Ecological Modelling 132 (2000) 115 – 124
www.elsevier.com/locate/ecolmodel
Modelling animal movement as a persistent random walk in
two dimensions: expected magnitude of net displacement
Hsin-i Wu a, Bai-Lian Li b,*, Timothy A. Springer c, William H. Neill d
a
Department of Industrial Engineering, Center for Biosystems Modelling, Texas A&M Uni6ersity, College Station,
TX 77843 -3131, USA
b
Department of Biology, Uni6ersity of New Mexico, 167 Castetter Hall, Albuquerque, NM 87131 -1091, USA
c
Wildlife International, 8598 Commerce Dri6e, Eastern, MD 21601, USA
d
Department of Wildlife and Fisheries Sciences, Texas A&M Uni6ersity, College Station, TX 77843 -2258, USA
Abstract
We present semi-empirical model of persistent random walk for studying animal movements in two-dimensions.
The model incorporates an arbitrary distribution for the angles between successive steps in the tracks. Inclusion of a
turning angle distribution enables explicit computation of the effect of persistence in the direction of travel on the
expected magnitude of net displacement of the animal over time. We employed a form-analogous approach to obtain
expressions for the expected net displacement and derived root mean square of the expected displacement of an
animal at the end of a multi-step random walk in which turning angles were drawn from the Lemicon of Pascal, the
elliptical, the von Mises, and the wrapped Cauchy distributions. The accuracy of these expressions for the expected
magnitude of net displacement was tested by comparison with simulated results of persistent random walks where
turning angles were drawn form the wrapped Cauchy distribution. Our results should be useful in predicting
two-dimensional distribution of moving animals for which frequency distributions of the turning angles can be
measured. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Animal movement model; Expected net displacement; Random walk; Root mean square of displacement; Turning angle
distribution
1. Introduction
Movement (i.e. spatial displacement) has come
to be recognized as a key element in the population biology of most organisms and the community dynamics of interacting species. Examples of
* Corresponding author. Tel.: +1-505-2775140; fax: +1505-2770304.
E-mail address: [email protected] (B.-L. Li).
recent studies are in wind dispersal of plant seeds
and pollen (Okubo and Levin, 1989), insect and
animal movements (Kareiva and Shigesada, 1983;
Li et al., 1987; Andow et al., 1990; Cain, 1991;
Turchin, 1991), and microbial transport (Li et al.,
1996). Quantifying movement patterns and relating them to the spatial distribution of organisms
have been approached from two different perspectives: (1) theoretical analyses using diffusion and
random walk models (Okubo, 1980; Levin, 1986;
0304-3800/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 3 0 4 - 3 8 0 0 ( 0 0 ) 0 0 3 0 9 - 4
116
H.-i. Wu et al. / Ecological Modelling 132 (2000) 115–124
Turchin, 1991; Li et al., 1996), and (2) empirical
studies employing computer simulations (Crist
and Wiens, 1995). Both types of approaches are
useful, but each also has its defects (Marsh and
Jones, 1988). Because detailed studies of organism
movement over long distances are technically and
logistically very difficult, the need to develop
quantitative methods for the description and prediction of such movements is of surpassing importance (Levin, 1987).
Random walk as a simple model of diffusion
processes has been adopted in study of many
biological movements (Levin, 1986; Li et al.,
1996). The classical random walk model deals
with a particle moving in a series of steps, where
the length of the step, time between steps, and
direction of the step are independent of each other
and those of preceding steps (Patlak, 1953).
Marsh and Jones (1988) classified random walk
models of animal movement into two classes, one
in which step length and direction of movement
are independent, the other in which they are not.
Each class is further divided into two subclasses,
‘oriented models,’ in which the direction of each
step is chosen relative to a fixed compass direction, and ‘unoriented models,’ in which the direction is chosen relative to the preceding step
(Marsh and Jones, 1988). Most of the commonly
used random walk models assume that the direction of travel during step i of a random walk is
independent of the direction taken at step i− 1.
