vol. 182, no. 3 the american naturalist september 2013
E-Article
Quantifying Behavioral Changes in Territorial Animals Caused
by Sudden Population Declines
Jonathan R. Potts,
1,2
Stephen Harris,
2,
* and Luca Giuggioli
1,2,3
1. Bristol Centre for Complexity Sciences, University of Bristol, Bristol, United Kingdom; 2. School of Biological Sciences, University of
Bristol, Bristol, United Kingdom; 3. Department of Engineering Mathematics, University of Bristol, Bristol, United Kingdom
Submitted January 29, 2013; Accepted March 19, 2013; Electronically published July 12, 2013
Online enhancements: appendix, videos. Dryad data: http://dx.doi.org/10.5061/dryad.5dn48
abstract: Although territorial animals are able to maintain exclu-
sive use of certain regions of space, movement data from neighboring
individuals often suggest overlapping home ranges. To explain and
unify these two aspects of animal space use, we use recently developed
mechanistic models of collective animal movement. We apply our
approach to a natural experiment on an urban red fox (Vulpes vulpes)
population that underwent a rapid decline in population density due
to a sarcoptic mange epizooty. By extracting details of movement
and interaction strategies from location data, we show how foxes
alter their behavior, taking advantage of sudden population-level
changes by acquiring areas vacated due to neighbor mortality, while
ensuring territory boundaries remain contiguous. The rate of ter-
ritory border movement increased eightfold as the population de-
clined and the foxes’ response time to neighboring scent reduced by
a third. By demonstrating how observed, fluctuating territorial pat-
terns emerge from movements and interactions of individual animals,
our results give the first data-validated, mechanistic explanation of
the elastic disc hypothesis, proposed nearly 80 years ago.
Keywords: animal movement, home range, red fox (Vulpes vulpes),
epizooty, territoriality, theoretical ecology.
Introduction
Much has been written about the factors affecting territory
size, such as allometry, resource dispersion and availability,
and population density (Kruuk and Parish 1982; Grant
and Kramer 1990; Jetz et al. 2004; Moorcroft and Barnett
2008; Van Moorter et al. 2009; Schradin et al. 2010). De-
spite this, we know remarkably little about how territories
form and change shape (Adams 2001) and how animals
interact to maintain these territories (Bo¨rger et al. 2008).
Providing this understanding is of great importance to
many areas of ecology, from conservation biology to wild-
life management and from predator-prey dynamics to ep-
* Corresponding author; e-mail: s.harris@bristol.ac.uk.
Am. Nat. 2013. Vol. 182, pp. E73–E82. 䉷 2013 by The University of Chicago.
0003-0147/2013/18203-54436$15.00. All rights reserved.
DOI: 10.1086/671260
idemiology (Beier 1993; Lewis Murray 1993; Kenkre et al.
2007; McCarthy and Destefano 2011). Though such ap-
plications often assume that territories are roughly sta-
tionary, population density can change rapidly in a variety
of situations, such as when a population is suffering an
epizooty of a terminal disease, thereby affecting the ter-
ritorial structure and behavior of the individual animals.
During 1994–1996, a sarcoptic mange epizooty decimated
the red fox (Vulpes vulpes) population in Bristol, causing
changes in both the movement of the foxes and territory
sizes (Baker et al. 2000). This provided support to the idea
that territories deform elastically, which was first observed
nearly 80 years ago by Huxley (1934) in coot (Fulica atra)
populations. While elastic territories have since been de-
tected in a variety of species, such as martins (Progne subis;
Stutchbury 1991), warblers (Acrocephalus arundinaceus;
Ezaki 1995), and lizards (Anolis aeneus; Stamps and Krish-
nan 1998), construction of a mechanistic theory that un-
derpins this elasticity has tended to remain elusive.
In the context of scent-marking animals, the process
with which animals respond to the information present in
scent deposited by a conspecific is key to the correct quan-
tification of territorial dynamics. From the movement per-
spective of the individual animal, this is a binary choice
between ignoring foreign scent, if the information con-
tained in it is either old or uninformative, or retreating.
