Gompertzmalli

Gompertzmalli, or the Gompertzian growth model, is a mathematical model used to describe sigmoidal growth in systems such as biological populations, tumors, and survival or mortality processes. It is defined by the differential equation dN/dt = r N ln(K/N), where N(t) is the quantity of interest at time t, K is the carrying capacity or asymptotic maximum, and r is a positive rate parameter.

The model implies that the relative growth rate decreases as N approaches K, yielding a characteristic S-shaped curve. The solution can be written as N(t) = K exp(-b e^{-r t}), with b = ln(K/N0) and N0 the initial value. Early in time, growth appears near exponential; as N grows, the logarithmic term slows further increases, producing a gradual approach to the carrying capacity without overshooting.

Historically, the Gompertz model was introduced by Benjamin Gompertz in 1825 to describe human mortality and later adapted to biology, ecology, and demography. It remains widely used to fit growth data where the rate of increase diminishes over time, including tumor growth, microbial culture growth under resource limitation, forestry and plant growth, and actuarial survival analyses.

Estimation and use of the model typically involve nonlinear regression or maximum likelihood methods to fit K, r, and N0 (or equivalent parameters) to observed data. Extensions of the basic form allow time-varying carrying capacity or mixed-effects modeling to account for heterogeneity. While useful, the Gompertz model may not capture all growth patterns, particularly when external factors alter carrying capacity or growth dynamics.

See also: Gompertz function, Gompertz law, logistic model.