![Fig. S5. A unimodel one-dimensional step-size distribution G i k (s) on [0, smax] is rotated through 360° to convert it to a spatially explicit two-dimensional distribution G i k (, ) defined over a featureless landscape grid C [i.e., all grid cells (a, b) have the same neutral state in terms of c i being defined to reflect a completely neutral landscape structure].](https://www.researchgate.net/profile/Wayne-Getz/publication/23627263/figure/fig4/AS:669132756377619@1536545089821/Fig-S5-A-unimodel-one-dimensional-step-size-distribution-G-i-k-s-on-0-smax-is_Q320.jpg)
Content uploaded by Wayne Getz
Author content
All content in this area was uploaded by Wayne Getz on Feb 17, 2015
Content may be subject to copyright.
Content uploaded by Wayne Getz
Author content
All content in this area was uploaded by Wayne Getz on Jan 01, 2014
Content may be subject to copyright.
A framework for generating and analyzing movement
paths on ecological landscapes
Wayne M. Getz
a,b,1
and David Saltz
c
a
Department of Environmental Science, Policy, and Management, University of California, Berkeley, CA 94720-3114;
b
Mammal Research Institute, University
of Pretoria, Pretoria 0002, South Africa; and
c
Ben-Gurion University of the Negev, Be’er Sheva 84105, Israel
Edited by Ran Nathan, The Hebrew University of Jerusalem, Jerusalem, Israel, and accepted by the Editorial Board June 2, 2008 (received for review
February 26, 2008)
The movement paths of individuals over landscapes are basically
represented by sequences of points (x
i
, y
i
) occurring at times t
i
.
Theoretically, these points can be viewed as being generated by
stochastic processes that in the simplest cases are Gaussian random
walks on featureless landscapes. Generalizations have been made of
walks that (i) take place on landscapes with features, (ii) have
correlated distributions of velocity and direction of movement in each
time interval, (iii) are Le´ vy processes in which distance or waiting-time
(time-between steps) distributions have infinite moments, or (iv)
have paths bounded in space and time. We begin by demonstrating
that rather mild truncations of fat-tailed step-size distributions have
a dramatic effect on dispersion of organisms, where such truncations
naturally arise in real walks of organisms bounded by space and, more
generally, influenced by the interactions of physiological, behavioral,
and ecological factors with landscape features. These generalizations
permit not only increased realism and hence greater accuracy in
constructing movement pathways, but also provide a biogeographi-
cally detailed epistemological framework for interpreting movement
patterns in all organisms, whether tossed in the wind or willfully
driven. We illustrate the utility of our framework by demonstrating
how fission–fusion herding behavior arises among individuals en-
deavoring to satisfy both nutritional and safety demands in hetero-
geneous environments. We conclude with a brief discussion of
potential methods that can be used to solve the inverse problem of
identifying putative causal factors driving movement behavior on
known landscapes, leaving details to references in the literature.
fission–fusion 兩 GPS 兩 landscape matrices 兩 random and Le´ vy walks 兩
dispersal 兩 movement ecology
T
he movement of all organism, but especially sentient an imals,
is a complex process that depends on both an indiv idual’s
abilit y to perform various t asks and the nature of the landscape
through which it moves (1–6). These tasks include the individ-
ual’s intrinsic ability to move in dif ferent ways (e.g., a horse
walks, trots, canters, and gallops), the individual’s internal state
to perform certain activities (e.g., forage, head home, flee, and
seek a mate), and the individual’s ability to sense its environ-
ment, remember landmarks, construct mental maps, and process
infor mation (7). Landscape variables that influence movement
include topography, abiotic variables (8, 9), location of resources
(10), conspecifics by gender and age, and heterospecific com-
petitors and predators.
Emerging digital and communications technologies have re-
fined our ability to measure movement at the resolution of
f ractions of sec onds w ith concomit ant spatial precision (11),
while kinematical (e.g., acceleration), physiological (e.g., heart
beat and temperature), and behavioral (e.g., vocalizations) in-
for mation are simultaneously recorded. The internal state driv-
ing movement, however, remains largely hidden: in animals, for
example, states of hunger, thirst, and fear are either inaccessible
or, at best, only indirectly inferable.
The sampling f requency of movement data affects our ability
to detect short-duration fundament al movement elements
(FME) (6), such as a lunge versus a step taken at normal speed.
Such fine-scale events are generally unrecoverable f rom sam-
pling movement at intervals coarser than the duration of these
events (12) [see supporting information (SI) Fig. S1]. Further-
more, activities such as foraging or heading home involve a mix
of FMEs such as being st ationary, ambling, and walk ing; and
these activities may differ only in the way the FMEs are str ung
together. If a string of normal steps interspersed with st ationary
periods, for example, is on the order of minutes for both foraging
and heading to a t arget, then movement paths sampled every 10
min during either of these t wo activities can be distinguished only
if they produce dif ferent characteristic ‘‘distance moved in each
sampling interval’’ distributions. This suggests that to appropri-
ately characterize movement components of an individual’s path
over time, we should endeavor to identify canonical activity
mode (CAM) distributions that emerge from the mix of FMEs
that characterize the activity in question: i.e., CAMS are com-
posites of the FMEs (Fig. S1), and their characteristic step size
and direction of heading distributions will depend on the length
of sampling intervals (Fig. S2) and scale of analysis (13).
Ideally, if one assumes that FMEs are characterized purely by
a fixed speed (because they relate to biomechanical traits of
individuals), then one could mechanistically construct a move-
ment path by specifying a sequence of FMEs with a direction of
heading according to know distributions of sequence lengths and
c orrelated heading directions for particular activities. Alterna-
tively, at fixed points in time one could specify the next location
of an individual by draw ing ‘‘distance moved’’ and ‘‘heading
direction’’ from empirical step-size and heading distributions (1,
6, 14–17). To date, such distributions are invariably derived from
sampling intervals c onsiderably longer than the shortest FME.
Thus, with most current data, it is not possible to construct
distributions in terms of strings of FMEs, but only in terms of
longer-lasting CAMs. Consequently, CAMs are currently the
preferred place to start developing a f ramework for movement
analysis, despite the fact that any set of CAMs is unlikely to
ac count for all of an individual’s time. In many cases, however,
a reasonable tradeoff may exist in defining several CAMS that
ac count for most of an individual’s time.
Movement Paths on Featureless Landscapes
Kinds of Data. The most basic set of dat a that can be collected
on the movement path of an individual is a sequence of positions
(x
i
, y
i
) at points t
i
, i ⫽ 0, 1, 2,...,n: that is, the set D ⫽
{u
0
,...,u
n
} with u
i
⫽ (x
i
, y
i
), where x
i
⫽ x(t
i
) and y
i
⫽ y(t
i
). For
Author contributions: W.M.G. and D.S. designed research, performed research, contrib-
uted new reagents/analytic tools, analyzed data, and wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission. R.N. is a guest editor invited by the Editorial Board.
1
To whom correspondence should be addressed. E-mail: getz@nature.berkeley.edu.
This article contains supporting information online at www.pnas.org/cgi/content/full/
0801732105/DCSupplemental.
© 2008 by The National Academy of Sciences of the USA
19066–19071
兩
PNAS
兩
December 9, 2008
兩
vol. 105
兩
no. 49 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0801732105
c onvenience, we set t
0
⫽ 0 but do not necessarily require all of
the time intervals
i
⫽ [t
i⫺1
, t
i
] to be of equal length. In fact, in
some analyses, the focus is on the distribution of waiting times
i
associated with the events of first being in position (x
i⫺1
, y
i⫺1
)
and then next in position (x
i
, y
i
) (14). These data can be
transfor med into polar coordinates (6) to obtain a set of vectors
z
i
⫽ (t
i
, d
i
,
i
) with
d
i
⫽
冑
共x
i
⫺ x
0
兲
2
⫹ 共y
i
⫺ y
0
兲
2
and
i
⫽ arctan
冉
y
i
⫺ y
0
x
i
⫺ x
0
冊
.
A c onsiderable body of diffusion and stochastic process theory
exists to analyze such data in the context of featureless land-
scapes, including uncorrelated and correlated random walks
(14–17) and super- and subdif fusive Le´v y walks (18–21) and
Le´vy modulated correlated walks (22).
Brief Review of Random Walks and Diffusion. Consider the distri-
bution of waiting times
i
, velocities v
i
, and absolute displace-
ments d
i
associated with a set of displacement data D.Ifthe
distributions of both waiting times and velocities have finite
mean and variance, then the stochastic process associated with
the data is said to be diffusive. From theory (16–20), this implies
that over an ensemble of K sets D
k
