Fractal measures of spatial pattern as a heuristic for return rate in vegetative systems.

Measurement of population persistence is a long-standing problem in ecology; in particular, whether it is possible to gain insights into persistence without long time-series. Fractal measurements of spatial patterns, such as the Korcak exponent or boundary dimension, have been proposed as indicators of the persistence of underlying dynamics. Here we explore under what conditions a predictive relationship between fractal measures and persistence exists. We combine theoretical arguments with an aerial snapshot and time series from a long-term study of seagrass. For this form of vegetative growth, we find that the expected relationship between the Korcak exponent and persistence is evident at survey sites where the population return rate can be measured. This highlights a limitation of the use of power-law patch-size distributions and other indicators based on spatial snapshots. Moreover, our numeric simulations show that for a single species and a range of environmental conditions that the Korcak-persistence relationship provides a link between temporal dynamics and spatial pattern; however, this relationship is specific to demographic factors, so we cannot use this methodology to compare between species.