Animal Behaviour 85 (2013) 269e280
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Animal Behaviour
journal homepage: www.elsevier.com/locate/anbehav
Commentary
Using conditional circular kernel density functions to test hypotheses on animal
circadian activity
L. G. R. Oliveira-Santos a, *, C. A. Zucco a, C. Agostinelli b
a
b
Ecology Department, Biology Institute, Federal University of Rio de Janeiro, Brazil
Department of Environmental Sciences, Informatics and Statistics, Ca’ Foscari University of Venice, Italy
a r t i c l e i n f o
Article history:
Received 24 May 2012
Initial acceptance 27 June 2012
Final acceptance 15 August 2012
Available online 7 November 2012
MS. number: AS-12-00394
Keywords:
activity overlap
activity pattern
activity range
camera trap
circular statistic
mammal
Activity is an important dimension of animal behaviour and
ecology. Few analytical advances, however, have been made in this
field, and activity data are typically analysed using linear or categorical approaches that allow only limited inferences. Here we
present a circular, continuous, nonparametric model of a conditional
circular kernel function to estimate the activity range of a species
and the activity overlap between species. We test the effect of the
model parameters (density isopleth and kernel smoothing parameter) on the estimates of the activity range and activity overlap of
mammals from the Pantanal wetlands of Brazil based on a cameratrap data set. We also present a complete activity case study of
two native peccary species and feral hogs living sympatrically in the
Pantanal to exemplify an ecology application of our model to
enhance current knowledge of animal activity. R codes for activity
ranges and activity overlap are provided in Supplementary Data and
are already incorporated in the package ‘circular’.
Community structure is most likely driven by a number of spatialand temporal-scale processes (Ricklefs 2008), but at the local scale,
population interactions following niche theory persist as a central
* Correspondence: L. G. R. Oliveira-Santos, Ecology Department, Biology Institute,
Federal University of Rio de Janeiro, Fundão Island, Rio de Janeiro, Rio de Janeiro
21941-902, Brazil.
E-mail address: [email protected] (L. G. R. Oliveira-Santos).
theme in the ecological sciences. Historically, the types of space and
food use based on exploitative competition have been considered to
be the most important niche dimensions for determining species
segregation, while time has been considered relatively less important
(Schoener 1974, 1983). It is important to stress, though, that some
degree of overlap in space and time dimensions is necessary for
interference through competition or predation to shape the structure
of communities. For territorial competitors, particularly those that
present defence and attack displays, temporal shifts in activity or the
use of a key resource may arise in response to interference, thus
enabling their coexistence (e.g. Carothers & Jaksic 1984; Valeix et al.
2007, 2009). Furthermore, complex spatiotemporal patterns of
behaviour can also be triggered by predator interference, as observed
for herbivores coexisting with lions (Valeix et al. 2009).
Several studies have investigated this temporal partitioning
(Kronfeld-Schor & Dayan 2003), and in many cases, time seems to
play a significant role, providing convincing evidence of niche
displacement at the temporal scale (Valeix et al. 2007; Di Bitetti
et al. 2009). Analyses of temporal segregation were first attempted using visual analysis of histograms, and researchers later began
to use linear frequency statistical procedures with the timescale
categorized in contingency tables (Jácomo et al. 2004; Lucherini
et al. 2009). However, as day and night are essentially a temporal
cycle, it is important to acknowledge that despite the linear 24 h
0003-3472/$38.00 2012 The Association for the Study of Animal Behaviour. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.anbehav.2012.09.033
270
L. G. R. Oliveira-Santos et al. / Animal Behaviour 85 (2013) 269e280
convention, time in this case has a circular nature. The origin
(0000 hours or midnight) is set arbitrarily, but it could be noon or
any other moment of the day. Additionally, the end of the scale
meets the origin, which means that events occurring at 2300 hours
are much closer to events at 0100 hours (2 h of distance), than
those at 1900 hours. A linear scale, in this case, would inevitably
produce misleading conclusions (Ridout & Linkie 2009).
More recently, ecologists have analysed circadian activity
through a circular inferential statistical approach (Oliveira-Santos
et al. 2008), which is still based on approaches analogous to the
comparison of frequencies between classes. Ridout & Linkie (2009)
proposed an innovative approach using a nonparametric circular
kernel density function to fit data from camera traps and then to
estimate a symmetrical coefficient of overlap between species
using a total variation distance function. Despite the meaningful
contribution of these authors, relevant biological information, such
as the hours of concentrated animal activity and the implications of
different kernel isopleths for activity and the overlap coefficient,
remain an open issue in statistical development.
Here, we present the use of a conditional circular kernel density
function to describe, analyse and test ecological hypotheses about
random samples of activity, such as those recorded through camera
traps, 24 h visual counts, or even time records of direct capture (e.g.
using live traps). Kernel estimates are nonparametric methods to
estimate densities directly from a data set, which can be a unidimensional time series of pictures taken by a camera trap, or
a bidimensional series, such as geographical coordinates of locations of an individual animal. Kernel density functions provide
a continuous measure of density of the data points throughout their
scale (Worton 1989).
Our approach follows the same mathematical basis used
throughout the last few decades to describe space use, distribution
and home range contours based on radiotelemetry and global
positioning system (GPS) data. Therefore, this activity circular
kernel has the same features as the home range kernel estimator,
such as a smoothing parameter and a conditional density isopleth.
