Gompertzian

Gompertzian refers to phenomena or models that follow the Gompertz form, named after Benjamin Gompertz who proposed the model in 1825. In growth and mortality studies, a Gompertzian pattern is a sigmoidal process in which early growth is rapid but decelerates over time, approaching a limiting value.

The Gompertz equation is commonly written as dN/dt = -a N ln(N/K), with N(t) the size at time t, a > 0 the rate constant, and K the carrying capacity. Its explicit solution is N(t) = K exp(-b e^{-a t}), where b is determined by the initial condition. The same family underlies the Gompertz function, used to describe tumor growth, microbial growth, and the aging-related increase in mortality risk, known as the Gompertz law of mortality, often expressed as hazard h(t) = h0 e^{β t}.

In oncology, Gompertzian growth describes tumors that grow quickly when small but slow as they become large, due to constraints like nutrient supply and diffusion limits. In demography and actuarial science, Gompertzian models capture how mortality rates escalate with age, typically exponentially with age. The term "Gompertzian" is also used in statistics (Gompertzian distribution) and mathematical biology to denote properties consistent with the Gompertz model.

Limitations include that real systems may exhibit time-varying parameters, multiple growth phases, or deviations from a pure Gompertz form, requiring extensions or alternative models. Despite limitations, Gompertzian models remain widely used for their simplicity and ability to capture early rapid growth and late slow growth in a variety of biological and demographic contexts.