Abdul Hussain Surosh, Reza Khoshsiar Ghaziani, and Javad Alidousti
Exploring The Influence of Time-Delay On The Dynamics of a Two Prey-One Predator System With Quadratic Self-Interaction
Dynamic Systems and Applications 31 (2022) 63-88

https://doi.org/10.46719/dsa202231.02.01

ABSTRACT.
This work is devoted to studying the effects of time-delay on the dynamics of a
two prey-one predator system with quadratic self-interaction. The essential dynamical structures
of the delayed system are analyzed by means of local stability analysis and bifurcation theory.
Taking the time lag as a free parameter, the necessary conditions for the existence of the Hopf
bifurcation around the interior equilibrium of the system has been derived both analytically and
numerically. It is observed that a Hopf bifurcation occurs when the bifurcation parameter crosses a
certain threshold value and it is found that the dynamics of the system can be effected significantly
by the time delay which has both stabilizing and destabilizing impacts depending on the magnitude
of the delay. Moreover, we derived the explicit formulas in order to determine the direction of the
Hopf bifurcation and examine the nature of the bifurcating periodic solution by using the normal
form method and the center manifold theorem. Eventually, some numerical simulations are given to
verify the derived theoretical analysis.

AMS (MOS) Subject Classification. 34L10, 34B24, 47E05.
Key Words and Phrases. Time delay system, Predator-prey, Center manifold theorem, Stability
analysis, Hopf bifurcation.