Stochastic prey–predator model with additional food for predator

Abstract

In this work we have studied a predator–prey model where the prey grows logistically in the absence of predator and the functional response of predator towards prey and additional food that are derived in the text. Prey’s growth rate and the predator’s death rate have been perturbed with Gaussian white noises which has been proved extremely useful to model rapidly fluctuating phenomena. These two parameters are the main terms subject to coupling of a prey–predator pair with its environment Dimentberg (1988). Existence and uniqueness of global positive solution of the system have been established under environmental noise. Then the conditions under which extinction of predator and prey populations occur have been established. In our analysis, it is found that the environmental noise plays an important role in extinction as well as persistence of prey and predator populations. We have also discussed about the persistence of the system under obtained conditions and how the solution of the underlying system is globally attractive in mean. To derive the theorems we have shown the uniform continuous behavior of the solutions. Although we have considered a prey–predator model, the survival of predator population is possible in absence of prey population, since the additional food is provided to predator. But it is found that the extinction of prey population drive predator population to extinction. Our analytical findings are explained through numerical simulation which show the reliability of our model from the ecological point of view. It is shown in numerical simulation that if the effectual food level of additional food which is provided to the predator is high, then the predator dominates the prey population.