Dynamical Study of a Discrete Predator–Prey Model Incorporating Smith Growth and Linear Functional Response

This study focuses on the dynamical mechanisms of a class of discrete predator–prey systems, aiming to uncover the nonlinear dynamic laws in ecological evolution. Based on the principles of biomathematical modeling, a continuous model incorporating the Smith-type population growth rate and a linear functional response was developed. Subsequently, this model was discretized using the forward Euler method, yielding a two-dimensional difference equation system with a clear ecological interpretation. Theoretically, the local stability criteria for all equilibria were comprehensively derived. Dynamically, the occurrence conditions for both flip bifurcations and Neimark–Sacker bifurcations were rigorously determined, and parameter thresholds that can be analytically verified were provided. Numerically, with the step-size parameter h taken as the bifurcation parameter, bifurcation diagrams and maximum Lyapunov exponent spectra were generated simultaneously, and both showed excellent consistency. This approach clearly demonstrated the evolutionary pathway of typical dynamical behaviors, such as period-doubling cascades, periodic oscillations, and chaos. Results indicate that h not only governs the system’s transition from stable equilibria to chaotic states but also highlights the sensitivity and criticality of ecological parameters in population regulation. This work not only enriches the theoretical basis of classical discrete ecological models but also provides a dynamical framework for field-based population monitoring and adaptive intervention strategies.