, k ⫽ 1,...,K, where each set
is one realization of the same stochastic process, the mean-
square displacement (msd)
i
of the series d
ik
averaged over all
K sets is asymptotically linear, that is
i
⫽ (1/K)¥
k⫽1
K
d
ik
2
⬃ t.By
c ontrast, if upon plotting this relationship we find that
i
⬃ t
p
for
p ⬎ 1or⬍ 1, then the walk is respectively referred to as
superdif fusion or subdiffusion. As we demonstrate below, using
simulated data drawn from a modified Pareto power law distri-
bution (23) (Fig. S3) and plotted on a log–log scale, such plots
can be misleading if time is not sufficiently large. The reason is
that, initially, the relationship is affected by the fact that the
dist ance moved in the first sampling interval is controlled by the
actual step-size distribution, but from the second step onwards
the distance moved from the origin is now also af fected by
turn ing angles.
Movement Pathways and Diffusion. Over the past decade, analyses
of the movement paths of several organisms, including albatrosses
(24) and spider monkeys (14), have concluded that the associated
movement proce sse s are superdiffusive, although a re-analysis of
these data refute this finding (ref. 1, but see ref. 25). A possible
source of error in estimating p in the msd relationship
i
⬃ t
p
(Fig.
1) is t must be sufficiently large for the asymptotic value to emerge.
Another source of error is that superdiffusion predicts fractal
looking movement paths (14, 20, 24). Using one of several different
methods to estimate the fractal dimension (26) of such paths, some
organisms have been pronounced as superdiffusive (14). In these
organisms, however, repeated fractal patterns occur at no more
than two or three particular scales and are more a feature of the way
resource s are distributed across the landscape than of a genuine
superdiffusive process. Thus, it is important to understand how
animal movement is influenced by landscape features and to asse ss
the extent to which step-size distributions are modified by landscape
heterogeneity.
We note here that because all bounded step-size distributions
produce Gaussian random walks when turning angles are un-
c orrelated, critical information in the step-size distribution, such
as multimodality arising from mixed distributions of FMEs, is
lost when using the statistics of emergent global characteristics
such as msd as a function of time. This stresses the importance
of knowing the actual step-size distributions when deconstruct-
ing movement paths and trying to understand the causal pro-
cesses creating local path structures.
Movement Paths on Structured Landscapes
FMEs and CAMs. The framework presented here is formulated in
the context of a group of N k nown individuals indexed by k. This
specificit y allows us to ac count for the following: (i) species,
gender, and age-specific differences; (ii) unique memory and
k nowledge of landscape; and (iii) cues and vectors individuals
use to select a new position on the landscape. Further, we assume
that each individual has n
m
k
FMEs (Table 1), each with its own
characteristic speed s
j
k
, j ⫽ 1,...,n
m
k
, f rom which its movement
track is generated. The set of FMEs constitute the basic motion
capacit y ⍀
k
(7). Furthermore, in any segment of the movement
path (Fig. S1), these FMEs are mixed in various proportions to
c onstitute a set of A
r
k
CAMs (Table 1), r ⫽ 1,...,n
a
k
best
characterized by distributions of speeds (equivalently distances)
with means and standard deviations s
r
and
r
for each r (Fig. S2).
We note, however, that when the sample intervals
i
k
⫽
k
are
fixed over all intervals i, for reasons discussed below only the
st andard deviation (and not the mean) depends on sampling
f requency 1/
k
.
As an example, ⍀
k
in horses has been defined in terms of five
FMEs or gaits (27) of increasing speeds: stationarity (s
1
⫽ 0),
walking (a four beat gait with s
2
⬇ 4 mph), trotting (a two beat gait
with s
3
⬇ 8 mph on average), cantoring (a three beat gait for which
s
4
⬎ s
3
), and galloping (a four beat gait for which s
5
⬇ 25–30 mph,
varying across horses). Beyond these natural gaits, some horses have
been bred to implement so-called ‘‘ambling gaits’’ that provide a
smoother ride, but not all horses can execute these, thereby
Fig. 1. The logarithm of mean-square displacements (msd)
versus the
logarithm of time t average over 10,000 simulations are plotted for a random
walk with step lengths drawn from modified Pareto distributions (Upper Left,
q ⫽ 2; Upper Right, q ⫽ 1.5; Lower Left, q ⫽ 1) and directions for each step
completely random. From the lines inserted by ‘‘eye’’ (red, small t; blue, large
t), Upper Left represents diffusion (P ⫽ 1 for large t), and Upper Right and
Lower Left represent super diffusion (respectively, P ⫽ 1.15 ⬎ 1 and P ⫽ 2 for
large t). In Lower Right, for the case q ⫽ 1, the length of these excursions are
truncated at a step size of 100 (biologically, an upper bound is set by the
maximum velocity of the organism multiplied by the length of the time
interval), which is far out in the tails of the distribution (two orders of
magnitude beyond the mode; see Fig. S3). In this case, the noisy superdiffusive
behavior is completely tamed even though, initially, it looks superdiffusive
(
⬃ t
1.4
for t ⱕ 2).
Getz and Saltz PNAS
兩
December 9, 2008
兩
vol. 105
兩
no. 49
兩
19067
ECOLOGY SPECIAL FEATURE
providing an example of selection at work on the motion capacity
component ⍀ of our guiding conceptual model (7).
For each of the n
a
k
, CAMs associated w ith indiv idual k, the
distribution of speeds (equivalently distances or ‘‘step sizes’’)
associated w ith a particular CAM is affected by the value of
:
if
is small an individual will generally be in one movement mode
or another in the mix that constitutes the activit y, whereas if
is
large an individual will more likely be executing some mix of
modes over one interval (Fig. S2). Thus, even though the
proportion of different movement mode events used to construct
the movement tracks may be quite st able with regard to a specific
activity, the directions of heading from one event to the next
make the relationship bet ween the length of a track and dis-
placement quite complicated. Only for very simple cases, such as
tracks generated from one type of FME, is the relationship
bet ween length-of-track and displacement easily cracked by
analy tical methods. For the rest, the easiest way to generate the
distribution of speeds (distances) A
r
(s) as a function of sample
interval size
is to use Monte Carlo simulation.
The proportions of FMEs in a particular CAM is likely to vary
with changes in landscape. For example, the mix of stationary,
walk ing, and running modes used by a foraging African antelope
will dif fer in an open savannah compared with thick bushveld.
This problem can be dealt w ith by assuming that the parameters
of a CAM distribution depend on external landscape factors in
addition to the frequency of sampling along a movement path.
Internal States and Goals. One of the goals in an analysis of
empirical data is to see how cleanly a set of CAM distribution of
step-size FMEs can be extracted from the movement track to
ex plain the actual activity producing different segments of the
movement track. Some segments may reflect a pure activity (e.g.,
foraging) while others are a mix of activities (e.g., foraging and
resting) or even a compromise between competing activities such
as an individual foraging as it heads to a known water source or
home. To be able to account for such mixed activities, as well as
assess factors that may lead to an individuals switching from one
CAM to another, we must infer that the individual has a
multidimensional internal state that drives the behav ior (7). The
current state of this driver can in turn be associated with a goal
emerging from an individual’s internal state, which in general
will vary with time. The relationship bet ween an indiv idual’s
internal state (i.e., the vector w—see Fig. S4) and its current goal
st ate can be treated in various ways. One way is to construct a
mapping of a continuous n-dimensional vector space to a discrete
space of g goals. Another is to consider the internal state as a
weighting vector
w
i
k
⫽ 共w
i1
k
, w
i2
k
, ···,w
n
a
k
k
兲
that produces a goal-modified ‘‘ideal’’ distribution G
i
k
(s)of
speeds (distances) s for individual k at time t
i
⫽ i
with the
weighted sum of its n
a
k
CAMs: i.e.,
G
i
k
共s兲 ⫽
冘
r⫽1
n
a
k
w
ir
k
A
r
k
共s兲.
In the case of organisms that have no internal mechanism for
generating goals (e.g., plants), the internal state may, for exam-
ple, represent elements controlling dehiscence of seed dispersal
str uctures.
Spatial Explication. The spatial infor mation embedded in the
distribution G
i
k
(s) needs to be made explicit before other
relevant spatially explicit geographical and biological informa-
tion can be incorporated into the movement process. The
distribution G
i
k
(s) is defined on s 僆 [0, s
max
]. On a flat struc-
tureless landscape—that is, in the absence of all cues and other
directional biasing factors beyond the c ontext of a correlated
walk—an individual is equally likely to move in any direction.
Thus, the ideal (i.e., featureless landscape) distribution G
i
k
(s)
can be given an explicit spatial dimension by rotating it around
the point [0, 0] on a plane parameterized by c oordinates (
␣
,

