Territorial dynamics and stable home range formation for central place foragers.

Abstract

Uncovering the mechanisms behind territory formation is a fundamental problem in behavioural ecology. The broad nature of the underlying conspecific avoidance processes are well documented across a wide range of taxa. Scent marking in particular is common to a large range of terrestrial mammals and is known to be fundamental for communication. However, despite its importance, exact quantification of the time-scales over which scent cues and messages persist remains elusive. Recent work by the present authors has begun to shed light on this problem by modelling animals as random walkers with scent-mediated interaction processes. Territories emerge as dynamic objects that continually change shape and slowly move without settling to a fixed location. As a consequence, the utilisation distribution of such an animal results in a slowly increasing home range, as shown for urban foxes (Vulpes vulpes). For certain other species, however, home ranges reach a stable state. The present work shows that stable home ranges arise when, in addition to scent-mediated conspecific avoidance, each animal moves as a central place forager. That is, the animal's movement has a random aspect but is also biased towards a fixed location, such as a den or nest site. Dynamic territories emerge but the probability distribution of the territory border locations reaches a steady state, causing stable home ranges to emerge from the territorial dynamics. Approximate analytic expressions for the animal's probability density function are derived. A programme is given for using these expressions to quantify both the strength of the animal's movement bias towards the central place and the time-scale over which scent messages persist. Comparisons are made with previous theoretical work modelling central place foragers with conspecific avoidance. Some insights into the mechanisms behind allometric scaling laws of animal space use are also given.

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Territorial Dynamics and Stable Home Range Formation
for Central Place Foragers
Jonathan R. Potts1,2*, Stephen Harris2, Luca Giuggioli1,2,3
1Bristol Centre for Complexity Sciences, University of Bristol, Bristol, United Kingdom, 2School of Biological Sciences, University of Bristol, Bristol, United Kingdom,
3Department of Engineering Mathematics, University of Bristol, Bristol, United Kingdom
Abstract
Uncovering the mechanisms behindterritory formation is a fundamentalproblem in behavioural ecology. The broad nature of
the underlying conspecific avoidance processes are well documented across a wide range of taxa. Scent marking in particular
is common to a large range of terrestrial mammals and is known to be fundamental for communication. However, despite its
importance, exact quantification of the time-scales over which scent cues and messages persist remains elusive. Recent work
by the presentauthors has begun to shed light on this problem by modelling animals as random walkers with scent-mediated
interaction processes. Territories emerge as dynamic objects that continually change shape and slowly move without settling
toa fixedlocation.Asaconsequence,the utilisationdistribution ofsuchananimalresultsina slowlyincreasing home range,as
shown for urban foxes (Vulpes vulpes). For certain other species, however, home ranges reach a stable state. The present work
shows that stable home ranges arise when, in addition to scent-mediated conspecific avoidance, each animal moves as a
central place forager. That is,the animal’s movement has a random aspect butis also biased towards a fixed location, such as a
den or nest site. Dynamic territories emerge but the probability distribution of the territory border locations reaches a steady
state, causing stable home ranges to emerge from the territorial dynamics. Approximate analytic expressions for the animal’s
probabilitydensity function arederived.A programmeisgivenforusingtheseexpressionstoquantifyboth thestrengthof the
animal’s movement bias towards the central place and the time-scale over which scent messages persist. Comparisons are
made with previous theoretical work modelling central place foragers with conspecific avoidance. Some insights into the
mechanisms behind allometric scaling laws of animal space use are also given.