However, the movements of many entities, including animals, exhibit persistence, i.e. the direction
of travel during step i is dependent of the direction travelled in step i −1. Environmental variation can lead to changes in the persistence of
movement and with a suitable model (e.g. Neill,
1979; Turchin, 1991), it is possible to relate
the dynamics of animal distribution to environmentally induced changes in persistence of movement.
The expected magnitude of the net displacement after n steps and the variance of displacement about the origin are both useful for
quantifying movements (Cain, 1991), but analytical difficulties led early researchers to restrict
study to the root mean squared displacement. A
one-dimensional random walk model with corre-
lation between step directions was developed by
Taylor (1921), who provided an expression for the
expected squared-displacement after n steps.
Goldstein (1951) expanded this model and corrected an error in Taylor’s expression. Okubo
(1980) also gave an expression for the expected
squared displacement. But his result appears to be
in error. Skellam (1973) derived a term for the
expected squared displacement after n steps in two
dimensions. As shown below, the correct expression for the two-dimensional case first derived by
Skellam (1973) and followed by Kareiva and
Shigesada (1983) is identical in form to the
correct solution for the one-dimensional case
(Goldstein 1951). Marsh and Jones (1988) also
derived the correct expression for the expected
squared displacement after n steps in two dimensions. Bovet and Benhamou (1988) developed an
approximation for the expected magnitude of net
displacement in 2D random walks, but their expression appears to be valid only if persistence is
low.
In this study, we propose a model of persistent
2D random walks that can be represented by
drawing turning angles from a circular distribution where the probability of turning at an angle
u between steps is proportional to the area swept
out within the differential rotation of the radial
axis about the origin. Four well-known circular
distribution functions, which can be used to correlate the turning angle between steps, were used in
this study. These distributions are the Lemicon of
Pascal, the elliptical, the von Mises, and the
wrapped Cauchy distribution. For each of these,
we obtained expressions for directional correlation in terms of the geometric parameters of the
distribution, and found how an expected net displacement and its variance can be calculated from
the geometric parameters via a correlation. The
wrapped Cauchy distribution is especially useful
for simulation because its random deviates are
easily obtainable. Our model could enable a new
approach for the description and prediction of
animal movements, especially those of organisms
like pelagic fish that seem to maintain persistent
courses of travel in a relatively featureless
environment.
&
&
H.-i. Wu et al. / Ecological Modelling 132 (2000) 115–124
=4
2. The mean squared displacement
p
f(u)cos udu
0
Let r i be the ith-step in a 2D random walk with
fixed step size l and time interval t. The resultant
displacement vector after n steps, Rb n, is
Rb n = %r i
(1)
i
The expected value for the square of the n-step
displacement, E[R 2n], can be computed as follows:
n
n
i=1
j=1
E[R 2n]= E % r i ’ % r j
n
n−1
n
n
n
=nl 2 + 2l 2 % % E[cos ui, j ],
(2)
i=1 j\i
where uij is the angle between step r i and r j.
In an ideal random walk, all turning angles
between steps i and j within the range −p to +p
are equally probable, so that E[cos ui,j ]= 0. In a
persistent random walk, the turning angle is
weighted and E[cos ui,j ]" 0. Under the simplifying assumption that angular correlation g exists
only between successive steps, i.e. E[cos ui,j ]= g,
&
p
−p
f(u)cos udu =g
(3)
with f(u) being the angular weighting function, or
the turning angle distribution.
For animals moving in a relatively featureless
environment, a symmetric weighting function is
desired. A symmetric weighting function is an
even function, thus Eq. (3) can be expressed
2
&
p
f(u)cos udu =g
(4)
0
The expected cosine for the two-step angle of
turning, E[cos ui,j + 2], can then be evaluated as
E[cos ui, j + 2]=
&
p
−p
du%
E[cos ui, j + s ]= g s
(6)
From Eq. (6), Eq. (2) can be simplified to
n−1
n−1
f(u)du2
i=0
n
&
p
−p
f(u%)cos(u + u%)
i=0
n
1+g
1− g n
n
−2g
, for 05 g5 1
1−g
(1−g)2
(7)
The above equation takes the same form as the
one-dimensional random walk with persistence
(Goldstein, 1951; Skellam, 1973; Okubo, 1980).
