  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.