Citation: Potts JR, Harris S, Giuggioli L (2012) Territorial Dynamics and Stable Home Range Formation for Central Place Foragers. PLoS ONE 7(3): e34033.
doi:10.1371/journal.pone.0034033
Editor: Frederick R. Adler, University of Utah, United States of America
Received December 14, 2011; Accepted February 20, 2012; Published March 30, 2012
Copyright: ? 2012 Potts et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was partially supported by the EPSRC grant number EP/E501214/1 http://www.epsrc.ac.uk (LG and JRP) and the Dulverton Trust http://www.
dulverton.org/ (SH). No additional external funding received for this study. The funders had no role in study design, data collection and analysis, decision to
publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: jonathan.potts.08@bristol.ac.uk
Introduction
Understanding the mechanisms behind animal territoriality is of
great importance to many areas of ecology [1], from conservation
biology [2] to epidemiology [3] to predator-prey dynamics [4]. A
species is called territorial if each animal, or group of animals,
constructs and defends a region of space from conspecific
neighbours or possible intruders. Maintaining a territory relies
on the animal’s ability to exclude conspecifics from the area it
occupies. Since the animal needs to spend time moving inside its
territory to carry out vital activities such as foraging, continuous
monitoring of territory boundaries is not possible. Therefore many
animals have evolved mechanisms whereby their territory is
identified by visual, auditory or olfactory signals [5], thereby
obviating the need for constant border patrolling.
In this paper we focus on a model where the signals are
olfactory (scent marks). It is based on an agent-based model of so-
called territorial random walkers, first introduced in [6], where animals
are modelled as lattice random walkers that deposit scent as they
move. The scent is only active for a finite amount of time, the so-
called active scent time, and if a lattice site contains active scent, no
other animal may move there. As a result, the terrain naturally
subdivides into territories demarcated by the absence of foreign
scent. Territories each have a boundary and if two boundaries are
juxtaposed, a border is formed. These borders never settle to a
stable state. Instead, they continually ebb and flow, albeit at a
much slower rate than the animals move. Specifically, the border
movement is subdiffusive (i.e. the variance of the border position’s
probability distribution increases sublinearly with time) since the
territories are undergoing an exclusion process [7,8], whereas the
animals move diffusively.
Here, we study a modified version of the territorial random walk
model where animals are random walkers with an attraction
towards a central place, such as a den or nest site where the
animals return occasionally [9], or a core area where animals tend
to spend most of their time [10]. Similar to the original territorial
random walk model, territories emerge whose borders are
continually fluctuating. However with central place attraction,
the mean square displacement (MSD), i.e. the variance of the
border position’s probability distribution, tends towards a finite
value, as confirmed by stochastic simulations. This causes stable
home range patterns to emerge from the territorial dynamics.
To understand better the precise nature of the emergent home
range patterns, we compare stochastic simulations of the many-
bodied non-Markovian central place attraction model with an
analytic approximation, following [11,12]. This exploits the time-
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scale disparity between the rate of animal movement and the
slower, subdiffusive territorial borders, to construct an adiabatic
approximation for the joint probability distribution of the animal
and territory border positions. The model is solved exactly in both
1D and 2D and the resulting marginal distribution for an animal’s
position allows the macroscopic properties of home range size and
overlap to be related to the microscopic details of the animals’
movement and interaction processes. In particular, our analytic
expressions can be used to infer the longevity of olfactory messages
purely by examining data on animal space use. Furthermore, since
various properties of space use are predicted to scale allometrically
[13], our theory can also be used to give insights into the
mechanisms behind these scaling laws. Our results are compared
with previous approaches to modelling conspecific avoidance with
reaction-diffusion formalisms [9].
Results
Agent-based simulations of territorial central place
foragers
Monte Carlo simulations of the territorial random walk system
were performed in both 1D and 2D where each animal has a bias
of moving towards a central place (CP) (see Methods for details).
The MSD of the territory border eventually reached a saturation
value that depended on both the strength of attraction towards the
CP and the dimensionless quantity Z1~TAS=TD1 in 1D or
Z2~TAS=TD2 in 2D, where TAS is the active scent time,
TD1~½Dr2?{1(TD2~½Dr?{1) is the diffusive time in 1D (2D)
representing the time it takes for an animal to move around its
territory, D is the animal diffusion constant and r the animal
population density. The parameter a~vL=D was used to measure
the dimensionless strength of CP attraction, where v is the drift
velocity towards the CP and L the distance between CPs of two
adjacent territories.
