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Incorporating stochasticity into population dispersal has been introduced by Hanski in 1991. Demographic sto-chasticity is found in events within the population that are random and unpredicted and are demonstrated by individual behaviors causing immigration and emigration into or out of the population. Another type of stochasticity is environmental stochasticity - events such as floods, droughts, and other catastrophes that may affect population spatial distribution. Environmental stochasticity has also been referred to as 'regional stochasticity' which more specifically relates to variation in population dynamics across patches for the purpose of 'risk spreading'. According to Hanski, species that show high rate of dispersal but low levels of regional stochasticity have a higher chance of long-term survival. It has also been proposed that dispersal is mostly advantageous to a population if patch variation (regional stochasticity) is low.

Figure 2 Spatial distribution of individuals within a habitat patch: (a) random, (b) clumped, and (c) uniform.

pattern of dispersion is random. If small- and large-sized groups occur more frequently than expected, and medium-sized groups occur less frequently than expected, the pattern of dispersion is clumped. If medium-sized groups occur more frequently than expected, and small-and large-sized groups occur less frequently than expected, the pattern of dispersion is uniform. Deviations from the normal distribution can be tested statistically:

where S2 is the variance from the sample mean (x) distribution. If I = 1, the pattern of dispersion is considered random; if I> 1, the pattern of dispersion is considered clumped (or aggregated); and if I< 1, the pattern of dispersion is considered uniform.

Another method (nearest-neighbor) measures the actual distance between randomly selected individuals in a population and their nearest neighbor. A plot of the frequency distribution of the square roots of these distances indicates the dispersion pattern. A bell-shaped distribution indicates a random dispersion pattern. A polygon skewed to the right indicates a uniform dispersion, and one skewed to the left indicates a clumped dispersion. Hopkins' test of randomness uses a more complex method by choosing random points in the habitat (group 1) and then independent of the above points, choosing random individuals in the habitat (group 2). If X represents the distance between each individual (i) and selected points (group 1), and r represents the distance between each individual and its nearest neighbor (group 2), then h = E(Xi )7X(ri )2 [15]

represents a measure of randomness. To identify the type of dispersion pattern, Ih, we write

If Ih = 1, the pattern of dispersion is clumped; if Ih = 0, the pattern of dispersion is uniform; and if Ih = 0.5, the pattern of dispersion is random.

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