Electronic supplementary material
Figure ESM 1. Phylogenetic tree of the 135 species of non-human social primate in our sample.
Figure ESM 2. Graphical representation of Bonferroni corrected (significance value set to p<
0.00833), univariate PGLS analyses to assess the effect of several socio-ecological variables on home
range overlap across non-human social primates.
Figure ESM 3. Since in this final analysis we only consider a subset of all species in our sample (those
living in mM-mF groups), one could argue we first need to re-establish that the ecological variables
(Activity period, Substrate use, Habitat type and Diet) omitted from follow-up analyses on the whole
dataset, can also justly be left out from this final test. Bonferroni corrected (significance value set to
p< 0.0125), univariate PGLS analyses show that none of these variables significantly affects home
range overlap in non-human social primates that live in mM-mF groups, thereby justifying their
omission.
Table ESM 1. Parameter estimates (B) and significance values as obtained from a PGLS analysis that
maximized sample size (n species= 126) by considering only those socio-ecological variables of home
range overlap that were found to be significant predictors in previous multi- and univariate PGLS
analyses.
Variable
B
SE t-value
p
Intercept
0.194 0.25
0.78 0.4396
Group size
0.201 0.04
4.67 <0.0001
Defendability -0.124 0.05
-2.44 0.0160
λML= 0.728; R2Adj.= 0.196, F(3,123)= 16.19, p< 0.0001
Table ESM 2. Both the number of males (a, n species= 125), and females (b, n species= 124) in a species’
modal grouping pattern were significantly and positively associated with home range overlap. This
provided a first suggestion that collective action problems may arise in either sex. Parameter
estimates (B) and significance values as obtained from PGLS analyses.
a)
Variable
Intercept
n Males
Controlling for:
Defendability
B
SE
0.524 0.26
0.019 0.01
t-value
2.03
3.40
p
0.0445
0.0009
-0.143 0.05
-2.72
0.0074
λML= 0.764; R2Adj.= 0.136, F(3, 122)= 10.76, p< 0.0001
b)
Variable
Intercept
n Females
Controlling for:
Defendability
B
SE
0.426 0.25
0.194 0.04
-0.127 0.05
t-value
p
1.72 0.0874
4.58 <0.0001
-2.46
0.0155
λML= 0.751; R2Adj.= 0.190, F(3, 121)= 15.41, p< 0.0001
Table ESM 3. Post-hoc PGLS pair-wise comparisons between species living in sM-sF groups (in which
a collective action problem among same-sex individuals cannot occur) and mM-mF species
characterized by dispersal and philopatry of the larger sex. The analysis shows that only mM-mF
species in which the larger sex disperses suffer from increased home range overlap, due to a
collective action problem. Parameter estimates (B) and significance values as obtained from PGLS
analyses.
Multiple comparisons (nspecies= 79)
B
Intercept
0.479
Social structure:
sM-sF
mM-mF Dispersal larger sex
0.469
mM-mF Philopatry larger sex
0.082
Controlling for:
Defendability
-0.067
SE t-value
p
0.07
6.84 <0.0001
0.08
0.12
0.06
5.62 <0.0001
0.67 0.5021
-1.13
0.2632
λML= 0.000; R2Adj.= 0.339, F(4, 75)= 14.36, p< 0.0001
Note to Table ESM 4
On the biological validity and practical utility of defendability indices
Originally conceived as a quantitative measure of the economic defendability [1] of a ranging area,
the Mitani-Rodman D-index was initially used to distinguish between territorial and non-territorial
primate species [2, 3]. Based on the assumption that economic defendability in non-volant species
depends on an animal’s (or group’s) ability to monitor the boundaries of its range in order to detect
potential intruders, the D-index is calculated as the ratio of the average daily path length and the
diameter of an (idealised) ranging area:
D= d/(4A/π)0.5
, where D is the defendability index, d the average daily path length (km), and (4A/π)0.5 the diameter
of a circle with a surface area A (km2) equal to that of the home range. As such, the D-index
expresses the number of times an agent (a solitary animal or group) can move across its range within
a day, where high values imply the possibility of frequent contact with boundaries at widely
dispersed points on the perimeter, and low values indicate that effective defence will be difficult as
distant points at the boundary cannot be monitored regularly [2]. In their original study, Mitani and
Rodman empirically established that territorial primate species invariably exhibit D-index values ≥
1.0, and thus that the defence of an area only becomes economically feasible if it can be crossed at
least once a day.