A given location thus either has or does not have an “active
scent,” that is, a scent that is responded to by conspecifics
as a fresh territory cue. The presence/absence nature of
the scent implies that the system is intrinsically stochastic,
so deterministic representation via reaction-diffusion for-
malisms may be unable to account for the discrete nature
of the interaction events (e.g., see Durrett and Levin 1994;
McKane and Newman 2004). A recent modeling frame-
work (Giuggioli et al. 2011a, 2011b, 2012; Potts et al. 2011,
2012) that accounts for the discrete nature of the scent-
mediated interaction events is employed here to interpret
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E74 The American Naturalist
observations of the Bristol fox population before and dur-
ing the 1994–1996 mange epizootic.
Some territorial animals have a strong drift tendency
toward the locations of their den sites and Moorcroft et
al. (2006) examined how these territorial patterns change
when a coyote (Canis latrans) pack is removed from a
population. Making use of a reaction-diffusion formalism
where the position densities of two neighboring packs are
coupled with density profiles of the scent marks and with
a predetermined knowledge of the locations of the coyote
den sites, Moorcroft and Lewis (2006) compared the steady
state position distributions of the animals before and after
the removal. This approach, however, cannot be used in
our context because the tendency to drift toward the den
site is either not present in Bristol’s foxes or not sufficiently
strong to generate steady state position distributions, since
the mean square displacement of the animals increases
with time and never settles (Giuggioli et al. 2011a). Neither
was the habitat spatially confined, as in Briscoe et al.
(2002). One of the fundamental advancements of the sto-
chastic framework proposed by Giuggioli et al. (2011a)is
the ability to quantify the (discrete) longevity of scent cues
and the movements in territory borders, a key notion im-
plicit in the elastic disc hypothesis. See Potts et al. (2012)
for a detailed comparison of the two approaches.
By using the more recent approach, we quantify elasticity
in territorial patterns by the use of a single parameter K,
the diffusion constant of the territory border, measuring the
rate at which the variance of border positions increases over
time. Variations in mean territory size, the amount of di-
rectional persistence in the animal movement process, and
the animal velocity are also quantified, enabling behavioral
changes due to a sudden population decline to be assessed.
Additionally, we analyze agent-based simulations of systems
of moving and interacting animals (Giuggioli et al. 2011a),
to determine the longevity of territorial scent marks, the
active scent time , from the information provided in theT
AS
parameter K. Animals are modeled to move at random
(Okubo and Levin 2002) but constrained to roam within
areas that do not contain scent of conspecifics. As each
animal moves, it deposits scent, but once the scent has been
present for time , it is no longer considered by othersT
AS
to be fresh and so is ignored. In the field, while scent marks
cannot persist after the chemicals have decayed or dispersed,
it may be beneficial for animals to intrude into a neigh-
boring territory if the odor of the scent mark they detect
is old, suggesting that the territory may no longer be de-
fended. To test whether this happens in the Bristol fox pop-
ulation, we showed that the active scent time decreased after
the outbreak of mange, demonstrating that must ariseT
AS
from a behavioral strategy rather than being solely a con-
sequence of the persistence of the chemicals in the envi-
ronment. This decrease might also be influenced by a ces-
sation of vocal cues from recently dead neighbors
(Newton-Fisher et al. 1993).
Methods
Stochastic Simulations
Stochastic simulations were performed based on the two-
dimensional (2-D) territorial random walk model of Giug-
gioli et al. (2011a) but employing one of two different
movement processes: nearest-neighbor random walks
(NNRWs) and ballistic walks (BWs). The NNRW process
is described in Giuggioli et al. (2011a), whereas ballistically
moving animals will always continue in a straight line,
unless they encounter foreign scent, causing them to turn
at random. For each simulation run, 25 animals were
placed on a 2-D square grid size with lattice spac-M # M
ing a and population density , so that the grid
2
r p 25/M
is approximately 25 times the size of a territory. While the
dominant pair share the same territory (Saunders et al.
1993) and territory configuration could be calculated from
any individual within a group (Baker et al. 2000), the
simulations only modeled one animal per territory, the
dominant male, so when fitted to the data, r was the
population density of dominant males.
All simulated animals moved with the same movement
process, NNRW or BW, constrained by the fact that each
animal could not enter a square that contained fresh scent
of a different animal, and so turned away from the square
in a random direction. Each animal deposited scent at
every square it visited, which remained for a finite time
, the active scent time. After this time period hadT
AS
elapsed, the scent was no longer recognized by conspecifics
as a fresh scent message and so was no longer present in
the simulation. The squares that contained active scent of
an animal constituted its territory and the locations where
two contiguous territories met made a territory border.
