Bifurcation and Chaos in a Discrete Fractional-Order Prey-Predator System Involving Allee Effect

Abstract

This chapter considers the discrete counterpart of a fractional order prey-predator ODE system involving Allee effect. Several more realistic models were proposed to describe nonlinear interactions between species by introducing different types functional responses and Allee effect. Non-local property of fractional differential equations is useful in modeling population interactions possessing memories. The model under investigation has three steady states, and the positive steady state exists under certain condition. Dynamic nature of the model is discussed through local stability analysis. Enquiry into the qualitative behavior of the model reveals rich and complex dynamics exhibited by the discrete-time model. Moreover, this model undergoes Neimark Sacker bifurcation when the chosen parameter passes through a critical value. The analytical results are strengthened with appropriate numerical examples. The computation of maximal Lyapunov exponents confirms the existence of chaos. Chaos control is achieved by linear feedback control and hybrid control methods.