Analysis of Oscillatory Patterns of a Discrete-Time Rosenzweig–MacArthur Model

Abstract

This article investigates the oscillatory patterns of the following discrete-time Rosenzweig–MacArthur model [Formula: see text] The system describes the evolution and interaction of the populations of two associated species (prey and predator) from generation to generation. We show that this system can exhibit co-dimension-1 bifurcations (flip and Neimark–Sacker bifurcations) as [Formula: see text] crosses some critical values and codimension-2 bifurcations (1:2, 1:3, and 1:4 resonances) for certain critical values of [Formula: see text] at the positive equilibrium point. The normal form theory and the center manifold theorem are used to obtain the normal forms. For codimension-2 bifurcations, the bifurcation diagrams are established by using these normal forms along the orbits of differential equations. Numerical simulations are presented to confirm the theoretical results.