Chevin et al 2015

Chevin et al 2015
O R I G I NA L A RT I C L E
doi:10.1111/evo.12741
Estimating the variation, autocorrelation,
and environmental sensitivity of phenotypic
selection
Luis-Miguel Chevin,1,2 Marcel E. Visser,3 and Jarle Tufto4
1
CEFE-CNRS, UMR 5175, 1919 route de Mende, 34293 Montpellier 05, France
2
3
E-mail: [email protected]
Department of Animal Ecology, Netherlands Institute of Ecology (NIOO-KNAW), Post Office Box 50, 6700AB
Wageningen, Netherlands
4
Centre for Biodiversity Dynamics/Department of Mathematical Sciences, Norwegian University of Science and
Technology, 7491 Trondheim, Norway
Received February 12, 2014
Accepted July 8, 2015
Despite considerable interest in temporal and spatial variation of phenotypic selection, very few methods allow quantifying this
variation while correctly accounting for the error variance of each individual estimate. Furthermore, the available methods do not
estimate the autocorrelation of phenotypic selection, which is a major determinant of eco-evolutionary dynamics in changing environments. We introduce a new method for measuring variable phenotypic selection using random regression. We rely on model
selection to assess the support for stabilizing selection, and for a moving optimum that may include a trend plus (possibly autocorrelated) fluctuations. The environmental sensitivity of selection also can be estimated by including an environmental covariate.
After testing our method on extensive simulations, we apply it to breeding time in a great tit population in the Netherlands.
Our analysis finds support for an optimum that is well predicted by spring temperature, and occurs about 33 days before a peak
in food biomass, consistent with what is known from the biology of this species. We also detect autocorrelated fluctuations in
the optimum, beyond those caused by temperature and the food peak. Because our approach directly estimates parameters that
appear in theoretical models, it should be particularly useful for predicting eco-evolutionary responses to environmental change.
KEY WORDS:
Fluctuating selection, Gaussian fitness function, generalized linear-mixed models, poisson regression.
The extent to which natural selection changes in time and space
has profound implications for the dynamics of adaptation, from
the maintenance of polymorphism (reviewed by Felsenstein 1976;
Hedrick et al. 1976; Hedrick 2006; Bell 2010) to the evolution of
bet hedging and/or phenotypic plasticity (Gillespie 1973, 1974;
Via and Lande 1985; Bull 1987; Gavrilets and Scheiner 1993;
Lande 2009; Svardal et al. 2011; Tufto 2015). Apparent temporal variation in selection is a common feature of several classic
examples of natural selection in the wild (reviewed in Bell 2010;
Calsbeek et al. 2012; Svensson and Calsbeek 2012), from beak
shape in Darwin’s finches (Grant and Grant 2002) to banding patterns in Cepea snails (Cain et al. 1990), or spine number in three-
C
2319
spined sticklebacks (Reimchen and Nosil 2002). Local adaptation
also is pervasive (Hereford 2009), indicating that natural selection often varies in space too. But despite a long-lasting interest
in variable selection, the latter is often reported mostly as a qualitative pattern, and there are few quantitative measurements of the
magnitude of variation in selection at the phenotypic level (but
see Calsbeek et al. 2012; Engen et al. 2012). In particular, the few
available methods do not treat phenotypic selection across time or
space as an actual stochastic process, but rather as a simple random variable, and thus provide no measure of its autocorrelation.
An important aspect of variation in phenotypic selection is the relationship between phenotypic selection and the
C 2015 The Society for the Study of Evolution.
2015 The Author(s). Evolution Evolution 69-9: 2319–2332
LUIS-MIGUEL CHEVIN ET AL.
environment. Conceptually, for a phenotypic trait to be involved
in adaptation to particular axes of the ecological niche (sensu
Hutchinson’s multidimensional niche, Holt (2009)), selection
on that trait needs to change with environmental variables that
define these axes. The causes of natural selection (or selective
agents) can thus be revealed by demonstrating a covariance
between measurements of selection and environmental variables
(Wade and Kalisz 1990), but a recent review highlighted that this
has seldom been performed so far (MacColl 2011), except for
a few notable examples (Kelly 1992; Ahola et al. 2012; Brown
et al. 2013; Reed et al. 2013b; Visser et al. 2015). Two obvious
reasons are the difficulty in identifying (or measuring) relevant
environmental variables, and having enough datapoints in time
or space to estimate the selection-environment covariance.
