Does Litter Size Variation Affect Models of Terrestrial Carnivore Extinction Risk and Management?

Abstract

Background
Individual variation in both survival and reproduction has the potential to influence extinction risk. Especially for rare or threatened species, reliable population models should adequately incorporate demographic uncertainty. Here, we focus on an important form of demographic stochasticity: variation in litter sizes. We use terrestrial carnivores as an example taxon, as they are frequently threatened or of economic importance. Since data on intraspecific litter size variation are often sparse, it is unclear what probability distribution should be used to describe the pattern of litter size variation for multiparous carnivores.

Methodology/Principal Findings
We used litter size data on 32 terrestrial carnivore species to test the fit of 12 probability distributions. The influence of these distributions on quasi-extinction probabilities and the probability of successful disease control was then examined for three canid species – the island fox Urocyon littoralis, the red fox Vulpes vulpes, and the African wild dog Lycaon pictus. Best fitting probability distributions differed among the carnivores examined. However, the discretised normal distribution provided the best fit for the majority of species, because variation among litter-sizes was often small. Importantly, however, the outcomes of demographic models were generally robust to the distribution used.

Conclusion/Significance
These results provide reassurance for those using demographic modelling for the management of less studied carnivores in which litter size variation is estimated using data from species with similar reproductive attributes.

Full text

Does Litter Size Variation Affect Models of Terrestrial
Carnivore Extinction Risk and Management?
Eleanor S. Devenish-Nelson
1
*, Philip A. Stephens
1
, Stephen Harris
2
, Carl Soulsbury
3
, Shane A. Richards
1
1School of Biological and Biomedical Sciences, Durham University, Durham, United Kingdom, 2School of Biological Sciences, University of Bristol, Bristol, United
Kingdom, 3School of Life Sciences, University of Lincoln, Lincoln, United Kingdom
Abstract
Background:
Individual variation in both survival and reproduction has the potential to influence extinction risk. Especially
for rare or threatened species, reliable population models should adequately incorporate demographic uncertainty. Here,
we focus on an important form of demographic stochasticity: variation in litter sizes. We use terrestrial carnivores as an
example taxon, as they are frequently threatened or of economic importance. Since data on intraspecific litter size variation
are often sparse, it is unclear what probability distribution should be used to describe the pattern of litter size variation for
multiparous carnivores.
Methodology/Principal Findings:
We used litter size data on 32 terrestrial carnivore species to test the fit of 12 probability
distributions. The influence of these distributions on quasi-extinction probabilities and the probability of successful disease
control was then examined for three canid species – the island fox Urocyon littoralis, the red fox Vulpes vulpes, and the
African wild dog Lycaon pictus. Best fitting probability distributions differed among the carnivores examined. However, the
discretised normal distribution provided the best fit for the majority of species, because variation among litter-sizes was
often small. Importantly, however, the outcomes of demographic models were generally robust to the distribution used.
Conclusion/Significance:
These results provide reassurance for those using demographic modelling for the management of
less studied carnivores in which litter size variation is estimated using data from species with similar reproductive attributes.
Citation: Devenish-Nelson ES, Stephens PA, Harris S, Soulsbury C, Richards SA (2013) Does Litter Size Variation Affect Models of Terrestrial Carnivore Extinction
Risk and Management? PLoS ONE 8(2): e58060. doi:10.1371/journal.pone.0058060
Editor: Martin Krkosek, University of Toronto, Canada
Received May 12, 2012; Accepted January 31, 2013; Published February 28, 2013
Copyright: ß2013 Devenish-Nelson 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: ESDN was funded by a Durham University Doctoral Fellowship and SH was funded by the Dulverton Trust. 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: e.s.nelson@durham.ac.uk
Introduction
Demographic variation, resulting from extrinsic and intrinsic
sources, fundamentally affects population dynamics and is
particularly important when assessing extinction risk for threat-
ened species [1,2]. Predictions of population dynamics depend on
the ability to attribute sources of stochasticity accurately in
population models [3,4]. Of particular importance is the
distinction between demographic stochasticity and demographic
heterogeneity. Demographic stochasticity is the random fate of an
individual arising from a chance event drawn from a specified
uniform vital rate, whereas demographic heterogeneity is the
variation in the underlying parameter value arising from within-
population variability in individual condition [5]. Both types of
demographic variation make important contributions to a
populations’ total demographic variance [3]. Indeed, accounting
for demographic stochasticity in fecundity can lead to increased
predictions of extinction risk; for example, overall demographic
variance is increased when this parameter is Poisson-distributed
[5]. Here, we focus on stochasticity in demographic fates, which
can easily be accounted for by drawing rates from appropriate
probability distributions [6,7].
