Scaling pattern of occupancy

Definition

Scaling pattern of occupancy (SPO, hereafter), also known as the area-of-occupancy, is a term in spatial ecology and macroecology, and decribes how species distribution changes across spatial scales. In physical geography and image analysis, it is similar to the Modifiable areal unit problem. Simon A. Levin (1992) states that the problem of relating phenomena across scales is the central problem in biology and in all of science. Understanding the SPO is thus one central theme in ecology.

Pattern description

This pattern is often plotted as log-transformed grain (cell size) versus log-transformed occupancy. William E. Kunin (1998) presented a log-log linear SPO and suggesting a fractal nature of species distribution. In reality, it has been shown to follow a logistic shape, reflecting a percolation process. Furthermore, the SPO is closely related to the intraspecific occupancy-abundance relationship. For instance, if individuals are randomly distributed in space, the number of individuals in an α-size cell follows a Poisson distribution, with the occupancy being P_α=1-Exp(-μ•α), where μ is the density. Clearly, P_α in this Poisson model for randomly distributed individuals is also the SPO. Other probability distributions, such as the negative binomial distribution, can also be applied for describing the SPO and the occupancy-abundance relationship for non-randomly distributed individuals.

Stephen Hartley and William E. Kunin (2003) further suggested that this AOO could be an important scaling pattern of species distribution and demonstrated the possibility of using AOO to estimate species abundance, although they suggested that the accurate abundance estimation should according to the occupancy-abundance relationship. Chris D. Wilson and his colleagues (2004) further using the slope of AOO to demonsrate the trend in abundance (rather say range size). Cang Hui and his colleagues (2006) presented a formula of species occupancy along scales using the Bayes' rule and named it the scaling pattern of occupancy. They also reported the scaling pattern of spatial correlation of species distribution. Their formula could imply that the scaling pattern of occupancy (or AOO) might be governed by the statistical and probability principles. Cang Hui and Melodie A. McGeoch (2007) further concluded the scaling pattern of occupancy into a general category of percolation. The scaling pattern of occupancy is also linked related to the occupancy frequency distribution and the occupancy-abundance relationship.

Explaination

Hui, McGeoch and Warren (2006) gave the following formula to describe the scaling pattern of occupancy and spatial correlation using pair approximation (e.g. in Dieckmann et al. [2000]) (joint-count statistics) and Bayes' rule:

This formula describe the

$p\,{{\left(4\,a\right)}_{+}}=1-{\frac {{\Omega }^{4}}{\mho }}$

$q\,{{\left(4\,a\right)}_{+/+}}={\frac {{\Omega }^{10}-2\,{\Omega }^{4}\,{\mho }^{2}+{\mho }^{3}}{{\mho }^{2}\,\left(-{\Omega }^{4}+\mho \right)}}$

where

$\Omega =p\,{(a)_{0}}-q\,{(a)_{0/+}}\,p\,{(a)_{+}}$

and

$\mho =p\,{(a)_{0}}{\left(1-{p\,{(a)_{+}}}^{2}\,{\left(2\,q\,{{\left(a\right)}_{+/+}}-3\right)}+p\,{{\left(a\right)}_{+}}\,{\left({q\,{{\left(a\right)}_{+/+}}}^{2}\,-3\right)}\right)}$

where $p\,{{\left(a\right)}_{+}}$ is occupancy; $q\,{{\left(a\right)}_{+/+}}$ is the conditional probability that a randomly chosen adjacent quadrate of an occupied quadrate is also occupied (Hui and Li 2004); the conditional probability $q\,{{\left(a\right)}_{0/+}}=1-\,q\,{{\left(a\right)}_{+/+}}$ is the absence probability in a quadrate adjacent to an occupied one; a and 4a are the grains. See detail explanation of this equation in Hui et al. (2006).

The key point of this formula is that the scaling pattern or characteristics of species distribution (measured by occupancy and spatial pattern) can be calculated across scales without any information of the biology of the species.