Role of fear in a predator-prey system with ratio-dependent functional response in deterministic and stochastic environment

In this article, we propose and analyse a predator-prey model where apart from direct predation the prey population is affected by the fear induced from predators. The reproduction of the prey population is reduced as a cost of fear. The predator is assumed to consume the prey according to ratio-dependent functional response and is also involved in intra-specific competition due to limited resources of food. Through model analysis, it has been observed that fear factor regulates the dynamics of the system in a completely different way than in the case where functional response is only prey dependent. Also, intra-specific competition among predators reduces the effect of fear and it forms a different pattern in the system dynamics than that of the effect of fear. Furthermore, the deterministic model has been extended to a stochastic model by perturbing the natural death rates of both prey and predators. It has been observed that the stochastic system possesses a unique positive solution that is globally stable with respect to anywhere in the interior of the positive quadrant. The stochastic extinction and persistence scenario for both the species have been analysed and a detailed comparison between the deterministic and stochastic models have been done through exhaustive numerical simulation. Finally, numerical simulation has been performed to figure out the impact of fear on the population dynamics.