Mean litter (or clutch) size has long been the focus of
evolutionary and population biologists concerned with causes of
interspecific variation [8–11], correlations with environmental
gradients [9,12–14] and optimality in this trait [15–18]. However,
intra-population variation in litter size has been largely overlooked
(but see [19]). Limited knowledge of the underlying measures of
empirical litter size distributions, such as the degree of dispersion,
hinders the accurate representation of the stochasticity of this
parameter in population models. Demographic stochasticity in
offspring number is most commonly modelled with Poisson or
normal distributions [6,20,21], although there is little theoretical
justification for these choices [19]. Furthermore, many demo-
graphic modelling programmes (e.g. RAMAS [7] and VORTEX
[21]) have limited provision for specifying distributions. Unlike
survival, which is a Bernoulli process [20], choosing a distribution
to describe variation in litter sizes in multiparous species can be
complex because the biology of reproduction differs substantially
among species and is ultimately limited by physiological capacity.
Standard probability distributions might lack the flexibility
required to account for litter size variation in many species.
In population modelling, the influence of distribution choice
has only been considered previously for demographic parameters
other than litter size, with a focus on environmental stochasticity.
Studies that modelled environmental stochasticity found that
population growth rate (l) estimates were underestimated as a
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result of inaccurately defined, symmetrical survival distributions
[22] and large differences in lestimates were found when
drawing recruitment rates from different distributions [23]. Yet,
the shape of the distribution may also be important for
populations that are susceptible to fluctuations in vital rates as
a result of demographic stochasticity, such as small populations.
Failing to account for demographic stochasticity in litter size may
lead to inaccurate predictions of extinction risk [19]. In this
context, it is useful to establish whether failing to incorporate an
appropriate theoretical distribution for litter size, to describe
demographic stochasticity, could lead to erroneous estimates of
model outputs.
Here, we examine the fit of specified candidate probability
distributions to empirical data on terrestrial carnivore litter size
frequencies. The Carnivora exhibit some of the most diverse life
history traits of all mammalian orders, as reflected in their broad
range of litter sizes [24]. While many carnivores are at increasing
risk of extinction [25], others are predators of economic
importance or important hosts of zoonotic and wildlife diseases
such as rabies [26]. Although data collection is often challenging
[27], both categories of carnivore are frequently the subject of
Table 1. Parameter values for the three population models.
