The logistic (Verhulst) model for sigmoid microbial growth curves revisited

The logistic (Verhulst) growth model and its various current modifications are all based on the notion that the isothermal momentary growth rate is proportional to the momentary population's size and the fraction of resources that are still available in the habitat. The model as originally formulated is inappropriate for growth curves that exhibit a long lag time when plotted on semi logarithmic coordinates. This problem can be eliminated by replacing the number of organisms as the independent variable in the equation by the logarithmic growth ratio. Moreover, the actual growth rate's dependence on the population's momentary size and residual resources might have scaling factors that are different from unity as the original model requires. When these are incorporated into the equation, the resulting generalized model can account for almost all observed sigmoid growth patterns, isothermal and non-isothermal, except for those that show a true peak. This is demonstrated with published growth data of various bacteria in different media, including some that exhibit a particularly long 'lag time'. Since most sigmoid isothermal growth curves can be described by empirical models that have only three adjustable parameters, the expanded model, like other modified versions of Verhulst's equation, does not abide by the principle of parsimony. However, its scaling factors provide a tentative kinetic explanation of a large class of observed growth patterns without invoking hypothetical universal relationships between growth parameters, as some current models do. (c) 2007 Elsevier Ltd. All rights reserved.