The smoothing parameter defines how smooth the shape of the
home range contours will be or how acute the peaks and valleys of
an activity function around the timescale will be. A conditional
density isopleth is the threshold of probability that specifies the
section of the function that accounts for a given proportion of the
whole probability function. In practical terms, a 95% isopleth in
a home range analysis may be understood as the smallest area that
accounts for 95% of its utilization distribution, or presumably
where the animal spent 95% of its time. In an activity analysis, a 95%
isopleth is the smallest time interval of the day that accounts for
95% of the whole activity function, or the time interval in which 95%
of the animal activity occurs.
Here, we propose the terms ‘activity range’ and ‘activity overlap’, both of which are conditioned on the choice of isopleths,
which we believe to be useful in developing knowledge regarding
animal activity. We tested the precision and robustness of the
activity range and the activity overlap under different combinations
of kernel parameters (isopleths and smoothing) and sample sizes
using real mammal activity data sets collected from camera traps in
the Brazilian Pantanal wetland. Finally, we present a complete case
study addressing the invasive feral hog and two peccary species,
investigating their patterns of activity as well as shifts in their
activity ranges and overlaps between seasons.
A Model to Estimate Activity Range and Activity Overlap
a data set (Silverman 1986; Wand & Jones 2005). Kernel density
estimators for circular/directional data sets are discussed in several
studies, such as those of Hall et al. (1987), Bai et al. (1988), Prayag &
Gore (1990), Hendriks (1990), Kim (1998) and Klemelä (2000).
Because we are interested in estimation on a circle, that is, circular
data, we can restrict our attention to the proposals presented in the
first two contributions.
Two approaches can be followed. However, they are both based
on the inner product between unit vectors on the circle, which is
the cosine of the angle between the two vectors. The first type is
defined as follows:
b
f ðq; kÞ ¼ n1 cðkÞ
n
X
Kðk cosðqi qÞÞ
i¼1
while the second type is
b
g ðq; uÞ ¼ n1 dðuÞ
n
X
Lðuð1 cosðqi qÞÞÞ
i¼1
where k and u are smoothing parameters, K and L are suitable
kernel functions and cðkÞ and dðuÞ are normalizing constants. The u
parameter has the usual interpretation (i.e. large values lead to
oversmoothing or loss of details in the function shape, while low
values lead to undersmoothing or maintenance of a sinuous
detailed shape). Note that k has the opposite meaning.
Optimal kernel function and smoothing parameter could be
chosen by minimizing a loss function, which is a function representing the ‘cost’ associated with the use of the estimate b
f ðq; kÞ
instead of the true value f ðqÞ. Two popular loss functions are the
squared error loss, L2 ðkÞ, and KullbackeLeibler loss, LKL ðkÞ, which
are defined as follows:
L2 ðkÞ ¼
2
Z2p
b
f ðq; kÞ f ðqÞ dq;
0
LKL ðkÞ ¼
Z2p
0
0
1
B f ðqÞ C
f ðqÞlog@
Adq;
b
f ðq; kÞ
Hall et al. (1987) reported that the large-sample formula of these
loss functions does not depend on the form of kernels K or L. In this
g ðq; uÞ are equivalent to a special
sense, all estimators b
f ðq; kÞ and b
case, that is, KðtÞ ¼ expðtÞ (this kernel corresponds to the Von
Mises density with the normalizing constant cðkÞ ¼ 1=ð2pI0 ðkÞÞ,
where I0 ðkÞ is the zeroth-order Bessel function of the first kind),
which is equivalent to LðtÞ ¼ expðtÞ.
The choice of smoothing parameter is crucial in both
noncircular (linear) and circular cases, and an optimal smoothing
parameter could be obtained by minimizing a loss function.
Hall et al. (1987) suggested using cross-validation to obtain unbiased estimates of the squared error loss and the KullbackeLeibler
loss. Let
b
f i ðq; kÞ ¼ ðn 1Þ1 cðkÞ
n
X
K k cos qj q ;
j ¼ 1;jsi
Circular kernel density estimation
Nonparametric kernel density estimation is a widely used
technique for nonparametric estimation of densities based on
be the density estimate constructed without the sample value qi,
and define
L. G. R. Oliveira-Santos et al. / Animal Behaviour 85 (2013) 269e280
CV2 ðkÞ ¼ 2n1
CVKL ðkÞ ¼
n1
n
P
b
f i ðqi ; kÞ i¼1
n
P
i¼1
Z2p
AR1 ¼ ½0; 2p (or any other interval that covers the whole circumference), while for q ¼ 0þ, the conditional activity range AR0 will
contain the mode(s) corresponding to a global maximum of the
density. An obvious estimate of ARq is based on the plugin principles
2
b
f ðq; kÞ dq;
0
b
log f i ðqi ; kÞ
R 2p
Then, c
L 2 ðkÞ ¼ CV2 ðkÞ þ 0 f ðqÞdq and c
L KL ðkÞ ¼ CVKL ðkÞþ
R 2p
0 f ðqÞlogðf ðqÞÞdq are unbiased estimates of L2 ðkÞ and LKL ðkÞ,
respectively; in the latter case, n is replaced by n 1 in LKL ðkÞ.
L KL ðkÞ is unknown since
Notice, that the second term in c
L 2 ðkÞ and c
it depends on f ðqÞ. However, this term does not depend on the
smoothing parameter k and hence it is irrelevant for finding the
optimal smoothing parameter.
