Dynamics and Strong Resonance Bifurcations in a Discrete Predator–Prey Model With Square Root Functional Response
Abstract
This paper investigates the dynamic behavior of a discrete‐time predator–prey model incorporating a square root functional response, which captures the herding behavior of prey populations. By applying Euler discretization to the continuous‐time model, we obtain a system of difference equations that serves as the foundation for a detailed bifurcation analysis. The study focuses on codimension‐two bifurcations, specifically the 1:2, 1:3, and 1:4 strong resonances, which are analyzed both analytically and numerically. Using center manifold theory and a series of affine transformations, we derive the normal forms associated with each resonance. These normal forms reveal the underlying bifurcation structures and provide conditions for the occurrence of complex dynamical phenomena. Numerical continuation methods, implemented via MATCONT, are employed to validate the analytical results and to visualize bifurcation curves and phase portraits. The findings contribute to the broader understanding of discrete dynamical systems in population ecology, highlighting the rich bifurcation scenarios that arise from parameter variations.