For a fixed a, the amount of border movement arises from the
ratio of the active scent time to the diffusive time, which is Z1in
1D or Z2in 2D (figure 1). Increasing a has the effect of reducing
the animal’s tendency to move into interstitial regions and claim
extra territory. This causes the borders to move less on average as
each animal keeps to a small core area well within its territory
most of the time. Consequently, when plotting the MSD saturation
value against Z1or Z2, we see that the curves for higher values of
a lie below those for lower values (figure 1).
Dynamics of a central place forager within its territory: a
reduced analytic model
By taking into account the fact that the border movement is
much slower than that of the animal, we employed an adiabatic
approximation to calculate the probability distribution of an
animal inside its fluctuating territory borders (see Methods). The
simulated animals are identical, so it is sufficient just to model one
animal. Since the MSD of each territority border saturates at long
times, the animal probability distribution reaches a steady state.
Movement in 1D.
By fixing the CP at the origin for
simplicity, we calculated the steady state 1D dimensionless joint
probability density function?P P1D(? x x,?L L1,?L L2) of the left (right) border
being at dimensionless positions?L L1~L1=L (?L L2~L2=L) and the
animal being at position ? x x~x=L at long times, where L1, L2and
x are dimensional parameters and L is the distance between CPs
of adjacent territories (see figure 2 for an illustration and table 1 for
details of notation). This is (see Methods for derivations, here and
elsewhere)
?P P1D(? x x,?L L1,?L L2)&½H(?L L1z1){H(?L L1)?½H(?L L2){H(?L L2{1)?
½H(? x x{?L L1){H(? x x{?L L2)?|? g g1(?L L1)? g g2(?L L2)?h h1D(? x xj?L L1,?L L2),
where H(z) is the Heaviside step function (H(z)~0 if zv0,
H(z)~1 if z§0), ? g g1(?L L1) (resp. ? g g2(?L L2)) is the probability
ð1Þ
Figure 1. Simulation output for systems of territorial central place foragers. The dependence of the saturation mean square displace-
ment (saturation MSD) Dx2
parameters a~vL=D and Z1~TAS=TD2 (Z2~TAS=TD2) from stochastic simulation output. The notation S...T denotes an ensemble average
over stochastic simulations. The border movement is non-dimensionalised by dividing by L, the average distance between central places of adja-
cent territories. Panel (a) shows output from 1D simulations and panel (b) from 2D simulations. The best-fit lines for the 2D plots are
Log10(Dx2
for a~1:6, Log10(Dx2
doi:10.1371/journal.pone.0034033.g001
b~S(xb{SxbT)2T (resp. Dx2
b~S(xb{SxbT)2T) of the dimensionless territory border position xb(xb) on the dimensionless
b)~0:22{0:071Z2for a~0:08, Log10(Dx2
b)~{0:54{0:20Z2for a~2:4, and Log10(Dx2
b)~0:33{0:14Z2for a~0:4, Log10(Dx2
b)~{0:93{0:19Z2for a~4.
b)~0:28{0:18Z2for a~0:8, Log10(Dx2
b)~0:08{0:19Z2
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Figure 2. Diagram of the reduced analytic 1D model of territorial dynamics. The CPs are fixed at positions A, B and C (left to right). The
territory borders are intrinsically subdiffusive and have positions L1and L2. Each animal moves diffusively with a constant drift towards the CP and
constrained to move between the two territory borders to its immediate right and left. The position of the animal studied in the main text is denoted
by x. The animals at xAand xCare drawn purely for illustrative purposes. In the Results section, B is assumed to be at 0 and B{A~C{B~L.
doi:10.1371/journal.pone.0034033.g002
Table 1. Notation glossary.
SymbolModelDimensionExplanation
TAS
S1,S2
T
Active scent time: time for which a scent mark is avoided by
conspecifics.
D
S1,S2,A1,A2
S2T{1
Animal diffusion constant.
r
S1,S2
S{d
Animal population density in dimension d.
v
S1,S2,A1,A2
ST{1
Drift velocity of the animal towards its central place (CP).
L
S1,S2,A1,A2
S
Distance between central places of adjacent territories.
L1, L2
A1
S
Positions of the left and right borders.