Initial parameter value Model 1. Island fox Model 2. Red fox Model 3. African wild dog
Quasi-extinction or disease
density threshold
50 87% of initial population One sex remains
Years 100 3 50
Time step Annual Monthly Annual
Age at first reproduction 2 1 3
Sex ratio at birth 0.5 0.5 0.55
Dispersal age 1 1 -
Dispersal probability 0.01 Female month 7–12: 0.03, 0.030, 0.136,
0.045, 0.045, 0.030
-
Male month 7–12: 0.68, 0.102,
0.182, 0.159, 0.102, 0.057
Dispersal survival 0.8 - -
Annual mortality rate pup 31.365.9 - 0.6860.20
Annual mortality rate juvenile male 25.266.0 Monthly: 0.137, 0.045, 0.040, 0.048, 0.036,
0.035, 0.044, 0.044, 0.039, 0.062, 0.032, 0.035
0.2060.03
Annual mortality rate
juvenile female
16.864.7 Monthly: 0.129, 0.052, 0.067, 0.037, 0.042,
0.037, 0.044, 0.032, 0.039, 0.025, 0.034, 0.030
0.2060.03
Annual mortality rate adult male 25.266.0 Monthly: 0.035, 0.039, 0.020, 0.028, 0.014,
0.039, 0.036, 0.046, 0.041, 0.121, 0.069, 0.029
0.1560.03
Annual mortality rate adult female 16.864.7 Monthly: 0.041, 0.055, 0.035, 0.025, 0.023,
0.034, 0.044, 0.049, 0.035, 0.062, 0.041, 0.036
0.1560.03
Probability of breeding 1 0.8 0.58 (dominant pairs only)
Density dependence in breeding
(% breeding at carrying capacity)
West subpopulation: 58.38 East
subpopulation: 55.03
--
Carry capacity, K West subpopulation: 300 - 20
East subpopulation: 1300
Initial population size West subpopulation: 90 1 male and 1 female per group, 20
East subpopulation: 63 additional male or female added with
probability of 0.80 and 0.58 additional individual
0.47 probability of being juvenile
Disease Introduction - September -
Incubation period - 1 month -
Probability of becoming
rabid once exposed
-0.42 -
Disease mortality - 1 -
Control - 40% control every 2 months, 3 months
after disease introduction
-
Catastrophes Frequency: 0.2 - Mild: Frequency: 0.05
Reduction in survival: 0.8 Survival reduction: 0.85
Reproduction reduction: 0.5
Severe: Frequency: 0.03
Survival reduction: 0.5
doi:10.1371/journal.pone.0058060.t001
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population models (e.g. [28–30]). Given the importance of
carnivore management and the sparseness of much of the data
used to model carnivore demography, it is useful to establish
whether the choice of distribution used to model demographic
stochasticity in litter sizes affects the inferences drawn from
models of carnivore population dynamics. To illustrate the
applied importance of using appropriate distributions, three
previously published population models are replicated to
determine the consequences of mis-specifying litter size distribu-
tions for inferences regarding extinction probabilities or disease
dynamics.
Methods
Probability distribution fitting
Litter size frequency data were collated for 32 terrestrial
multiparous carnivore species, from 63 published studies of 73 wild
populations, to reflect the diversity of life history within the order.
Each species has a single annual breeding attempt. None of the
studies included litters of zero; modelling litter size inherently
assumes that an individual has bred. If studies presented data for
multiple conspecific populations or for multiple methods of litter
size determination, these were analysed as discrete datasets. For 15
species, data were obtained for between two and ten populations.
For three species, data from multiple methods of litter size
determination (e.g. placental scars and direct counts) were
available. We thus also considered whether there was strong
support for genuine underlying difference in litter size distributions
between conspecific populations or between data determined by
different methods (again, for a given population) (see Appendix S1
for details of the analyses).
Twelve probability distributions were selected based on a
review of previous studies. Specifically, four discrete distributions
were chosen: the Poisson distribution [6]; the generalised Poisson,
which has a wide-ranging suitability for describing litter size
frequencies [19]; the binomial distribution, previously fitted
successfully to carnivore litter data [19]; and the negative
binomial, widely used to describe ecological processes (e.g. [31]).
For each discrete distribution, both a ‘right shifted’ and ‘zero-
truncated’ form were fitted (Appendix S2), to exclude litter sizes
of zero. For zero-truncation, the probability mass function was
scaled by the exclusion of predicted zeros. Shifting involved
moving the entire distribution one interval to the right. Three
continuous probability distributions were chosen: the normal and
lognormal distributions are both widely used [6], although log-
transformation is not recommended for count data [32]; and the
stretched beta (two and three parameter forms), as proposed by
Morris and Doak [6]. Appendix S2 provides details of how these
continuous distributions were converted into discrete forms.