More recently, Taylor (2008) proposed a rule of thumb for the
choice of the smoothing parameter, k, which is the concentration
parameter in the von Mises kernel. Taylor (2008) showed that if the
data arose from a von Mises distribution with a concentration
parameter, n, then the asymptotic integrated mean squared error
(AMISE) was minimized by letting the smoothing parameter be
equal to
k ¼
3nn2 I2 ð2nÞ
!2=5
n ¼ max nj ; j ¼ 1; .; J ;
where the estimates nj are solutions of the equations
Aj ðnÞ ¼
bq
q : bf ðq; kÞ h
n
1X
cos jqi mbj ;
n
j ¼ 1; .; J;
i
in which Aj ðnÞ ¼ Ij ðnÞ=I0 ðnÞ and mbj are the directions of jth sample
trigonometric moment (see Jammalamadaka & SenGupta 2001,
section 1.3). If the data are from a von Mises distribution with
concentration parameter n, then n1 is the maximum likelihood
estimate of n and values of nj for j > 1 should be similar. However, if
the underlying distribution is not a von Mises distribution, the
estimators nj are expected to differ from each other (Taylor 2008). A
value of J ¼ 3 proved to be effective in most situations.
Conditional activity range estimation
For a given absolutely continuous random variable on a circle with
density f ðqÞ, we might consider the set Mh : ¼ fq : f ðqÞ hg for a fixed
threshold h. The set Mh is referred to as a ‘modal region’ in the linear
(noncircular) space by Sager (1979), and its boundary is an isopleth.
Modal regions can be used to characterize the modes of a given
density function, in the present case, moments throughout the
timescale where activity is maximum. In fact, for decreasing h, if the
family of sets fMh : 0 h Ng is connected and nested, the density is
strongly unimodal. Furthermore, set Mh is characterized as the one
with the highest probability among sets of the same length. To define
an activity range, we can consider a redefining parameters of the
modal regions as follows. Let 0 q 1 be a quantile order and then
ARq : ¼ M hq ¼ q : f ðqÞ hq :
R
where hq is such that Mðhq Þ f ðtÞdt ¼ q. We refer to ARq as the
‘conditional activity range of order’ q. For q ¼ 1, we have
R
f ðt; kÞ dt ¼ q. In a similar fashion we
where b
h q is such that c b
AR
q;k
define the estimate of ARq based on b
g ðq; uÞ. The statistical properties of the plugin estimator have been investigated in several
studies (e.g. Polonik 1995; Tsybakov 1997; Baíllo et al. 2001; Cuevas
et al. 2006).
Conditional overlap coefficient between multiple species
Let Q and J denote two random circular variables with corresponding absolutely continuous distribution functions F and G and
densities f and g. The coefficient of overlap (OVL) introduced by
Weitzman (1970) and investigated in several studies, including
Schmid & Schmidt (2006) and references therein, can be easily
extended to the circular context as follows:
OVLðQ; JÞ ¼
where Ir ð$Þ is the modified Bessel function of order r. Taylor (2008)
then considered various ways of calculating a suitable estimate of n
to be used in formula (1) when the underlying distribution may not
be a von Mises distribution. A simple approach that generally
worked quite well in Taylor’s simulations was to let
c
AR
q;k : ¼
(1)
4p1=2 I0 ðnÞ2
271
Z2p
minðf ðtÞ; gðtÞÞdt:
0
The overlap coefficient has a range between 0 and 1; OVLðQ; JÞ ¼
1 if and only if F ¼ G, while OVLðQ; JÞ ¼ 0 if and only if the
functions f and g have no intersection at all. Therefore, the overlap
coefficient can be interpreted as a measure of agreement between
the two distributions. Recently, Ridout & Linkie (2009) and Linkie &
Ridout (2011) proposed the use of OVL to compare kernel density
functions of animal activity. Here, though, we prefer to consider the
complement to one of the overlap coefficients corresponding to the
well-known total variation distance
TVðQ; JÞ ¼ sup
Pr ðAÞ Pr ðAÞ
;
J
Q
A˛A
where A denotes the Borel sets on the circle. TV represents the
degree of dissimilarity between two activity density functions. Its
outcome lies in the interval 0 TV 1, where 0 denotes that the
functions are completely identical (exactly the same activity
pattern) and 1 denotes that the functions have no overlap (e.g.
a strictly diurnal pattern compared with a strictly nocturnal
pattern). In this case OVLðQ; JÞ ¼ 1 TVðQ; JÞ.
The total variation distance is invariant (Witting 1985) with
respect to any strictly increasing transformation, h, of Q and J; that
is, differentiability of h is not required. The total variation distance
arises in the asymptotic statistics (Pollard 2003) and in the stability
of estimators under misspecification (i.e. in robust statistics;
Donoho & Liu 1988). When densities exist, we have
TVðQ; JÞ ¼
1
2
Z2p
jf ðtÞ gðtÞjdt
0
0 2p
1
Z
Z2p
1@
þ
A
½f ðtÞ gðtÞ dt þ
½f ðtÞ gðtÞ dt
¼
2
0
Z2p
¼
0
½f ðtÞ gðtÞþ dt:
0
272
L. G. R. Oliveira-Santos et al. / Animal Behaviour 85 (2013) 269e280
and an estimator is readily available using the plugin method and
a nonparametric kernel density estimator.
