  The growth function of populations is central in biomathematics. The main
dogma is the existence of density dependence mechanisms, which can be modelled
with distinct functional forms that depend on the size of the population. One
important class of regulatory functions is the $\theta$-logistic, which
generalises the logistic equation. Using this model as a motivation, this paper
introduces a simple dynamical reformulation that generalises many growth
functions. The reformulation consists of two equations, one for population
size, and one for the growth rate. Furthermore, the model shows that although
population is density-dependent, the dynamics of the growth rate does not
depend either on population size, nor on the carrying capacity. Actually, the
growth equation is uncoupled from the population size equation, and the model
has only two parameters, a Malthusian parameter $\rho$ and a competition
coefficient $\theta$. Distinct sign combinations of these parameters reproduce
not only the family of $\theta$-logistics, but also the van Bertalanffy,
Gompertz and Potential Growth equations, among other possibilities. It is also
shown that, except for two critical points, there is a general size-scaling
relation that includes those appearing in the most important allometric
theories, including the recently proposed Metabolic Theory of Ecology. With
this model, several issues of general interest are discussed such as the growth
of animal population, extinctions, cell growth and allometry, and the effect of
environment over a population.
