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== Pattern description ==
== Pattern description ==


This pattern is often plotted as log-transformed grain (cell size) versus log-transformed occupancy. Kunin (1998)<ref>Kunin, WE. 1998. Extrapolating species abundance across spatial scales. Science, 281: 1513-1515.</ref> 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]]<ref>Hui, C., McGeoch, MA. 2007. Capturing the "droopy tail" in the occupancy-abundance relationship. Ecoscience, 14: 103-108.</ref>. 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<sub>α</sub>=1-Exp(-μ•α), where μ is the density<ref>Wright, D.H. 1991. Correlations between incidence and abundance are expected by chance. Journal of Biogeography, 18: 463-466.</ref>. Clearly, P<sub>α</sub> 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<ref>He, F., Gaston, K.J. 2000. Estimating species abundance from occurrence. American Naturalist, 156: 553-559.</ref>. Other occupancy-abundance models that can be used to describe the SPO includes Nachman's exponential model<ref>Nachman, G. 1981. A mathematical model of the functional relationship between density and spatial distribution of a population. Journal of Animal Ecology, 50: 453-460.</ref>, Hanski and Gyllenberg's [[metapopulation]] model<ref>Hanski, I., Gyllenberg, M. 1997. Uniting two general patterns in the distribution of species. Science, 284: 334-336.</ref>, He and Gaston's<ref>He, F., Gaston, K.J. 2003. Occupancy, spatial variance, and the abundance of species. American Naturalist, 162: 366-375.</ref> improved negative binomial model by applying [[Taylor's power law]] between the mean and variance of species distribution<ref>Taylor, L.R. 1961. Aggregation, variance and the mean. Nature, 189: 732-735.</ref>, and Hui and McGeoch's droopy-tail percolation model<ref>Hui, C., McGeoch, MA. 2007. Capturing the "droopy tail" in the occupancy-abundance relationship. Ecoscience, 14: 103-108.</ref>.
This pattern is often plotted as log-transformed grain (cell size) versus log-transformed occupancy. Kunin (1998)<ref>Kunin, WE. 1998. Extrapolating species abundance across spatial scales. Science, 281: 1513-1515.</ref> 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<sub>α</sub>=1-Exp(-μ•α), where μ is the density<ref>Wright, D.H. 1991. Correlations between incidence and abundance are expected by chance. Journal of Biogeography, 18: 463-466.</ref>. Clearly, P<sub>α</sub> 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<ref>He, F., Gaston, K.J. 2000. Estimating species abundance from occurrence. American Naturalist, 156: 553-559.</ref>. Other occupancy-abundance models that can be used to describe the SPO includes Nachman's exponential model<ref>Nachman, G. 1981. A mathematical model of the functional relationship between density and spatial distribution of a population. Journal of Animal Ecology, 50: 453-460.</ref>, Hanski and Gyllenberg's [[metapopulation]] model<ref>Hanski, I., Gyllenberg, M. 1997. Uniting two general patterns in the distribution of species. Science, 284: 334-336.</ref>, He and Gaston's<ref>He, F., Gaston, K.J. 2003. Occupancy, spatial variance, and the abundance of species. American Naturalist, 162: 366-375.</ref> improved negative binomial model by applying [[Taylor's power law]] between the mean and variance of species distribution<ref>Taylor, L.R. 1961. Aggregation, variance and the mean. Nature, 189: 732-735.</ref>, and Hui and McGeoch's droopy-tail percolation model<ref>Hui, C., McGeoch, MA. 2007. Capturing the "droopy tail" in the occupancy-abundance relationship. Ecoscience, 14: 103-108.</ref>.
<br />

