The impact of Time Delay on a Prey-Predator Model with Crowley-Martin Response Type and Migration in the Predator.

This article examines the dynamics of one prey and two predator populations. The predator populations migrate between two areas. The model incorporates a Crowley-Martin response and a time delay. This study reveals several novel dynamical behaviors of populations, with and without time delay. The existence of extinction and coexistence equilibrium points is investigated using a linearization approach and the Routh-Hurwitz criteria to determine the stability. This study explores the impact of time delay by identifying the delay margin at which the time delay destabilizes the equilibrium point. Under appropriate parameter values, a stable interior equilibrium point occurs when there is no time delay. When a time delay is incorporated, certain conditions indicate that the time delay does not affect the stability of the equilibrium point. Under different parameters, the presence of a delay margin can induce harmonic oscillations and trigger a Hopf bifurcation. The results demonstrate that time delay may either preserve or alter the stability of the equilibrium point. In some cases, the presence of the time delay may even lead to population extinction over a prolonged period. Numerical simulations are provided to illustrate the population dynamics under scenarios with and without time delay.