Eq. (7) is also the same equation in the section
Model Ib of the Appendix A of Marsh and Jones
(1988) (p. 126) when E[l]= l. (Note that Eq.
1+g
1− g n
(5.20) E[R 2n]= l 2 n
−2g 2
on p. 70
1−g
(1−g)2
in Okubo (1980) is in error: the parameter g of the
second term in the square brackets is squared.)
3. Evaluation of E[cos ui,j ]
(5)
since p− p f(u)sin udu =0 for f(u)sin u is an odd
function. Likewise, the multi-step expected cosine
for the angle of turning, E[cos ui,j + s ], can be
evaluated by the same procedure to give the
multi-step correlation parameter
=l2
i=1 j\i
n−1
f(u%)cos u%du%= g 2,
0
E[R 2n]= nl 2 + 2l 2 n % g i − n− % ig i
= % E[r 2i ]+ 2 % % E[r i ’r j ]
i=1
117
p
n
4. Evaluation of E[Rn ]
An expression for the expected net displacement after n-steps, E[Rn ], would be useful (for
example, see Cain (1991)). In evaluating E[R 2n],
one evaluates the average of R 2n over n turning
angles. One might attempt to evaluate E[Rn ] by
simply averaging the cosine of the turning angles
that appear in a radical, but determining the
average of n turning angles over functions inside a
radical is impossible in general. Therefore, an
alternative approach must be employed.
Bovet and Benhamou (1988) derived an approximated expression (their Eq. (5)) for the expected magnitude of net displacement by
decomposing tracks into x and y components and
applying statistical distribution theory to the distributions. Their approximation,
EBB[Rn ]: l
'
0.79n
1+b
,
1−b
(8)
118
H.-i. Wu et al. / Ecological Modelling 132 (2000) 115–124
has shortcomings that can be easily demonstrated.
In Eq. (8), we have replaced g by b to distinguish
the difference between the two parameters: b
refers to the correlation parameter for the expected displacement while g denotes the correlation parameter for the expected square
displacement. Suppose that an animal moves in a
persistent (or ‘correlated,’ in terminology of Bovet
and Benhamou (1988)) random walk with steps of
unit length. After n =1 step, the animal will always be a step away from the origin. For n \ 1,
E[Rn ] must always be less than n since E[Rn ]= n
implies straight-line movement. If EBB[Rn ]/n is
calculated for various parameter values of b, a
pattern emerges as shown in Fig. 1. If b \ 0.11,
then EBB[Rn ]\n for small n. As b increases, the
value of n for which EBB[Rn ]\ n also increases.
When b= 0.99, EBB[Rn ]\ n is for n up to 157
steps, and when b =0.999, EBB[Rn ]\ n is for n up
to 1580 steps. Clearly, this approximation is of
questionable value for highly persistent walks.
McCulloch and Cain (1989) used Taylor series
expansion of E[R 2n], which involves variance
(Rn ), to obtain an approximated expression for
E[Rn ]. Their approximation, except for the first
few steps, is in very good agreement with a simulated result, as shown in their Fig. 2. They have
successfully removed the deficiency encountered
with the formulation of Bovet and Benhamou.
However, their approximation requires evaluations of higher-orders of E[R m
n ] with m] 4 because variance of at least R 2n is needed for the
dominating expansion term. Our attempt in this
study is to present an alternative approximation
that requires only an evaluation of the successive
step correlation parameter.
