Scaling pattern of occupancy
Encyclopedia
In spatial ecology
and macroecology
, scaling pattern of occupancy (SPO), also known as the area-of-occupancy is the way in which 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.
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(−μα), where μ is the density. 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. Other occupancy-abundance models that can be used to describe the SPO includes Nachman's exponential model, Hanski and Gyllenberg's metapopulation
model, He and Gaston's improved negative binomial model by applying Taylor's power law between the mean and variance of species distribution, and Hui and McGeoch's droopy-tail percolation model. One important application of the SPO in ecology is to estimate species abundance based on presence-absence data, or occupancy alone. This is appealing because obtaining presence-absence data is often cost-efficient. Using an dipswitch test consisting of 5 subtests and 15 criteria, Hui et al. confirmed that using the SPO is a robust and reliable for assemblage-scale regional abundance estimation. The other application of SPOs includes trends identification in populations, which is extremely valuable for biodiversity
conservation
.
model, the cross-scale model and the Bayesian estimation model. The fractal model can be configured by dividing the landscape into quadrats of different sizes, or bisecting into grids with special width-to-length ratio (2:1), and yields the following SPO:
where D is the box-counting fractal dimension. If during each step a quadrat is divided into q sub-quadrats, we will find a constant portion (f) of sub-quadrats is also presence in the fractal model, i.e. D = 2(1 + log ƒ/log q). Since this assumption that f is scale independent is not always the case in nature, a more general form of ƒ can be assumed, ƒ = q−λ (λ is a constant), which yields the cross-scale model:
The Bayesian estimation model follows a different way of thinking. Instead of providing the best-fit model as above, the occupancy at different scales can be estimated by Bayesian rule based on not only the occupancy but also the spatial autocorrelation
at one specific scale. For the Bayesian estimation model, Hui et al. provide the following formula to describe the SPO and join-count statistics of spatial autocorrelation:
Spatial ecology
Spatial ecology is a specialization of ecologyand geography that is concerned with the identification of spatial patterns and their relationships to ecological events. In spatial ecology, ecological events can be explained through the detection of patterns at a given spatial scale; local,...
and macroecology
Macroecology
Macroecology is the subfield of ecology that deals with the study of relationships between organisms and their environment at large spatial scales to characterise and explain statistical patterns of abundance, distribution and diversity...
, scaling pattern of occupancy (SPO), also known as the area-of-occupancy is the way in which species distribution changes across spatial scales. In physical geography
Physical geography
Physical geography is one of the two major subfields of geography. Physical geography is that branch of natural science which deals with the study of processes and patterns in the natural environment like the atmosphere, biosphere and geosphere, as opposed to the cultural or built environment, the...
and image analysis
Image analysis
Image analysis is the extraction of meaningful information from images; mainly from digital images by means of digital image processing techniques...
, it is similar to the modifiable areal unit problem
Modifiable Areal Unit Problem
The modifiable areal unit problem is a source of statistical bias that can radically affect the results of statistical hypothesis tests. It affects results when point-based measures of spatial phenomena are aggregated into districts. The resulting summary values are influenced by the choice of...
. Simon A. Levin
Simon A. Levin
Simon Asher Levin is an American ecologist. He is a Moffett Professor of Biology in the Department of Ecology and Evolution at Princeton University. He specializes in using mathematical modeling and empirical studies in the understanding of macroscopic patterns of ecosystems and biological...
(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. 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 logisticLogistic
Logistic may refer to:* Logistics, the management of resources and their distributions** Logistic engineering, the scientific study of logistics** Military logistics, the study of logistics at the service of military units and operations...
shape, reflecting a percolation
Percolation
In physics, chemistry and materials science, percolation concerns the movement and filtering of fluids through porous materials...
process. Furthermore, the SPO is closely related to the intraspecific occupancy-abundance relationship
Occupancy-abundance relationship
In macroecology, the occupancy-abundance relationship is the relationship between the abundance of species and the size of their ranges within a region. This relationship is perhaps one of the most well-documented relationships in macroecology, and applies both intra- and interspecifically . In...
. For instance, if individuals are randomly distributed in space, the number of individuals in an α-size cell follows a Poisson distribution
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...
