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Inbreeding depression and the detection of changes in the intrinsic rate of increase from time series

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1. Demographic changes can decrease the intrinsic population growth rate of species. Detecting these changes from ecological time series, however, is particularly challenging for small and declining populations, precisely the type of population that might urgently need targeted management responses. 2. Here, we present a statistical method to detect changes in the intrinsic rate of increase, r(t), using time series of count data. We focus on inbreeding depression as an endogenous driver that can exacerbate the extinction risk of endangered populations. We first use simulations to assess power and type I errors. We then analyse simulated inbred age-structured populations with life history parameters representing felid and ungulate species. As a case study, we analyse the wolf population on Isle Royale from 1959 to 1998, when inbreeding accumulated dramatically. 3. The method has good type I error rates for time series length ≥ 30 years, and statistical power can be high for time series of 40 years. We further found that constraining measurement error reduces type I errors. For the wolf population on Isle Royale, the model detected a strong decrease in r(t), consistent with inbreeding depression after accounting for changes in prey abundance and environmental conditions. 4. The approach we present offers a statistical way to detect time-varying demographic rates, incorporating solutions to problematic data features such as temporal confounding effects and measurement error. The statistical method can be tailored to different organisms and types of data and is a useful addition to a conservation scientist’s quantitative toolbox.
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Inbreeding depression and the
detection of changes in the intrinsic
rate of increase from time series
Technical Report
Claudio Bozzuto
Wildlife Analysis GmbH
Anthony R. Ives
University of Wisconsin-Madison
Zurich, 27 October 2020
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Inbreeding depression and the detection of changes in the
intrinsic rate of increase from time series
Technical report
Authors
Claudio Bozzuto Anthony R. Ives
Wildlife Analysis GmbH University of Wisconsin-Madison
Oetlisbergstrasse 38 Department of Integrative Biology
8053 Zurich Madison, WI 53706
Switzerland United States
bozzuto@wildlifeanalysis.ch arives@wisc.edu
Date published | 27 October 2020.
How to cite this document | Bozzuto, C., Ives, A.R. (2020): «Inbreeding depression and the
detection of changes in the intrinsic rate of increase from time series». Technical Report
Wildlife Analysis GmbH, Zurich, Switzerland. DOI: 10.13140/RG.2.2.23514.57289/1.
Copyright notice | The authors are the copyright holders, licence CC-BY-NC-ND 4.0.
Cover illustration | Andrea Klaiber, Doppelkopf Grafik & Illustration, Schaffhausen.
Keywords: felids, inbreeding depression, Isle Royale, population dynamics, state-space
model, time-series analysis, ungulates, wolves.
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Acknowledgements | We thank Karen B. Strier (University of Wisconsin-Madison) for the
idea of using wolves on Isle Royale as a case study. CB is grateful to Lukas F. Keller
(University of Zurich) for partially funding this project; ARI received funding from NSF DEB-
1052160 and DEB-1240804.
Author contributions | Both authors conceived the study, performed the analyses, and
wrote the report.
Data and code accessibility | All relevant data can be found in this document. The function
tviri(), written in the programming language R, along with the documentation is available
upon request from the authors.
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Contents
Abstract ………………….………………….………………….………………….……………
4
Introduction …………….………………….………………….………………….…………….
5
Materials and methods …………….………………….………………….…………………..
8
Model description …………….………….………………….……………………..
8
Simulation tests for power and type I errors .……..………….…………………
12
Simulations of inbreeding depression …………….………………….………….
13
Case study: wolves on Isle Royale ……………………………………………...
14
Results …………….………………….………………….………………….…………………..
16
Simulation tests for power and type I errors .……..………….…………………
16
Simulations of inbreeding depression …………….………………….………….
21
Case study: wolves on Isle Royale ……………………………………………...
22
Discussion …………….………………….………………….………………….……………...
25
References …………….………………….………………….………………….……………..
28
Supplementary Material …………….………………….………………….…………………
30
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Abstract
Demographic changes can decrease the intrinsic population growth rate of species. Detecting
these changes from ecological time series, however, is particularly challenging for small and
declining populations, precisely the type of population that might urgently need targeted
management responses.
Here, we present a statistical method to detect changes in the intrinsic rate of increase, !(!),
using time series of count data. We focus on inbreeding depression as an endogenous driver
that can exacerbate the extinction risk of endangered populations. We first use simulations to
assess power and type I errors. We then analyse simulated inbred age-structured populations
with life history parameters representing felid and ungulate species. As a case study, we
analyse the wolf population on Isle Royale from 1959 to 1998, when inbreeding accumulated
dramatically.
The method has good type I error rates for time series length 30 years, and statistical power
can be high for time series of 40 years. We further found that constraining measurement error
reduces type I errors. For the wolf population on Isle Royale, the model detected a strong
decrease in !(!), consistent with inbreeding depression after accounting for changes in prey
abundance and environmental conditions.
The approach we present offers a statistical way to detect time-varying demographic rates,
incorporating solutions to problematic data features such as temporal confounding effects
and measurement error. The statistical method can be tailored to different organisms and
types of data and is a useful addition to a conservation scientist’s quantitative toolbox.
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Introduction
Many questions arising in conservation biology revolve around the persistence of small,
endangered populations and the factors threatening their persistence (Morris and Doak 2002,
Ewen et al. 2012). Understanding persistence requires understanding the dynamics of the
studied population, in particular understanding possible changes in the synoptic measure of
population growth, the intrinsic rate of increase. To emphasize this point, we denote the
intrinsic rate of increase of a population, !(!), as a function of time. Important questions
include (i) whether the current !(!) allows persistence, (ii) how intrinsic and/or extrinsic
factors affected !(!) in the past, and (iii) how mitigation actions have affected, and hopefully
increased, !(!). Most studies on small populations take a static view of !(!); this is often
sufficient to assess the current status of a population, and is often necessitated by the
absence of good data. Nonetheless, studying how !(!) changed in the past and is expected
to change in the future offers richer insights and impetus for management.
