This study develops a semi-discretized time system from the continuous-time Rosenzweig–-MacArthur model via the method of piecewise constant argument—a discretization approach that preserves both mathematical rigor and biological interpretability. For the proposed system incorporating constant-effort harvesting on both prey and predator populations, we present rigorous quantitative derivations for the existence and local stability of non-negative equilibrium. Furthermore, we investigate complex dynamical behaviors, including transcritical and Neimark–Sacker bifurcations, induced by parameter variations. We specifically focus on calculating the first Lyapunov coefficient to determine the stability of closed orbits emerging from the Neimark–Sacker bifurcation. Numerical validation of chaotic dynamics is conducted using the computed Maximum Lyapunov Exponent spectrum. Numerical simulations not only confirm consistency with analytical results but also reveal key ecological dynamics of the system: (i) the paradox of enrichment—a classic ecological phenomenon—persists even under constant-effort harvesting; (ii) appropriate tuning of harvesting parameters enables the coexistence of prey and predator populations in a stable closed orbit, resulting in cyclic coexistence.