Modelling the effect of resource subsidy on a two-species predator-prey system under the influence of environmental noises

Our ecosystem is full of various types of food chain systems. Most of the species have more than one food source with various kinds of possible interactions. In this work, we have studied a predator-prey-subsidy model in a rapidly fluctuating environment. As per our consideration predator exploits both a prey population and an allochthonous resource which is provided as a subsidy to the system exogenously, with Holling Type II response functions. Furthermore, we have introduced Gaussian white noise to the main parameters subject to coupling of a prey-predator pair with its environment. We have examined the existence of unique global positive solutions for both the deterministic and stochastic systems. Boundedness, uniform continuity, global attractiveness in mean of solutions of the stochastic system are proved. Conditions for extinction of prey and predator population are derived and the importance of environmental noise as well as subsidy for extinction is discussed through mathematical results and numerical findings. We have derived conditions for persistence of the proposed stochastic system. The results of persistence and extinction have been justified by numerical simulations. Numerically we have found that presence of subsidy can be the cause of survival of predator even there is no prey available for consumption which fact is different from the studies of those systems where additional food is provided for predator. We have also observed that very high input of subsidy can be a cause for extinction of prey population.