We employ a form-analogous approach to approximate an expression for the expected displacement that is applicable to walks of any persistence
(i.e. correlation) and works well for small n. As Rb n
is the resultant displacement vector at the end of
n steps, the magnitude of Rb n can be expressed as
Fig. 1. The expected range of Bovet and Benhamou (1988), E[Rn ]/n, in relation to random walks of n steps. For persistent random
walks where the correlation, b, between direction of movement in step i and step i +1 is high, the approximation proposed by Bovet
and Benhamou clearly fails.
H.-i. Wu et al. / Ecological Modelling 132 (2000) 115–124
119
suggest with the idea of testing the validity of Eq.
(10). This is in parallel to an approach often used
in the development of physical sciences: when
detailed knowledge is lacking or when one encounters an intractable mathematical difficulty,
simple expressions are adopted and tested against
data. The Dirac delta function first introduced in
1930 (Marsden, 1974) and the Lennard–Jones
potential for describing the interaction between
two molecules (Reif, 1965; Barrow, 1988) are the
two well-known examples.
The expected value for the magnitude of the
resultant after two steps, E[R2] can be evaluated
E[R2]= l
Fig. 2. With E[R2] being the expected magnitude of resultant
displacement after two steps with equal length l, the angle u0 is
the expected angle of deviation between successive steps.
Rn + l
'
n−1
n
n+2 % % cos ui, j
(9)
i=1 j\i
Notice the analogy between Eqs. 2 and 9. If
cos ui,j is replaced with multi-step correlation
parameter g j − i in Eq. (9), then the result would
be the expected root mean square of Rb n, given
that the expression of Eq. (9) is identical to Eq.
(7). The successive-step angular correlation g for
the expected square displacement is defined in Eq.
(3), which led to the expression for E[R 2n]. If we
could find a successive-step correlation parameter
b, then by the analogy to the form of Eq. (7), the
expected displacement after n-steps would
become:
E[Rn ]: l
'
1+b
1 −b n
− 2b
, for 0 5 b5 1
1−b
(1 − b)2
(10)
n
This approximate expression for E[Rn ] was not
derived through rigorous mathematical arguments, nor do we claim that the expression
'
E l
−1
n
n+2ni =
1 j \ i cos ui, j
proximated by l
'
n
is accurately ap-
−1
n
E[n]+ 2ni =
1 j \ i E[cos ui, j ].
Only because there is need for a simple applicable
expression for the expected displacement, do we
&
p
&
−p
= 4l
p
0
where
&
u0
cos = 2
2
p
0
f(u)
2 + 2cos udu
u
u0
f(u)cos du =2lcos ,
2
2
(11)
u
f(u)cos du
2
The angle u0 can be interpreted as the expected
angle of deviation between successive steps as
shown in Fig. 2. Let b be the cosine of the
expected turning angle between successive steps,
b= cos u0 = cos[2cos − 1
E[R2]
E[R2]2
]=
− 1;
2l
2l 2
(12)
this will be used as the successive-step angular
correlation parameter for the expected displacement. Since it is impossible to evaluate the expected angle of deviation for a multi-step angle of
turning, we follow the result given in Eq. (6) and
use the form-analogous approach to propose that
b s be interpreted as cos u0 j,j + s, or
b s : cos u0 j, j + s
(13)
Once again, Eq. (13) was not obtained through
rigorous mathematical derivation. The relationship between E[Rn ] and b for a wrapped Cauchy
distribution is shown in Fig. 3. Note that E[Rn ]
rendered from Eq. (10) is always less than n, for
n\ 1.
To test the validity of the approximated expression for E[Rn ], we compared results from an
unbiased 2D random walk, E[Rn ]= np/2 (that
120
H.-i. Wu et al. / Ecological Modelling 132 (2000) 115–124
is same as Eq. (3) in Cain (1991)(p. 2138), and Eq.
(10). For a uniform turn-angle distribution,
f(u) = 1/2p, b= − 0.18943 is obtained from Eqs.
