An Adaptive Milne Predictor-Corrector Method for Harvested Logistic Growth Model: Performance Compared with Standard Approaches

Abstract

Population models that incorporate harvesting are essential for the sustainable management of renewable biological resources such as fisheries, forests, and wildlife populations. The harvested logistic growth model dx/dt=rx(1-x/K)-H, provides a simple yet effective framework for examining how populations respond to continuous exploitation, including conditions leading to persistence, equilibrium, or collapse. Although analytical solutions exist under ideal assumptions, practical applications often involve long simulation periods and dynamics near critical thresholds, thereby necessitating robust numerical techniques. This study develops an adaptive Milne predictor–corrector method equipped with local error estimation and automatic step-size control, and evaluates its performance against the classical fourth-order Runge–Kutta (RK4) method and the fixed-step Milne predictor–corrector scheme. The proposed method was implemented in MATLAB and tested using two biologically realistic parameter sets representing slow- and fast-growing populations under low, critical, and above-critical harvesting regimes. Numerical results show that all three methods achieved comparable levels of accuracy when validated against high-precision reference solutions. However, the adaptive Milne method consistently reduced the number of integration steps by approximately 50% in smooth regimes and by an even greater margin during population collapse, thereby improving computational efficiency. The study also reveals that the classical fixed-step Milne method may exhibit step-size dependent instability, particularly for systems with higher intrinsic growth rates, whereas the adaptive approach effectively mitigates this limitation through automatic adjustment of the step size. Furthermore, all methods predicted extinction times with errors of less than 1% relative to the reference solutions. These findings demonstrate that adaptive multistep methods provide an accurate, efficient, and numerically stable alternative for simulating harvested populations, particularly in applications involving repeated scenario analysis and decision-making near ecological thresholds.