Effects of prey refuge with Holling type IV functional response dependent prey predator model

Abstract

ABSTRACT In the present study, we have proposed two mathematical models for predator-prey interactions incorporating prey refuge having significant role in curbing the dynamical behaviour of both the system of prey-predator interactions from both the ecological and mathematical point of view. Here one introduces the prey refuge which depends on the encounters between prey and predator with Holling type IV functional response. Intra-specific competition among predators in searching food is considered in the second model as an additional effect to the first one. After the formulation of the models, the points of equilibria with its stability are systematically analyzed and the existence of bifurcation at the points of equilibrium has been duly carried out through their graphical representations with appropriate discussion in order to validate the applicability of both the system under consideration. It is found that the Hopf bifurcation of both the systems, which is of sub-critical type. The predation rate bears an important role in changing the stability of predator-free equilibrium to co-existence equilibrium. It is also observed that the biomass of prey density increases and predator density decreases with respect to gradual increment of the coefficient of prey refuge. Existence of paradox of enrichments are examined as well in both the model system.