The logistic growth model assumes that a population N(t) of
individuals, cells, or inanimate objects grows or diffuses at an
exponential rate until the approach of a limit or capacity
slows the growth, producing the familiar symmetrical
S-shaped curve. This model can be expressed mathematically by the following ordinary differential equation (ODE)
which specifies the growth rate
as a nonlinear function of
:
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While can be easily seen in a graph,
and
cannot. Accordingly, we replace them with two related metrics, the
midpoint and growth time. We define the growth time,
, as the length of the interval during which growth progresses
from 10% to 90% of the limit
. Through simple algebra, the
growth time is
. We define the
midpoint as the time
where
.
Again simple algebra shows
, which is also the
point of inflection of
, the time of most rapid growth, the
maximum of
.
The three parameters ,
, and
define the
parameterization of the logistic model used as the basic building
block for Loglet analysis