, with the occupancy being Pα = 1 − exp(−μα), where μ is the density. Clearly, Pα in this Poisson model for randomly distributed individuals is also the SPO. Other probability distributions, such as the negative binomial distribution
Negative binomial distribution
In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified number of failures occur...
, can also be applied for describing the SPO and the occupancy-abundance relationship for non-randomly distributed individuals. Other occupancy-abundance models that can be used to describe the SPO includes Nachman's exponential model, Hanski and Gyllenberg's metapopulation
Metapopulation
A metapopulation consists of a group of spatially separated populations of the same species which interact at some level. The term metapopulation was coined by Richard Levins in 1970 to describe a model of population dynamics of insect pests in agricultural fields, but the idea has been most...
model, He and Gaston's improved negative binomial model by applying Taylor's power law between the mean and variance of species distribution, and Hui and McGeoch's droopy-tail percolation model. One important application of the SPO in ecology is to estimate species abundance based on presence-absence data, or occupancy alone. This is appealing because obtaining presence-absence data is often cost-efficient. Using an dipswitch test consisting of 5 subtests and 15 criteria, Hui et al. confirmed that using the SPO is a robust and reliable for assemblage-scale regional abundance estimation. The other application of SPOs includes trends identification in populations, which is extremely valuable for biodiversity
Biodiversity
Biodiversity is the degree of variation of life forms within a given ecosystem, biome, or an entire planet. Biodiversity is a measure of the health of ecosystems. Biodiversity is in part a function of climate. In terrestrial habitats, tropical regions are typically rich whereas polar regions...
conservation
Conservation biology
Conservation biology is the scientific study of the nature and status of Earth's biodiversity with the aim of protecting species, their habitats, and ecosystems from excessive rates of extinction...
.
Explanation
Models providing explanations to the observed scaling pattern of occupancy include the fractalFractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
model, the cross-scale model and the Bayesian estimation model. The fractal model can be configured by dividing the landscape into quadrats of different sizes, or bisecting into grids with special width-to-length ratio (2:1), and yields the following SPO:
where D is the box-counting fractal dimension. If during each step a quadrat is divided into q sub-quadrats, we will find a constant portion (f) of sub-quadrats is also presence in the fractal model, i.e. D = 2(1 + log ƒ/log q). Since this assumption that f is scale independent is not always the case in nature, a more general form of ƒ can be assumed, ƒ = q−λ (λ is a constant), which yields the cross-scale model:
The Bayesian estimation model follows a different way of thinking. Instead of providing the best-fit model as above, the occupancy at different scales can be estimated by Bayesian rule based on not only the occupancy but also the spatial autocorrelation
Autocorrelation
Autocorrelation is the cross-correlation of a signal with itself. Informally, it is the similarity between observations as a function of the time separation between them...
at one specific scale. For the Bayesian estimation model, Hui et al. provide the following formula to describe the SPO and join-count statistics of spatial autocorrelation:
-
where Ω = p(a)0 − q(a)0/+p(a)+ and
= p(a)0(1 − p(a)+2(2q(a)+/+ − 3) + p(a)+(q(a)+/+2 − 3)). p(a)+ is occupancy; q(a)+/+ is the conditional probability that a randomly chosen adjacent quadrat of an occupied quadrat is also occupied. The conditional probability q(a)0/+ = 1 − q(a)+/+ is the absence probability in a quadrate adjacent to an occupied one; a and 4a are the grains. The R-code of the Bayesian estimation model has been provided elsewhere http://www.unc.edu/~dmcglinn/local_density_script.R. The key point of the Bayesian estimation model is that the scaling pattern of species distribution, measured by occupancy and spatial pattern, can be extrapolated across scales. Later on, Hui provides the Bayesian estimation model for continuously changing scales:
where b, c, and h are constants. This SPO becomes the Poisson model when b = c = 1. In the same paper, the scaling pattern of join-count spatial autocorrelation and multi-species association (or co-occurrenceCo-occurrenceCo-occurrence or cooccurrence can either mean concurrence / coincidence or, in a more specific sense, the above-chance frequent occurrence of two terms from a text corpus alongside each other in a certain order. Co-occurrence in this linguistic sense can be interpreted as an indicator of semantic...
) were also provided by the Bayesian model, suggesting that "the Bayesian model can grasp the statistical essence of species scaling patterns." -