K
S1,S2,A1,A2
S2T{1
Territory border generalised diffusion constant.
c
A1,A2
T{1
Rate at which territory sizes tend return to the mean size.
x
A1
S
Position of the animal in 1D.
(r,h)
A2
(S,none)
Position of the animal in 2D polar coordinates.
s
A2
S
Radius of the territory.
a
S1,S2
S
Lattice spacing.
F
S1,S2
T{1
Rate of jumping to the nearest neighbour.
p
S1,S2noneProbability of an animal moving towards its CP next jump.
TD1
S1
T
TD1~½Dr2?{1is the 1D diffusive time.
TD2~½Dr?{1is the 2D diffusive time.
Normalised TASfor 1D simulations, Z1~TAS=TD1.
TD2
S2
T
Z1
S1none
Z2
S2noneNormalised TASfor 2D simulations, Z2~TAS=TD2.
a
S1,S2,A1,A2noneNormalised drift velocity a~vL=D.
k
A1,A2none
Normalised territory border MSD, k~K=(cL2).
?L L1,?L L2
A1noneDimensionless positions of the left and right boundaries,
?L L1~L1=L and?L L2~L2=L:
? x x
A1none Dimensionless position of the animal in 1D, ? x x~x=L.
? r r
A2noneDimensionless radial component of the animal position in
2D, ? r r~r=L.
? s s
A2none Dimensionless radius of the territory, ? s s~s=L.
Glossary of the various symbols used throught the text. The second column details whether the symbol is used in the 1D simulation model (S1), the 2D simulation
model (S2), the 1D analytic model (A1) or the 2D analytic model (A2). The third column gives the dimensions of the parameter, or ‘none’ if it is dimensionless, where S
stands for space and T for time.
doi:10.1371/journal.pone.0034033.t001
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distribution of the left (right) border and?h h1D(? x xD?L L1,?L L2) is the
probability distribution of an animal being at position ? x x, given that
the borders are at?L L1and?L L2. The border probability distributions
are given by the following expressions
? g g1(?L L1)~1z2
X
?
n~1
({1)ne{4p2n2kcos½2pn(?L L1{1)?,{1ƒ?L L1ƒ0 ð2Þ
? g g2(?L L2)~1z2
X
?
n~1
({1)ne{4p2n2kcos½2pn?L L2?,0ƒ?L L2ƒ1
ð3Þ
where k~K=(cL2), K is the border generalised diffusion
constant, representing the amount the borders tend to move (see
methods), and c is the rate at which the territory size tends to
return to the expected average value. To visualise these
distributions, notice that when k is relatively large, the nw1
terms are negligible, so that ? g g1(?L L1)&1{2e{4p2kcos½2p(?L L1{1)?
and ? g g2(?L L2)&1{2e{4p2kcos½2p?L L2?.
The probability distribution of an animal being at position ? x x,
given that the borders are at ?L L1 and ?L L2, is the following
normalised version of a Laplacian distribution with average
displacement 1=a
?h h1D(? x xD?L L1,?L L2)~
aexp {aD? x xD
ðÞ
2{exp(a?L L1){exp({a?L L2):
ð4Þ
Movement in 2D.
circular, the CP is at the centre of the circle and the border
movement is modelled by fluctuations in the territory radius. The
steady state dimensionless joint probability density function
?P P2D(? r r,h,? s s) for the territory and the animal at long times is
In 2D we assumed that the territory is
?P P2D(? r r,h,? s s)&½H(? s s){H(? s s{1)?½H(? r r){H(? r r{? s s)?? g g(? s s)?h h2D(? r r,hD? s s), ð5Þ
where ? s s~s=L is the dimensionless radius, (? r r,h)~(r=L,h) are the
dimensionless polar coordinates of the animal, s is the radius and r
is the radial component of the animal’s coordinates. Here,
? g g(? s s)~1z2
X
?
n~1
({1)ne{4p2n2kcos½2pn(? s s)?,
ð6Þ
is the probability distribution of the territory radius and
?h h2D(? r r,hD? s s)~
a2E1(a? r r)
p½1{2a? s sE2(a? s s){2E3(a? s s)?,
ð7Þ
is the probability distribution of the animal being at position (? r r,h)
inside a territory of radius ? s s, where En(z) is the special function
defined by En(z)~Ð?