For any random variable Q with density f and 0 q 1, where 1
is the indicator function, let
0
B
fq ðqÞ ¼ @
Z
11
C
f ðtÞdt A
f ðqÞ1ARq ðqÞ
ARq
¼ q1 f ðqÞ1ARq ðqÞ
be the density of the conditional random variable Qq : ¼ QjQ˛ARq .
This conditional random variable contains valuable information
about the modal behaviour of the original randomvariable Q. We refer
to TVðQq ; Jq Þ as the conditional (modal region) total variation
distance between Q and J of order q, and correspondingly,
OVLðQq ; Jq Þ ¼ 1 TVðQq ; Jq Þ is the conditional overlap coefficient.
Testing the Effect of the Parameters on the Activity Range and
Overlap
The analytical methods described here were applied to data
obtained through camera-trap monitoring conducted at the Nhumirim Research Station (18 590 S, 56 370 W), in the Pantanal
wetlands of Brazil. The study was approved under license number
ICMBio-21560-1 issued by Brazil’s federal environmental agency.
We established a 50-trap-station grid, with camera traps (Tigrinus
Equipment Research Ltd, Timbó, Santa Catarina, Brazil) systematically placed 1.5e2 km apart, covering 80 km2. The entire grid was
sampled in the rainy (March to May 2009) and dry seasons (August
to October 2009), during which each station was monitored 24 h/
day over 30 days. To avoid the inclusion of dependent records, we
discarded all photographs taken of the same species at the same
station within 1 h. The entire sampling effort covered 3000 cameratrap days. A more detailed description of the study area and the
sampling design can be found in Oliveira-Santos et al. (2011).
We obtained 2917 photographs of 27 mammal species during
the entire study period. To investigate the effects of the choice of
smoothing parameter on the values of the activity range conditioned to different probability density isopleths, we selected data
sets for five species showing contrasting general activity patterns
for which there were large numbers of photographs in our data set
(Table 1). For each species, we estimated the activity range conditioned on 10 isopleth values (from 0.5 to 0.95, by 0.05) using 20
smoothing parameters (k from 0.5 to 10, by 0.5).
The activity overlap between agouti (the most diurnal species)
and the four other species were also calculated for all tested
combinations of the smoothing parameter and isopleth values. We
then graphically evaluated the curve of the activity range and the
activity overlap estimation conditioned on each given isopleth as
a function of the smoothing choice.
We identified the smallest k value for which the activity range
estimation would stabilize, regardless of the isopleth considered to
be the best smoothing choice. Because smoothing parameter
selection is a key aspect of the proposed method, we report the
results obtained for selected smoothing parameters under four
methods (‘NRD Ml’ and ‘NRD Robust’ based on Taylor (2008); ‘CV
MSE’ and ‘CV Ml’ following Hall et al. (1987) in the Appendix,
Table A1).
Results
The graphic representation of the activity patterns fitted using
our circular kernel approach described the raw data distribution in
accordance with our previous expectation, both from field
Table 1
Number of independent photographs (n) and general activity pattern of five species
in the Pantanal wetland of Brazil
Species
n
General pattern
Tapir, Tapirus terrestris
169 Primarily nocturnal
Crab-eating fox, Cerdocyon thous 308 Primarily nocturnal with
crepuscular peaks
Cattle, Bos indicus
408 Cathemeral with crepuscular peaks
Coati, Nasua nasua
330 Primarily diurnal
Agouti, Dasyprocta agouti
480 Strictly diurnal with crepuscular peaks
experience and from the exploratory analysis of the raw data
(Fig. 1c, d; Appendix, Fig. A2). The estimated activity range conditioned to any given isopleth was smaller for higher k values
(overfitted functions) for all species (Fig. 1a, b; Appendix, Fig. A3).
However, this tendency was stronger for species showing more
concentrated activity, such as agouti and coati, than for species with
a more cathemeral pattern of activity, such as cattle (Appendix,
Fig. A3). The activity range varied more for k values between 0.5
and 5 (Fig. 1a, b) and stabilized for higher values. Thus, we chose
k ¼ 5 as a reference smoothing value to perform further analyses.
The activity density function fitted using this value returned
a pattern in agreement with the literature of the studied species,
without several peaks and valleys. For the species showing more
concentrated activity, it represented conspicuous activity peaks
within the kernel density 50% isopleth. For cattle, the most cathemeral of the species analysed, the density function never reached
values below 0.05, which means that the species showed no zero
probability of activity at any time in the daily cycle. Bimodal
activity, with split activity cores around sunset and sunrise, was
detected only with k > 3.2 for agouti and k > 3.4 for coati.
Performing comparisons between species showing concentrated activity values using low values of k, which oversmooth the
pattern, tended to result in relatively higher overlap coefficients
(Fig. 1e, f). However, when a species presented a cathemeral pattern
of activity with several disjointed activity peaks, the overlap coefficient did not behave monotonically for lower isopleths. This is
because changes in the k value may result in shifts in the length of
the activity range conditioned to a specific isopleth (Appendix,
Fig. A4).