)
to obtain a radially symmetric distribution G
i
k
(
␣
,

) that has a
top-half-of-a-donut-like structure (Fig. S5). The distribution can
now be relocated so that its center of symmetry is the current
point of location u
i
k
⫽ (x
i
k
, y
i
k
) of organism k.
Landscape Raster. We incorporate landscape features into each
individual’s goal as follows. First, we cover the landscape with a
raster of rectangular cells C
␣
. Second, we locate the position u
i
k
⫽
(x
i
k
, y
i
k
) of each individual within the cell containing this point: in
general, this allows several individuals of one or more types to be
located in each cell. Also, we associate an n
c
-dimensional external
state to account for all factors relevant to the movement of each of
the N individuals on the landscape, including other individuals.
Third, the one-dimensional CAM distributions A
r
k
(s) are used, as
described above, to generate two-dimensional radially symmetric,
but discretized, distributions A
r
k
(
␣
,

) ⱖ 0 with ¥
(
␣
,

)
A
r
k
(
␣
,

) ⫽ 1
for all k and r ⫽ 1,...,n
a
k
. Fourth, we incorporate the landscape
effects through a set of n
a
k
landscape modifier matrices (LMM) L
ir
k
(i.e., specific to individual k, their activity r, and time i
)with
elements ᐉ
ir,
␣
k
constructed from those elements of the external
state vector that are applicable for the activity in que stion. These
elements ᐉ
ir,
␣
k
are used to modify the CAM distributions to reflect
the preference that each individual has for each of the landscape
cells while involved in one of its CAMs (e.g., one cell may be the
most desirable from a foraging point of view, whereas another is
desirable as a target when heading for water). These LMMs are
used to modify the movement distribution G
i
k
(
␣
,

