Scaling pattern of occupancy

William E. Kunin (1998) [Kunin, WE. 1998. Extrapolating species abundance across spatial scales. Science, 281: 1513-1515.] presented a method to estimate species relative abundance by using the presence-absence distribution map. In his paper, he plotted the range size of the species against the grain (the resolution, the grid size) and named it 'the area-of-occupancy' or AOO. Stephen Hartley and William E. Kunin (2003) [Hartley, S., Kunin, WE. 2003. Scale dependence of rarity, extinction risk, and conservation priority. Conservation Biology, 17: 1559-1570.] 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) [Wilson, RJ., Thomas, CD., Fox, R., Roy, RD., Kunin, WE. 2004. Spatial patterns in species distributions reveals biodiversity change. Nature, 432: 393-396.] further using the slope of AOO to demonsrate the trend in abundance (rather say range size). Cang Hui and his colleagues (2006) [Hui, C., McGeoch, MA., Warren, M. 2006. A spatially explicit approach to estimating species occupancy and spatial correlation. Journal of Animal Ecology, 75: 140-147.] 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) [Hui, C., McGeoch, MA. 2007. Capturing the "droopy tail" in the occupancy-abundance relationship. Ecoscience, 14: 103-108.] 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.

Formula

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] [Dieckmann, U., Law, R. & Metz, JAJ. (2000) The geometry of ecological interactions: simplifying spatial complexity. Cambridge University Press, Cambridge.] ) (joint-count statistics) and Bayes' rule:

This formula describe the
$p,$left( 4,a ight) }_+}=1 - fracOmega }^4}{mho }
$q,$left( 4,a ight) }_{+/+=fracOmega }^{10} - 2,{Omega }^4, {mho }^2 + {mho }^3} mho }^2, left( -{Omega }^4 + mho ight) }
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 ight) }_{+/+ -3 ight)}+ p,left( a ight) }_+},{left( {q,left( a ight) }_{+/+}^2,-3 ight)} ight)}

where $p,$left( a ight) }_+} is occupancy; $q,$left( a ight) }_{+/+ is the conditional probability that a randomly chosen adjcent quadrate of an occupied quadrate is also occupied (Hui and Li 2004 [Hui, C. and Li, Z. (2004) Distribution pattern of metapopulation determined by Allee effects. Population Ecology, 46: 55-63.] ); the conditional probability $q,$left( a ight) }_{0/+=1-,q,left( a ight) }_{+/+ 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.

References