A Ratio-Dependent Nonlinear Predator-Prey Model With Certain Dynamical Results

Abstract

In order to see the dynamics of prey-predator interaction, differential or difference equations are frequently used for modeling of such interactions. In present manuscript, we explore some qualitative aspects of two-dimensional ratio-dependent predator-prey model. Taking into account the non-overlapping generations for class of predator-prey system, a novel consistency preserving scheme is proposed. Our study reveals that the implemented discretization is bifurcation preserving. Some dynamical aspects including local behavior of equilibria, phase-plane analysis and emergence of Hopf bifurcation for continuous predator-prey model are studied. Moreover, existence of biologically feasible fixed points, their local asymptotic behavior and phase-plane classification of interior (positive) fixed point are carried out. Furthermore, bifurcation theory of normal forms is implemented to prove that proposed discrete-time model undergoes Neimark-Sacker bifurcation around its unique positive fixed point. Taking into account the bifurcating and fluctuating behaviour of discrete system, three chaos control strategies are implemented. Numerical simulations are provided to illustrate the theoretical discussion and effectiveness of introduced chaos control methods.

Keywords

• Prey-predator interaction
• stability analysis
• Neimark-Sacker bifurcation
• chaos control