Maximum-likelihood parameters, denoted ^
hh,were estimated
using the ‘optim’ function in R 2.14.0 (R Development Core
Team 2011). Here, the multinomial log-likelihood defined by h
and given all the data is:
LL(hDdata)~C(Nz1)zX
xmax
i~1
Niln Pi(h){C(Niz1)½ð1Þ
where Nis the total number of litters observed, N
i
is the number of
litters observed of size i,P
i
is the predicted litter size probability
determined by a given distribution (Appendix S2), x
max
is the
maximum litter size, and C(x) is the complete gamma function.
The fits for each probability distribution were compared using
Akaike’s Information Criterion (AIC); all distributions having a
DAIC#6 of the best fitting distribution (i.e. lowest AIC) were
considered to have some support [33]. To check that our best-
fitting models were consistent with the data, and because of the
small sample sizes of the predicted frequencies, we performed
goodness-of-fit tests using Fisher’s Exact Test. Variance-mean
ratios [34] were determined to measure the dispersion of the
empirical and fitted distributions.
Carnivore population models
Published stochastic population models for three management
scenarios were used to illustrate the broader applied significance
of this study. The Canidae were chosen because they provide the
widest range of litter sizes within the Carnivora [24]. Models
were chosen to depict a range of conservation and management
scenarios that could be replicated from published data; the
intention was to identify whether the choice of distribution used
to represent litter sizes influences predicted model outcomes. By
‘‘outcomes’’, we refer to a major emergent parameter from the
models, on which further inference would be based (see below).
The emergent parameter of interest varied because the three
models were created for different applications. Using the
parameters that were estimated by maximum likelihood as
described above, 10,000 stochastic replicates of the models were
simulated drawing litter sizes from each of the 12 probability
distributions. This enabled calculation of 95% confidence
intervals around mean outcome values. For each case study,
disparities were determined between the outcome values of the
12 model versions. This allowed us to evaluate the effect on each
model of employing different litter size distributions, in relation
to the degree of empirical support for those distributions. See
Appendix S3 for full descriptions of each case study model and
Table 1 for the initial parameter values for the three models.
First, we investigated the island fox Urocyon littoralis, which
reached near extinction on Santa Catalina Island due to an
outbreak of canine distemper virus [35]. We conducted a density-
dependent population viability analysis (PVA) for two subpopu-
lations, based on Kohlmann et al. [28]; the outcome of interest
was the probability of quasi-extinction, defined in this model as
the probability of the population declining to 50 individuals, due
to a disease epidemic. Second, we investigated the red fox Vulpes
vulpes, a locally abundant carnivore that is the focus of much
attention due to its economic importance as a predator and role
in the spread of rabies [36]. A density-dependent model
simulating control after a rabies outbreak [29] was replicated
to illustrate, as the outcome of interest, the probability of
successful disease control. Finally, we investigated the African
wild dog Lycaon pictus, which is restricted throughout much of its
range and susceptible to several diseases, including rabies [37]. A
density-dependent PVA [30] for small wild dog populations was
reproduced to determine quasi-extinction probabilities (the
outcome variable), defined here as the probability of only one
sex remaining. Following Vial et al. [37], we also investigated the
effects of including a component Allee effect (a positive
relationship between population size and a measurable compo-
nent of fitness [38]) with respect to recruitment. Here, rather
than reducing pup mortality, individual litter size was assumed to
be an increasing function of group size, sensu Vial et al. [37].
These three investigations illustrate canids with small, medium,
and large mean litter sizes, respectively (Table S1). All modelling
and analyses were conducted in R 2.14.0 (R Development Core
Team 2011).