Many factors can influence !(!), ranging from intrinsic factors such as inbreeding depression
to extrinsic factors such as habitat degradation and increased predation pressure (possibly
after the invasion of exotic species). It is beyond the scope of this report to study all possible
factors influencing !(!), and we focus on inbreeding depression for two reasons. First,
inbreeding is potentially an important cause of populations declining to extinction (Allendorf et
al. 2013, Bozzuto et al. 2019). Second, even though detrimental inbreeding effects on
individual fitness are relatively easy to document (Keller & Waller 2002), effects on individual
fitness do not necessarily imply a detrimental effect on the population as a whole (e.g.
Wootton and Pfister 2013). This is because there can be compensating factors, such as
intraspecific competition (Saccheri & Hanski 2006) or purging (Hedrick & Garcia-Dorado
2016), that (partially) counteract the individual-level detrimental effects of inbreeding.
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Assessing the possible population-level effects of inbreeding can be much more difficult than
assessing the individual-level effects (Keller et al. 2007), and we have targeted this more
difficult challenge.
Our goal is to discuss a method for detecting changes in !(!) from time series of population
count data. Time-series data from populations, including endangered populations, are much
more common than data sets combining information on both inbreeding and demographic
rates. Furthermore, for the majority of endangered populations, mostly count data are
available, as opposed to (st)age-structured data (Morris et al. 2002). To detect changes in
!(!), we therefore developed a statistical method to fit time series of population count data
and explanatory variables, if available. The statistical method allows testing for time-
dependent changes in !(!) and, if there are, reconstructing these changes.
Population count data, and time-series data more generally, present statistical challenges
because temporally autocorrelated data contain less information than is often assumed and is
always desired (Ives and Zhu 2006). When there is autocorrelation, the value of one data
point gives information about the value of a temporally nearby data point. This implies that the
two data points in combination contain less information than they would contain in the
absence of autocorrelation. This issue can be seen in the hypothetical case of two
populations that are declining towards extinction: in the first population (Fig. 1a, p. 18), !(!) is
initially greater than zero but decreases, leading to a decrease in the population size. In the
second case (Fig. 1b), !(!) is zero throughout the time series, and the population declines
from its initial value. Although the two time series look similar, knowing that there is a change
in !(!) in the first case could prompt management actions to increase genetic diversity if
inbreeding were a possible cause. Or it could prompt land protection or other management
actions to counteract recent changes that caused the decrease in !(!). We are not saying
that the second case (Fig. 1b, a declining population with !!0) is not a cause for concern
and management action. Nonetheless, statistically showing that !(!) is declining implies an
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immediacy of the intrinsic (e.g. inbreeding depression) or extrinsic (e.g. environmental
factors) changes that are driving a population extinct.
Studies have previously addressed the analysis of ecological time series by allowing model
parameters to vary through time (e.g. Zeng et al. 1998; Ives & Dakos 2012; Solbu et al.
2015). These authors adopt a hierarchical modelling approach, as we do in the present
article, and they discuss different functional forms to capture changes in the time-varying
parameters. What is missing, to our knowledge, is to cast these insights in a form relevant to
conservation scientists, for example in order to perform a count-based population viability
analysis. Thus, in the present study we go beyond presenting a general time-series model
with time-varying parameters by specifically addressing the needs of conservation scientists
to analyse data of small, declining populations. To this end, we pay particular attention to
characteristics of this kind of data, such as low population abundances, the effect of
measurement error on estimation precision, how rapidly the intrinsic rate of increase changes
through time, and the ability to detect changes in short time series.
After presenting the statistical model, we explore its type I errors and power to detect
changes in !(!). We then apply it to simulated time series of stylized ungulate and felid
populations based on a well-known age-structured model including inbreeding depression
(Mills & Smouse 1994). Finally, as a case study we analyse the wolf population on Isle
Royale, where inbreeding has been accumulating dramatically since their arrival on the
island.
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Materials and methods
Model description
The idea behind our approach is to model population dynamics with extrinsic environmental
variables if they are available and account for observation error (Dennis et al. 2006), while
letting some parameters vary through time. The dynamics of the population are described by
an autoregressive process with time lag !, AR !, which for !=1 is referred to as a
Gompertz model in the ecological literature. Previous studies have extended this framework
for the Gompertz and other lag-1 and lag-2 models by considering time-varying coefficients
according to a random walk, and considering time-varying coefficients for the general AR !
case with the additional inclusion of measurement error (Zeng et al. 1998, Ives & Dakos
2012). The random-walk variances of the coefficients are estimated from the data, and
estimates greater than zero imply that the dynamics given by the AR ! are changing. Here,
we formulate the AR ! process to have a density-independent coefficient corresponding to
the intrinsic rate of increase, !!, of the population on a log scale. We further tailor the error
variances to be appropriate for count data from small populations. As we describe below,
obtaining good statistical properties in statistical tests requires care in how the error variance
is treated. The function tviri() (time-varying intrinsic rate of increase) in the programming
language R is provided for fitting the model.
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The general specification of the model is
!!=!!1+!!!1!!1++!!!1!!!+
!" !1+!!+!!
(eq. 1a)
!!=!!1+!!!
(eq. 1b)
!!!=!!!1+!!,!!
(eq. 1c)
!!=!!+!!
