Stability, Bifurcation Analysis and Chaos in a Discretized Fractional-Order Predator–Prey System with Nonlinear Functional Response

This study examines a discrete fractional-order predator–prey system incorporating a Holling type-III functional response. The Caputo fractional derivative is employed because it naturally incorporates memory and hereditary effects while preserving biologically meaningful initial conditions. The system is formulated from a biologically relevant continuous fractional-order framework through the application of the piecewise constant argument approach, enabling an analysis of how memory-dependent effects and discrete dynamics influence predator–prey interactions. The existence and local stability of fixed points are determined by using the Jacobian matrix and eigenvalue conditions. The bifurcation of the positive fixed point is analyzed by using the center manifold and normal form methods. Numerical simulations, including bifurcation diagrams, phase portraits, and maximum Lyapunov exponent plots, confirm our analytical results and reveal periodic, quasiperiodic, and chaotic behavior. The findings of this study reveal that the combined influence of memory-dependent dynamics, nonlinear predator–prey interactions, and discrete-time effects can generate rich and complicated behaviors in fractional-order predator-prey systems.