Input - Genetic Diversity in Establishing Plant Populations: Founder Number and Geometry - page 45

Individuals of a species usually occur at varying densities due to
differences in resource availability across sites. In cases where offspring or
pollen maximum dispersal distance can be thought of as being more or less
absolute (e.g., seed of a wind-dispersed plant can generally only travel 150
m), the number of safe sites within dispersal distance will decrease as the
density of the species supportable at a site decreases. One way to model
this with NEWGARDEN is to decrease the dispersal distance (fewer safe
sites will be available within the square of available grid points) relative to
increased dispersal distance representing a site that can support a higher
density of individuals. Another way to think about this variation is to
assume that the distance frames merely defi ne the number of safe sites
or paternal individuals in the pool available at specifi ed distances. The
number can be altered with the dispersal distance percentage assignations.
Input methods by which patchy habitats can be constructed are discussed
below.
Dispersal distance is a complex topic. For example, if dispersal can only
occur in one cardinal grid direction (e.g., increasing x or y values from an
individual), given a maximum dispersal distance = d grid points from an
individual, the average will be as follows:
(d +1)/ 2 for annuals, since an annual's dispersule can establish on the
point it occupies, and
d/2 for perennials, since dispersules cannot establish where a perennial is
already established
if dispersal to all of the points along the specifi ed line is equally
likely.
However, note that the average potential dispersal distance of offspring
or microgametes is not just that given above, since, when NEWGARDEN
establishes the two-dimensional potential grid points for dispersal, the
number of grid points increases as a square with increasing distance along
the x or y axis. As dispersal distance becomes greater, the number of points
included in the eligible zone increases as a square. Under this simple model
of offspring and pollen dispersal, for a given annual individual, the average
distance (number of grid points away from a particular individual in the x
and y direction) of offspring dispersal or to eligible pollen donors is:
(4/3)d(d+1)/(2d+1)
where d = the maximum number of grid units for dispersal as entered in
the NEWGARDEN input statements for offspring and pollen distances.
Figure 4.1A shows how average dispersal distance (given in average
density dispersal grid units) increases with maximum dispersal distance.
The lower line is the increase if dispersal is only along one axis, while the
upper line shows the average increasing according to the formula above
when dispersal can be in two dimensions. Figure 4.1B shows how the