Previous experimental and theoretical studies suggest that predator’s interference in predator-prey relationship provides better descriptions of predator’s feeding over a range of predator-prey abundances. Also biological delays and environmental stochasticity play an important role to describe the system and its values. In this present study, I consider a Gaussian white-noise induced stochastic predator-prey model with the Beddington-DeAngelis functional response and gestation delay. Stochastic stability is measured by second order moment terms by calculating the nonequilibrium fluctuation of the nondelayed system and Fourier transform technique depicts the fluctuation of stochastic stability by introducing time lag. Different dynamical behaviors for both situations have been illustrated numerically also. The biological implications of my analytical and numerical findings are discussed critically.

In ecology, predation describes a biological interaction where a predator (an organism that is hunting) feeds on its prey (the organism that is attacked). Predators may or may not kill their prey prior to feeding on them, but the act of predation often results in the death of its prey and the eventual absorption of the prey’s tissue through consumption. Predators can have profound impacts on the dynamics of their prey that depend on how predator consumption is affected by prey density (the predators functional response). Understanding the prey-predator relationship is the central goal in ecology and a very significant component of this is the predators rate of feeding upon prey. The simplification of the food web structure due to predator interference allows qualitative predictions concerning the response of a food web to an external perturbation [

Delay is frequently used in a predator-prey model to make the model biologically more realistic. Recently, researchers are using more than one delay to study the effect of past history of the system populations [

Deterministic model in ecology does not usually incorporate environmental fluctuation; they are often justified by the implicit assumption that, in large populations, stochastic deviations are small enough to be ignored. Deterministic model will prove ecologically useful only if the dynamical patterns they reveal are still in evidence when stochastic effects are introduced. For terrestrial system, the environmental variability is large at both short and long time periods and could be expected to develop internal mechanisms to the system which would cope with short term variability and minimize the effects of long term variations; hence analysis of the system with white noise gives better results. Uncertain growth of populations is usually considered as an effect of environmental stochasticity. Reproduction of species depends on various factors, such as temperature, humidity, parasites and pathogens, and environmental pollution [

In the next section, we perform the stochastic scenario of nondelayed system of the model system (

I have so far considered an exact deterministic representation for the external force which derives the birth and death rates. In this section, I introduce the environmental stochasticity in the form of Gaussian white noise in the growth terms of both prey and predator equations. The model system (

To study the behavior of the system (

A method [

Eliminating

In this case, roots of (

In this case, roots of (

Consider

Taking Fourier transform of both sides of each of the equations in (

The system of equations (

Similarly the fluctuation intensity of

Let

Then after some calculation, I obtain the fluctuation intensities of

Let

In the numerical experiments of the stochastic system, I consider the set of parameter values taken as

Behavior of the model system (

(a)-(c) and (b)-(d) depict, respectively, the time evolution and corresponding phase trajectory of the stochastic model system (

Behavior of the model system (

(a)-(c) and (b)-(d) depict, respectively, the time evolution and corresponding phase trajectory of the stochastic model system (

Understanding the relationship between predator and prey is central goal in ecology. Predators rate of feeding upon prey is one of the most significant components in the study of predator-prey relationship. Predator's interference in prey-predator interaction is supposed to have significant role in the stability of their interaction. Also the time lag between prey capture and its corresponding positive feedback to predator's growth rate is a very significant part to realize the system. The purpose of this work is to observe the extent to which predator's interference and gestation delay drive the population dynamics of a predator-prey interaction under fluctuating environment. Results show that predator's interference plays a significant role in changing the stochastic stability of the system. To study the effect of environmental fluctuation on the time-delayed predator-prey system (

The author declares that there is no conflict of interests regarding the publication of this paper.

Research is supported by UGC (Dr. D. S. Kothari Postdoctoral Fellowship), India, no. F.4-2/2006(BSR)/13-1004/2013(BSR).