Dynamics of generalist predator in a stochastic environment: Effect of delayed growth and prey refuge

Modified ratio-dependent Holling-Tanner model with prey refuge is considered.Delays are taken in logistic growth of prey as feedback mechanism and gestation period of predator.Model system exhibits Hopf-bifurcation when the delay parameters cross their critical values.Stochastic analysis of the model system is carried out by incorporating Gaussian white noise.Using Fourier transform technique, fluctuation and stability of the stochastic model is studied. In this paper, an attempt has been made to understand the dynamics of a prey-predator system with multiple time delays where the predator population is regarded as a generalist type. In this regard, we consider a modified 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 prey-predator relationship provides better descriptions of predator's feeding over a range of prey-predator abundances, so the predator's functional response is considered to be Type II ratio-dependent and foraging efficiency of predator largely varies with the refuge strategy of prey population. In accordance with previous studies, it is observed that delay destabilizes the system, in general and stability loss occurs via Hopf-bifurcation. In particular, we show that there exists critical values of the delay parameters below which the coexistence equilibrium is stable and above which it is unstable. Hopf bifurcation occurs when the delay parameters cross their critical values. Also, environmental stochasticity in the form of Gaussian white-noise plays a significant role to describe the system and its values. Numerical computation is also performed to validate and visualize different theoretical results presented. The analysis and results in this work are interesting both in mathematical and biological point of views.

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