Another reason is that the importance of the covariance between
environment and phenotypic selection has perhaps not been fully
appreciated until recently, at least outside of the theoretical literature. This situation is changing, however, and the environmental
sensitivity of selection (change in an optimum phenotype with
the environment), a key component of theories on phenotypic
adaptation and phenotypic plasticity, is increasingly perceived as
an important determinant of the interplay between genetic evolution, phenotypic plasticity, and population growth in a changing
environment (Chevin and Lande 2010; Gienapp et al. 2013;
Vedder et al. 2013; Michel et al. 2014). A third impediment is the
lack of a robust and general method for estimating the variance
of phenotypic selection, and its covariance with environmental
variables. One of our aims is to provide such a method.
Most measurements of phenotypic selection in a given
generation rely on methods stemming from the Lande and
Arnold (1983) approach, which is based on classic linear regression. Extensions of this method allow treating more realistic
distributions of fitness residuals and trait-fitness relationships
by use of generalized linear models. For instance, Janzen
and Stern (1998) proposed using logistic regression (with a
logit link function and Bernoulli or Binomial responses) for
analysis of survival selection. More recently, aster modeling
has been developed to decompose fitness into components with
known distributions, and fit these components jointly to infer
selection (Shaw and Geyer 2010). The Lande–Arnold approach
also has been extended recently by Morrissey and Sakrejda (2013)
to allow for measurements of selection gradients from arbitrary
fitness functions estimated by spline-based methods (Schluter
1988), rather than by the original quadratic approximation.
In contrast, there are fewer developments on measuring the
variability of phenotypic selection. Even when phenotypic selection is shown to vary significantly through space or time, the
magnitude of this variation often is not quantified (Kelly 1992;
Ahola et al. 2012; Brown et al. 2013). When measurements of
phenotypes and fitnesses are available for different times (or
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EVOLUTION SEPTEMBER 2015
locations), the most common procedure has been to estimate each
selection gradient separately, and then compute the variance of selection gradients from the variance of each estimates (as reviewed
by Siepielski et al. 2009). This is however problematic, as it conflates sampling variance with the actual variance of the process, as
highlighted by Morrissey and Hadfield (2012) (see also Siepielski
et al. (2013)). A method has recently been proposed to estimate
fluctuating selection in age-structured populations (Engen et al.
2012), which directly estimates the variance of selection gradients by maximum likelihood from the phenotypes and fitnesses in
all years, correctly accounting for measurement error. However,
this method assumes that selection gradients are independent and
identically distributed, and therefore does not allow for autocorrelation (the same holds for a nonparametric approach proposed
by Calsbeek et al. (2012)). Yet theory has shown that the temporal autocorrelation in phenotypic selection is at least as important
as its variance for the long-term evolution and demography of a
population (Charlesworth 1993; Lande and Shannon 1996; Bürger
and Gimelfarb 2002; Chevin 2013; Zhang 2012; Tufto 2015), so
it is necessary to treat phenotypic selection explicitly as a time
series.
One of the most appealing properties of the directional selection gradient is that, in a given generation, it directly relates
to the quantity that predicts evolutionary change of the mean
phenotype for a normally distributed trait, namely the slope of
the adaptive surface relating (log) mean fitness to the mean
trait (hence the name gradient, Lande (1976); Lande and Arnold
(1983)). But in a temporally variable environment, although the
variance of directional gradients can be used to predict the magnitude and variance of responses to selection (conditional on the
additive genetic variance), it cannot reveal whether the genetic
response to selection in a given generation is likely to be increase
fitness in the next (as discussed in Chevin (2013)), which will
often be of interest. This more complete description of the evolution of quantitative traits in fluctuating environments is afforded
by models of a moving optimum, where fitness is maximized at
an intermediate phenotypic value that changes with the environment (Charlesworth 1993; Lande and Shannon 1996; Bürger and
Gimelfarb 2002; Chevin 2013; Tufto 2015). In such models, directional selection is caused by a mismatch between the optimal
and mean phenotype, and the long-term evolution (and demography) of the population depends on patterns of movements of
the optimum, which are not fully captured by temporal change
in directional selection gradients (Chevin and Haller 2014). Furthermore, fluctuating selection is likely to be a strong driver of
the environmental stochasticity of population growth, but computing the contribution of fluctuating selection on some measured trait(s) to the overall observed fluctuations in population
size requires more information than is provided by the temporal
distribution of selection gradient (which only describes the local
E S T I M AT I N G F L U C T UAT I N G S E L E C T I O N
slope of the fitness function). Hence, if we are to estimate parameters of fluctuating phenotypic selection that can be related to
theoretical predictions on expected fitness and population growth
in a random environment, we should focus as much as possible on
changes in the actual (absolute) fitness surface, rather than just on
selection gradients. If the fitness function includes an optimum
phenotype, then the location, width and height of the fitness peak
should be estimated.
