Snapshot 1: growth curve generated with the Gompertz model before being fitted with the Weibull model (as seen in the Thumbnail)

Snapshot 2: growth curve generated with the Weibull model and fitted with the non-exponential model

Snapshot 3: growth curve generated with the Gompertz model and fitted with the logistic model

Snapshot 4: growth curve generated with the power model and fitted with the shifted logistic model

Snapshot 5: growth curve generated with the logistic model and fitted with the Gompertz model after several fit attempts, where the final fitted

value is outside its slider's range

Experimental sigmoid growth curves have been described by a variety of models. Among them are the Gompertz model

, the Weibull or "stretched exponential" model

, the non-exponential model

, the power model

, the logistic model

, and the shifted logistic model

.

In these equations,

stands for the linear or logarithmic growth ratio

or

, respectively, where

is the momentary growing entity (e.g., the number of individual organisms), and

is the initial number. In all these models,

is the asymptotic growth level, that is,

. The parameters

,

,

, and

or their combination account for the growth curve inflection point’s location and its slope at this point, which can be considered a growth rate measure.

In this Demonstration you can select the generation model, its parameters and the number of points to be generated, and the model to fit the data. Upon clicking the green setter button the program attempts to fit the data with the chosen model, starting with initial parameter values taken from the sliders located below. If the first fit attempt fails, you can use the sliders to modify the parameters' initial estimates (starting with

is recommended) until the gray curve roughly matches the generated data points; then you can click the green button to attempt a new fit. If the new fit is still unsatisfactory, you can click the "last fitted values" button to reset the initial parameters to the last fitted values. This procedure can be repeated until a satisfactory fit is obtained. (Note that the best-fitted parameter values after several attempts might be found outside the limits of the initial parameters' sliders, in which case a slider's button will be shadowed in red, as seen in Snapshot 4.) The fitted model’s parameters are displayed with the corresponding

above the plot of a colored fitted curve superimposed on the generated data. The fitted curve is displayed in a different color for each fitted model. You can modify the plot's axes using the two sliders at the bottom.

Notice that the initial values are

,

,

, and

by definition, as it should be, while

and

. This has little effect on the goodness of fit, but can become a serious problem when the Gompertz or logistic model is used for dynamic growth, where the boundary condition is that

, that is,

.