Modeling microbial populations with the original and modified versions of the continuous and discrete logistic equations

The life histories of microbial populations, under favorable and adverse conditions, exhibit a variety of growth, decay, and fluctuation patterns. They have been described by numerous mathematical models that varies considerably in structure and number of constants. The continuous logistic equation alone and combined with itself or with its mirror image, the Fermi function, can produce many of the observed growth patterns. They include those that are traditionally described by the Gompertz equation and peaked curves, with the peak being symmetric or asymmetric narrow or wide. The shape of survival and dose response curves appears to be determined by the distribution of the resistance's to the lethal agent among the individual organisms. Thus, exponential decay and Fermian or Gompertz-type curves can be considered manifestations of skewed to the right, symmetric, and skewed to the left distributions, respectively. Because of the mathematical constraints and determinism, the original discrete logistic equation can rarely be an adequate model of real microbial populations. However, by making its proportionality constant a normal-random variate it can simulate realistic histories of fluctuating microbial populations, including scenarios of aperiodic population explosions of varying intensities of the kind found in food-poisoning episodes.