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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.

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The paper investigates mixed-culture lactate (LA) fermentation of cheese whey (CW) in order to verify the possibility of using waste materials as feedstock to produce a product with high economic potential. The fermentation performance of two reactors operating in repeated-batch mode under uncontrolled pH conditions and various hydraulic retention time and feeding conditions was evaluated in terms of LA production. Five experimental phases were conducted. The hydraulic retention time (HRT) was varied from 1 to 4 days to verify its effect on the process performance. The best results, corresponding to the maximum LA concentration (20.1 g LA/L) and the maximum LA yield (0.37 g chemical oxygen demand (COD)(LA)/g COD(CW)), were reached by feeding the reactors with cheese whey alone and setting the HRT to 2 days. The maximum productivity of lactic acid (10.6 g LA/L/day) was observed when the HRT was decreased to 1 day.