Equilibrium Preservation and Finite Memory Bias in Fractional Monod Chemostat Models

Abstract

Fractional chemostat models distinguish transient memory from the equilibrium geometry of microbial growth, but a truncated numerical history can alter both. We analyze a two-state Monod chemostat with Caputo derivatives and develop exact diagnostics for this distinction. The positive equilibrium meets washout at the order-independent dilution . Its linearization has no characteristic root in the closed right half-plane for any pair of orders in , while a common order also yields an exact Mittag–Leffler balance for substrate-equivalent mass. For an explicit Grünwald–Letnikov recurrence applied to uncorrected states, the missing Caputo initial correction produces immediate drift from a nonzero equilibrium. A finite cutoff adds a stationary loss whose exact coefficient is derived. The resulting Monod branch equation is quadratic and may introduce two artificial collision points; its upper displacement admits an exact decomposition. At a fixed retained time horizon, this loss persists under grid refinement. Independent deterministic computations using corrected full-history Grünwald–Letnikov and Adams predictor–corrector methods verify equilibrium preservation, exact-solution accuracy, and decreasing cross-method discrepancies. For the stated kinetics and mixed orders , the uncorrected upper collision approaches  h⁻¹ when a 7.5-hour history is retained, rather than the continuous value  h⁻¹. Correcting initialization alone also leaves a nonzero truncation floor in scalar relaxation. These results provide a reproducible basis for evaluating washout detection and fractional-order estimation without attributing numerical memory loss to biological mechanisms.