A Seascape Origin of Richards Growth

TL;DR

This paper demonstrates that Richards growth laws naturally arise from spatially distributed populations with migration and stochastic growth rate variability, linking empirical models to underlying analytical mechanisms.

Contribution

It provides a theoretical derivation of Richards growth laws from generic population dynamics with migration and stochasticity, connecting empirical fitting to fundamental processes.

Findings

Richards growth laws emerge from spatial diffusion and stochastic growth.

Stochasticity leads to power-law distribution in local populations.

The model offers a testable link between empirical laws and population heterogeneity.

Abstract

First proposed as an empirical rule over half a century ago, the Richards growth equation has been frequently invoked in population modeling and pandemic forecasting. Central to this model is the advent of a fractional exponent $\gamma$, typically fitted to the data. While various motivations for this non-analytical form have been proposed, it is still considered foremost an empirical fitting procedure. Here, we find that Richards-like growth laws emerge naturally from generic analytical growth rules in a distributed population, upon inclusion of {\bf (i)} migration (spatial diffusion) amongst different locales, and {\bf (ii)} stochasticity in the growth rate, also known as "seascape noise." The latter leads to a wide (power-law) distribution in local population number that, while smoothened through the former, can still result in a fractional growth law for the overall population. This justification of the Richards growth law thus provides a testable connection to the distribution of constituents of the population.