However, the biological validity of the D-index is not limited to merely making this potentially misleading- dichotomy between territoriality and non-territoriality. Instead, it is far more
informative (and true to biological reality) to regard defendability as a continuous variable along an
axis of “the monopolizability of space”. Indeed, the extent to which a range can be defended is
modelled by the D-index as a monotonically increasing function of the agent’s mobility in relation to
its range size requirements. In more recent studies then, the D-index has also successfully been used
to e.g. characterise seasonal variation in home range defendability within populations [4, 5], and
even as a measure of space use intensity to predict animal parasite loads [6]. The biological validity
and utility of the D-index as a continuous measure of economic defendability are thus wellestablished by multiple empirical studies.
As an extension of the original defendability index, the Lowen-Dunbar M-index [3] was
defined to express economic defendability as a function of the frequency of boundary collisions
(similar to the D-index, although now defendability decreases quadratically with range size, rather
than linearly), as well as the length of the boundary that needs to be monitored to obtain effective
defence against intruders. In addition, the M-index allows for multiple, independently moving
foraging parties, as might occur in species with high fission-fusion dynamics. It is defined as:
M= N(sv/d2)
, where M is the defendability index (or “fractional Monitoring rate”), N the number of
independently moving foraging parties, s the detection distance or length of the home range
boundary that can be monitored during each visit (km), v the daily path length (km), and d2 the
squared diameter of a circle with surface area equal to that of the home range (km2). Thus defined,
economic defendability scales with the proportion of the boundary that each independent sub-group
can effectively monitor every time it reaches the border of the home range. Both the D- and M-index,
therefore, reflect a monotonically increasing propensity to defend an area as it becomes easier to
patrol. Lowen and Dunbar, moreover, pointed out that both indices are strongly rooted in the kinetic
theory of ideal gases [3]. A recent and independent review on the use of ideal gas models to analyse
animal encounters concluded that this framework continues to provide a useful approximation of
biological reality [7].
Although the M-index has some theoretically desirable properties over the D-index, the two
indices are conceptually very similar and were indeed highly interrelated in the original Lowen and
Dunbar study (log-transformed values from Tables 2 and 3 were strongly correlated [rPearson= 0.886,
nspecies= 49, p< 0.0001]). Unfortunately, however, the two additional parameters required by the Mindex (N and s) have never explicitly been quantified for any primate species. Moreover, the implicit
assumptions that foraging parties move truly independently, and are equally capable of effective
range defence (not all sub-groups may be sufficiently large or contain members of the appropriate
age-sex class to keep intruders out) severely undermine the practical value of the M-index. Despite
its theoretical refinements then, in practice the M-index may not really provide an improved
estimate of economic defendability, as it is simply not possible yet to correctly parameterise it based
on empirical data. This stands in contrast to the proven usefulness and analytical power of the readily
calculated D-index. For this reason, we based our analyses and conclusions in the main body of our
paper on the D-index, rather than the M-index. As we show below, however, our results and
conclusions are very robust against the choice of the measure of economic defendability used.