11 and 12. With this b value, Eq. (10) over-estimates the value obtained from the ideal 2D random walk by less than 7%. After the first couple
of steps, Eq. (10) starts to under-estimate the ideal
2D random walk. The relative error increases as n
increases and the under-estimation is capped by
6.85% as n approaches infinity. A computer simulation was also employed to generate persistent
random walks from which values of E[Rn ] were
calculated and averaged. Simulations were performed using the Pascal/VS programming language on an AMDAHL 470V8 computer and
Mathematica on a SUN SPARC 1+ workstation. Random angles were generated by random
deviates drawn from a wrapped Cauchy distribu-
Fig. 4. The relative percent difference of the expected magnitude of the net displacement after n-steps, E[R2], calculated
using Eq. (10), under-estimates the average simulated magnitude of the net displacement by at most 8%. Relative percent
difference is defined as the difference between calculated and
simulated results divided by the average magnitude of simulated displacement. Angles were drawn from a wrapped
Cauchy distribution, with parameter o, which is equal to b.
The number of steps covered by the walk is denoted by n.
Symbols , , , and are for b=0.01, 0.7, 0.9, and 0.99,
respectively.
tion and the walks proceeded with constant unit
step size. The simulated tracks were sampled to
give 2500 resultants for each combination of n
and b. The averaged magnitude of these resultants is compared with the calculated values of
E[Rn ] in Fig. 4. The relative small errors and their
lack of tendency to increase as n increases indicate
that the proposed relationship of Eq. (10) is a
reasonable approximation. Estimates of the variance of the magnitude of the net displacement
V(Rn )= E[R 2n]− E[Rn ]2
Fig. 3. In a persistent random walk, the expected magnitude of
the net displacement after n-steps, E[R2], increases as the
angular correlation between steps, b, increases. There is good
agreement between the calculated expected displacement from
Eq. (10) (solid circles) and estimates of Bovet and Benhamou
(1988) (solid triangles) for small b; agreement becomes increasingly poor as b increases. Expected net displacements by
simulation (open circles) agree well with both approximations
for small b but diverge from Bovet and Benhamou estimates
as b increases.
(14)
are sensitive to errors in estimation of E[Rn ]. Fig.
5 compares estimates of one standard deviation
using the approximated expression for E[Rn ] with
estimates from a simulation. As Eq. (10) tends to
under-estimate displacements given by the simulation that leads to larger estimates of variance than
the simulation. Hence, statistical results using values from Eqs. 10 and 14 are conservative.
5. Symmetrical turning angle distributions
Several useful symmetrical turning angle distributions (also known as circular distributions
(Batschelet, 1981)) are reviewed. From a statistical
H.-i. Wu et al. / Ecological Modelling 132 (2000) 115–124
121
point of view, the von Mises distribution is the
circular distribution function of choice. However,
only the wrapped Cauchy turning angle distribution provides a closed-form cumulative distribution function. For illustration, we present the case
for the wrapped Cauchy distribution function
here, which is modified from the elliptical function. Other distributions are included in the
appendix.
The geometric form of a wrapped Cauchy function, shown in Fig. 6, is expressed in polar coordinates as
r=a
'
1− o 2
, 0 Bo B1
1+ o −2ocos u
2
(15)
The total area swept by this radius is pa 2; thus,
the probability density function for a wrapped
Cauchy distribution is
f(u) =
1− o 2
, 0B o B 1
2p(1+ o 2 − 2ocos u)
(16)
The correlation parameter gC, where the subscript C denotes Cauchy, equals o. The expression
for E[R2] can be obtained by using Eqs. 11 and
16,
E[R2]=
2l(1− o 2)
p
&
p
0
cos(u/2)
du
1 + o 2 −2ocos u
Fig. 6. Equi-areal plots of the Cauchy distribution function
with o =0.2 and 0.8. Although the total area enclosed by each
figure is the same, the areas sustained by a given angle of
deviation u are clearly different.
