Stabilizing Effect of Prey Refuge and Predator’s Interference on the Dynamics of Prey with Delayed Growth and Generalist Predator with Delayed Gestation

Abstract

In the present paper, I study a prey-predator model with multiple time delays where the predator population is regarded as generalist. For this regard, I consider a Holling-Tanner prey-predator system where a constant time delay is incorporated in the logistic growth of the prey to represent a delayed density dependent feedback mechanism and the second time delay is considered to account for the length of the gestation period of the predator. Predator’s interference in predator-prey relationship provides better descriptions of predator's feeding over a range of prey-predator abundances, so the predator's functional response here is considered to be Type II ratio-dependent. In accordance with previous studies, it is observed that delay destabilizes the system, in general, and stability loss occurs via Hopf bifurcation. There exist critical values of delay parameters below which the coexistence equilibrium is stable and above which it is unstable. Hopf bifurcation occurs when delay parameters cross their critical values. When delay parameters are large enough than their critical values, the system exhibits chaotic behavior and this abnormal behavior may be controlled by refuge. Numerical computation is also performed to validate different theoretical results. Lyapunov exponent, recurrence plot, and power spectral density confirm the chaotic dynamical behaviors.