) to account for
Table 1. Definition of parameters and variables
Frames (indices) Mathematical objects Descriptions
Time (i ) t
i
i
Current time, variable and fixed inter-interval size
Random walks d
i
i
i
Step size, heading, mean-square displacement (msd)
v
i
w
i
Linear and radial velocities at time t
i
Dataset D
k
of individuals (k) N u
i
k
⫽ (x
i
k
, y
i
k
) Number, Cartesian location for each k updated each t
i
Fundamental movement elements (FMEs) ( j ) n
m
k
s
j
k
⍀
k
Number, characteristic speed, ⬙capacity⬙ for each k
Canonical activity modes (CAMs) (r) n
a
k
A
r
k
A
r
k
Number, 1D and 2D distance distributions
Internal state of individual k w
i
k
w
ir
k
Vector and elements (r ⫽ 1,䡠䡠䡠, n
a
k
) updated each t
i
Landscape grid C
␣
(
␣
rows,

columns)
c
i
␣
⫽ 共c
i1
␣
,...,c
in
c
␣
兲⬘
Vector state of grid cell (
␣
,

) updated each t
i
Landscape modifier matrices (LMMs) L
ir
k
ᐉ
ir,
␣
k
Modifier matrix and elements on (
␣
,

) updated each t
i
Ideal movement distributions (fixed
) G
i
k
(s) G
i
k
(
␣
,

) Continuous 1D and binned 2D representations
Realized movement distributions (fixed
) M
i
k
(
␣
,

) 2D histograms for each k updated each t
i
19068
兩
www.pnas.org兾cgi兾doi兾10.1073兾pnas.0801732105 Getz and Saltz
the state of the landscape, as well as individual specific information
that inter alia relates to the location of places [remembered, sensed,
and inferred through landscape cues, including the earth’s magnetic
field (28)] associated with heading activities, the distribution of
resource s across the landscape, and the location of other individuals
on the landscape; and hence the elements of these matrices must be
updated each time step.
The simplest way to incorporate this landscape information
into the movement process is to use the activity-specific LMM
elements ᐉ
ir,
␣
k
(Table 1) as a way to weight the terms in the
idealized goal determined movement distribution G
i
k
(s) to pro-
duce a composite realized movement distribution, with spatial
matrix elements
M
i
k
共
␣
,

兲 ⫽
1
M
i
k
冘
r⫽1
n
a
k
w
ir
k
ᐉ
ir,
␣
k
Ꮽ
r
k
共
␣
,

兲
for all
␣
,

, i, and k ⫽ 1, where
M
i
k
⫽
冘
␣
,

冘
r⫽1
n
a
k
w
ir
k
ᐉ
ir,
␣
k
Ꮽ
r
k
共
␣
,

兲
nor malizes the discrete elements of the probability distribution
over the cells C
␣
for each individual k and all time i.
Decision Mechanism. The final component of the movement
process is how the organism selects the particular cell (
␣
,

)
that will deter mine its next position u
i⫹1
k
⫽ (x
i⫹1
k
, y
i⫹1
k
) at time
t ⫽ (i ⫹ 1)
. The simplest rule is for an organism to move to cell
(
␣
,

) with largest value M
i
k
(
␣
,

), effectively without making any
interim decision on its way to the new cell. For the case of an
organ ism driven purely by stochastic landscape processes (such
as wind), movement to the next cell can be regarded as random
with the probability of selecting a particular cell (
␣
,

) equal to
the values M
i
k
(
␣
,

). A third possibility is to select cells with a
probabilit y that is proportional to some power of M
i
k
(
␣
,

), a
solution that is intermediate between the first two if this power
is ⬎1. If this power is ⬍1, the solution is intermediate between
the second mechanism and a purely random solution that gives
no weight to the relative values M
i
k
(
␣
,