(eq. 1d)
Here, !! is the unobserved, log-transformed population size at time !, and !! is the
observed population size that depends on the observation uncertainty described by the
random variable !!. The latent state variable !! is the intrinsic rate of increase whose
changes through time are modelled as a random walk with the normal random variable !!!
having variance !!
!. Thus, the key statistical test to identify changes in !! is based on the
null hypothesis H0:!!!
!=0. The autoregressive coefficients !!! (!=1,,!) are also
modelled as random walks with normal error terms !!,!! having variances !!,!
!. The state
variable !! can potentially depend on !!1, a measured environmental variable whose
effect on !! is given by the coefficient !; although written with only a single environmental
variable !!1, more environmental variables can be included in the obvious way. Further,
!! depends on !! and !!, two random variables capturing demographic and
environmental variation, respectively. Throughout the analyses we present here, for simplicity
we assume that !=1, although our findings apply to the more general case with added lags.
The simulated population trajectories shown in Fig. 1 were generated from this model with
!=1, !!0, and !!! constant.
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The choice to model !! as a random walk in equation (1) has both a structural and a
statistical motivation. In terms of the structural motivation, our preliminary analyses of the
time series for inbred felid and ungulate populations from the nonlinear stage-structured
model of inbreeding depression (Methods: Simulations of inbreeding depression) showed that
observed temporal changes in !! can be accurately described by an autoregressive
process with time lag 1 (Fig. S1). This result supports the structure of the model in which !!
is modelled as an autoregressive process (equation 1b). Modelling !! as an AR 1 process
assumes a constant underlying mechanism responsible for the accumulation of inbreeding
over time, in the present case hard selection (Supplementary Material). However, it is known
that environmental conditions can exacerbate inbreeding depression (Fox and Reed 2010).
Thus, modelling !! as a random walk allows capturing such interactions that would
otherwise be difficult to be modelled explicitly.
In terms of the statistical motivation, an alternative to treating !! as an autoregressive
process is to treat it as a deterministic function of time; for example, inbreeding depression
could be modelled as a decreasing linear time trend. We investigated the statistical properties
of such a model, but it invariably gave inflated type I errors (results not shown). The origin of
the inflated type I error rates can be seen by comparing Fig. 1a-b. When there is no change
in !! and the population is moving to a new stationary state at zero (Fig. 1b), the mean of
the transition distribution approaches zero geometrically. If instead !! decreased linearly,
the time-dependent change in the mean of the stationary distribution also approaches zero
roughly geometrically. This causes an identifiability problem in which both patterns, geometric
approach of the mean to its stationary value and geometric changes in the mean of the
stationary distribution, cannot be statistically distinguished; in an ecological context, this has
been referred to as a temporal confounding effect (Hefley et al. 2016). In our simulation
studies, maximum likelihood estimation often fit models with a linear trend in !! in favour of
fitting a constant !! with a population moving to a stationary distribution at zero, leading to
inflated type I errors for the hypothesis that !! was not changing. When we treat !! as a
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random walk, however, type I error control is much better, giving a practical solution to the
temporal confounding effect. A more detailed analysis of identifiability and type I errors is
presented in the Results section.
We assume that the abundance data follow a lognormal-Poisson distribution. Thus, the
expectation of abundance at time ! is exp !!, and the variance is proportional to this
expectation. On the log-transformed scale of !!, this implies that !! has mean zero and
variance approximately ln !!
!exp !!+1, where !!
! scales the variance (Fig. S2a): if
the data followed a Poisson distribution, then !!
!=1, while values of !!
! greater than one
imply over-dispersion. The variance of the random variable !!, capturing demographic
stochasticity, is also assumed to be proportional to the mean, as in a lognormal-Poisson
distribution, so the variance of !! is ln !!
!exp !!+1. In contrast to demographic
stochasticity, we assume that the variance of the environmental stochasticity !!, !!
!, is
constant on the logarithmic scale of !!.
We fit the model using a Kalman filter to compute the maximum likelihood (e.g. Harvey 1989;
Dennis et al. 2006; Wang 2009). The function tviri() allows treating the initial value of
!!, !!, in two ways. First, !! is estimated, and the variance in !! is assumed to be given by
the stationary distribution of !! conditional on the initial estimates of the AR ! coefficients.
Second, !! is assumed to be the first observed value, !1, and the variance is assumed to
be zero. For our simulations and application, we estimated !! under the assumption that its
variance given by the stationary distribution of !! at !=1. This is a conservative approach,
in the sense that it is assuming less information about !! is known than if !! were taken as its
observed value; we have found that this can lead to a loss of statistical power, although it can
also increase statistical power. For !!, and !!! if they are allowed to vary, the initial value
is estimated, and the variance is set equal to !!
! and !!,!
!, respectively.
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Simulation tests for power and type I errors
To explore statistical power and type I errors, we simulated data from equation 1 and then fit
the simulated data with the tviri model. To simulate demographic stochasticity, we computed
!! from equation 1a with !!0, and then pulled the simulated population size from a
Poisson distribution with mean exp !!, which corresponds to !!
!=1. To simulate
measurement error, we sampled from a Poisson distribution with mean equal to the "true"
simulated population size; this allows the observed population size to exceed the true
population size, as might happen if individuals were sampled multiple times. Environmental
variability was given by !!
!=0.2.