=
=
4l(1−o 2)
p
2l(1+o)
p
o
&
1
0
dx
(1− o)2 + 4ox 2
tan − 1
2
o
,
1− o
(17)
and thus the correlation parameter of angular
deviation between successive steps for a wrapped
Cauchy distribution is
bC =
Fig. 5. The variance of the expected magnitude of the net
displacement, V[Rn ], predicted by Eq. (14) (solid circles) exceeds the variance obtained by simulation (open circles) for
random walks with high persistence (upper pair of curves,
b= 0.9) and low persistence (lower pair of curves, b =0.01).
2(1+ o)2
2
o
tan − 1
op 2
1−o
2
−1
(18)
The cumulative distribution FC(u) for this geometric function can be obtained by
FC(u)=
&
u
−p
f(u%)du%
122
H.-i. Wu et al. / Ecological Modelling 132 (2000) 115–124
1+o
u
1
1
tan +
= tan − 1
1−o
2
p
2
(19)
The random deviate n is then
n= 2tan − 1
n
1− o
1
tan RN − p ,
1+ o
2
(20)
where 05RN 51 refers to random numbers
drawn from a uniform distribution. From Eq.
(20), the generated nth step coordinates xn and yn
are,
xn = xn − 1 + l cos n
yn = yn − 1 +l sin n
(21)
6. Conclusion and discussion
The usefulness of the expected displacement in
studying the spread of biological organisms is well
recognized (McCulloch and Cain, 1989; Cain,
1991). Unfortunately, an exact expression for the
expected displacement has not been derived successfully thus far. Many attempts to obtain an
accurate approximation have been made in the
past. We have succeeded in developing an approximation that is simple and easy to apply.
In deriving E[R 2n], the expected squared displacement for an n-step 2D random walk, we
assumed that angular correlation exists only between successive steps. This assumption results in
a multi-step correlation that is simply the correlation parameter raised to sth power, where s is the
number of intervening steps. This leads to an
expression for E[R 2n], which has the same form as
the E[R 2n] in the one-dimensional case. The
derivation of E[Rn ] is not that mathematically
fortunate, however. Yet an expression for E[Rn ]
would be useful in many ecological studies. As we
were unable to establish the relationship between
the angular deviation for multiple steps and b, we
followed the result obtained in Eq. (6) and propose that, using a form-analogous approach, the
expected multi-step correlation for angular deviation can roughly be expressed as b s where s is the
number of steps. The expected magnitude of Rb n,
approximated by Eq. (10), and the calculated
variance given in Eq. (14) are compared with
simulation results for the wrapped Cauchy distri-
bution using various values of the parameter b.
The agreement between simulation and analytical
results reasonably supports the validity of the
relationship proposed in Eq. (10). Our expression
for E[Rn ] extends the range of b over which
estimates of the expected magnitude of the net
displacement can be made, and is reasonably accurate even when n is small.
McCulloch and Cain (1989) used Taylor series
expansion of Eq. (10) to obtain their approximation while retaining the definition of g intact.
Mathematically, their approach is very rigorous.
However, evaluations of higher-orders of E[R m
n]
for m] 4 poses some difficulties for most of the
statistical distribution functions. One alternative
is to use a numerical approach as they suggested
to generate required probability distributions. Our
form-analogous approach has provided another
alternative. The result is simple and only one
parameter needs to be evaluated.
In this study we have assumed fixed step
lengths for simplification. Although step lengths
for moving animals are highly variable, we believe
this approximation is not sensitive to variable step
lengths if n is sufficiently large. A variable length
simulation is currently planned to study this
effect.
There are at present very few studies on the
detailed movement patterns of animals over long
distances. It is very difficult to be certain that the
appropriate model has indeed been chosen in such
a study (Marsh and Jones, 1988). Our formulation provides an alternative way for modeling
animal movements with persistent random walks
over long distances. Our results can also be very
useful in estimating the expected net displacement
of animal movements when the turn-angle distributions are the well-established functions. These
results also are useful as a potential forecasting
tool for the fishing industry and management
agencies, to locate large aggregated fish schools in
the relatively featureless ocean.
