Gompertzmodel

The Gompertz model is a mathematical model of growth that yields a sigmoidal, or S-shaped, trajectory in which the growth rate decreases as size increases. It is named after Benjamin Gompertz, who introduced the form in the 1820s to describe human mortality and later found use in biology and other fields. The model is widely used to describe processes that approach an upper bound but slow growth as they near that limit.

A common form of the Gompertz function for size or population at time t is N(t) = K exp(-b e^{-r t}), where K > 0 is the carrying capacity or asymptotic maximum, r > 0 is the intrinsic growth rate, and b > 0 is a shape parameter. An equivalent differential form is dN/dt = r N ln(K/N), which makes explicit how the growth rate depends on how far the current size N is from the carrying capacity. At t = 0, N(0) = K e^{-b}.

Applications of the Gompertz model span biology and beyond. It is commonly used to describe tumor growth, microbial and plant growth under resource limitation, and other processes with rapid early growth that slows over time. In actuarial science and demography, Gompertz-type formulations are used to characterize age-dependent mortality, known as the Gompertz law of mortality, where hazard increases exponentially with age.

Fitting the model usually involves non-linear regression or maximum likelihood estimation, sometimes with censored data in survival contexts. Limitations include potential poor fit in certain phases of growth and identifiability issues for parameters, leading practitioners to compare with alternative models such as the logistic or generalized Gompertz forms.