An alternative way of describing spatial distributions is to move the focus to the position of separate individuals in space. When viewed from above, the distribution of individuals over the landscape can be visualized as a pattern of tiny dots dispersed over a blank area. This pattern may in theory be completely random, but usually individuals are either clumped together, or at the opposite, spread out in a regular fashion (see Figure 3).
Empirically, dispersion patterns can be characterized by the average number of individuals in randomly sampled units of area within the landscape (x) and the variance between number of individuals occurring per sampling unit (s2). The ratio s2/x is commonly known as the index of dispersion. In randomly distributed populations, the index of dispersion is approximately equal to 1 (see Box 1).
Clearly, dispersion patterns are strongly dependent upon the scale at which they are perceived - for instance, the distribution of an insect species living on leaves of trees can be studied on many different spatial scales, such as the distribution on each leaf, among leaves within a branch, among branches within a tree, and between trees within a forest.
From the beginning, it seems obvious that there should be some clumping of individuals - organisms do not appear out of the thin air. Each individual originates from a parent, and as such appears in close vicinity to other individuals, which in most cases are relatives. After birth, or at a later stage in the life cycle, they undergo some juvenile dispersal (see Dispersal-Migration) before settling down in a favorable location, but some degree of aggregation is still retained. Another factor which predisposes for aggregation is that individuals of the same species are attracted to a set of conditions and resources, which are themselves patchily distributed.
Aggregated dispersion patterns are very common in nature. An example of an extremely aggregated species is
Box 1 Analysis of spatial data
As the null hypothesis, it is assumed that individuals in a population are randomly distributed among the n sampling units of a sample. If this is the case, it is expected that the variance should equal the average so that the 'index of dispersion', s2/x, is approximately equal to 1. If the ratio exceeds 1, it indicates that the population has a patchy (or clumped) distribution whereas a value less than unity indicates an even (or regular) distribution. However, since data originate from sampling, they will always be associated with some variation, so it is likely that some deviation in s2/x from unity will occur even if the underlying distribution is random. Especially if the sample size n is small, s2/x will exhibit large variation due to sampling noise. A x2-test can be used for testing whether s2/x deviates significantly from 1 since x2= (n-1)s2/x with n-1 degrees of freedom. It should be noted that the test is two-tailed (in contrast to the majority of cases where X2-tests are used) since values significantly smaller or larger than n-1 can lead to rejection of the null hypothesis.
Though the index of dispersion indicates whether a population is evenly, randomly, or patchily distributed in space, it does not explicitly reveal information about the underlying spatial distribution. This requires that the empirical distribution of sampling units with x individuals can be fitted by a theoretical 'probability function' called P(x), which denotes the probability that a randomly selected spatial unit contains exactly x individuals. As all probability functions, P(x) for all possible integer values of x equal to or larger than 0 should sum to unity.
The Poisson distribution is used to describe the underlying distribution when it is random, the positive binomial distribution when it is even and the negative binomial distribution when it is clumped. However, other less frequently used distributions are also available to model clumped populations, for example, the Thomas distribution, the logarithmic series distribution, the Polya-Aeppli distribution, and Neyman's type A, B, and C distributions. Once an adequate probability function has been identified and fitted to data, the quality of the fit can be assessed by means of a goodness-of-fit test, usually a x2 one-sample test or a Kolmogorov-Smirnov one-sample test.
A problem often encountered in analyzing spatial data statistically is the fact that they do not represent independent observations. Thus, if sampling unit i is separated from sampling unit j by a distance dj, it seems likely that x/ will be more similar to Xj, the smaller the dj is. This phenomenon is known as 'spatial autocovariation'. Spatial autocovariance is often depicted as a so-called 'semivariogram' where the 'semivariance' (7d) is plotted against d. The semivariance at distance d is calculated as 7d=^Si(X/+d - xi)2/2nd, wherexi+d is the value of x measured at distance d from another measurement x/ and nd is the number of measurements separated by distance d.
Spatial patterns can be depicted graphically by means of a technique known as 'kriging'. The principle is to place a large number of points spaced out over the entire area under study. Each point is characterized by its coordinates in the two-dimensional x-y space, and by the value of a given attribute (for instance the population density in the area around the point). The value of the attribute is denoted the z coordinate, which represents a height above the x-y plane. Hence, small and large values of z will appear as troughs and peaks in a three-dimensional (3-D) landscape. 3-D landscapes can be projected into two-dimensional (2-D) landscapes by means of contour plots where points with identical z-values are connected with lines (isoclines), similar to how temperature and atmospheric pressure are depicted in meteorological maps. The more fine-grained the information is, the more precise the map will be. Various algorithms have been developed to interpolate values between neighboring points so as to estimate z in points that have not been sampled, and to smooth out the landscape by removing local peaks and troughs caused by sampling noise.
Since kriging is computationally demanding, various specialized software products exist to perform it, for example, easy_krig, DACE, GS+. In addition, kriging can be handled by some statistical packages, such as R.
the Kashmir cave bat (Myotis longipes), which is only known from nine localities in the Himalaya region, each home to populations of a thousand individuals or more.
A regular or even distribution, on the other hand, is the result when individuals compete for limited resources.
This type of pattern is commonly exhibited by many sessile organisms such as trees, which space themselves evenly as a result of competition for water or sunlight. Also many animals have approximately regular distributions - a familiar example is the territories of songbirds.
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