For the power simulations, we mimicked dynamics of small, introduced populations that
would be most susceptible to inbreeding depression. The strength of density dependence in
the tviri model is determined by the autoregressive coefficient !, and as we show below
(Results), the value of ! affects both power and type I errors. Therefore, we simulated values
with != 0.2, 0.4, 0.6, and 0.8, implying strong to weak density dependence. For each value
of !, we selected an initial value of !! such that the mean population size conditioned on
!1 is 200 individuals; these were !1= 1.06, 2.12, 3.18, and 4.24, respectively. We then
assumed that !! decreased geometrically to a value of !1/2 over 40 time steps, the
length of the simulated time series. The ability to detect changes in !! also depends on the
initial population size, since initial increases from low values will make it statistically easier to
estimate values of !! early in the time series. Therefore, we simulated time series with
initial abundances of exp !!= 20, 50, 100, and 200. Figure 1a gives an example time
series from this simulation, in which != 0.2 and exp !!= 200.
To test the null hypothesis of no change in !! (i.e. !!
!=0), we fit the full model and the
reduced model in which !!
!0, and performed a likelihood ratio test (LRT) with one degree
of freedom at the nominal level !=0.05. When testing variances which are constrained to
be positive, LRTs follow a mixture of !! distributions, and Ives and Dakos (2012) showed that
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for generalized AR p models with time-varying parameters, these mixture distributions gave
good statistical tests. Here, however, we use standard LRTs with a !! distribution having
degrees of freedom equal to the difference in number of parameters between models;
because the tviri model in some situations can have inflated type I error inflation, using a
standard LRT is more conservative.
The power tests identified the case of weak density dependence (!= 0.8) and high initial
population size (200 individuals) as the greatest challenge for detecting changes in !!.
Therefore, we built our simulations to explore type I errors around these conditions. We
assumed a constant !!0, so that the population would eventually go deterministically
extinct. To give very weak density dependence, and hence a long expected time until
extinction, we set != 0.92, 0.93, ... 0.99. To standardize the lengths of the time series, only
simulated time series that were non-zero at the last time step were included. Figure 1b gives
an example time series from this simulation, in which != 0.98.
Simulations of inbreeding depression
To investigate the statistical properties of the model when applied to data from a population
experiencing inbreeding depression, we simulated time series of inbred populations using the
stochastic age-structured model of Mills and Smouse (1994), parameterized for stylized felids
and ungulates (Table S1), where we additionally included density-dependent reproduction.
Both species had the same number of age classes, but they differed in the maximum
multiplicative growth rate: exp !
!= 1.05 and 1.24 for ungulates and for felids, respectively.
We started all populations with 50 individuals, and we set the strength of density dependence
so that both ungulates and felids would reach the same carrying capacity of approximately
230 animals in the absence of inbreeding depression. We let survival follow a beta
distribution and fecundity a Poisson distribution; we let survival and fecundity be correlated,
so that for example “good” environmental conditions produced both high survival and high
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fecundity. We added observation error by assuming that each animal in the population had a
yearly probability of detection of 90%, with no chance of repeated sampling (Fig. S2b). We
set all inbreeding coefficients for all age classes at the beginning of the time series to zero.
For simplicity, we assumed that inbreeding costs for survival and fecundity were both age-
independent and of equal value. For both species, we chose no, low (ungulates: 0.75; felids:
0.25), and high (ungulates: 1.5; felids: 0.50) inbreeding costs to assess how detection
depends on the strength of inbreeding depression (Supplementary Material); despite the
much higher costs for ungulates, the simulated time series for both species were comparable
in shape. This comparison was designed to highlight the difficulty of detecting inbreeding
depression for a species with a "slow life history" by comparing ungulates to felids. For each
combination of life history parameters and inbreeding costs, we simulated 200 time series of
lengths 20, 30, and 40 years, and for low inbreeding costs also of length 50 and 60 years; for
high inbreeding costs numerous populations went extinct after 50 or more years. For each
simulation, we summed animals regardless of age to give count time series. Finally, each
simulation also produced time series of inbreeding coefficients !! and !! that were saved
to analyse the results from the fitted models. Further implementation details are given in the
Supplementary Material.
Case study: wolves on Isle Royale
Around 1950, wolves colonised Isle Royale, Lake Superior, from the nearby Canadian
mainland. Research on this system now spans almost six decades, with monitoring starting in
1959. The wolves prey heavily on moose, which make up the majority of their diet on Isle
Royale, generating tight dynamical coupling in this predator-prey system (Peterson & Page
1988).
The small wolf population had an estimated effective population size of 3.8, and the observed
high incidence of congenital bone deformities can be attributed to inbreeding (Räikkönen et
al. 2009). By the late 1990s, the population-level inbreeding coefficient was estimated to be
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!= 0.81, and 85% of the wolves had congenital bone deformities, compared to 1% in
outbred populations (Adams et al. 2011). Nonetheless, demographic rates appeared to be
comparable to outbred wolf populations, and evidence for inbreeding depression affecting
population dynamics was only circumstantial (Räikkönen et al. 2009). In 1998, a single male
wolf (M93) immigrated from the Canadian mainland and generated a “genomic sweep”: !
decreased to below 0.2 within a few years (Adams et al. 2011). Nonetheless, this genetic
rescue was of short duration, and ! then started to increase again (Hedrick et al. 2014). The
wolves experienced a steady decline in abundance with only two wolves left in the winter
season 2017/18 (Peterson, Vucetich and Hoy 2018). Given the impact of M93 on the level of
inbreeding, we analysed data from 1959 to 1998 (Fig. 3a; http://www.isleroyalewolf.org,
download June 2015).
To model wolf dynamics (eq. 2), we used equation 1a with two independent variables: !!!
for the log-transformed population of moose older than nine years (Vucetich & Peterson
2004) and !!! for the impact of a canine parvovirus epizootic (Fig. 3b). Although wolves
show a clear preference for moose calves and senescent moose, given the explicit time
series of senescent moose in the data file, we included this prey stage group as covariate.
!!=!!1+!" !1+!!!!!1+!!!!!1+!!+!!
