Modeling biological systems with an improved fractional Gompertz law

TL;DR

This paper introduces a fractional generalization of the Gompertz law using a Caputo-like fractional derivative, providing a new model that captures long-term behavior in biological systems like fermentation and microalgae growth.

Contribution

It presents a novel fractional Gompertz model with unique asymptotic properties, validated on biological systems relevant to biophysics and environmental engineering.

Findings

Model exhibits distinct long-term asymptotic behavior.

Successfully applied to dark fermentation, photofermentation, and microalgae growth.

Provides a new framework for fractional modeling in biological systems.

Abstract

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.