Gompertz growth function

Computes the value of the Gompertz curve parameterised in terms of starting value, asymptote, growth rate and lag: $$G(x) = y_0 + (y_{\max} - y_0)\, \exp\!\left[-\exp\!\left(\frac{k\,e\,(\mathrm{lag} - x)}{y_{\max} - y_0} + 1\right)\right].$$

Usage

Gompertz(x, y0, ymax, k, lag)

Arguments

x: Numeric vector. The independent variable (e.g. time).
y0: Lower asymptote.
ymax: Upper asymptote.
k: Maximum specific growth rate.
lag: Lag time.

Value

A numeric vector the same length as x.

Details

This parameterisation, often called the "Zwietering Gompertz" form after Zwietering et al. (1990, eq. 3), gives directly interpretable parameters: y0 is the lower asymptote, ymax the upper asymptote, k the maximum growth rate (the slope of the curve at its inflection point), and lag the lag time (where the tangent at the inflection point crosses y0). The factor $e$ is what makes k the maximum slope; versions of shewhartr up to 1.3.0 omitted it, so their curves had maximum slope $k/e$.

References

Gompertz, B. (1825). On the Nature of the Function Expressive of the Law of Human Mortality. Philosophical Transactions of the Royal Society of London, 115, 513-583.

Zwietering, M. H., Jongenburger, I., Rombouts, F. M., & van 't Riet, K. (1990). Modeling of the Bacterial Growth Curve. Applied and Environmental Microbiology, 56(6), 1875-1881. doi:10.1128/aem.56.6.1875-1881.1990

Examples

x <- seq(0, 30, by = 0.5)
y <- Gompertz(x, y0 = 0, ymax = 100, k = 5, lag = 5)
plot(x, y, type = "l", main = "Gompertz growth curve")