Memory and Time-Varying Activity in Batch Microbial Growth: A ψ-Fractional Monod Model Tested on Substrate–Biomass and Open Growth Data

Batch cultures often show features that the Monod model cannot reproduce: lag phases, residual substrate and loss of cellular activity. Fractional-order extensions are increasingly proposed to capture such effects, but they are rarely tested against data. We formulate a ψ-fractional Monod model in which the order θ represents memory of the cells’ physiological history and a parameter β represents a time-varying activity. The model conserves mass, keeps concentrations positive, contains Monod kinetics as a special case and, without cell decay, predicts residual substrate if and only if activity declines (β > 0). We compare it with the Monod, Baranyi–Roberts, Weibull and tailing models on synthetic substrate–biomass data, a tequila batch fermentation and 43 open growth and survival curves, using information criteria, profile likelihoods and forecasting. With joint substrate–biomass data, the model recovers memory and deactivation where the Monod model returns biased kinetic constants. In the tequila fermentation, which stalls with 5.5% of the sugar unconsumed, it reproduces the stall with an identifiable order (θ = 0.54; 95% interval 0.40–0.67), but Monod kinetics with ethanol inhibition is equally supported. With biomass data alone, the order is not identifiable and the Baranyi–Roberts model forecasts better. We conclude with practical guidelines for model selection.