Database Reference
In-Depth Information
1.000
10000
1000
0.100
100
0.010
10
0.001
1
1
10
100
1000
Move length, d
(a)
(b)
Figure 13.5 Rank-frequency plots using log 10 axes. (a) Models' prediction for a Brownian
walk (grey dots) and a Levy walk (triangles). (b) The observed behavior of animals in
group (black) and solitary (grey dots). Given a sample of move length d 1 ,d 2 ,...d n sort
these value in increasing order and rank them from 1 to n . For each d i compute the number
of distances d i ( s i ). Finally, s i is plotted as a function of d i in double-logarithmic plot.
As noted above, we have assumed till now that the probability density function
(PDF) of move length, p ( d ), exhibits finite variances. Recent literature, however,
has been shifting toward distributions that have a long-fat tail (see Chapter 15 ).
Some authors have conjectured that organismal movement is so generally heavy-
tailed that the moments of the PDF are no longer finite. Levy distributions have
figured prominently in such treatments. Levy walks are random movements
where the probability of a displacement d is p ( d )
cd μ
=
for d>d min where
1) d μ 1
c
min .Levy behavior applies only to the tail of the distribution, and
P ( d ) is valid only beyond some minimal value of d ; the investigator must select
an appropriate value of d min for a particular data set. The scaling parameter μ
has the remarkable property of being independent of the measurement units,
so direct comparison can be made across studies. Application of the central
limit theorem shows that for 1 3, a sum of Levy distributed moves is
also Levy distributed. Conversely, for μ> 3 the distribution of the sum of
such moves converges to a Gaussian distribution, recovering Brownian motion.
Obviously, sample variances are always finite and some authors have invoked
the use of truncated Levy distribution as more realistic for actual animals.
The basic differences between a Brownian and a Levy walk is presented in
Figure 13.5 a, using rank-frequency plots. The rank-frequency plot is recom-
mended to discriminate between Brownian and Levy walks. The plot demon-
strates that in a Brownian walk the fraction of very long moves falls rapidly to
0 while in a Levy walk such a decrease is much slower and follows a linear
pattern, indicating that a Levy distribution is characterized by a “fat” tail.
The presence of CCRW and LW in fallow deer has been studied by Focardi
et al. (2009) and it was shown that solitary fallow deer adopted a LW tactic while
=
( μ
Search WWH ::




Custom Search