(eq. 2)
where !!! is an indicator variable identifying years 1980-1981 when canine parvovirus had
the most pronounced effect on the population (Wilmers et al. 2006).
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Results
Simulation tests for power and type I errors
We assessed the power of a likelihood ratio test (LRT) to reject the null hypothesis H0:
!!
!=0 by counting the proportion of 500 simulated data sets for which !!
! differed from zero
at the !=0.05 significance level (Table 1). Overall, the tviri model had good power to detect
changes in !! when the initial population size was 20 individuals, especially when there
was intermediate density dependence (!= 0.4 and 0.6; Table 1). However, as the initial
population size increased, power dropped, with low power when exp !!= 200. This is not
surprising; to illustrate the challenge, Fig. 1a shows the case of weak density dependence
(!= 0.8) and a population starting at exp !!= 200. As !! decreases from 1.06 to 0.53,
the population decreases roughly monotonically. The time series looks similar to the case
with constant !!0 and very weak density dependence (!= 0.98), leading to a nearly
random walk towards zero (Fig. 1b). The similarity of these two time series with and without
changing !! suggests that it should be difficult to reject the null hypothesis that !! shows
no temporal trend. Thus, it is not surprising that for the tviri model fit to the time series, the
maximum likelihood (black line, Fig. 1c) occurs with ! very close to 1 (near-random walk
behaviour), where the estimate of the variance in !!, !!
!, is zero (red line, Fig. 1e).
Examination of the fit of the tviri model across values of !, however, reveals a pattern that
may help to identify changes in !!. At the ML estimate of != 0.986 (black line, Fig. 1c), the
estimate of !!
! is 2.35 (black line, Fig. 1e). Because the observation of individuals in the
population was assumed to be Poisson, the expected value of !!
! is 1. If the tviri model is fit
to this data set while constraining !!
! to the value one, the ML estimate of ! is 0.073, and
! ! ! ! !
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the estimate of !!
! is 0.18. This global maximum in the log likelihood coincides with a local
maximum for the unconstrained tviri model, and in the unconstrained model the estimate of
!!
! is close to one (black line, Fig. 1e). In a LRT, the null hypothesis H0:!!!
!=0 is rejected
(!!
! = 7.91, P = 0.0049). Therefore, constraining !!
! = 1 leads to the rejection of the null
hypothesis. For the simulated time series used to evaluate statistical power with the full tviri
model we also calculated the proportion of simulations rejected by the constrained tviri model
(!!
! = 1; Table 1). Power was still low for exp !!= 200, although for initial sizes of 100 and
200 individuals power was considerably higher than for the unconstrained tviri model.
Detecting a significant change in !! under the constraint !!
! = 1 is possible because it
forces the model to fit the long-term patterns in population fluctuations. When the true
dynamics show very weak density dependence (Fig. 1b), there is little variation in
abundances between time steps; very weak density dependence leads to very high
autocorrelation, and most of the observed step-to-step variation is caused by measurement
variation. Measurement variation is relatively small, with the ML estimate of !!
! close to one
(black line, Fig. 1f). In comparison, when the decline in the population is caused by a
changing !! (Fig. 1a), there is greater variation in abundances between time steps. This
indicates either stronger density dependence and fluctuations around a changing mean
population caused by changes in !!, or short-term effects of measurement error. The
measurement error required for this magnitude of variation between time steps, however, is
high (!!
!> 2). If we know that sampling follows a Poisson distribution and use the resulting
expected value of !!
! = 1, then we can exclude measurement variation as the cause of
variation in abundance between time steps. Hence, we conclude that !! changes.
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Figure 1 | Two simulated time series of small and declining populations. In panel (a), the intrinsic rate of
increase, !!, decreases from 1.06 to 0.53 (red circles) and the population drops from 200 to 16 individuals, while
in panel (b) !!0 and the population drops from 200 to 5. The tviri model (eq. 1) fit to !! and the population
size under the constraint !!
!= 1 are given by red and black lines. Panels (c) and (d) give the log-likelihood values
as a function of the density dependence parameter ! for the unconstrained model (black line) and constrained
model with !!
!= 1 (red line). Panels (e) and (f) give the ML estimates of !!
! (red line) and !!
! (black line) from the
unconstrained tviri model conditional on the value of !.
r(t)%
r(t)%
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Table 1 | Proportion of 500 simulated data sets for which the null hypothesis H0: !!
! = 0 was rejected by a
likelihood ratio test (LRT) based on a !!
! distribution with a significance level of ! = 0.05. The results are
ordered by the strength of density dependence !, by the initial population size, and by how measurement error
was treated (either unconstrained and estimated, or fixed to !!
! = 1); all simulations with ! > 0.9 were started with
an initial population size of 200 individuals. Results for ! > 0.9: type I errors, i.e. in the absence of a decrease in
!!; all other results: power analysis, i.e. in the presence of a decrease in !!.
!
Initial population size
20
50
100
200
!!
! estimated
0.99
-
-
-
0.101
0.98
-
-
-
0.068
0.97
-
-
-
0.062
0.96
-
-
-
0.053
0.95
-
-
-
0.015
0.94
-
-
-
0.017
0.93
-
-
-
0.011
0.92
-
-
-
0.015
0.80
0.72
0.63
0.51
0.16
0.60
0.97
0.95
0.53
0.04
0.40
0.88
0.86
0.55
0.05
0.20
0.58
0.53
0.31
0.00
!!