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Results
Probability distribution fitting
Variance-mean ratios of observed litter size frequencies (mean
= 0.41, SD 60.40) indicated that empirical distributions tend to be
underdispersed (Table S1). While the majority of datasets each
represented one population (96%), most data were pooled over
multiple years (97%) (Table S1). Best fitting distributions differed
substantially between datasets (Tables 2 and S2), although all
distributions with DAIC #6 provided fits consistent with the
Figure 1. Observed litter size frequencies with fitted distributions with DAIC
#
6. The top two panels show for a range of sample sizes (of
litters sampled), mean litter size, and carnivore families. The third panel from the top shows three populations of Vulpes vulpes with litter size
determined by placental scars and the bottom panel illustrates three different methods for determining litter size of a Bristol population of V. vulpes
(Harris, unpublished data). (A) Lycaon pictus, n = 36 [53]; (B) Crocuta crocuta, n = 108 [54]; (C) Panthera tigris altaica, n = 16 [55]; (D) Ursus arctos, n = 303
[56]; (E) Meles meles, n = 37 [57]; (F) Lontra canadensis, n = 9 [58]; (G) V. vulpes, n = 112 [59]; (H) V. vulpes, n = 506 [60]; (I) V. vulpes, London, n = 158
(Harris, unpublished data); (J) V. vulpes, placental scars, n = 340; (K) V. vulpes, embryos, n = 60; (L) V. vulpes, direct counts, n = 191. See Table S1 for
details of datasets. Distribution abbreviations: observed frequencies (Obs); shifted Poisson (SP); ZT Poisson (ZTP); discretised normal (DN); discretised
lognormal (DLN); discretised stretched beta –2 parameter form (DSB2); discretised stretched beta 3 parameter form (DSB3); shifted generalised
Poisson (SGP); ZT generalised Poisson (ZTGP); shifted binomial (SB); ZT binomial (ZTB); shifted negative binomial (SNB); ZT negative binomial (ZTNB).
doi:10.1371/journal.pone.0058060.g001
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empirical data (Table S3). For 97% of all datasets, several of the
12 candidate distributions (mean = 6.54, SD 63.38) could not be
discounted based on their AIC values (Table S2 and Fig. 1A–F for
examples). The most widely applicable distribution was the
discretised normal, with DAIC #6 for 95% of datasets; all other
distributions were selected for between 22% and 87% of datasets.
The ‘‘right shifted’’ method consistently performed better than
zero-truncation for all distributions (Table S2), being on average
1.32 (SD 60.16) times more likely to have a DAIC #6. While
there was little support for intraspecific differences between red fox
populations, distinct probability distributions best described litter
size data determined by pre- and post-birth methods (Appen-
dix S1 and Table S1).
Carnivore population models
The demographic modelling showed that the distribution
chosen to represent litter size uncertainty in the three canid
models has limited impacts, regardless of the fit of the
distributions. PVA models for island foxes showed that estimating
extinction probability was largely unaffected by the choice of
distribution, with less than 1% difference in quasi-extinction
probabilities between models that used the best and worst fitting
litter size distributions (Fig. 2A, B). Similarly, regardless of whether
the litter size distributions used in the model provided a good fit to
empirical litter size data, there was only a 2% difference in the
probability of successful disease control in the rabies model for red
foxes (Fig. 2C, D). Likewise, quasi-extinction probabilities for
Table 2. Model selection results for fitting probability distributions to carnivore litter size frequencies.