).
Food, Safety, and Fission–Fusion Dynamics
The following example illustrates the application of our frame-
work to simulating the movement of a herd of social ungulates
foraging on a heterogeneous landscape with conflicting needs to
both assuage hunger and remain safe by staying close to other
individuals.
In social ungulates, the existence of groups is presumably the
result of safety offered by the group (29). However, membership
in a group comes at a c ost, namely c ompetition for limited
resources (usually food). These conflicting needs presumably
drive the observed fission–fusion dynamics typical of many social
ungulates (30). Thus, the goal for each individual is feeding while
remain ing near conspecifics, and the main internal states for this
goal are levels of hunger and safety, with each individual seeking
to assuage hunger while remaining safe. The CAM in this
example is pure foraging, and the FMEs allow either mov ing
bet ween patches constrained by a maximum distance traversed
in each time set, or remaining in the current patch while feeding
(see SI Te xt). Navigation in our example is relatively simple:
individuals move directly using visual information to a cell
selected within a fixed radius interpreted as the observable
range. The distribution of food patches and the location of other
herd members represent the external states that a given herd-
member responds to in its choice of direction. The external states
are combined by using LMMs to obtain the realized movement
distribution from which an individual selects its next target.
Our model is spatially explicit with each grid cell, in this particular
case roughly the size of an individual. For each time step an animal
may perform one of three activities: (i) moving toward a selected
patch (one cell per time step), (ii) feeding in a patch (consuming
one unit per time step), or (iii) resting in a patch when it has a
full gut. There are two matrices, each representing one of the
external states: a food (vegetation) matrix and a safet y matrix
(Fig. 2 Upper). The vegetation matrix is static and is generated at
the start of each simulation by using a combination of random
procedures to create a patchy heterogeneous landscape (see Ma-
terials and Methods and SI Text). The safety matrix is a function of
the spatial location of all group members (Fig. 2), risk doe s not vary
with landscape features, and there are no visible predators. In each
time step, a safety score is calculated to each cell as the sum of the
inverse of the distance of all herd members to that cell (see Materials
and Methods). The LMM is a weighted sum of the vegetation and
safety matrices, where the weightings
␣
and (1 ⫺
␣
) depend on the
internal driver (state)
␣
僆 [0, 1] reflecting the current priorities to
the individual in trading safety against hunger. When an animal
stays within a patch to feed, its value of
␣
increases and when it
moves its value of
␣
decreases so that the relative importance of
food to safety oscillates up and down as the animal approaches
satiation or becomes increasingly hungry (see Materials and Meth-
ods and SI Text).
By playing around with simulation parameters, it is possible to
test how various movement-related questions—such as fission–
fusion dynamics, subgroup structure, and trajectory patterns—
are affected by patch size, gut size, and feeding strategies. For
example, we explored how patch density affects fission–fusion
and subgroup structure as follows: 20 animals with equal gut
sizes were initially spread randomly over a 20 ⫻ 20 cell section
of a 220 ⫻ 1,100 cell grid, with movement rules det ailed in SI
Text. We assumed that the vegetation in each cell is not
Fig. 2. Each of the five panels is an extract from a much larger mapping of
the values of elements in the landscape modifier matrices (LMMs) for the
vegetation and safety landscapes of the realized discretized movement dis-
tributions constructed from these matrices. The focal individual is represented
by the small red squares in each of the five panels, with the positions of its
conspecifics represented by other small squares in Upper Left and Lower.In
the distributions represented in Lower, the most attractive areas are the
lighter areas (but ignoring the small dark blue squares, which are just con-
specific position markers for reference). The relative weighting of safety over
resources ranges from safety being the only consideration (Lower Left), safety
and resources being equally important (Lower Center), and resources being
the only consideration (Lower Right). Imposed upon Lower is a circle repre-
senting the maximum possible movement displacement in one time step.
Getz and Saltz PNAS
兩
December 9, 2008
兩
vol. 105
兩
no. 49
兩
19069
ECOLOGY SPECIAL FEATURE
renewable, so that depleted patches act as repulsive regions. The
narrow landscape channels the herd into a directional movement
along the long axis, but the short axis is sufficiently wide to
enable fission events (Fig. 3).
The simulation was repeated with the same starting locations
with two different vegetation matrices, each with a mean patch size
of 38 cells. The first, representing ‘‘good’’ habitat, produced a
landscape where ⬇75% of the cells were positive (Fig. 3), whereas
the second, representing ‘‘poor’’ habitat, produced a landscape with
⬇65% positive cells. We ran each simulation for 10,000 time steps.
Group structure was analyzed every 500 steps, starting with the
1,000th step, using cluster analysis with a two-group restriction (see
SI Text). We then calculated the mean individual distance for the
entire herd and for each subgroup. We considered a ratio of 0.5,
between the sum of the two subgroup mean distance s to the mean
distance of the entire herd, as an indicator of fission. Simulating the
herd’s movement by using the ‘‘poor’’ habitat matrix produced three
fission events compared with none for the ‘‘good’’ habitat (Fig. 3).
All fissions were followed by fusion after 1,000–1,500 time steps.
We note that the herd was generally cohe sive, with fission and
fusion events emerging rather than explicitly constructed, despite
that fact that the movements of the individuals themselves were
completely deterministic. Our approach highlights the importance
of quantifying the internal drivers in understanding animal move-
ments. In the context of our specific simulation, empirical data on
giving-up densities (31) appear to be a good tool for quantifying
internal drive s and understanding movement patterns.
Conclusion
The framework presented here provides a mathematically detailed
exposition of the conceptual model developed by Nathan et al. (7)
that can be used for both the construction and deconstruction of the
pathway of organisms on structured landscape, the former through
simulations and the latter through state-space methods (32) for
fitting parameters. We have not dealt in any detail with the problem
of movement mode identification that is the key to the deconstruc-
tion component other than stressing that the frequency with which
data are collected limits our ability to identify CAMs and their
underlying FMEs (12). To undertake such an analysis is not a trivial
problem: it requires computationally complex methods that can
only be successfully applied to high-resolution data, but a start has
been made (33–35).
Before automated GPS data collection, VHF telemetry posi-
tion data were typically collected too infrequently and were also
not sufficiently accurate to be useful for reconstructing path-
ways; but these data were suitable for constructing home ranges
or types of utilization distributions at a seasonal scale, using both
parametric (36–38) and nonparametric (39, 40) kernel methods
as the preferred methodolog y. Since the mid-1990s (24) move-
ment data have been fitted to Le´vy models to evaluate the extent
to which this movement is superdiffusive with, as we have
discussed, mixed success (1) and also to assess the degree to
which path characteristics can be fitted by correlated random
walks (17) or mixtures of random walks (4), but these analyses
do not explicitly incorporate landscape structure or factors.
In terms of general methods for deconstructing movement
pathways, various time series and frequency domain techniques
can be brought to bear on the problem under the rubric of
ex ploratory data analysis (EDA) (41). Furthermore, stochastic
dif ferential equation methods (42) can be used to construct
vector fields from data on the contemporaneous movement of
many indiv iduals, and then thin plate splines can be used to fit
potential fields to these vector fields to identify regions on the
landscape that are either repealing or attracting the indiv iduals
at a particular time of day (2). Recently, techniques new to the
field of movement ec ology, such as wavelet analysis (33) and
artificial neural net works (34), are being applied to obtain
insights into the effects that the internal and environment al
st ates of a system have on movement paths. Beyond these, as the
resolution of movement and landscape data improves dramati-
cally over the next decade, we should expect to see the appli-
cation of state-space estimation methods (32) that can t ake
advant age of formulations such as ours, because our formulation
per mits the inclusion of det ailed landscape information.
Materials and Methods
Details are elaborated on in SI Text.
The Vegetation Matrix. This matrix is the only component in the model that has
stochastic elements. It consists of a series of patches set up using Monte Carlo
methods, with a parameter controlling patch density and a beta distribution
controlling patch size. The quality of resources in the center of each patch was
then assigned a number at random between 1 and 10 with values declining to
the edge of the patch. The resource value of each cell was reduced by a set
amount in each time step for which the patch was occupied by a feeding