The speed of each animal was , and was the timev t p a/v
it took for an animal to move distance a.
While certain animals are so-called borderland markers,
for example, badgers (Meles meles; Hutchings et al. 2001),
who actively patrol their borders to discourage invaders,
foxes are hinterland markers (Macdonald 1980), meaning
they deposit scent marks evenly throughout their territory.
Our previous studies have shown that greater patrolling
of the borders can reduce the amount they shift (Giuggioli
et al. 2011a), and we conjectured that foxes might be
adopting such a strategy. However, closer analysis of the
relative amount of time foxes spend near their territory
borders reveals that this is not the case (see “Methods for
Inferring Time Spent at the Territory Border,” available
online). Therefore, we have not included active border
patrolling in this model.
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Behavior Changes in Population Declines E75
A Reduced Analytic Approximation of
the Simulation Model
To fit data to the model efficiently, we used an analytic
approximation from Giuggioli et al. (2012) that models
the movement of a correlated random walker (CRW; an-
imal) inside a moving square territory of average width
L, which is equal to (see fig. 1a for a pictorial
1/2
1/(r)
explanation). To measure the amount of time an animal
has directional persistence, we used the correlation time
(Viswanathan et al. 2005). This has the following prac-T
tical interpretation. If position fixes from a CRW with
correlation time were taken at time intervals greater thanT
, analyzing the turning angles between the position fixesT
would suggest that the animal was moving in an uncor-
related fashion. Conversely, if the fixes were taken at time
intervals less than , turning angle analysis would suggestT
that the movement was correlated.
As territories exclude one another, the time dependence
of the territory border mean square displacement (MSD),
that is, the variance of the occupation probability, is
slightly sublinear. This occurs due to an exclusion process,
well studied in the statistical physics literature, for ex-
ample, Liggett (1985), but only recently introduced into
the literature on animal territoriality (Giuggioli et al.
2011a). An exclusion process is one where there are mul-
tiple moving objects, in our case territories, that cannot
either overlap or occupy the same place at the same time.
Since the territories hamper each others’ movement, the
MSD does not grow linearly in time (free diffusion). In
fact, Landim (1992) has indicated that 2-D exclusion pro-
cesses have an asymptotic (i.e. long-time) MSD propor-
tional to . Therefore, we assume that the output oft/ln(t)
the territory border movement in the simulation model
is asymptotically equal to . The reason for di-2Kt/ln(t/t)
viding by the constant t here is to ensure K has the units
of a diffusion constant (space
2
/time), which becomes con-
venient later when we compare it with the diffusion con-
stant of the animal (e.g., see fig. 2). The analytic ap-
2
v t
proximation model has the territory border diffusion
constant K as an input parameter.
Since larger-than-average territories in the simulation
model tend to shrink but smaller-than-average ones tend
to grow, the analytic model contains a rate parameter g
that measures the strength of this tendency (fig. 1a). We
have summarized the various parameters in the model in
table 1. The data were fitted to the probability density
function for the animal to be at coordinatesP(x, y, tFV)
relative to its home range center, defined as the(x, y)
centroid of all the measured positions of the animal over
the study period, at time t, given the input parameters
(see “Analytic Expression of the Prob-V p (v, K, T, g, L)
ability Distribution for an Animal in a Slowly Moving
Territory,” available online for the full analytic expression
of and its derivation).P(x, y, tFV)
Calculating Home Range Overlap from
Territory Border Movement
The amount of overlap between neighboring home ranges
was calculated by assuming that the two territories share
a common edge. In the model, this edge is continuously
moving, creating an overlap in the space used by the two
adjacent animals when measured over a time interval .T
*
For example, if an edge between two territories, each mod-
eled as a square of side L (see fig. 1), moves a distance of
in the perpendicular direction during the time , thenDLT
*
the overall area shared by these animals in two territories
during this time is . In practice, represents theL # DLT
*
time window during which location data are collected. As
the territory borders move during this time window, we
observe overlaps in the home ranges measured from the
locations of animals in neighboring territories (fig. 1b).
In our model, the MSD in the perpendicular direction
of the shared edge between two adjacent territories is
. Therefore, the width of the overlapping stripKt/ln(t/t)
between the two neighboring home ranges is equal to the
mean absolute displacement, which is .