Distribution
Family Species SP ZTP SB ZTB SNB ZTNB SGP ZTGP DN DLN DSB3 DSB2
Canidae Vulpes velox 1/1 - 1/1 - - - 1/1 - 1/1 1/1 1/1 1/1
Vulpes macrotis --1/2--- --2/2 1/2 1/2 2/2
Vulpes vulpes 5/12 2/12 4/12 4/12 2/12 - 4/12 2/12 11/12 3/12 6/12 7/12
Urocyon littoralis 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 2/2 2/2 2/2
Urocyon cinereoargenteus 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 2/2 2/2 2/2 2/2
Alopex lagopus - - 1/3 - 1/3 - 1/3 1/3 2/3 2/3 3.3 3/3
Canis lupus 2/2 2/2 1/2 1/2 1/2 2/2 2/2 2/2 2/2 1/2 2/2 2/2
Lycaon pictus 1/4 1/4 - - 1/4 1/4 3/4 3/4 4/4 2/4 4/4 3/4
Nyctereutes procyonoides 1/1 1/1 1/1 1/1 - - 1/1 - 1/1 1/1 1/1 1/1
Hyaenidae Crocuta crocuta - - 1/3 1/3 - - - - 3/3 3/3 3/3 2/3
Procyonidae Procyon lotor 1/1 1/1 1/1 1/1 - - 1/1 1/1 1/1 1/1 1/1 1/1
Felidae Acinonyx jubatus - - 1/1 - - - - - 1/1 1/1 1/1 1/1
Felis concolor 1/3 - 2/3 2/3 - - - - 3/3 3/3 3/3 3/3
Felis iriomotensis 1/1 1/1 - - 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1
Lynx pardinus --1/1--- --1/11/1 1/1 1/1
Panthera tigris altaica 1/1 1/1 - 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1
Panthera onca 1/1 1/1 - 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 -
Panthera leo 2/6 - 3/6 1/6 - - 2/6 - 6/6 5/6 6/6 6/6
Panthera pardus --1/1--- --1/11/1 1/1 1/1
Leopardus pardalis 1/1 1/1 1/1 1/1 1/1 1/1 1/1 - 1/1 1/1 1/1 1/1
Ursidae Ursus maritimus ------ --4/4 4/4 4/4 4/4
Ursus arctos --2/4 --- --2/4 3/4 31/4 4/4
Ursus americanus 2/7 2/7 6/7 3/7 1/7 - 2/7 1/7 7/7 5/7 4/7 5/7
Mustelidae Lutra lutra 4/7 2/7 3/7 4/7 3/7 1/7 4/7 1/7 7/7 7/7 7/7 4/7
Lontra canadensis 2/2 2/2 2/2 2/2 2/2 1/2 2/2 2/2 2/2 2/2 2/2 2/2
Mustela erminea 1/1 1/1 1/1 1/1 - - 1/1 1/1 - 1/1 1/1 1/1
Mustela nigripes --1/1--- --1/1 1/1 1/1 1/1
Martes pennanti ------ --1/11/1 1/1 -
Martes americana 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1
Spilogale putorius 1/1 1/1 1/1 1/1 1/1 - 1/1 1/1 1/1 1/1 1/1 1/1
Gulo gulo - - 1/1 1/1 - - - - 1/1 1/1 1/1 -
Meles meles - - 1/2 1/2 - - - - 1/2 2/2 2/2 2/2
The number of datasets tested for each species (denominator, see Table S1 for details) and the number of datasets that were adequately fitted by a given distribution
(numerator, see Table S2 for details). Bold indicates distributions that were most parsimonious for at least one dataset. SP: Shifted Poisson; ZTP: Zero-truncated Poisson;
SB: Shifted binomial; ZTB: Zero-truncated binomial; SNB: Shifted negative binomial; ZTNB: Zero-truncated negative binomial; SGP: Shifted generalised Poisson; ZTGP:
Zero-truncated generalised Poisson; DN: Discretised normal; DLN: Discretised lognormal; DSB3; Discretised stretched-beta (3 parameter form); DSB2; Discretised
stretched-beta (2 parameter form).
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Figure 2. Model outcomes for 12 probability distributions against the variance (left panel) and skew (right panel) of distributions,
showing quasi-extinction probabilities and probability of successful disease control, with 95% confidence intervals. (A, B) Island fox
Urocyon littoralis PVA: west and east subpopulations; (C, D) red fox Vulpes vulpes; (E, F) African wild dog Lycaon pictus PVA without an Allee effect; (G,
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African wild dogs showed only a 1% difference among models that
employed different litter size distributions (Fig. 2E, F). When litter
size was reduced as a function of group size, to simulate an Allee
effect, the influence of the distributions was slightly greater
(Fig. 2G, H), with an increase of approximately 4% between quasi-
extinction probabilities for the best and worst-fitting distributions.