Stephen Hartley and William E. Kunin (2003)<ref>Hartley, S., Kunin, WE. 2003. Scale dependence of rarity, extinction risk, and conservation priority. Conservation Biology, 17: 1559&ndash;1570.</ref> 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)<ref>Wilson, RJ., Thomas, CD., Fox, R., Roy, RD., Kunin, WE. 2004. Spatial patterns in species distributions reveals biodiversity change. Nature, 432: 393&ndash;396.</ref> further using the slope of AOO to demonsrate the trend in abundance (rather say range size). [[Cang Hui]] and his colleagues (2006)<ref>Hui, C., McGeoch, MA., Warren, M. 2006. A spatially explicit approach to estimating species occupancy and spatial correlation. Journal of Animal Ecology, 75: 140&ndash;147.</ref> 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)<ref>Hui, C., McGeoch, MA. 2007. Capturing the "droopy tail" in the occupancy-abundance relationship. Ecoscience, 14: 103&ndash;108.
Stephen Hartley and William E. Kunin (2003)<ref>Hartley, S., Kunin, WE. 2003. Scale dependence of rarity, extinction risk, and conservation priority. Conservation Biology, 17: 1559&ndash;1570.</ref> 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)<ref>Wilson, RJ., Thomas, CD., Fox, R., Roy, RD., Kunin, WE. 2004. Spatial patterns in species distributions reveals biodiversity change. Nature, 432: 393&ndash;396.</ref> further using the slope of AOO to demonsrate the trend in abundance (rather say range size). [[Cang Hui]] and his colleagues (2006)<ref>Hui, C., McGeoch, MA., Warren, M. 2006. A spatially explicit approach to estimating species occupancy and spatial correlation. Journal of Animal Ecology, 75: 140&ndash;147.</ref> 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)<ref>Hui, C., McGeoch, MA. 2007. Capturing the "droopy tail" in the occupancy-abundance relationship. Ecoscience, 14: 103&ndash;108.
</ref> 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]].
</ref> 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]].

Revision as of 17:04, 12 January 2010

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)[1] 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. Kunin (1998)[2] 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 Pα=1-Exp(-μ•α), where μ is the density[3]. Clearly, Pα 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[4]. Other occupancy-abundance models that can be used to describe the SPO includes Nachman's exponential model[5], Hanski and Gyllenberg's metapopulation model[6], He and Gaston's[7] improved negative binomial model by applying Taylor's power law between the mean and variance of species distribution[8], and Hui and McGeoch's droopy-tail percolation model[9].
Stephen Hartley and William E. Kunin (2003)[10] 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)[11] further using the slope of AOO to demonsrate the trend in abundance (rather say range size). Cang Hui and his colleagues (2006)[12] 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)[13] 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][14]) (joint-count statistics) and Bayes' rule:

This formula describe the

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle p\,{{\left( 4\,a \right) }_+}=1 - \frac{{\Omega }^4}{\mho }}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle q\,{{\left( 4\,a \right) }_{+/+}}=\frac{{\Omega }^{10} - 2\,{\Omega }^4\, {\mho }^2 + {\mho }^3} {{\mho }^2\, \left( -{\Omega }^4 + \mho \right) }}

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \Omega=p\,{(a)_0} - q\,{(a)_{0/+}}\,p\,{(a)_+}}

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle p\,{{\left( a \right) }_+}} is occupancy; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 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[15]); the conditional probability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 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.

References

  1. ^ Levin, SA. 1992. The problem of pattern and scale in ecology. Ecology, 73, 1943-1967.
  2. ^ Kunin, WE. 1998. Extrapolating species abundance across spatial scales. Science, 281: 1513-1515.
  3. ^ Wright, D.H. 1991. Correlations between incidence and abundance are expected by chance. Journal of Biogeography, 18: 463-466.
  4. ^ He, F., Gaston, K.J. 2000. Estimating species abundance from occurrence. American Naturalist, 156: 553-559.
  5. ^ Nachman, G. 1981. A mathematical model of the functional relationship between density and spatial distribution of a population. Journal of Animal Ecology, 50: 453-460.
  6. ^ Hanski, I., Gyllenberg, M. 1997. Uniting two general patterns in the distribution of species. Science, 284: 334-336.
  7. ^ He, F., Gaston, K.J. 2003. Occupancy, spatial variance, and the abundance of species. American Naturalist, 162: 366-375.
  8. ^ Taylor, L.R. 1961. Aggregation, variance and the mean. Nature, 189: 732-735.
  9. ^ Hui, C., McGeoch, MA. 2007. Capturing the "droopy tail" in the occupancy-abundance relationship. Ecoscience, 14: 103-108.
  10. ^ Hartley, S., Kunin, WE. 2003. Scale dependence of rarity, extinction risk, and conservation priority. Conservation Biology, 17: 1559–1570.
  11. ^ Wilson, RJ., Thomas, CD., Fox, R., Roy, RD., Kunin, WE. 2004. Spatial patterns in species distributions reveals biodiversity change. Nature, 432: 393–396.
  12. ^ 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.
  13. ^ Hui, C., McGeoch, MA. 2007. Capturing the "droopy tail" in the occupancy-abundance relationship. Ecoscience, 14: 103–108.
  14. ^ Dieckmann, U., Law, R. & Metz, JAJ. (2000) The geometry of ecological interactions: simplifying spatial complexity. Cambridge University Press, Cambridge.
  15. ^ Hui, C. and Li, Z. (2004) Distribution pattern of metapopulation determined by Allee effects. Population Ecology, 46: 55–63.
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