1/2
[KT /ln(T /t)]
**
The width of the home range is then
1/2
1/(r) ⫹
, owing to the fact that the width of each
1/2
[KT /ln(T /t)]
**
territory is . Therefore, the fraction of a home range
1/2
1/(r)
that overlaps with this particular neighbor is
1
HRO p 1 ⫺ .(1)
冑
1 ⫹ KT r/ln(T /t)
**
Notice that as increases, the fraction of overlap increasesT
*
toward the theoretical maximum value of 1, where the two
home ranges coincide. However, since territory border
movement is typically very slow, the time it would take
to get close to this situation is likely to be far longer than
the lifetime of the animal. Therefore, complete overlap is
highly unlikely to be observed in reality.
Data Collection and Analysis
Movement data were taken from a long-term study of the
red fox population in the Bristol urban area. Our data are
available in the Dryad Digital Repository, http://dx.doi
.org/10.5061/dryad.5dn48 (Potts et al. 2013). Radio fixes
with a spatial resolution of were taken every25 m # 25 m
5 min between 20:00 and 04:00 GMT, which encompasses
most fox activity (Saunders et al. 1993), so throughout
this article “1 day” is equal to8hoffoxlocation data.
Radio telemetry data from 22 different territorial adult
foxes (i.e.,
11 year old) monitored between 1990 and 1995
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E76 The American Naturalist
Figure 1: Representation of the approximate analytic model of animal movement within a dynamic territory. Territory borders move
randomly with a mean square displacement of around an average width of , where r is the population density. The
1/2
2Kt/ln(t/t) L p 1/(r)
process that keeps the territories at this average width is represented by two springs, one vertical and one horizontal, each having a spring
constant of g. The higher g, the greater the tendency for larger- or smaller-than-average territories to move back toward an average size.
The animal, represented by a filled circle in a, moves as a correlated random walker with speed and correlation time (see “Methods”).v T
Panel a represents this setup, while b demonstrates how overlaps between adjacent home ranges arise from this model as animal positions
are measured over time. In b, the mean position of the territory border is represented by the solid black line, while the average extent of
the movement of this border to the left and right is represented by the dashed gray lines.
were analyzed. Data from both males and females were
used because the space use distributions of a male and a
female from the same group are very similar (Baker et al.
2000). Each fox was tracked during one, two, or three
seasons (spring, March–May; summer, June–August; au-
tumn, September–November; winter, December–Febru-
ary; see table A1, available online). Starting in the summer
of 1994, a mange epizootic spread through Bristol’s foxes
causing the population density to decline rapidly, even-
tually killing almost all the foxes in the city (Baker et al.
2000). When analyzing the data, we split them into two
sets: premange, before summer 1994 when the population
density was relatively stable, and postmange, after summer
1994, when the population was rapidly declining. The pre-
mange data set contained data points from 18N p 8,693
different foxes, postmange from 4 foxes (seeN p 2,313
table A1). The last fox in the study area died in spring
1996 (Baker et al. 2000).
The log maximum likelihood method was employed to
fit data to the theoretical probability distribution
. In particular, the Nelder-Mead simplex al-P(x, y, tFV)
gorithm (Lagarias et al. 1998) was used to find the max-
imum of for each set of pa-L(V) p
冘 ln [P(x , y , t FV)]
nnn
n
rameter values V, where the sum is taken over 99% of
position-time locations that attain the highest(x , y , t )
nnn
values. We excluded 1% of outliers to ensureP(x , y , t FV)
nnn
the results were not biased by anomalous behavior (Harris
et al. 1990; Kenward et al. 2001). The Nelder-Mead al-
gorithm requires a starting value for V, called V p
0
, that is expected to be close to the max-(v , K , T , g , L )
00000
imum. We set to be the total distance moved by thev
0
foxes divided by the total time moved. Term was ob-T
0
tained by the formula min, whereT p ⫺5/ ln [A cos (v)S]
0
is the mean of the cosines of the turning anglesA cos (v)S
v from the data (Viswanathan et al. 2005). The factor of
5 comes about since location measurements were taken
every 5 min. Since the long-time MSD of the animal is
taken to be (Giuggioli et al. 2012, eq. 3.3),2Kt/ln(t/T)
was obtained by fitting a curve toKA⫹ 2Kt/ln(t/T )
000
the fox MSD against time t for day, using the leastt 1 1
squares method, where A is a fitting constant.