En(z)*e{z=z so the limit as z?? is 0 for every n.
1ds½exp({zs)?=sn. The limit as z?0 of
En(z) is infinite for n~1 and finite for nw1. For large z,
The marginal distribution of the animal
Equations (4) and (7) enabled us to calculate the marginal
probability distribution of the animal in both 1D and 2D
scenarios, where the territory can be anywhere else, by integrating
over all possible positions for the territory border. In 1D the
dimensionless marginal distribution of the walker at long times is
M1D(? x x)~
ðminf? x x,0g
{1
d?L L1
ð1
maxf? x x,0g
d?L L2? g g1(?L L1)? g g2(?L L2)?h h1D(? x xD?L L1,?L L2), ð8Þ
and in 2D, this is
M2D(? r r,h)~
ð1
? r r
d? s s? g g(? s s)?h h2D(? r r,hD? s s):
ð9Þ
The effects that the two parameters a and k have on the
marginal distribution (figure 3) can be characterised by observing
that a tends to govern the shape of the density function towards
the centre of the territory, whereas k governs the degree to which
the distribution tails off sharply (high k) or with a shallow gradient
(low k).
Expressions (8) and (9) are directly compared with those
measured from territorial central place forager simulations. It
turns out that the 1D case gives an excellent agreement for all
parameter values we tested (figures 3(a–d)). In 2D, a qualitatively
close fit is attained only when k and a are sufficiently low. For
higher k or a the borders are moving too fast for the adiabatic
approximation to be accurate (e.g. figure 3h). However for lower k
and a, the terrain contains very little interstitial area at any point
in time, so the territories are forced to tesselate the plane.
Therefore they each form more of a hexagonal than a circular
shape (e.g. figure 3e).
Obtaining active scent time from animal position data
To make use of the present theory, data must be gathered over a
sufficiently long period for the animal MSD to saturate. For
certain species, the saturation value fails to be reached during the
maximal biologically relevant time-window. Male red foxes (Vulpes
vulpes), for example, may spend parts of the autumn and winter
moving outside their territories to cuckold or disperse [14], so
territorial dynamics can only be measured reliably from the animal
positions during spring and summer when the males tend to stay
within their territories. During those two seasons, the tendency to
return to the CP is so weak that the animal MSD continues to
increase slowly, never settling [6]. In such cases, it is necessary to
use methods developed in [6] to analyse the animal territorial
system.
However, if the animal MSD does saturate then the marginal
distribution (9) can be fitted to the non-dimensionalised distribu-
tion of position locations from the data in order to obtain the
parameters a and k. From the theory, the saturation MSD
D? s s2(k)~S(? s s{S? s sT)2T of the territory radius can then be derived
from the equation
D? s s2(k)~
ð1
0
d? s s(? s s{1
2)2? g g(? s s)~1
12z
X
?
n~1
({1)nexp({4p2n2k)
p2n2
, ð10Þ
which allows the MSD of the territory radius ? s s to be computed
from k. The MSD of ? s s is the analogue, in the analytic model, of
the dimensionless territory border MSD Dx2
from the simulation model, so we equate Dx2
using the appropriate curve from the simulation output (figure 1b)
related to the value of a calculated from the data, a value for
Z2~TASDr is obtained, from which TAScan be derived.
In summary, the active scent time may be obtained from data
on animal locations by using the following programme.
b~S(xb{SxbT)2T
band D? s s2(k). By
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1. Fit equation (9) to the data in order to obtain values of a and k.
2. Use this value of k to find the theoretically expected saturation
value of the MSD D? s s2(k) via equation (10).
3. Note that D? s s2(k) from the analytic model is equal to Dx2
the simulation model.
4. Identify the best-fit line from figure 1b for the value of a found
in step 1.
5. Use this line, together with the value of Dx2
determine the Z2-value from figure 1b for the data being
studied.