Here, we introduce a method to estimate variation in phenotypic selection using measurements of traits and fitness (or components thereof) across time (or space). This method responds
to the requirements detailed above, as well as several other important ones. First, fitness is treated as an expected property of
individuals, rather than a realized number of offspring, for instance. This corresponds to the propensity definition of fitness,
as generally accepted in population genetics (Mills and Beatty
1979), and implemented in analysis of phenotypic selection for
instance by Shaw and Geyer (2010). Considering fitness or its
components as parameters of distributions of random variables
(survival probability, expected number of offspring), rather than
realizations of the variables themselves, allows separating the
actual selection gradient/differential from the random component of phenotypic change caused by demographic stochasticity
(Engen and Sæther 2014). The latter contributes to random genetic drift rather than change caused by natural selection (Rice
2004, pp. 181–187). Second, we use phenotypes and fitnesses
from all years (or locations) to estimate variation in selection using random regression, rather than estimating selection separately
within each year (or location). This allows a more accurate estimation of parameters, ruling out for instance apparent fluctuating
selection caused by estimation error across years (Siepielski et al.
2009). This is especially relevant for quadratic terms describing
stabilizing (if negative) or disruptive (if positive) selection. These
two forms of selection have important qualitative differences in
terms of the evolutionary predictions they generate (Bulmer 1974;
Lande 1976; Dieckmann and Doebeli 1999; Bürger and Gimelfarb 2002), so it is of considerable biological importance to be
able to assess whether a trait is always under stabilizing selection, even though some individual estimates may be compatible
with disruptive selection owing to estimation error. Third, we
compare different models (with and without stabilizing selection,
environmental fluctuations, trends, and effects of environmental
covariates) using a model selection procedure (Spiegelhalter et al.
2002).
In the following, after introducing the underlying biological model and its statistical formulation, we use simulations of a
randomly fluctuating optimum to assess the performance of our
approach in detecting key aspects of the model (stabilizing selection, fluctuating selection, autocorrelation of the optimum). We
then apply our method to a longitudinal dataset of breeding time
in great tits (Visser et al. 1998; Reed et al. 2013a), illustrating how
our approach can shed new light on classic examples of selection
in response to climate change.
Methods
MODEL
We present our approach in the context of temporally changing selection, and address spatial variation in the Discussion. Our
method can be applied to any episode of viability or fertility selection, or to overall selection through lifetime fitness for organisms
with discrete nonoverlapping generations. For the presentation of
the method, we mostly focus on models for fecundity components
of fitness, such as number of surviving offspring. Such discrete
nonnegative variables are best modelled by distributions such as
the Poisson, or Poisson mixtures. Within the framework of generalized linear models, the logarithm is a commonly used link
function relating the expected value of such response variables to
the linear predictor containing the effects of covariates of interest.
If including both linear and quadratic effects of the traits, this
leads to a Gaussian model of stabilizing selection.
Writing the model first in terms of parameters of biological
interest, the fitness (expected number of offspring) in year t of
individuals with phenotype z is given by
(z − θt )2
Wt (z) = Wmax,t exp −
,
(1)
2ω2
where θt is the optimum phenotype at time t, and ω is the width of
the fitness peak (smaller ω causes stronger stabilizing selection).
We shall assume that θt is linearly dependent on one or several
observed or latent environmental processes. Letting xt denote an
observed environmental variable, and t a latent environmental
process,
θt = A + Bxt + t .