! = 1
0.99
-
-
-
0.092
0.98
-
-
-
0.054
0.97
-
-
-
0.046
0.96
-
-
-
0.030
0.95
-
-
-
0.008
0.94
-
-
-
0.010
0.93
-
-
-
0.006
0.92
-
-
-
0.012
0.80
0.72
0.66
0.42
0.19
0.60
1.00
0.98
0.72
0.15
0.40
1.00
0.95
0.91
0.27
0.20
0.68
0.97
0.74
0.29
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Table 2 | Proportion of simulated times series of lengths 20-60 for which the null hypothesis H0: !!
! = 0 was
rejected by a likelihood ratio test based on a !!
! distribution with a significance level of ! = 0.05. Data were
simulated using the age-structured model of Mills and Smouse (1994) parameterized for felid and ungulate life
histories, including density-dependent reproduction. Inbreeding costs were set to zero, low, and high. The
statistical fitting was performed either estimating !!
! or setting !!
! = 1. Two hundred simulations were performed
for each parameter combination.
Time series length (!)
Felids
Ungulates
zero
low
high
zero
low
high
!!
! estimated
20
0.02
0.04
0.08
0.07
0.06
0.14
30
0.01
0.01
0.32
0.05
0.06
0.34
40
0.02
0.06
0.77
0.06
0.13
0.55
50
0.02
0.16
0.04
0.20
60
0.01
0.51
0.02
0.39
!!
! = 1
20
0.01
0.01
0.01
0.03
0.03
0.06
30
0.01
0.01
0.25
0.04
0.02
0.26
40
0.00
0.02
0.78
0.02
0.12
0.56
50
0.01
0.14
0.05
0.18
60
0.01
0.47
0.03
0.36
We tested the tviri model for type I errors, focusing on the case of very weak density
dependence and exp !!= 200, because this is the case that poses the greatest challenges.
When there was weak density dependence (! 0.96), the test with unconstrained !!
! had
inflated type I errors, with a greater proportion of simulated data sets rejected than the
nominal level of != 0.05 (Table 1). The constrained case with !!
! = 1 had better type I error
control, although very weak density dependence (!= 0.99) led to inflated type I errors. The
likely explanation for this is similar to the explanation for the low power to detect changes in
!! when there is weak density dependence: when density dependence is weak, the
population dynamics approach a random walk. In this case, the model can (correctly)
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estimate a high value of ! that attributes this behaviour to weak density dependence, or it can
(incorrectly) estimate a high value of !!
!, which implies that !! is a random walk that drives
the random walk of the population density. Thus, in the limit as ! approaches 1, there is an
identifiability problem, because in the tviri model both the dynamics of !! conditional on
!! give a random walk, and the dynamics of !! give a random walk.
Simulations of inbreeding depression
The model had a good type I error control for felid time series, while for ungulate time series
type I error was inflated especially for shorter time series (Table 2). In the presence of
inbreeding costs, the power to reject the null hypothesis expectedly increased with time
series length. For felids, characterized by a fast life history, time series of length 40 were
sufficient to reject the null hypothesis for up to almost 80% of the time series with high
inbreeding costs, although time series of length 60 were needed to reject the null hypothesis
in approximately 50% of the simulations with low costs. Rejection rates for the slower growing
ungulate populations were lower, indicating lower power to statistically detect changes in
demographic rates, in line with the findings from the generic simulations. Finally, constraining
measurement error (!!
! = 1) did not appreciably improve results.
Because the population sizes in simulated time series varied stochastically, independently of
the effects of inbreeding depression, the magnitude of inbreeding varied among time series
even under the same model parameters. We therefore further investigated whether the level
of inbreeding and the final population size in the simulations were associated with the
rejection of H0: !!
!= 0. In general, compared to ungulates time series, felid time series of
length 40 were characterised by less variation in both final abundance and final inbreeding
coefficient value (Fig. 2); Fig. S3 shows the variation over the whole time series length.
Furthermore, felid populations tended to achieve higher final inbreeding coefficient values
(Fig. 2). These characteristics together help to explain the greater power to reject H0: !!
!= 0
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for populations with a fast life history (Table 2): a more dynamic abundance trajectory due to
a higher intrinsic growth rate (Table S1, Fig. S3), together with a faster accumulation of
inbreeding that rapidly reduces !! (Fig. S4), helps to better discern the contribution of a
changing !! to the dynamics. In contrast, populations with a slow life history are
characterized by a less dynamic trajectory (Fig. S3), also due to a weaker density
dependence (Supplementary Material), and it is more difficult to detect changes in population
dynamics and thereby reject H0: !!
!= 0.
Case study: wolves on Isle Royale
Because we expected measurement error to be very low (Vucetich & Peterson 2004), we
fitted the tviri model without constraining !!
!. The null hypothesis of no change in !! through
time was rejected (!!
! = 13.8, P = 0.0002), thus supporting the hypothesis of inbreeding
depression reducing the wolf population’s growth rate; all parameter estimates are reported in
Table S2. Fig. 3b shows how exp !! decreased by approximately 50% over 40 years.
Further, notice how this reduction was exacerbated during the years coinciding with canine
parvovirus, in line with previous findings (Wilmers et al. 2006).
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Figure 2 | Final inbreeding coefficient, !!" , versus final abundance, !!" , scaled by the carrying
capacity (! = 232), of all simulated time series of inbred populations of length 40 years (Table 2). Results
are ordered by inbreeding costs (columns) and life history (rows). Within each panel, analyses leading to
significant results are shown in red.
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Figure 3 | Dynamics of the wolf population on Isle Royale. Panel (a) shows the wolf data from Isle Royale and
the fitted abundance trajectory !! from the tviri model (eq. 2), including 95% confidence interval. Panel (b) gives
the estimate of exp !!, including 95% confidence interval (left y-axis), years with canine parvovirus (CPV)
(1980 and 1981; grey bars, left y-axis), and the population of moose over 9 years old (right y-axis).
