Gompertz and Verhulst frameworks for growth and decay description

Verhulst logistic curve either grows OR decays, depending on the 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.