Dynamical analysis of a delayed ratio-dependent prey-predator model within fluctuating environment

In this paper, we consider a two-dimensional ratio-dependent prey-predator model involving discrete time delay within fluctuating environment. Firstly, we discuss basic dynamical properties of non-delayed and delayed model system in brief, within deterministic environment. Next we construct the stochastic differential equation model and stochastic delay differential equation model to study the effect of environmental driving forces on the dynamical behaviour. Stochastic models are extension of deterministic models, by perturbing two demographic parameters, namely, birth rate of prey population and death rate of predator population with white noise terms characterized by Gaussian distribution having zero mean and unit spectral density. We calculate population fluctuation intensity (variance) for prey and predator species by Laplace transform methods for stochastic differential equation model and Fourier transforms method for stochastic delay differential equation model. Numerical simulations are carried out to substantiate the analytical findings. Significant outcomes of our analytical findings and their interpretations from ecological point of view are provided in concluding section.