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Discussion
Directly testing for possible inbreeding effects at the population level is challenging, because
it requires, at the minimum, genetic information on inbreeding and data on demographic
changes in populations. We have presented a statistical approach to test whether there are
time-dependent intrinsic changes in the dynamics of populations using only time-series count
data, the most common type of data available for populations of concern. While the approach
does not test directly for inbreeding effects, it nonetheless can detect the hallmark of
inbreeding depression at the population level. We show this using simulated data to validate
the statistical properties of the approach, and then applying the model to a case study of
wolves on Isle Royale to show strong statistical evidence of population dynamics consistent
with inbreeding depression.
We focused on small and declining populations, and we asked what characteristics of the
data, as well as of the species, increase power to detect a potentially declining growth rate.
An expected finding, albeit not that helpful for situations calling for prompt actions, is that
longer time series increase power to detect changes in the intrinsic rate of increase, !!
(Table 2). More helpful is the effect of life history (Table 2): we showed that, all else being
equal (equal carrying capacity and inbreeding costs affecting the dynamics in a similar way),
identifying changes in !! was facilitated by a fast life history with a high overall intrinsic rate
of increase (felids) compared to a slow one (ungulates). A fast life history will cause a
growing population to quickly approach carrying capacity, and at the same time it will cause a
faster accumulation of inbreeding that in turn could speed the decline in !!; the speed of
this decline is additionally affected by inbreeding costs that might or might not correlate with
life history (Mills and Smouse 1994; Hedrick and Garcia-Dorado 2016 and references
therein). In general, a faster life history will speed the demographic processes underlying a
time series, and the resulting dynamism of the trajectory will increase the information
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available for inferring changes in !! (Fig. S3). Thus, the greatest challenges for applying
the tviri model will probably arise for long-lived species with slow life histories and weak
density dependence.
In addition to explicitly considering measurement error, a recommended and now-standard
practice (e.g. Dennis et al. 2006), the tviri model also includes demographic stochasticity, an
important model ingredient for small populations (Engen et al. 1998). In principle, it is
possible to disentangle and estimate all variances in equation 1. In practice, however, this is
likely to be difficult given widespread data constraints in ecology and especially conservation
biology. We found that fixing, rather than estimating, measurement error can increase power
to detect changes in !! (Table 1, Fig. 1); this is a good strategy when the true
measurement error variance can be estimated by independent means not involving the time
series being analysed. We caution that in real data, it can be ill advised to make a priori
assumptions about the measurement variation. For example, there are many ways in which a
random survey design will lead to measurement error with greater-than-Poisson variance.
Nonetheless, if measurement error can be quantified independently, such as by replicating
survey samples, the estimated measurement variation can lead to greater power to detect
changes in !! and also better type I error control (Table 1).
The tviri model is designed to detect changes in !! caused by any endogenous changes to
the population, or due to exogenous but unmeasured changes. We have focused on
inbreeding depression, although the tviri model could be used in other contexts. For example,
a small population might be vulnerable to an increase in predation (including an increase in
harvesting), an increase in the prevalence of an endemic disease, or a decrease in
availability of vital resources. Furthermore, a deteriorating environment could change other
demographic properties in addition to !! (e.g. Solbu et al. 2015; Hefley et al. 2016). In such
cases, it is advisable to fit time-varying autoregressive coefficient(s) !! as well as !! (eq.
1). If changes in !! and/or !! are found, this would help conservation scientists to decide
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whether detailed information should be collected and analysed for possible candidate
endogenous or exogenous drivers.
Our approach gives a statistical tool for conservation management that widens traditional
count-based population viability analyses (PVA). The approach can include all of the
components found in a traditional PVA, such as density dependence, demographic and
environmental stochasticity, measurement error, and covariates reflecting introductions or
removals (e.g. Morris and Doak 2002). The additional ability in tviri of letting !!, and !! if
appropriate, vary through time makes it possible to determine whether there are on-going
changes that are driving decreases in a population. Identifying on-going changes affecting a
population could lead to different management strategies than those that would be enacted if
current declines in a population were the consequence of long-term problems. From a
statistical perspective, tviri also helps to overcome the temporal confounding effect in which
monotonic changes in environmental drivers cannot be separated from directional changes in
the time series. For example, a tviri model could be fit that does not include an environmental
driver in question, and if there are statistically significant changes in !!, then the
reconstructed trajectory of !! could be compared to the environmental driver in question to
determine if it could be responsible for the reconstructed !!. This two-step approach could
avoid the statistical identifiability problem and resulting inflated type I errors of models that
include the environmental driver (Hefley et al. 2016). In general, the reconstructed trajectory
of !! could a posteriori be used for a time-varying phenomenological function to be fitted to,
in order to project the most likely evolution of !! and !!. A further advantage of the tviri
approach is that models can be specifically tailored for the time series in question; we did this
for the Isle Royale wolf data, in which moose abundance and a canine parvovirus epizootic
were included in the model to absorb variance associated with these factors and improve the
fit of the model to the wolf time series. Thus, the flexibility of the tviri approach allows its use
in numerous ways, whenever detecting changes in the dynamics of populations from time
series is desired.
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Peterson, and J. A. Vucetich. 2011. Genomic
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Bozzuto, C., I. Biebach, S. Muff, A. R. Ives, and L. F.
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Ecological Studies of Wolves on Isle Royale:
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29
Solbu E. B., S. Engen, and O. H. Diserud. 2015.
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Influence of prey consumption and
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growth of Isle Royale wolves Canis lupus.
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ecological data using Bayesian and non-
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Wilmers, C. C., E. Post, R. O. Peterson, and J. A.
