A Seascape Origin of Richards Growth*

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.