3.5. Density-Invariant Indexes of Aggregation

   Simple indexes of aggregation are specific to a particular population sampled at
   particular time. They cannot be extrapolated neither in space nor in time, and
   this is their major limitation. In order to overcome this limitation, several
   density-invariant indexes has been developed.

   1. The most frequently used is the "power law" (Taylor 1961):

                                    [eqtaylor.gif]

   Coefficient b is considered as species-specific. This equation was shown to work
   well in the wide range of species density. Of course, it is hard to expect that b
   will be constant in any kind of environment, but for populations in similar
   environments it is usually stable.

   2. Mean crowding regression (Iwao 1968):

                                     [eqiwao.gif]

   Regression coefficients can be used to distinguish between different patterns of
   spatial distribution:

                                      [iwao.gif]

   3. Negative binomial k. This is not a good index because usually it is not
   density-invariant.

Effect of Quad Size on Aggregation Indexes

   Different scales of spatial distribution should be considered. The distribution
   may be random at small scales and aggregated at larger scales. Thus, it is
   important to examine the distribution using different quad sizes (quad size =
   spatial resolution).

   For example, the distribution is random, if and only if the coefficient of
   dispersion (CD) is equal to 1 for all the range of quad sizes. Thus, if CD=1 for
   a specific quad size, then we cannot conclude that the distribution is random, we
   need to test other quad sizes.

   Any index of aggregation can be plotted against quad size. However, there are
   specialized indexes designed for multiple quad sizes. For example, ro index (Iwao
   1972) was defined as:

                                     [eqiwao1.gif]

   where subscripts 1,2,...i stand for successively increasing sizes of quads. Ro
   index is used to determine characteristic distances in a spatial distribution:

                                      [iwao1.gif]

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     ____________________________________________________________________________

   [4]Alexei Sharov 12/28/95

References

   1. http://alexei.nfshost.com/PopEcol/lec3/agindex.html
   2. http://alexei.nfshost.com/PopEcol/lec3/spatdist.html
   3. http://alexei.nfshost.com/PopEcol/lec3/geostat.html
   4. http://alexei.nfshost.com/~sharov/alexei.html