A robust multi-step predictor-corrector method for solving fractional order biological models

This paper develops a new multi-step predictorcorrector numerical scheme for the approximate solution of fractional-order biological models with Caputo derivatives of order 0 < α ≤ 1. The proposed method extends the classical Adams-Bashforth-Moulton framework to the fractional setting by replacing the standard Lagrange polynomial interpolation with binomial coecients derived from Euler's gamma function. A three-step fractional Adams-Bashforth predictor is coupled with a two-step fractional Adams-Moulton corrector to form a unied scheme. Existence and uniqueness of solutions are established using xed-point arguments, while consistency, stability, and convergence of the numerical method are rigorously analysed. It is shown that the combined scheme is stable and convergent with order O(h 2α). Truncation and global error bounds are derived, and convergence rates are veried numerically using the double mesh principle. The performance of the method is assessed on three representative fractional-order biological models, including ecological predator-prey dynamics and epidemiological SIR and SEIR systems. Numerical experiments demonstrate that the scheme is robust, accurate, and well suited for nonlinear models exhibiting memory eects. The results further indicate that smaller fractional orders are particularly eective in capturing fast epidemic dynamics, while larger orders are more appropriate for ecological systems with longer memory.