Modeling biological systems with an improved fractional Gompertz law

• Published in: Communications in nonlinear science & numerical simulation 2019-07, Vol.74, p.260-267
• Main Authors: Frunzo, Luigi, Garra, Roberto, Giusti, Andrea, Luongo, Vincenzo
• Format: Article
• Language: English
• Subjects: Fractional calculus; Fractional derivative of a function with respect to another function; Gompertz growth law; Mittag–Leffler functions
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2019.03.024

•Fractional derivative of a function with respect to another function.•Fractional generalization of the Gompertz law.•Special functions: Mittag–Leffler functions.•Model validation: biological systems (Dark Fermentation, photo fermentation and microalgae biomass growth). The aim of this paper is to provide a fractional generalization of the Gompertz law via a Caputo-like definition of fractional derivative of a function with respect to another function. In particular, we observe that the model presented appears to be substantially different from the other attempt of fractional modifications of this model, since the fractional nature is carried along by the general solution even in its asymptotic behavior for long times. We then validate the presented model by employing it as a reference frame to model three biological systems of peculiar interest for biophysics and environmental engineering, namely: dark fermentation, photofermentation and microalgae biomass growth.