The smoothing parameter selection methods NRD Ml and NRD
Robust yielded k values equivalent to our reference value of k ¼ 5,
whereas the methods CV MSE and CV Ml tended to return much
higher values. However, it is clear that the more concentrated the
activity pattern, the higher the value of the selected k (Appendix,
Table A1). This finding is supported by the fact that kernel function shapes with more concentrated patterns (e.g. highly crepuscular species) tended to be sensitive to low k values (k < 5;
Appendix, Fig. A3), and thus, a more robust pattern is only
returned with higher values. We present the R code to estimate
both the activity range (function modal.region.circular) and
activity overlap (function totalvariation.circular) in the
Supplementary Data. These functions are also included in the
package ‘circular’ available at the Comprehensive R Archive
Network (cran.r-project.org).
Case Study: Activity Ranges and Overlap among the Invasive Feral
Hog and Two Peccary Species
The feral pig, Sus scrofa, has been introduced worldwide and is
considered to be one of the most harmful invasive species in the
world, causing damage to populations, communities and ecosystems (Lowe et al. 2000). As it was introduced in the Pantanal
approximately 200 years ago and shares several ecological,
L. G. R. Oliveira-Santos et al. / Animal Behaviour 85 (2013) 269e280
(a)
273
(b)
Isopleth
20
0.95
0.9
15
Isopleth
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
10
5
Activity range estimated
Activity range estimated
20
0
0.85
10
5
0
0
2
10
4
6
8
Smoothing parameter (κ )
0
(c)
2
Kernel density estimates
0.3
0.2
0.1
0.2
0.1
0
0
0
3
6
9
12
15
18
21
24
0
3
Time of day
18
21
24
9
12
15
6
18
Smoothing parameter κ =5
21
24
6
9
12
15
Time of day
1 (e)
(f)
0.3
0.6
Isopleth
0.95
0.9
0.8
0.7
0.4
0.6
Kernel density estimates
0.8
Coefficient of overlap
10
4
6
8
Smoothing parameter (κ )
(d)
0.3
Kernel density estimates
0.8
0.75
0.7
0.65
0.6
0.55
0.5
15
0.2
0.1
0.2
0.5
0 0
1
2
3 4 5 6 7 8 9
Smoothing parameter (κ )
10
0
0
3
Figure 1. Circular kernel density function application on camera-trap data. Effects of smoothing values (k ranging from 0.5 to 10, by 0.5) on activity range estimation conditioned to
different isopleth levels (ranging from 0.5 to 0.95, by 0.05) for agouti (a) and cattle (b). Visual representation of 95% (hatched area) and 50% (black area) isopleth on kernel fitted
with k ¼ 5 for agouti (c) and cattle (d). Effect of smoothing values on overlap coefficient between agouti and cattle activity for different isopleths (e). For q ¼ 0.95, graphical
representation of the overlap between agouti and cattle (black area) and twice the total variation distance (dashed area) (f).
274
L. G. R. Oliveira-Santos et al. / Animal Behaviour 85 (2013) 269e280
behavioural (Kiltie 1982; Desbiez et al. 2009), morphological and
physiological (Bodmer 1991; Sicuro & Oliveira 2002; Elston et al.
2005) features with the South American peccary species, there is
currently much speculation about whether the hog causes negative
impacts on the native peccary. In this case study, we investigated
the activity patterns of the invasive feral hog and two native
peccary species as well as the shifts in activity overlap between
these species during two contrasting seasons.
The data used here came from the same camera-trap monitoring
efforts described in the previous section. We obtained 185 and 123
independent records of collared peccaries, Tayassu tajacu, 30 and 69
records of white-lipped peccaries, Tayassu pecari, and 122 and 97
records of feral hogs during the rainy and dry seasons, respectively.
The activity ranges and activity range cores of the three species
were described by season using kernel isopleths of 95% and 50%,
respectively. The activity range overlap and activity range core
overlap between these species were also measured within each
season under kernel isopleths of 95% and 50%, respectively. All
analyses were run with k ¼ 5. Confidence intervals were calculated
for activity ranges via bootstrapping 200 samples with the original
sample size, performing replacement following Ridout & Linkie
(2009).
These three species were cathemeral, presenting a wide activity
range (1900e2200 hours) and activity range core (0700e
1000 hours) in both seasons, with slight differences being detected
between the seasons for white-lipped and collared peccaries
(Table 2, Fig. 2). The collared peccary was active all day in both
seasons, but it showed a shift in activity range core from a unimodal
pattern with a duskeevening peak in the rainy season to a bimodal
pattern with both dawn and duskeevening peaks in the dry season.
The white-lipped peccary avoided activity during the night; its
activity range core extended from midday through dusk during
both seasons. Feral hogs avoided the hours around midday in both
seasons, shifting their activity range core from dusk-night in the
rainy season to exclusively nocturnal in the dry season.
Generally, the three species showed a high activity overlap
(Table 3, Fig. 3). The feral pig presented a higher activity overlap and
activity core overlap with the collared peccary than with the whitelipped peccary in both seasons. However, considering the activity
range core alone, the overlap of the feral hog with both collared and
white-lipped peccaries was higher in the rainy season than in the
dry season. Overlap between the native peccary species was as high
as that observed between them and feral hogs in both seasons.
Discussion
Activity range and activity overlap models
The major advantages of using kernel functions to fit activity
data are related to their continuous and circular structure. These
factors allow us to avoid problems associated with the arbitrariness
of the timescale categorization and definition of the origin (usually
noon or midnight in linear approaches). Additionally, as nonparametric functions, they can adequately fit either a bimodal or
multimodal pattern, which are both widespread among animals.
