  Verhulst logistic curve either grows OR decays, depending on the {\it growth
rate} parameter value. A similar situation is found in the Gompertz law about
human mortality. However, growth can neither be infinite nor reach a finite
steady state at an infinite asymptotic time. Moreover before some decay, some
growth must have occurred. For example, some levelling-off could occur at
finite time, followed either by some growth again or then by some decay.
Numerous examples show that the Verhulst and Gompertz modelisations are too
reductive (or restrictive) descriptions of their original purpose. It is aimed,
in the present note, to encompass into ONE simple differential equation the
growth AND decay features of, e.g., population sizes, or numbers, but also of
many other measured characteristics found in social and physical science
systems. Previous generalisations of Verhulst or Gompertz functions are
recalled. It is shown that drastic growth or decay jumps or turnovers can be
readily described through drastic changes in values of the growth or decay
rate. However smoother descriptions can be found if the growth or decay rate is
modified in order to take into account some time or size dependence. Similar
arguments can be carried through, but not so easily, for the so called carrying
capacity, indeed leading to more elaborate algebraic work.
