A distance measure of spatial pattern in a biological population

Summary

Suppose thatn individuals locate independently and randomly on a segment of line of finite length (habitat). Let the theoretical and observed ranges of the sites of the individuals on the segment be μ _n-1 andr _n-1, respectively. Then, the degree of dispersion of the individual sites is measured by the ratio, T _n =n _n-1/μ _n-1, as follows: A random spatial pattern forI _r−1 =1 An aggregated spatial pattern for 0≤I _r <1 A uniform spatial pattern for (n+1)/(n−1)≥I _r >1.

Another method was derived. Let the probability that an observed range is less thanr _n−1 beI _p , under the hypothesis of a Beta distribution. Then$I_p = \frac{{n(n - 1)}}{{\theta ^n }}r_{n - 1}^{n - 1} \left[ {\frac{\theta }{{n - 1}} - \frac{{r_{n - 1} }}{n}} \right]$ indicates A random spatial pattern forI _p =1/2 An aggregated spatial pattern forI _p <1/2 A uniform spatial pattern forI _p >1/2.

The first index can be used for comparing populations having the same number of individuals, whereas the second one can be used for comparing populations with different numbers of individuals.