individual. Fig. 2 Upper Left depicts the result of one such construction.
The Safety Matrix. For each of the cells containing an individual (i.e., focal
individual), a matrix of values for all of the remaining cells was constructed.
The values associated with each of these remaining cells is based on the sum
of the inverse distances of all of the remaining organisms to these cells. Thus,
cells of highest values are those closest in an integrative sense to the organisms
as a group that excludes the focal individual (see Fig. 2 Upper Right).
The Navigation Matrix. All points within a fixed distance of an individual (and
only these points) were regarded as selectable targets to move to next (circles
in Fig. 2). An individual then moves toward the cell that has the highest value
of a weighted sum of the vegetation and safety matrix values for that cell.
Feeding and Energetics. Each animal has an energy bank that determines its
level of hunger and, hence, its weighting of vegetation and safety matrix
values. During each time step, an animal can either feed and increase its
energy bank or move and decrease its energy bank. Once an individual has
consumed the vegetation locally, it moves to a patch within navigation range
that maximizes its current tradeoff for resources versus safety.
ACKNOWLEDGMENTS. We thank the members of the Institute of Advanced
Studies movement ecology group for ideas that we have absorbed into our
text during the course of our weekly meetings over many months, and Leo
Polansky for comments on the manuscript. This work was supported by the
Institute of Advanced Studies at the Hebrew University of Jerusalem and by a
James S. McDonnell 21st Century Science Initiative Award.
1. Edwards AM, et al. (2007) Revisiting Le´ vy flight search patterns of wandering alba-
trosses, bumblebees and deer. Nature 449:1044 –1048.
2. Preisler HK, Ager AA, Johnson BK, Kie JG (2004) Modeling animal movements using
stochastic differential equations. Econometrics 15:643– 657.
Fig. 3. Simulation of fission–fusion behavior as a function of vegetation
quality. Open squares, high quality; filled diamonds, low quality. Population
defined to be in a two-herd (one herd) state when the fission index is ⬍0.5
(⬎0.5). See main text and SI Text for details.
19070
兩
www.pnas.org兾cgi兾doi兾10.1073兾pnas.0801732105 Getz and Saltz
3. Zhang X, Johnson SN, Crawford JW, Gregory PJ, Young IM (2007) A general random
walk model for the leptokurtic distribution of organism movement: Theory and
application. Ecol Modell 200:79 – 88.
4. Morales JM, Haydon DT, Frair JL, Holsinger KE, Fryxell JM (2004) Extracting more out
of relocation data: Building movement models as mixtures of random walks. Ecology
85:2436–2445.
5. Morales JM, Fortin D, Frair JL, Merrill EH (2005) Adaptive models for large herbivore
movements in heterogeneous landscapes. Landsc Ecol 20:301–316.
6. Root RB, Kareiva PM (1984) The search for resources by cabbage butterflies (pieris-
rapae)—Ecological consequences and adaptive significance of Markovian movements
in a patchy environment. Ecology 65:147–165.
7. Nathan R, et al. (2008) A movement ecology paradigm for unifying organismal
movement research. Proc Natl Acad Sci USA 105:19052–19059.
8. Mandel JT, Bildstein KL, Bohrer G, Winkler DW (2008) The movement ecology of
migration in turkey vultures. Proc Natl Acad Sci USA 105:19102–19107.
9. Revilla E, Wiegand T (2008) Individual movement behavior, matrix heterogeneity, and the
dynamics of spatially structured populations. Proc Natl Acad Sci USA 105:19120 –19125.
10. Wright SJ, et al. (2008) Understanding strategies for seed dispersal by wind under
contrasting atmospheric conditions. Proc Natl Acad Sci USA 105:19084–19089.
11. Cooke SJ, et al. (2004) Biotelemetry: A mechanistic approach to ecology. Trends Ecol
Evol 19:334 –343.
12. Jerri AJ (1977) The Shannon sampling theorem—Its various extensions and applica-
tions: A tutorial review. Proc IEEE 65:1565–1596.
13. Fryxell JM, et al. (2008) Multiple movement modes by large herbivores at multiple
spatiotemporal scales. Proc Natl Acad Sci USA 105:19114 –19119.
14. Ramos-Ferna´ ndez G, Mateos JL, Miramontes O, Cocho G (2004) Le´vy walk patterns in
the foraging movements of spider monkeys (Ateles geoffroyi). Behav Ecol Sociobiol
55:223–230.
15. McCulloch CE, Cain ML (1989) Analyzing discrete movement data as a correlated
random walk. Ecology 70:383–388.
16. Bartumeus F, Da Luz MGE, Viswanathan GM, Catalan J (2005) Animal search strategies:
A quantitative random-walk analysis. Ecology 86:3078 –3087.
17. Dai X, Shannon G, Slotow R, Page B, Duffy KJ (2007) Short-duration daytime move-
ments of a cow herd of African elephants. J Mammal 88:151–157.
18. Shlesinger MF, Klafter J, West BJ (1986) Le´ vy walks with applications to turbulence and
chaos. Phys A 140:212–218.
19. Klafter J, Zumofen G, Shlesinger MF (1993) Le´ vy walks in dynamical systems. Phys A
200:222–230.
20. Metzler R, Klafter J (2004) The restaurant at the end of the random walk: Recent
developments in the description of anomalous transport by fractional dynamics. J Phys
A Math Gen 37:R161–R208.
21. Bartumeus F, Catalan J, Fulco UL, Lyra ML, Viswanathan GM (2002) Optimizing the
encounter rate in biological interactions: Le´ vy versus Brownian strategies. Phys Rev
Lett 88:097901.
22. Bartumeus F, Levin SA (2008) Fractal reorientation clocks: Linking animal behavior to
statistical patterns of search. Proc Natl Acad Sci USA 105:19072–19077.
23. Reed WJ (2001) The Pareto, Zipf and other power laws. Econ Lett 74:15–19.
24. Viswanathan GM, et al. (1996) Le´vy flight search patterns of wandering albatrosses.
Nature 381:413– 415.
25. Sims DW, et al. (2008) Scaling laws of marine predator search behavior. Nature
451:1098–1102.
26. Nams VO (1996) The VFractal: A new estimator for fractal dimension of animal
movement paths. Landscape Ecol 11:289 –297.
27. Harris SE (1993) Horse Gaits, Balance and Movement (Howell Book House, New York),
p 178.
28. Lohmann KJ, Putman NF, Lohmann MF (2008) Geomagnetic imprinting: A unifying
hypothesis of long-distance natal homing in salmon and sea turtles. Proc Natl Acad Sci
USA 105:19096 –19101.
29. Roberts G (1996) Why individual vigilance declines as group size increases. Anim Behav
51:1077–1086.
30. Rubenstein DI (1986) Ecology and sociality in horses and zebra. Ecological Aspects of
Social Evolution: Birds and Mammals, eds Rubenstein DI, Wrangham RW (Princeton
Univ Press, Princeton), pp 282–302.
31. Kotler BP, Brown JS, Bouskila A (2004) Apprehension and time allocation in gerbils: The
effects of predatory risk and energetic state. Ecology 85:917–922.
32. Patterson TA, Thomas L, Wilcox C, Ovaskainen O, Matthiopoulos J (2008) State-space
models of individual animal movement. Trends Ecol Evol 23:87–94.
33. Wittemyer G, Polansky L, Douglas-Hamilton I, Getz WM (2008) Disentangling the
effects of forage, social rank, and risk on movement autocorrelation of elephants using
Fourier and wavelet analyses. Proc Natl Acad Sci USA 105:19108 –19113.
34. Dalziel BD, Morales JM, Fryxell JM (2008) Fitting probability distributions to animal
movement trajectories: Dynamic models linking distance, resources, and memory. Am
Nat, in press.
35. Forester JD, et al. (2008) State-space models link elk movement patterns to landscape
characteristics in Yellowstone National Park. Ecol Monogr 77:285–299.
36. Worton BJ (1989) Kernel methods for estimating the utilization distribution in home-
range studies. Ecology 70:164 –168.
37. Seaman DE, Powell RA (1996) An evaluation of the accuracy of kernel density estima-
tors for home range analysis. Ecology 77:2075–2085.
38. Horne JS, Garton EO (2006) Likelihood cross-validation versus least squares cross-
validation for choosing the smoothing parameter in kernel home-range analysis. J
Wildl Manage 7:641– 648.
39. Getz WM, Wilmers CC (2004) A local nearest-neighbor convex-hull construction of
home ranges and utilization distributions. Ecography 27:489 –505.
40. Getz WM, et al. (2007) LoCoH: Nonparametric kernel methods for constructing home
ranges and utilization distributions. PLoS ONE 2:e207.
41. Brillinger DR, Preisler HK, Ager AA, Kie JG (2004) An exploratory data analysis (EDA) of
the paths of moving animals. J Stat Plann Infer 122:43– 63.
42. Brillinger DR, Preisler HK, Ager AA, Kie JG, Stewart BS (2002) Employing stochastic
differential equations to model wildlife motion. Bull Brazil Math Soc 33:93–116.
Getz and Saltz PNAS
兩
December 9, 2008
兩
vol. 105
兩
no. 49
兩
19071
ECOLOGY SPECIAL FEATURE
Supporting Information
Getz and Saltz 10.1073/pnas.0801732105
SI Text
This text c ontains details of the simulation methods used to
generate results pertaining to Figs. 2 and 3.
The Vegetation Matrix. The vegetation matrix was set up to
provide a high level of flexibility in its design so habitat structure
c ould be easily changed for future analyses. It is the only
c omponent in the model that has stochastic elements. However,
the matrix is formed at the onset of the model and, except for
depletion due to consumption by the animals, it remains un-
changed throughout a given model run (i.e., there is no plant
regrowth). The vegetation matrix consists of a series of patches
and its attributes include patch density, patch size, patch quality,
edge effect (i.e., reduced qualit y of cells at the edges of the
patch), and some level of randomization in cell quality within the
patch.
Patch location was probabilistic based on a predeter mined
densit y. Each cell c ould become a center of a patch using a
Monte Carlo draw. In our case, the probability of a cell becoming
a center of a patch was 0.02 (i.e., in a matrix of 1,000 cells, the
ex pected number of patches would be 50) in the ‘‘good quality’’
habit at as opposed to 0.017 in the ‘‘poor qualit y’’ habitat. Patches
were square with the side of the square ranging from 1 to 10 cells
drawn from a beta distribution with parameters
␣
⫽ 2.3 and