For our purposes, the automated smoothing parameter selectors
proposed by Taylor (2008) behaved better than those proposed by
Hall et al. (1987). While the selectors NRD Ml and NRD Robust
corroborated the results found during the test of the k effect on the
kernel fitting, the excessive overfitting of the CV MSE and CV Ml
selectors gave rise to several detailed peaks in the activity patterns,
suggesting an inadequacy for biological data, which usually consist
of small random samples of activity. As we are accumulating
information across several cycles (days) of an intrinsically variable
process, we caution against the use of overfitted functions to
describe animal activity. We recommend the use of an ad hoc
approach for smoothing choice, taking into account the sample
size, the question being addressed, the type of pattern being
described and the prior idea of the desired level of detail in the
description of the activity density pattern. Narrow gaps along the
activity core and activity range (shorter than 1 h) might not hold
any ecological meaning, and thus researchers should be careful in
their interpretation. When they occur, narrow gaps may be ignored
or circumvented by a slight decrease in the smoothing value that
can eliminate them.
Our methodology allows the identification of periods of
concentrated activity following the same conceptual framework
that has already been consolidated in spatial ecology through the
use of kernel functions to extract home range contours and utilization distributions based on GPS and radiotelemetry data. The
symmetric activity overlap presented here outperforms the asymmetrical overlap coefficients commonly used in ecological studies
(e.g. home range and diet), allowing further use as an association
(distance) measure in multivariate statistical procedures. However,
the nonmonotonic behaviour of the overlap index in relation to the
isopleth value could increase the complexity of data interpretation.
Given that the unconstrained total variation (100% isopleth) is
already balanced by the shape of the density function, we understand that it carries more information than any conditioned overlap
measure without being biased. However, the conditioned overlap
will permit the researcher to verify whether the overlap is more
concentrated in the activity cores of the species. Furthermore, we
emphasize that the overlap coefficient alone hides some ecological
meaningful information like the hours when overlap occurs.
Therefore, it is also advisable, in behavioural studies, to address the
extent to which the activity ranges coincide and to assess the
moment in the cycle in which this overlap occurs.
We think that the activity range and activity overlap, in both
conceptual and statistical terms, present new possibilities for
Table 2
Activity range conditioned to kernel isopleth 95% (activity range) and kernel isopleth 50% (activity range core) for the feral hog, collared peccary and white-lipped peccary in
rainy and dry seasons
Season
Species
Activity range (h)
Kernel 50%
Hour intervals
Rainy
Collared peccary
White-lipped peccary
Dry
Feral hog
Collared peccary
White-lipped peccary
Feral hog
*
16.60e00.36
10.60e12.00
12.50e18.96
00.32e03.84
15.80e21.15
05.80e08.23
16.81e23.75
11.14e19.52
20.80e05.50
Kernel 95%
*
Range (CI 95%)
Hour intervals*
Range (CI 95%)
7.78 (7.34e8.23)
14.20e11.77
21.56 (21.12e22.01)
7.83 (6.84e8.80)
19.29 (17.92e20.65)
8.90 (8.42e9.38)
9.35 (8.85-9.84)
05.42e00.72
09.75e12.97
13.34e07.96
14.40e11.33
21.84 (21.39e22.28)
20.92 (20.50e21.34)
8.38 (7.48e9.27)
8.54 (7.93e9.14)
04.63e02.45
14.36e11.73
21.82 (21.23e22.41)
21.37 (20.62e22.11)
Intervals are given in decimal hours. Confidence intervals are based on 200 bootstrap samples.
L. G. R. Oliveira-Santos et al. / Animal Behaviour 85 (2013) 269e280
0.3
0.3
(a)
Kernel density estimates
Kernel density estimates
0.2
0.15
0.1
0.2
0.15
0.1
0.05
0.05
0
0
0
0.3
3
6
9
12
15
Time of day
18
21
0
24
0.3
(c)
0.25
3
6
9
12
15
Time of day
18
21
24
3
6
9
12
15
Time of day
18
21
24
3
6
9
12
15
Time of day
18
21
24
(d)
0.25
Kernel density estimates
Kernel density estimates
(b)
0.25
0.25
0.2
0.15
0.1
0.2
0.15
0.1
0.05
0.05
0
0
0
3
6
9
12
15
18
21
24
0
Time of day
0.3
0.3
(e)
0.25
(f)
0.25
Kernel density estimates
Kernel density estimates
275
0.2
0.15
0.1
0.2
0.15
0.1
0.05
0.05
0
0
0
3
6
9
12
15
Time of day
18
21
24
0
Figure 2. Density activity estimates of collared peccary (a, b), white-lipped peccary (c, d) and feral hog (e, f) in rainy (left panel) and dry (right panel) seasons. Hatched area: activity
range (95% isopleth); black area: activity range core (50% isopleth).
L. G. R. Oliveira-Santos et al. / Animal Behaviour 85 (2013) 269e280
Table 3
Coefficient of overlapping conditioned to kernel isopleth 95% (activity range) and
kernel isopleth 50% (activity range core) between feral hog (FH), collared peccary
(CP) and white-lipped peccary (WLP) in rainy and dry seasons
0.82
0.66
0.65
0.87
0.63
0.69
(0.37e0.68)
(0.19e0.57)
(0.06e0.48)
(0.13e0.50)
(0.00e0.12)
(0.12e0.42)
(0.67e0.96)
(0.46e0.86)
(0.58e0.73)
(0.71e1.00)
(0.50e0.75)
(0.63e0.75)
Confidence intervals are based on 200 bootstrap samples.