Even in this case, only models employing the worst-fitting
distributions differed substantially in their predictions from those
of models employing other distributions. The variation in the skew
and variance of the fitted distributions (Fig. 2) may be attributed to
process and sampling error in the data, as well as properties of the
distributions such as the tendency to favour overdispersion, e.g. the
negative binomial. However, for all parsimonious distributions,
these measures were generally consistent with the empirical
distributions for all models except island foxes (Fig. 2). In this
latter case, the variation in agreement between distributions with
DAIC #6 and the empirical properties (Fig. 2A, B) is probably due
to the small sample size increasing the uncertainty of the observed
parameter estimates, translating into the selection of multiple
distributions. Despite the widely varying variance, the resultant
model outcomes were in general unaltered by the choice of
distribution. Coefficients of variation (CV) were small for all model
outcomes (Table S4), with the greatest variation in the African
wild dog model with an Allee effect; the best-fitting distribution
(CV = 0.712) was 1.07 times more variable than for the worst
fitting model (CV = 0.668).
Discussion
Multiple distributions were shown to be consistent with the data
for describing litter size frequencies for a range of carnivore
species. However, the outcomes of demographic models appear
robust to the choice of litter size distribution. These findings are
discussed in light of the biological implications of litter size
distribution choice and the applied importance of incorporating
suitable probability distributions in demographic models.
Model selection for describing litter size variation
Unlike many biological parameters, offspring number is often
underdispersed [39,40] and positively skewed [41,42]. Litter size
frequencies are best fitted by probability distributions able to
describe the biological constraints on the upper limit of offspring
production. While the Poisson distribution is most commonly used
for fitting count data in general, it does not allow for under-
dispersion. In contrast, the generalised Poisson separates the
variance from the mean [19], allowing greater flexibility, but at the
cost of additional parameters. Of the continuous functions, the
discretised normal distribution is the most flexible and is suitable
for data characterised by low variance.
In a recent model of vertebrate reproductive success, the zero-
truncated generalised Poisson was consistently the best-fitting of
several parametric distributions fitted to litter size [19]. However,
that study only included one carnivore population, (lion, Panthera
leo), which was fitted solely by the zero-truncated-binomial. In our
study, that distribution performed less well, perhaps because more
competitive functions were considered (including shifted discrete
distributions and discretised continuous distributions) that were
not assessed in the earlier study [19]. The better fit of shifted forms
over zero-truncation suggests that further work is needed to
determine whether there is an underlying probabilistic mechanism
in the distribution of litter size.
The lack of evidence for intraspecific variation in underlying
litter size distributions (Appendix S1) could indicate that biological
limitations on reproduction allow for little intraspecific variation in
this trait. The known biases associated with litter size determina-
tion methods for red foxes [43,44] probably explain the observed
differences in litter size distributions (Appendix S1), although the
results of the management scenarios analysed in this study (see
next section) suggest that this finding is unlikely to be of
consequence for future modelling efforts. Given the pooling of
litter size datasets in this study over multiple years, due to
insufficient data, the results must be interpreted with caution in
light of potential temporal variation.
These analyses assumed that individuals had the same
underlying expected reproductive capacity. However, demograph-
ic heterogeneity in offspring production is influenced by many
factors, including female age, body condition or social status
[45,46], as well as maternal versus offspring trade-offs in
reproductive success [47,48]. The methods in these analyses could
be incorporated into population models that address such intrinsic
individual variation, as well as those modelling environmental
stochasticity.
Applied importance of litter size distributions
Despite interspecific variability in the consistency of distribu-
tions to describe litter size data, we have shown that model
outcomes of applied management scenarios, e.g. extinction risk,
may be robust to the distribution chosen to represent litter sizes.
The lack of any apparent effect of litter size distribution choice in
carnivore models might be because mammalian litter sizes are
generally small due to physiological limitations. Underdispersion
will promote sampling of offspring closer around the mean;
therefore, sampling variation will only weakly impact model
outcomes. There are indications that the distribution choice could
be important in limited circumstances. In the case of African wild
dog populations, the example presented here illustrates how
modelling a component Allee effect in reproduction using an ill-
fitting, underdispersed distribution can result in an overestimation
of extinction risk (see Fig. 2E–H).
