Gompertz growth function

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

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), gives directly interpretable parameters: y0 is the lower asymptote, ymax the upper asymptote, k the maximum specific growth rate, and lag the lag time before exponential growth.

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")