Extending the Monod Model of Microbial Growth with Memory

The candidate reproduces the paper’s core derivations: (a) solving the Riemann–Liouville integral formulation to obtain µ(R) = µmax/Γ(1+α)·(R/(K+R))^α matches Eq. (7) after assuming µ(0)=0 and 0<α≤1 (see the Riemann–Liouville set-up and solution in the paper) ; (b) the half-saturation R = M(α)K with M(α) = [(2/Γ(1+α))^{1/α}−1]^{-1} matches Eq. (8) ; (c) the chemostat equilibrium R* = ((dΓ(1+α)/µmax)^{1/α}/(1−(dΓ(1+α)/µmax)^{1/α}))K and X* = d(Rs−R*)/ρ(R*) matches the equilibrium given in the paper ; and (d) the estimator for α by matching to Droop and taking time-averaged logs matches Eqs. (12)–(13) and the accompanying interpretation via uptake-rate normalization . A minor clarity note: when moving from Eq. (12) to the “uptake over max uptake” form, one should explicitly allow Kρ≠K by writing the denominator as log(ρ/Vmax), as the candidate prudently notes.