Logistic growth
A key insight of Darwin in
formulating his Theory of Natural Selection was the recognition
that, as Malthus had argued, all species' numbers tend to
increase geometrically, whereas resources increase arithmetically
at best. In modern ecological theory, in the absence of checks
to natural increase, population size N would increase geometrically over
time at some intrinsic growth rate r,
so that dN/dt = rN.
If the environment imposes a upper limit K (carrying
capacity) to population size, N increases
by a logistic growth
curve towards K, such that the limit to dN/dt
=
rN (K-N) / K : the rate of growth slows as it approaches K.
Carrying capacity of the environment may be a consequence of biotic
and (or) abiotic factors, for example the presence of
predators or the amount of rainfall.
In the example above, r = 2 such that the population tends
to double every generation, and K = 10,000. The black
curve shows the population size N at any point, and
the blue curve shows N as the
discrete value of dN/dt ,
the change in N per generation