⫽
1.5 on the interval [0, 10]. In this manner, patches with 7- to
8-un it sides were the most c ommon (⬇35%), followed by sides
of 3–6 units (⬇25%). Small (1–2 units on the side) and large
(9–10 units on the side) patches were rare (acc ounting for a total
of ⬇15%).
Patch quality was determined randomly assigning a value
ranging f rom 1 to 10 (drawn from un iform distribution) to the
center cell. These values reflect units of energy that could be
c onsumed by the feeding animals. The quality of the remaining
cells around the center cell declined as a function of the distance
f rom it. Specifically, this was calculated as one minus the
proportion of the distance from the maximum possible distance
in a specific patch with an exponent of 0.3. Thus, the decline in
qualit y of the cells relative to the center of the patch was sharp
at the edges whereas the more central cells were relatively
similar. To generate w ithin-patch variability, the actual quality
assigned to each cell was then multiplied by a random number
generated from a beta distribution with
␣
⫽ 100 and

⫽ 1, so
that most cells received values very near their original one
(⬎0.95) but on rare occasion could drop by as much as 15%.
Patches could overlap, and cells falling within more than one
patch received the values generated for the last patch being
c onstructed in the code. The vegetation g rid was surrounded by
a 40-cell-w ide band with zero values. In this manner, individuals
did not bounce off of the edge but rather simply avoided it
because of lack of resources.
The Time Step. Time in this model is not explicitly defined and can
be v iewed as a sampling time step (
). The time step is short and
represents points at which the individual may make decisions. In
each time step, an animal could: move, feed, or remain standing.
The decision the animal makes between these three behaviors
depends on the animal’s energetic status, its location, and the
location of other group members (see below). However, in this
specific model, after a target is selected and the animal decides
to move there, the decision remains unchanged until the animal
arrives at that t arget—i.e., the animal can make only one choice
and then c ontinue moving in the same direction until arrival at
the selected target. However, a decision between several choices
in each movement step easily can be incorporated (e.g., reas-
sessing the target after other herd members have changed their
position).
Feeding and Energetics. Each animal has an energy bank that has
an upper limit (E
s
). This upper limit may vary between individ-
uals, although in the our example all animals were assigned the
same upper limit of 100 units. For each time step, an animal loses
a given percentage of its energy reserves (i.e., a basal cost) set
in our case to 0.006E
s
. When moving, there is an additional
energetic cost of E
m
⫽ 0.003E
s
per time step. When feeding,
upt ake is constant at a rate of one unit of energy per time step.
A nimals consider the value of the veget ation in patches of 3 ⫻
3 cells. An animal arriv ing at a cell in a given patch remains in
that cell and in each time unit consumes one veget ation unit
f rom the total number of units in the 3 ⫻ 3 cell matrix
surrounding it. If its energy bank is full, the animal remains in
place but does not feed. Each time step a given proportion
(0.006E
s
) of the energy bank is emptied. Thus, an animal in a
patch with a full energy bank (gut) will alternate between feeding
and resting in consecutive time steps.
Movement Rules. Once the n ine cells around a given individual are
c onsumed, the animal checks its surroundings w ithin a given
perception radius and selects a new target to move to. This is
done by the animal evaluating both the vegetation and the safety
matrices (the latter, as described in the main text, is a score for
each cell calculated as the sum of the inverse of the dist ance of
all herd members to that cell—see defin ition of S
p
below)
provided by each cell within the radius, based on the animals’
energetic status (i.e., the realized movement distribution—see
Fig. 2). In this model, each cell of the realized movement matrix
receives a score reflecting the veget ation content (i.e., energetic
value) of the 3 ⫻ 3 cell patch around it (E
p
)
,
the level of safety
(S
p
) offered by the cell in terms of its spatial location relative to
other herd members, and the cost of moving to the new cell based
on the distance to it (D
p
) and the cost of movement (E
m
). S
p
is
given in ter ms of the distances D
i
is the dist ance from the cell to
herd member i (i covers all members of the herd excluding the
one currently evaluating the patch) by the formula
S
P
⫽
冘
i⫽1
# ind
冉
1
D
i
冊
x
,
where the exponent x modifies the shape of the safety curve as
a function of distance from an individual from a sharp decline
for x ⱖ 1, to a more gradual decline for x ⬍ 1. In our specific
model, this value was fixed to 1 but can be used as a function of
the level and type of risk. Only one an imal can feed in a cell at
any one time, and the cell and the eight cells surrounding it are
not accessible to other members of the group.
We then weighted the veget ation qualit y and safety values for
each cell by the internal drivers to produce the actual scores for
the realized movement distribution (see below). In our case,
there are two drivers, hunger and safety. Hunger (
␣
) is that
proportion of the bank of energy reserves that is empty. The
drive for safety is determined as 1 ⫺
␣
.
Getz and Saltz www.pnas.org/cgi/content/short/0801732105 1of7
The score of each cell of the realized movement distribution
within the perception range of individual i is calculated as
Sc ore⫽ (E
P
⫺E
m
⫻ D
P
) ⫻
␣
⫹S
P
⫻ (1⫺
␣
).
Thus, a satiated animal has a zero hunger drive and is interested
only in safety. As the amount of reserves decline, the animal is
driven more by hunger and less by safety.
Once a target is selected, the animal moves toward it on a
straight line one cell each time step traveling along the imaginary
line f rom the center of the cell the animal currently occupies to
the center of the selected target. During movement, the animal
does not feed and does not consider new targets.
We note that to keep our simulation simple, we have not made
ex plicit use of step size distribution A
k
(s) for the two activities of
foraging within a patch (k ⫽ 1) and foraging between patches
(k ⫽ 2). Rather, the distributions A
k
(s) are implicitly embedded
in our movement rules that keep individuals within a 3 ⫻ 3 block
of cells (a patch) until the patch is depleted or must be exited
ac cording to the rules specified, and then an individual must
move to a new patch that may be as close as contiguous with the
current patch or as far as the edge of circle defined by the
perceptual radius centered on middle cell of the current patch.
In essence, we have not specified the landscape matrices and
movement distributions separately, but we have combined the
quantities
ᏸ
ri,
␣
k
Ꮽ
r
k
共
␣
,

