THE TIME DELAY EFFECT AND HARVESTING ON THE PREDATOR-PREY: ANALYSIS AND SIMULATION

Time-delay differential equations-based mathematical modeling is a significant tool for understanding the effect of delay in biological systems and analyzing how it affects the dynamics of those systems' asymptotic behavior. The prey-predator model described in this paper includes disease in the prey species, harvesting in each population, and time delays in predation and gestation of the predator. The solutions of the model are positive and bounded for all times within a realistic region. The existence of all fixed points has been established. When a time delay is present, the essential requirements for the local stability of the positive equilibrium and the occurrence of Hopf bifurcation can be determined by analyzing the associated characteristic equation. The characteristics of the Hopf bifurcation are obtained by utilizing both normal form theory and the center manifold theorem. Finally, we employ numerical simulations to validate our analytical findings. A Hopf bifurcation in the system occurs when the delay surpasses a particular threshold.