In tables ESM 4.1 - 4.3, we offer results from the analyses presented in Tables 1 - 3, after
substituting the D-index by a “biologically most plausible guess” at the value of the M-index for each
species in our sample. In a first trial, we parameterised the M-index as suggested in the original
paper [3], assuming a value of s= 0.05, and N=1 (as even in fission-fusion taxa, effective defence is
only possible if a sufficiently large number of individuals of the appropriate age-sex class are within a
single party). A strong correlation existed within our dataset between log-transformed values of the
D-index and this variant of the M-index (rPearson= 0.891, nspecies= 126, p< 0.0001). Parameter estimates
and associated standard errors were virtually identical in all analyses, and predictor variables
identified as significant were always the same as in our original analyses. In a second trial, we
attempted to add more biological realism to the M-index by doubling the value of s for species living
in open habitats as well as those with loud inter-group vocalisations [primary source: 8] and, in
addition, doubling the value of N for species with high fission-fusion dynamics. Again, the correlation
between log-transformed D- and M-index values was very high (rPearson= 0.898, nspecies= 126, p<
0.0001). Results from these analyses are presented in Tables ESM 4.1 - 4.3, and are fully in line with
the original analyses and conclusions. No noticeable improvements in model performance were
detected (ΔAIC’s with the original model were always less than 2).
We conclude that, regardless of the index used to model economic defendability, social
variables always accounted for a significant (and previously ignored) proportion of the variation in
home range overlap across the primate taxon. Thus, our findings provide evidence for the
interpretation that primate territory economics are strongly affected by a collective action problem,
more so than by range defendability. This implies that non-human social primates perform suboptimal in a cooperative task, as has also been described for human groups [9].
Literature cited
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Lowen C., Dunbar R.I.M. 1994 Territory size and defendability in primates. Behav Ecol Sociobiol 35(5), 347-354.
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Wich S.A., Nunn C.L. 2002 Do male "long-distance calls" function in mate defense? A comparative study of longdistance calls in primates. Behav Ecol Sociobiol 52(6), 474-484. (doi:10.1007/s00265-002-0541-8).
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Table ESM 4 Outcome of PGLs analyses presented in the main body of our article in Tables 1, 2 and 3
respectively, using an alternative measure of economic defendability, the Lowen-Dunbar M-index
(see introductory note for information on parameter values). All results and conclusions presented in
the main body of our article are corroborated by the independent set of analyses presented here.
ESM 4.1
Variable
Intercept
Group size
Defendability
Activity period
Substrate use
Habitat type
Diet
B
0.031
0.207
-0.070
-0.045
-0.088
0.058
-0.061
SE t-value
p
0.27
0.12 0.9066
0.05
3.93 <0.0001
0.03
-2.43 0.0167
0.18
-0.25 0.8059
0.11
-0.77 0.4416
0.15
0.38 0.7042
0.13
-0.49 0.6282
λML= 0.544; R2Adj.= 0.212, F(7, 109)= 6.16, p< 0.0001
ESM 4.2
a)
Variable
B
SE t-value
p
Intercept
0.333 0.23
1.47 0.1449
Social structure:
sM-sF
sM-mF
0.065 0.10
0.62 0.5338
mM-sF
-0.125 0.14
-0.90 0.3718
mM-mF
0.258 0.10
2.64 0.0095
Controlling for:
Defendability -0.087 0.03
-3.37 0.0010
λML= 0.688; R2Adj.= 0.175, F(5, 121)= 7.63, p< 0.0001
b)
Variable
B
SE t-value
p
Intercept
0.321 0.24
1.31 0.1916
ID larger sex:
Male
Female
-0.070 0.09
-0.76 0.4464
n larger sex:
One
Multiple
0.171 0.07
2.50 0.0139
Controlling for:
Defendability -0.078 0.03
-2.77 0.0065
λML= 0.702; R2Adj.= 0.125, F(4, 108)= 5.49, p< 0.0005
ESM 4.3
Variable
B
SE t-value
p
Intercept
0.817 0.10
8.05 <0.0001
Dispersal pattern:
Dispersal
Philopatry
-0.377 0.12
-3.05 0.0038
Controlling for:
Defendability
-0.051 0.04
-1.40 0.1695
λML= 0.000; R2Adj.= 0.171, F(3, 46)= 5.94, p< 0.005