Exploring The Influence of Time-Delay On The Dynamics of a Two Prey-One Predator System With Quadratic Self-Interaction

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 τ 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.