A flock as shy as the shiest

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

(15.1)

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 non-negative random variables, that the expected values equals the integral (c.q.sum) over the survivor function. The expected critical distance is thus

(15.2)

Since , we know that must be a non-decreasing 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 species-specific "shyness-parameter"). Then,

So

(15.5)

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

(15.6)

we find

(15.7)

So

(15.8)

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.