Acknowledgements
We thank W.M. Childress for his comments on
an early version of this paper. Signed reviews by
H.-i. Wu et al. / Ecological Modelling 132 (2000) 115–124
M.L. Cain and three anonymous reviewers were
helpful. This work was partially supported by the
National Science Foundation under the grants
BSR-91-09240, DEB-93-06679 and DEB-9411976, and DOE/Sandia National Laboratories
under contract BE-0229. This is Sevilleta LTER
publication no. 138.
Useful symmetrical turn-angle distributions for
which cumulative distribution functions can be
generated from random numbers (Cain, 1985) are
presented below:
The polar coordinate expression for the Lemicon
of Pascal (Eq. 11.32, p. 44 of Spiegel, 1968) is
given as
(A1)
The total area A enclosed by the boundary is
&
1
A=
2
p
−p
r 2du = pa 2
o2
1+
2
(A2)
The turning angle distribution is obtained by taking the ratio of the area swept-out per unit angular displacement to the total area A,
f(u) =
r 2 (1+ocos u)2
=
2A
p(2+o 2)
gL =
&
−p
f(u)cos udu =
2o
2+o
(b) Elliptical distribution
r=
a(1− o 2)
, 0BoB 1,
1− ocos u
and the area pa 2
1 − o 2, the elliptical turning
angle distribution function is
f(u)=
(A4)
4l
p(2+o 2)
=
8l
p(2+o 2)
&
&
u
(1 + ocos u)2cos du
2
p
0
1
(1 + o −2ox 2)2dx,
0
where x =sin u/2, thus
(1− o 2)3/2
, 0BoB 1
2p(1− ocos u)2
(A6)
The turning angle correlation parameter for an
ellipse, ge, is equal to the parameter of conic
section, or eccentricity, o,
ge =
&
p
−p
f(u)cos udu= o
(A7)
The E[R2] value can be evaluated as follows:
u
cos
2
2l(1−o 2)3/2 p
E[R2]=
du
2
p
0 (1−ocos u)
&
(A3)
The correlation parameter of angular deviation
between successive steps bL can be evaluated by
mean of E[R2]. From Eq. (11), one can evaluate
E[R2]=
(A5)
The expression for bL can then be obtained by
substituting Eq. (A5) into Eq. (12).
=
The correlation parameter gL, here again the subscript refers to the distribution function employed, can then be expressed in terms of the
parameter of the Lemicon of Pascal, o,
p
8l
2
7
1+ o − o 2 .
p(2+ o 2)
3
15
From the polar coordinate expression for an
ellipse,
Appendix (a) Lemicon of Pascal distribution
r = a(1+o cos u) 0 B o B1
E[R2]=
123
4l(1−o 2)3/2
p
'
&
1
0
dx
(1− o+ 2ox 2)2
2l
=
1 − o 2
p
+
(1+o 2)3
tan − 1
2o
' n
2o
,
1− o
(A8)
and be can be obtained by using Eq. (12).
(c) von Mises distribution
The von Mises probability density function f(u)
is the circular analog of the normal distribution
and is frequently used in studying stochastic angular processes (Batschelet, 1981). It is given by
f(u)=
ekcos u
, k =0,
2pI0(k)
(A9)
H.-i. Wu et al. / Ecological Modelling 132 (2000) 115–124
124
where k is the von Mises parameter and I0(k) is
the modified Bessel function of order zero. The
von Mises correlation parameter gM is given as
gM =
1
pI0(k)
&
p
ekcos ucos udu =
0
I1(k)
,
I0(k)
(A10)
since
&
p
ekcos ucos udu =pI1(k)
0
(Eq. 9.6.19, p.376 of Abramowitz and Stegun,
1972), where I1(k) is the modified Bessel function
of order one. The evaluation of E[R2] is given as
E[R2]=
2l
pI0(k)
&
0
k
=
=
p
4le
pI0(k)
'
ekcos ucos
&
p
u
du
2
2
e − 2kx dx
0
2 le k
erf(
2k),
pk I0(k)
(A11)
which leads to the expression for bM.
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