"The Ecological Footprint of Taylor's Universal Power Law"
Abstract
In 1961 L. R. Taylor proposed a power law for the relationship between
the spatial mean and variance of population abundance in biology,
which over time has been observed for a large number of species, as
well as in many other epidemiological, social and physical systems.
Many disparate explanations for the power law have been proposed in
the literature, but no consensus on their adequacy has emerged. We
review the historical background for the power law and some the
controversies surrounding it. We then turn to a possible theoretical
explanation based on the power variance functions of the so-called
Tweedie distributions, along with a new spatial self-similarity
hypothesis. The theory suggests that there are actually two separate
power laws in swing, one describing the spatial variance as a function
of population density, and the other involving the size of the sampled
area. This is confirmed empirically, and in turn leads us to propose a
new log-linear spatial modelling framework with long-range dependence,
which allows simultaneous estimation of the regression parameters and
the two power parameters. We discuss some of the possible
ramifications of these ideas for studying animal abundance data and
other spatial phenomena exhibiting clustering and long-range dependence.