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arXiv · 2108.10419

A Seascape Origin of Richards Growth

Daniel Swartz·Bertrand Ottino-Löffler·Mehran Kardar
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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.

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Daniel Swartz, Bertrand Ottino-Löffler, Mehran Kardar. 2021-08-23. A Seascape Origin of Richards Growth. https://doi.org/10.1103/physreve.105.014417

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