Vucetich. 2006. Predator disease out-break
modulates top-down, bottom-up and climatic
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!
!
!
!
!
!
!
! ! ! ! !
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Supplementary Material
Supplementary Methods ……………………………………………………………………….
31
Supplementary Tables (S1 S2) ……………………………………………………………..
34
Supplementary Figures (S1 S3) …………………………………………………………….
35
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Supplementary Methods
We give a very brief summary of the model presented in Mills and Smouse (1994) and used
in the present study as simulation model; for additional information please refer to the original
work. In addition, we also present how we included density dependence, and we present
additional implementation details for generating the stochastic test time series.
The deterministic version of the age-structured demographic model can be written as
!!+1=!!!!, where !! is a !!×!! pre-breeding projection matrix that is time-
dependent through the ! age classes’ inbreeding levels, and !! is a !!×!1 column vector
with the abundances of ! age classes; equation S1 gives an example for three age classes.
!!
=
!!!
!exp !!!!!!!!!!!!!
!exp !!!!!3!!!!!3
!!exp !!!!!00
0!!exp !!!!!20
(S1)
Here, !!=!!/1!(!), !!! and !!! are inbreeding costs on newborn survival and
survival of age class n, respectively, and !!! are age-dependent inbreeding costs on fertility;
in Mills and Smouse (1994) there is only a distinction between juvenile and post-juvenile
inbreeding costs on survival and no age-specificity of inbreeding costs on fecundity. The
function !! arises when inbreeding depression is caused by synergistic effects.
The inbreeding coefficient of newborns at time t, !
!(!), is calculated as
!
!(!)=11!(!)1
1
2!!(!)
(S2)
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Here, !(!) is a weighted average of the !-values for the previous ! cohorts contributing
offspring to the new generation at time t, and effective population size !!(!) is expressed in
terms of population size, vital rates, and inbreeding costs (Mills and Smouse 1994).
We assumed that density dependence caused a reduced number of newborns in year ! as a
function of the total abundance in the previous year, !!1. For instance, the number of
newborns at time t and born to parents in age-class ! is !!!!!!=!!!
!exp !!!!!
!!!!!!!!" !1!!!1, where ! measures the strength of density
dependence. Finally, to calculate the time-varying intrinsic rate of increase !! we log-
transformed the dominant eigenvalue of the density-independent projection matrix in a given
year prior to applying density-dependent reproduction in that year.
As mentioned in the main text, we chose 50 animals as our starting abundance: higher
(lower) values would lead to a slower (faster) increase in inbreeding and thus in a reduced
(enhanced) effect of inbreeding depression on population dynamics. Further, all simulations
were started with an initial inbreeding level of zero, noting that increasing this value leads to
more pronounced inbreeding effects. Finally, we set the density dependence parameter ! =
0.001 for ungulates and ! = 0.0037 for felids, leading the same carrying capacity (k) of
approximately 230 animals without inbreeding depression (k = 232).
As introduced in the main text, we simulated inbred populations with low and high inbreeding
costs: 0.75 and 1.50 (ungulates), and 0.25 and 0.50 (felids). We based our choice of these
values on the simulation outputs and preliminary analyses. One constraint was the production
of time series length of 40 years: given the different density-independent asymptotic growth
rates (Table S1), the chosen high costs led to similar (extended) times to generate enough
time series (200 each). Mills and Smouse (1994; see also Hedrick and Garcia-Dorado 2016)
reviewed published values and found that synergistic effects already factored in the
median inbreeding costs for juvenile survival is 0.8 (upper quartile = 1.4), post-juvenile
survival is 1.0, and fecundity approaches 1.0.
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Finally, we implemented stochasticity similar to Mills and Smouse (1994), albeit with a lower
coefficient of variation (CV = 0.2 instead of 0.3). We implemented stochastic survival using a
beta distribution, with means given in Table S1 and variances calculated using the means
and the CV. For stochastic fecundity, we used a Poisson distribution with means given in
Table S1. Finally, following Mills and Smouse (1994), we also correlated these random
variables: a good (bad) year is not only good (bad) for survival but also for reproduction.
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Supplementary Tables and Figures
Table S1. Life history parameters for a stylized ungulate and felid population: survival (S), fecundity (M), the stable
age distribution (STAD), and the normalized reproductive value (RV); survival and fecundity values are from Mills
and Smouse (1994). Rows correspond to age classes, and the last row, !, contains the respective leading
eigenvalue of the population projection matrix.
Ungulate population
Felid population
S
M
STAD
RV
S
M
STAD
RV
0-1
0.65
0
0.177
1.000
0.60
0
0.371
1.000
1-2
0.90
0
0.152
1.167
0.75
0
0.225
1.649
2-3
0.95
0.15
0.138
1.187
0.80
1.00
0.146
1.798
3-4
0.95
0.40
0.124
1.039
0.80
1.20
0.094
1.879
4-5
0.95
0.50
0.113
0.807
0.85
1.50
0.065
1.674
5-6
0.95
0.50
0.102
0.549
0.85
1.50
0.045
1.377
6-7
0.95
0.50
0.092
0.265
0.80
1.50
0.029
1.003
7-8
0.80
0.30
0.070
0.105
0.70
1.50
0.016
0.485
8-9
0.40
0.15
0.027
0.031
0.60
1.00
0.008
0.000
9-10
0.20
0.05
0.005
0.000
0.30
0.00
0.002
0.000
10-11
0
0
-
-
0
0
-
-
!
1.0502
1.2364
Table S2. Parameter estimates and P-value for the wolf population using the tviri model. P-value: LRT, ! = 0.05.
!(!=!)
3.0001
!!
!
0.0053
!(