Open Access

ARTICLE

Stability and Bifurcation of a Prey-Predator System with Additional Food and Two Discrete Delays

Ankit Kumar, Balram Dubey^*
Department of Mathematics, BITS Pilani, Pilani Campus, Pilani, 333031, India
* Corresponding Author: Balram Dubey. Email: -pilani.ac.in

Computer Modeling in Engineering & Sciences 2021, 126(2), 505-547. https://doi.org/10.32604/cmes.2021.013206

Received 29 July 2020; Accepted 22 October 2020; Issue published 21 January 2021

Abstract

In this paper, the impact of additional food and two discrete delays on the dynamics of a prey-predator model is investigated. The interaction between prey and predator is considered as Holling Type-II functional response. The additional food is provided to the predator to reduce its dependency on the prey. One delay is the gestation delay in predator while the other delay is the delay in supplying the additional food to predators. The positivity, boundedness and persistence of the solutions of the system are studied to show the system as biologically well-behaved. The existence of steady states, their local and global asymptotic behavior for the non-delayed system are investigated. It is shown that (i) predator’s dependency factor on additional food induces a periodic solution in the system, and (ii) the two delays considered in the system are capable to change the status of the stability behavior of the system. The existence of periodic solutions via Hopf-bifurcation is shown with respect to both the delays. Our analysis shows that both delay parameters play an important role in governing the dynamics of the system. The direction and stability of Hopf-bifurcation are also investigated through the normal form theory and the center manifold theorem. Numerical experiments are also conducted to validate the theoretical results.

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Keywords

Additional food; Gestation delay; Hopf-bifurcation; Prey-predator

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