Further work is required to determine the potential influence of
temporal variation in the underlying litter size distribution on
predictions of extinction risk. This is particularly important given
that temporal or environmental variability means that combining
data over time will inflate estimates of litter size variation, leading
to erroneous predictions of extinction risk. In spite of these
concerns, the lack of available data meant that pooling data was
necessary for our purposes; consequently, our results are indicative
only of how mis-specified distributions could affect model
predictions. As in [19], we stress that determining appropriate
distributions is a step towards a more mechanistic understanding
of litter size variability that could provide insight into a species’
response to selective pressures or management actions.
That litter size distributions have limited effects on the outcomes
of management models may also reflect the relative contributions
of life history traits to population growth. For long-lived species
such as carnivores [49], the elasticity of adult survival typically
H) African wild dog PVA with an Allee effect included as a decrease in litter size as a function of group size. Solid error bars indicate distributions with
DAIC #6. .indicates the estimate from the previously published model, with the empirical litter size variance in the left panels and empirical litter
size skew in the right panels (except G and H, for which there is no previous model estimate).
doi:10.1371/journal.pone.0058060.g002
Carnivore Litter Size Variation
PLOS ONE | www.plosone.org 7 February 2013 | Volume 8 | Issue 2 | e58060

contributes more to population growth than fecundity. Indeed,
variance in demographic parameters with low elasticities will have
little effect on the variance of the population growth rate, due to
the near linear relationship between population growth and vital
rates [50]. Notably, for all three canid populations in the models
presented here, the elasticity of survivorship is as high or higher
than fecundity [28,30,51], which is consistent with the limited
impact of litter size variation observed in the case studies.
Although this study focused on the Carnivora, our findings
should apply to taxa with multiparous females, including other
mammals, birds and lizards. While it is hard to determine the
exact ecological and physiological mechanisms generating a litter
size distribution, insight into the drivers of these empirical
distributions could aid our understanding of the adaptation of
reproductive strategies to extrinsic and intrinsic population
pressures. Recent work demonstrating that female red foxes
exhibit sex-biased investment in offspring as a function of body
mass and population density suggests that altering litter size
composition rather than litter size could be an alternative
mechanism for increasing fitness [52]. Ultimately however, applied
models for carnivores appear to be robust to choice of litter size
distribution, which has positive implications for modelling species
with limited data.
Supporting Information
Table S1 Summary of terrestrial carnivore litter size
data from published studies.
(DOC)
Table S2 Model selection for 12 probability distribu-
tions fitted to carnivore litter size frequencies, with
DAIC values.
(DOC)
Table S3 Results of the Fisher Exact test goodness-of-fit
of probability distributions to empirical carnivore litter
size frequencies.
(DOC)
Table S4 Coefficient of variation for model outcomes of
quasi-extinction probabilities and probability of suc-
cessful disease control, for 12 probability distributions.
(DOC)
Appendix S1 Testing for intraspecific variation in litter
size distributions, using the red fox Vulpes vulpes as an
example.
(DOC)
Appendix S2 Functional forms for the 12 probability
distributions fitted to empirical litter size frequency
data.
(DOC)
Appendix S3 Model descriptions for the three canid
management scenarios used to illustrate the conse-
quence of using different distributions to model litter
size.
(DOC)
Acknowledgments
We thank Helen Whiteside for assistance with the Bristol red fox litter size
data, Tim Coonan for providing litter size data for the island fox, the
Durham University Ecology Group for insightful discussion and one
anonymous reviewer for their constructive comments.
Author Contributions
Collated the data: ESDN SH. Conceived and designed the experiments:
ESDN SAR PAS. Performed the experiments: ESDN PAS. Analyzed the
data: ESDN SAR PAS. Wrote the paper: ESDN SH CS SAR PAS.
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