Qualitative Analysis of a Density-Dependent Prey–Predator Model with Holling Type III Functional Responses

This research examines the behavioral shifts within a discrete-time predator–prey framework, constructed by applying the forward Euler discretization to a continuous model. The system incorporates Smith’s growth dynamics for the prey population alongside a Holling type III functional response to characterize predator behavior. Through bifurcation analysis, it is demonstrated that the interior fixed point undergoes stability loss via Neimark–Sacker and period-doubling transitions, leading to the emergence of quasiperiodic oscillations and chaos. Furthermore, the application of normal-form theory verifies the nondegeneracy of these bifurcations and establishes the direction of the resulting orbits. We use phase portraits, Lyapunov exponents, and bifurcation diagrams to confirm the model’s rich dynamics. These numerical tools demonstrate how the system moves from stable equilibria to more intricate behaviors. The application of partial rank correlation coefficients reveals the most influential parameters governing the system’s asymptotic population levels, providing a global perspective on parameter sensitivity. The Ott–Grebogi–Yorke (OGY) chaos control strategy is employed to suppress unwanted bifurcations and stabilize chaotic oscillations within the system. These results underscore the role of nonlinear interactions and discrete-time frameworks in precipitating unpredictable population fluctuations while simultaneously offering a suite of mechanisms for enhancing the stability of ecological networks.