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Walking along the beach in winter, one is likely to meet a flock of
sandpipers. When approaching, one bird usually grows nervous and the
flock flies off. The minimum distance tends to increase with the size
of the flock. Approach at close range seems only possible for
solitairy individuals. Social effects, like reassurance for being only
one of the many potential victims of the approaching 'predator', can
only be studied after we know what to expect when the individuals are
independent.
Suppose the critical distance, , below which an individual will
fly off in a certain circumstance is characteristic for that
individual, and can be conceived as a random trial from some
probability density, . The probability, , that
the flock of size does not fly off at distance equals the
probability that all critical distances less than , so
where is the distribution function of the critical distance
for a single individual, i.e.
. The
derivation of densities of extremes is easy by using distribution c.q.
survivor functions. When evaluating expected values, we use the
property for nonnegative random variables, that the expected values
equals the integral (c.q.sum) over the survivor function. The expected
critical distance is thus
Since
, we know that must be a
nondecreasing function of . We have to specify , however,
before we can tell more about the behaviour of as
function of .
Suppose that follows a nearest neighbour distribution, so
where is a parameter (the speciesspecific "shynessparameter"). Then,
So
A plot of against is given in Fig.15.1. We see
that, although does not have an asymptote, because
approaches 1 asymptotically, it increases very slowly for larger
than 8.
If we choose an exponential distribution for , so we
replace (15.3) by
we find
So
This one has not an asymptote either, but it levels off less rapidly.
(see Fig.15.1) This is because the exponential density has
a thicker tail compared to the nearest neighbour one. The first one is
proportial to , while the second one is proportial to
. Tail thickness is thus closely connected with the
expected value of the extreme, as function of the number over which
the extreme is taken. Indeed, for
, we have
, irrespective of .
When walking on the beach, we fail to notice that large flocks are
much shier than small ones, this does not automatically indicate that
their is social interaction, apart from flie when the shiest flies. In
order to detect such interaction, we first have to identify the
critical distance distribution for single individuals.
Figure 15.1:
The expected critical distance as a function of flock size in units
of that of a single individual. The critical distance of a single
individuals is assumed to follow a nearest neighbour or an
exponential distribution.

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Theoretische Biologie
20020501