Taylor’s law for exponentially growing local populations linked by migration

We consider the dynamics of a collection of $n>1$ populations in which each population has its own rate of growth or decay, fixed in continuous time, and migrants may flow from one population to another over a fixed network, at a rate, fixed over time, times the size of the sending population. This model is represented by an ordinary linear differential equation of dimension $n$ with constant coefficients arrayed in an essentially nonnegative matrix. This paper identifies conditions on the parameters of the model (specifically, conditions on the eigenvalues and eigenvectors) under which the variance of the $n$ population sizes at a given time is asymptotically (as time increases) proportional to a power of the mean of the population sizes at that given time. A power-law variance function is known in ecology as Taylor’s Law and in physics as fluctuation scaling. Among other results, we show that Taylor’s Law holds asymptotically, with variance asymptotically proportional to the mean squared, on an open dense subset of the class of models considered here.