Permanency in predator–prey models of Leslie type with ratio-dependent simplified Holling type-IV functional response

Abstract

We consider a predator–prey model of Leslie type with ratio-dependent simplified Holling type-IV functional response. The novelty of the model is that the functional response simulates group defense of the prey kind, which in turn, affects permanency of the system and existence of limit cycles. We show that permanency of the system holds automatically for some values of parameters and provides sufficient conditions for global stability of interior equilibrium by constructing a Lyapunov function. We prove that for some values of parameters the system exhibits a Hopf cycle and provides conditions by which the corresponding stable Hopf cycle is the only cycle that model may have. Numerical simulations show that if the conditions are broken, the model may have more than one limit cycle, which is phenomenal among the predator–prey models with one interior equilibrium.