Nonlinear Dynamics

, Volume 96, Issue 4, pp 2653–2679 | Cite as

Dynamics of prey–predator model with stage structure in prey including maturation and gestation delays

  • Balram DubeyEmail author
  • Ankit Kumar
Original Paper


This study proposes a three-dimensional prey–predator model with stage structure in prey (immature and mature) including maturation delay in prey population and gestation delay in predator population. It is assumed that the immature prey population is consumed by predators with Holling type I functional response and the interaction between mature prey and predator species is followed by Crowley–Martin-type functional response. We analyzed the equilibrium points, local and global asymptotic behavior of interior equilibrium point for the non-delayed system. Hopf-bifurcation with respect to different parameters has also been studied for the system. Further, the existence of periodic solutions through Hopf-bifurcation is shown with respect to both the delays. Our model analysis shows that time delay plays a vital role in governing the dynamics of the system. It changes the stability behavior of the system into instability, even with the switching of stability. The direction and stability of Hopf-bifurcation are also studied by using normal form method and center manifold theorem. Finally, computer simulation and graphical illustrations have been carried out to support our theoretical investigations.


Prey–predator Stage structure Hopf-bifurcation Maturation delay Gestation delay Switching stability 



Authors are thankful to anonymous reviewers for careful reading and constructive suggestions that improved the quality and presentation of the paper. The author (AK) acknowledges the Junior Research Fellowship received from University Grant Commission, New Delhi, India.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsBITS Pilani, Pilani CampusPilaniIndia

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