6. Assuming the user also has values for D and r from the data,
TAScan then be derived from Z2~TASDr.
bfrom
bfrom step 3, to
Home range patterns and relations to allometry
Since the animal probability distribution reaches a steady state,
it is possible to calculate both the size of the resulting home ranges
and the degree to which they overlap. By using the 95% MCP
method [15], the dimensionless radius of the home range, after
dividing by the mean distance between CPs, is given by R95%,
implicitly defined by the following equation
2p
ðR95%
0
d? r r? r rM2D(? r r,h)~0:95:
ð11Þ
This allowed us to plot R95%as various functions of k, one for
each a (figure 4a). Each of these can be approximated by a
sigmoidalfunctionof
K~Log10(k).
Qzn=f1zexp({f½K{g?)g,
n&0:278{0:048a, f&3:18 and g&3:139{0:025a (figure 4a).
For certain values of k and a, the value of R95%is less than 0:5,
Specifically,
Q&0:495{0:010a,
R95%&
where
meaning that gaps arise between adjacent territories. These so-
called buffer zones have been observed between wolf (Canis lupus)
territories [4] as a safe place for wolf prey, such as white-tailed
deer (Odocoileus virginianus), to occupy.
The allometric predictions of [13] show that the fraction of
exclusively used area E is approximately proportional to M{j
where j&1=4 and M is the mass of a single animal. In our model
E~(1{R95%)2=R2
95%
so
(1{R95%)=R95%!M{j=2. By using the values of j fitted from
the large data sets in [13], the value of R95%can be estimated for
an animal of given mass. Using the trend lines from the simulation
plots in figure 1b and equation (10) allows k and a to be related to
Z2, thus estimating how Z2scales with M, as shown in figure 4b.
In [13] the tendency for larger animals to have a lower
proportion of exclusive area in their home ranges was explained
intuitively, by noticing that they are less efficient than smaller
animals in patrolling their territory to deter conspecifics. That is,
the time it takes for a larger animal to get around its territory is
greater than that of a smaller animal. In our model, this means the
diffusive time, TD2, increases with mass. Our results show that this
ability to deter conspecifics is also driven by an additional factor:
the active scent time. The ability to maintain exclusive area in fact
arises from the ratio of TASto TD2. Figure 4b shows that a smaller
animal’s ability to maintain a higher proportion of exclusive space
use arises from maintaining a higher ratio of TASto TD2, not just a
lower diffusive time.
allometric studiespredict
Comparison with previous approaches
Territoriality in animals with central place attraction has been
studied previously in [4] using a reaction-diffusion formalism,
which was developed further in [9]. Although both that model and
the one presented here use conspecific avoidance mediated by
scent marking as the mechanism of territory formation, the present
Figure 3. Comparison of the many-bodied simulation system and the reduced analytic model. Saturation marginal probability
distributions from simulations of systems of territorial central place foragers are overlaid on the same distributions (equations 8 and 9) from the
reduced analytic models. Panels (a–d) compare the two distributions for the 1D system. Dashed lines denote the simulation output and solid lines the
analytic approximation. The animal’s central place (CP) is at position 0, whereas CPs of conspecifics exist at positions 21 and 1. The distribution
decays to 0 at the conspecific CPs, where the animal cannot tread. The values used were (a) k~0:0017, a~4:0, (b) k~0:0014, a~0:80 (c) k~0:00033,
a~4:0 and (d) k~0:0040, a~0:80. Panels (e–g) compare the two distributions for the 2D system. The black contours show the deciles (i.e. 10%, 20%,
30% etc.) of the height of the probability distribution for the simulation system. The red contours show the same quantities for the analytic
approximation. The values used were (e) k~0:011, a~0:80, (f) k~0:0014, a~4:0, (g) k~0:022, a~0:80 and (h) k~0:016, a~4:0. As we increase k or
a, the effect of the adiabatic approximation becomes more apparent, since each red contour is further away from the respective black contour. This is
due to the fluctuations of the territory border being more pronounced for higher k or a.
doi:10.1371/journal.pone.0034033.g003
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