Citation Quotient (MCQ) 2017: 0.71

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Dynamical analysis of a predator-prey interaction model with time delay and prey refuge

Jai Prakash Tripathi
  • Department of Mathematics, Central University of Rajasthan, NH-8, Bandarsindri, Kishangarh-305801, Distt.-Ajmer, Rajasthan, India
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Swati Tyagi / Syed Abbas
Published Online: 2018-11-10 | DOI: https://doi.org/10.1515/msds-2018-0011


In this paper, we study a predator-prey model with prey refuge and delay. We investigate the combined role of prey refuge and delay on the dynamical behaviour of the delayed system by incorporating discrete type gestation delay of predator. It is found that Hopf bifurcation occurs when the delay parameter τ crosses some critical value. In particular, it is shown that the conditions obtained for the Hopf bifurcation behaviour are sufficient but not necessary and the prey reserve is unable to stabilize the unstable interior equilibrium due to Hopf bifurcation. In particular, the direction and stability of bifurcating periodic solutions are determined by applying normal form theory and center manifold theorem for functional differential equations. Mathematically, we analyze the effect of increase or decrease of prey reserve on the equilibrium states of prey and predator species. At the end, we perform some numerical simulations to substantiate our analytical findings.

Keywords: Time delay; Stability; Center manifold theorem; Hopf bifurcation; functional response


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About the article

Received: 2017-03-13

Accepted: 2018-10-10

Published Online: 2018-11-10

Published in Print: 2018-10-01

Citation Information: Nonautonomous Dynamical Systems, Volume 5, Issue 1, Pages 138–151, ISSN (Online) 2353-0626, DOI: https://doi.org/10.1515/msds-2018-0011.

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© by Jai Prakash Tripathi, et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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