Term was found by taking the square root of theL
0
mean 100% minimum convex polygon (MCP) home range
area (Harris et al. 1990). While the MCP method is in
general not the most accurate for finding home range sizes
(Fieberg and Bo¨rger 2012), we use it only to find a starting
point for running the Nelder-Mead algorithm. Though a
more accurate method, for example, kernel density esti-
mation (KDE; Laver and Kelly 2008), may cause the al-
gorithm to converge slightly more quickly, the estimation
method we use to obtain the algorithm’s initial condition
makes no difference to the outcome of the algorithm. Since
MCP has the advantage of being very simple to measure
from the data and is known to be an accurate measure of
home range area and territory boundaries in this particular
fox population (Saunders et al. 1993), this is the method
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Behavior Changes in Population Declines E77
0 5 10 15
10
−8
10
−6
10
−4
10
−2
10
0
Normalized active scent time
Normalized territory border movement
Nearest Neighbour RW
Ballistic walk
0 5
0
20
40
60
Population density (km
−2
)
Percentage overlap
Figure 2: Simulation output showing the dependence of territory boundary movement on the active scent time for nearest-neighbor random
walks (NNRWs) and ballistic walks (BWs). The vertical axis is , and the horizontal is (see table 1 for definitions of terms).
22
K/v t Z p Tvtr
AS
The crosses and circles show values from the simulation output. The solid line is a best fit for the NNRW case
2
log (K/v t) p 0.085 ⫺
10
, and the dashed line is the best fit for the BW case . Inset, the percentage of each home range that
2
0.247Z log (K/v t) p 0.467 ⫺ 0.709Z
10
overlaps with neighboring ranges. The solid (NNRW) and dashed (BW) lines use fixed values of , , , and taken from theTvT t p T
AS *
premange data, whereas the dotted (BW) and dot-dashed (NNRW) lines use postmange data.
we choose here. Finally, to find , the maximum ofg
0
was calculated for as g variesL(V) V p (v , K , T , g, L )
000 0
across parameter space from to . Error
⫺34
g p 10 g p 10
bars for the best fit were obtained using the bootstrap
algorithm for variance calculation (see, e.g., Wasserman
2004) by resampling each data set 100 times.
Results
The Effect of Movement Processes on Territorial Dynamics
For both of the movement processes simulated, NNRW
and BW, the borders of the emergent territories each had
an MSD that increased asymptotically as . The2Kt/ln(t/t)
magnitude of the diffusion constant K depended on the
directionality of the animal’s movement, the population
density r and the active scent time (fig. 2). For eachT
AS
movement process, K depended upon the ratio Z p
between the active scent time and ,
2
T /TTp 1/(v tr)
AS C C
the territory coverage time, representing how long it takes
for an animal to move around its territory (Potts et al.
2012). For the NNRW case, is closely related to theT
C
mean time for an animal in a confined region to return
to the place from which it started (Condamin et al. 2007).
For either movement process, K decreased exponentially
as Z increased. The rate of decrease was much greater in
the BW case since each animal tended to move across the
territory in a much shorter timescale than the NNRW case.
This caused each point on the territory border to be res-
cented at shorter time intervals so that the borders moved
less. As an indicator of how these timescales differ, if the
territories were immobile and square, the territory width
would be and the area would be . Therefore,
1/2
1/(r)1/r
the respective timescales would be proportional to
for the BW case, the time it takes for a ballistic
1/2
1/(r)
walker to move a distance of , compared to for
1/2
1/(r)1/r
NNRW, the first passage time for such a walker to traverse
a distance of (see, e.g., Redner 2007, section 2.4).
1/2
1/(r)
Territorial Dynamics of a Red Fox Population before and
after an Outbreak of Sarcoptic Mange
During the 1994–1996 mange epizootic, the red fox pop-
ulation density declined rapidly, causing the foxes to both
extend their home ranges and move faster (Baker et al.
2000). To investigate further the effect of population de-
cline on the animals’ movements and interactions, we fit-
ted the data to the model of animal movement in a ter-
ritory of fluctuating size and position (Giuggioli et al.
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