兲
through a single definition, and some computational effort, not
required for our simulations, is needed to separate movement
distributions from the landscape matrices.
Cluster Analysis. The (x, y) locations at the given time step were run
through cluster analysis limited to two groups. Clusters were
identified by using the ‘‘clusdata procedure’’ in MATLAB with the
default setting of Euclidean distance among points. Once points
had been assigned to a cluster, the mean distance among points in
a cluster was calculated as well as the mean distance among all
20-group members of the entire herd. The sum of the two means
for the subgroups divided by the mean for the entire herd was then
calculated and plotted as the ratio (y-axis)inFig.3.
Getz and Saltz www.pnas.org/cgi/content/short/0801732105 2of7
Fig. S1. An idealized movement track consists of an ordered sequence of events selected from a set of fundamental movement elements (FMEs). A canonical
activity mode (CAM) is a mixture of FEMs. Sample points t
i
, i ⫽ 0,1,2,...,aretypically independent of event start and stop times and intervals
i⫺1
⫽ [t
i⫺1
, t
i
]
may mix parts of two or more CAMs.
Getz and Saltz www.pnas.org/cgi/content/short/0801732105 3of7
Fig. S2. These distribution illustrate how the same canonical activity mode distribution made up of two fundamental movement elements, one stationary
element s
1
⫽ 0 and one with a characteristic speed s
2
⫽ 4, changes when the sampling interval is several times longer (Left) or an order of a magnitude shorter
(Right) than the characteristic lengths of individual movement mode events. (Note the different vertical scales because the area below both curves is 1.)
Getz and Saltz www.pnas.org/cgi/content/short/0801732105 4of7
Fig. S4. An illustrative example of the different layers and flows of information needed to construct goal-modified movement distributions and analyze
movement data. The yellow arrows relate to movement path constructions, beginning with an identification of goal states based on the internal state w of the
individual and the landscape matrices incorporating the external state r and the navigation capacity ⌿ discussed in ref. 1. These goal states, in turn, determine
a mix of CAM distributions that are modified (weighted) using the values in the associated landscape matrix (representing a distribution of landscape values
across a covering of cells). The CAM distributions are themselves constructed from FMEs—constituting the motion capacity ⍀ introduced in the general
conceptual model (1)—in a way that depends on the simulation time step (Fig. S2). In terms of data analyses, state space (2), exploratory data (3), and other
suitable methods may be used to identify segments of movement tracks that are generated under different CAMs and mixes of CAMs.
1. Nathan R, et al. (2008) A movement ecology paradigm for unifying organismal movement research. Proc Natl Acad Sci USA 105:19052–19059.
2. Patterson TA, Thomas L, Wilcox C, Ovaskainen O, Matthiopoulos J (2008) State-space models of individual animal movement. Trends Ecol Evol 23:87–94.
3. Brillinger DR, Preisler HK, Ager AA, Kie JG (2004) An exploratory data analysis (EDA) of the paths of moving animals. J Stat Plann Infer 122:43– 63.
Getz and Saltz www.pnas.org/cgi/content/short/0801732105 6of7
Fig. S5. A unimodel one-dimensional step-size distribution G
i
k
(s)on[0,s
max
] is rotated through 360° to convert it to a spatially explicit two-dimensional
distribution G
i
k
(
␣
,

) defined over a featureless landscape grid C
␣
[i.e., all grid cells (a, b) have the same neutral state in terms of c
i
␣
being defined to reflect a
completely neutral landscape structure].
Getz and Saltz www.pnas.org/cgi/content/short/0801732105 7of7
