This Demonstration displays linear and semi-logarithmic plots of one of the generalized versions of the logistic (Verhulst) equation formulated as
is the momentary number and
are constants, with the boundary condition
. This equation can be used to model isothermal microbial growth in foods by slider-controlled manipulation of five growth parameters. The equation depicts a scenario where the momentary growth rate scales with the momentary population size
as a power that need not be 1 and with the habitat's carrying capacity by the same or a different power. The plotted population size
versus time relationship is controlled by setting with sliders the population's initial size,
, the two scaling factors,
, the proportionality constant,
, and the asymptotic level of the population's size,
. Two other sliders control the maximum settings of the time (in arbitrary units) and number of cells axes.
Although the scales are set so that most of the generated curves will correspond to observable microbial growth patterns, the program can also be used to model growth phenomena in other areas such as economics.
For more on the generalized Verhulst model see:
A. Tsoularis, "Analysis of Logistic Growth Models," Res. Lett. Inf. Math. Sci.
, 2001 pp. 23–46.
On the ratio-based modified version of the Verhulst model and its potential applications in food microbiology see:
M. Peleg, M. G. Corradini, and M. D. Normand, "The Logistic (Verhulst) Model for Sigmoid Microbial Growth Curves Revisited," Food Res. Internl.
(7), 2007 pp. 808–818.