0.8
investigations and for testing hypotheses concerning seasonal
animal activity shifts, the presenceeabsence of predators or
competitors and many other experimental treatments. These
concepts and estimates provide new opportunities for advancing
understanding of animal activity through comparisons of several
pairs of species and enabling identification of meaningful ecological
traits (e.g. body mass, food habitats, locomotion and habitat type)
across a phylogenetic history that determine widespread activity
patterns in the natural world.
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Low overlap
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Kernel isopleths
1
(b)
0.9
Coefficient of overlap
0.8
0.7
0.6
0.5
0.4
0.3
0.2
FH−CP
CP−WL
FH−WL
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Kernel isopleths
1
(c)
0.9
0.8
Coefficient of overlap
The feral hog and the two native peccaries
The models applied to the feral hog and peccary activity data were
useful for describing the activity patterns of each species and for
identifying the location of their main activity peaks. The activity
overlap was also a powerful tool for measuring the daily activity
overlap between the species, describing the species that showed
higher overlap and whether this overlap was more pronounced
during activity peaks. Seasonal changes in activity ranges and activity
overlaps were also identified, enabling a researcher to make inferences regarding factors such as environmental shifts or the effects
caused by the presence or absence of predators or the introduction or
exclusion of competitors. Indeed, the possibility of calculating
confidence intervals via bootstrapping for both the activity range and
overlap metrics could be useful in making statistical inferences from
comparisons between species and treatments.
The three investigated suiform species were active throughout
the entire day, corroborating the hypothesis of van Schaik &
Griffiths (1996) that large mammals are expected to be cathemeral. The white-lipped and collared peccaries have been
described as diurnal species in the Neotropics (Gómez et al. 2005;
Tobler et al. 2009), while the feral hog has been identified as
nocturnal (Ilse & Hellgren 1995). However, in this case study, the
collared peccary displayed a different activity pattern, concentrating its activity in the nocturnalecrepuscular hours. This
uncommon pattern could be linked to the high temperatures
associated with the sandy soils found in the study area, resulting in
more nocturnal behaviour in collared peccaries, similar to what is
found in arid sites in Texas (Ilse & Hellgren 1995).
The high activity overlap between feral hogs and the two
peccary species, although mainly for the collared peccary, along
with the high vagility of these species greatly increases the chance
of interspecific encounters. Feral hogs have difficulty maintaining
their water balance (MacNab 1970; Gabor et al. 1997) and, thus,
may be taking advantage of the greater availability of waterholes
during the rainy season, allowing them to become more diurnal
and concentrating their activity from the middle of the afternoon
until dusk. This activity shift was responsible for the increase in the
activity overlap core between the two peccary species. However, no
evidence of spatial avoidance among feral hogs and peccary species
was observed in this study area (Oliveira-Santos et al. 2011).
Although the feral hog is widespread in the Pantanal wetland,
High overlap
la
p
Kernel 95%
0.52
0.38
0.27
0.31
0.00
0.27
ov
er
Dry
Kernel 50%
ia
te
FH-CP
FH-WLP
CP-WLP
FH-CP
FH-WLP
CP-WLP
ed
Rainy
(a)
0.9
Activity overlap (CI 95%)
rm
Pair of species
Coefficient of overlap
Season
1
In
te
276
0.7
0.6
0.5
0.4
0.3
0.2
0.1
FH−CP
CP−WL
FH−WL
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Kernel isopleths
Figure 3. Theoretical interpretation of the coefficient of overlap, conditional total
variation distance, between two species (a) and between the feral hog (FH), the whitelipped peccary (WLP) and the collared peccary (CP) in the rainy season (b) and in the
dry season (c). The vertical dashed line represents the activity core (kernel 50%)
relative to each species.
L. G. R. Oliveira-Santos et al. / Animal Behaviour 85 (2013) 269e280
white-lipped and collared peccaries continue to maintain densities
as high as those observed in other areas that are free of hogs
(Desbiez et al. 2010; Oliveira-Santos et al. 2011).
L.G.R.O.S and C.A.Z are supported by National Council for Scientific and Technological Development (CNPq). We acknowledge the
Brazilian Agricultural Research Corporation (Embrapa Pantanal) for
logistic support, the CNPq, the Coordination for the Improvement of
Higher Level Personnel (CAPES) and the Mato Grosso do Sul State
Research Foundation (FUNDECT) for financial support.
Supplementary Material
Supplementary data for this article is available, in the online
version, at http://dx.doi.org/10.1016/j.anbehav.2012.09.033.
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Appendix
Smoothing Parameter Selection
Smoothing parameter selection is an important issue to address in
the application of the introduced methodology. In Table A1 we report
the estimated smoothing parameters for the data sets analysed. We
use the methods proposed in Taylor (2008) (NRD Ml and NRD Robust)
and Hall et al. (1987) (CV MSE and CV Ml). The results show high
variability in the estimates using these different methods.
