Stability and Local Bifurcations of a Discrete Predator–Prey System with Allee Effect and Holling-IV Functional Response

Abstract

This paper investigates the local dynamics of a discrete predator–prey system incorporating an Allee effect and a Holling-IV functional response. After nondimensionalization, a positivity-preserving discrete model is obtained by the semi-discretization method. The non-negative fixed points are classified and their local stability is established. The center manifold theorem and local bifurcation theory are then used to prove two transcritical bifurcations at the boundary fixed points E1 and E2 and a saddle-node bifurcation at the positive critical fixed point E*. Numerical equilibrium branch diagrams and phase portraits illustrate these local results. The possible Neimark–Sacker bifurcation at E3 and flip bifurcation at E4 are beyond the present scope of this study.