Table A1
Estimated smoothing parameters using several methods for the data sets analysed:
agouti, cattle, coati, crab-eating fox, tapir, collared peccary (CP), white-lipped
peccary (WLP) and feral hog (FH)
Data set
NRD Ml
NRD Robust
CV MSE
CV Ml
Agouti
Cattle
Coati
Crab-eating fox
Tapir
CP e Rainy
CP e Dry
WLP e Rainy
WLP e Dry
FH e Rainy
FH e Dry
9.147
1.426
6.810
1.109
4.958
1.661
0.624
1.501
1.083
1.039
1.631
44.886
12.881
24.754
6.890
19.847
8.805
12.136
7.041
12.696
15.886
5.012
120.521
38.020
57.691
15.712
22.924
8.362
11.198
1.503
6.012
121.005
18.761
146.146
26.203
19.405
16.086
11.926
8.081
16.567
2.024
2.393
46.586
17.591
278
L. G. R. Oliveira-Santos et al. / Animal Behaviour 85 (2013) 269e280
(a)
(b)
0.3
Kernel density estimates
Kernel density estimates
0.3
0.2
0.1
0
0.2
0.1
0
0
3
6
9
12
15
Time of day
18
21
24
0
(c)
6
9
3
6
9
12
15
Time of day
18
21
24
18
21
24
(d)
0.3
0.3
Kernel density estimates
Kernel density estimates
3
0.2
0.1
0
0.2
0.1
0
0
3
6
9
12
15
Time of day
18
21
0
24
12
15
Time of day
(e)
Kernel density estimates
0.3
0.2
0.1
0
0
3
6
9
12
15
18
21
24
Time of day
Figure A1. Density estimates of daily activity patterns of agouti (a), cattle (b), coati (c), crab-eating fox (d) and tapir (e) based on camera-trap data from the central Pantanal of
Brazil. Hatched area: activity range (95% isopleth); black area: the activity range core (50% isopleth).
L. G. R. Oliveira-Santos et al. / Animal Behaviour 85 (2013) 269e280
279
Cattle
Agouti
(b)
20
15
Isopleth
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
10
5
Activity range estimation (hours)
Activity range estimation (hours)
(a)
Isopleth
0.95
20
0.9
0.85
0.8
15
0.75
0.7
0.65
0.6
0.55
0.5
10
5
0
0
0
2
4
6
8
Smoothing parameter (κ )
10
0
2
Coati
(d)
20
Isopleth
15
0.95
10
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
5
Activity range estimation (hours)
Activity range estimation (hours)
10
Crab−eatingfox
(c)
Isopleth
0.95
20
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
15
10
5
0
0
2
4
6
8
10
Smoothing parameter (κ )
4
6
8
Smoothing parameter (κ )
Feral hog
Tapir
(e)
0
0.95
20
0.9
0.85
15
0.8
0.75
0.7
0.65
0.6
0.55
0.5
10
5
0
2
10
(f)
Isopleth
Activity range estimation (hours)
0
Activity range estimation (hours)
4
6
8
Smoothing parameter (κ )
20
Isopleth
15
0.95
10
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
5
0
0
2
4
6
8
Smoothing parameter (κ )
10
0
2
4
6
8
10
Smoothing parameter (κ )
Figure A2. Effects of smoother values (ranging from 0.5 to 10, by 0.5) on activity range estimation conditioned to different isopleth levels (ranging from 0.5 to 0.95, by 0.05). Species
are ordered from concentrated to cathemeral activity from top left to bottom right. Species: agouti, Dasyprocta azarae (strictly diurnal with crepuscular peaks), coati, Nasua nasua
(primarily diurnal), crab-eating fox Cerdocyon thous (primarily nocturnal with crepuscular peaks), tapir, Tapirus terrestris (primarily nocturnal), feral hog Sus scrofa (cathemeral), and
cattle, Bos indicus (cathemeral with crepuscular peaks).
280
L. G. R. Oliveira-Santos et al. / Animal Behaviour 85 (2013) 269e280
Agouti X Cattle
1
(a)
0.8
Coefficient of overlap
Coefficient of overlap
1
Agouti X Coati
0.6
Isopleth
0.95
0.9
0.8
0.7
0.4
0.6
0.2
0
2
4
6
8
10
Smoothing parameter (κ )
0.95
0.9
0.8
0.7
0.6
0.5
0.8
0.6
0.4
0.2
0.5
0
Isopleth
(b)
0
12
0
2
Agouti X Crab−eating fox
12
Agouti X Tapir
1 (d)
(c)
0.8
Coefficient of overlap
Coefficient of overlap
1
4
6
8
10
Smoothing parameter (κ )
0.6
Isopleth
0.95
0.4
0.9
0.8
0.7
0.6
0.5
0.2
0.8
0.6
0.4
0.8
0.2
0.7
0.95
0.6
0
Isopleth
0.9
0 0.5
0
2
4
6
8
10
Smoothing parameter (κ )
12
0
2
4
6
8
10
Smoothing parameter (κ )
12
Figure A3. Effect of the smother choice on the activity overlap conditioned to six kernel isopleths (50%, 60%, 70%, 80%, 90% and 95%). The activity overlap was measured between
a typical bimodal diurnal species with crepuscular peaks (agouti, Dasyprocta azarae) and other five mammals with contrasting activity patterns: coati Nasua nasua (primarily
diurnal), crab-eating fox, Cerdocyon thous (primarily nocturnal with crepuscular peaks), tapir, Tapirus terrestris (primarily nocturnal), and cattle Bos indicus (cathemeral with
crepuscular peaks).