(2)
We assume that t is an autoregressive process with autocovariance function Cov(t , t+h ) = α|h| σ2 and mean E(t ) = 0. (We
also briefly consider a random walk of the optimum, see below).
The primary parameters of fluctuating selection that we need
to estimate are thus the width ω the fitness function, and parameters of movements of the optimum θt , namely A, B, σ2 , and α.
These parameters relate to classic theory of evolution under stabilizing selection. For instance, the strength of stabilizing selection
can be measured by the reduction in phenotypic (and additive genetic) variance within a generation, which is proportional to the
opposite of the curvature of the log-mean fitness landscape. For
a Gaussian fitness function, this equals S = 1/(ω2 + σ2z ), where
σ2z is the phenotypic variance before selection (Lande and Arnold
(1983); Phillips and Arnold (1989a); note that S should not be
confused with the selection differential, sometimes denoted S or
EVOLUTION SEPTEMBER 2015
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LUIS-MIGUEL CHEVIN ET AL.
s). It can be shown that 1/S = ω2 + σ2z is the squared width of
the fitness landscape representing mean fitness as a function of
the mean phenotype (Lande 1976). The quadratic selection gradient estimated by the Lande and Arnold (1983) approach is (in
univariate form) γ = S + β2 . Thus, unlike S, γ does not measure
the strength of stabilizing selection directly but must be combined
with the estimated selection gradient β, as previously pointed out
by other authors (Lande and Arnold 1983; Schluter 1988; Phillips
and Arnold 1989b; Arnold et al. 2001). With a Gaussian fitness
function, the directional selection gradient as estimated by the
Lande and Arnold (1983) approach is β = S(θ − z̄), so the response to selection by the mean phenotype is proportional to the
strength of stabilizing selection and the deviation of the mean
phenotype from the optimum. Predictions for the lag load or evolution of plasticity in a fluctuating environment in turn depend
on parameters of fluctuations in θ (e.g., Charlesworth (1993);
Lande and Shannon (1996); Zhang (2012); Chevin (2013); Tufto
(2015)).
For the statistical estimation of these primary parameters, it
is convenient to rewrite equations (1) and (2) as
ln Wt (z) = μt + (βz + ζt )z + βzz z 2 + βx z xt z,
(3)
where the expected number of offspring Wt (z) is treated as the
mean parameter of a Poisson distribution, or related distributions such as the negative binomial (a gamma-Poisson mixture).
This corresponds to a generalized linear-mixed model (hereafter
GLMM) with a logarithmic link function. The linear predictor
of this GLMM includes (i) a fixed effect of t, treated as a factor
(μt ); (ii) a linear effect of z, with regression slope that includes
both a fixed effect βz and a random effect ζt ; (iii) a quadratic
effect of z; and (iv) an interaction between xt and z. The random
effect on the regression slope of z is assumed to have autocovariance function Cov(ζt , ζt+h ) = α|h| σζ2 . Equation (3) thus describes
a form of random regression, where the random effects on the
slope are structured according to a first-order autoregressive process (AR1). Equation (3) also is a latent Gaussian model, a class
of hierarchical models where an unobserved latent Gaussian process, with distribution described by some hyperparameters, gives
rise to observations that are independent conditional on the latent
process. Here, the latent process is fluctuating selection, that is,
the changing trait-fitness relationship described by the slopes in
equation (3) and defined at the population level, whereas the observations are individual phenotypes and fitnesses at each time
point.
The primary parameters of fluctuating selection can be recovered from this GLMM through the transformations
ω=
−
1
,
2βzz
2322
θt = −
β z + β x z x t + ζt
,
2βzz
B=−
EVOLUTION SEPTEMBER 2015
βx z
,
2βzz
σ2
=
σζ2
4β2zz
,
Wmax,t
(βz + βx z xt + ζt )2
= exp μt −
. (4)
4βzz
In our analyses, we also fit a GLMM similar to (3) but
with no quadratic term (βzz = 0), in order to compare models with and without stabilizing selection. This GLMM corresponds to an exponential fitness function, for example, Wt (z) =
Wmax,t exp {μt + (βz + ζt )z}, for which the selection gradient in
year t is βz + ζt regardless of the phenotype distribution. Such
a fitness function is thus a good null model for fluctuating selection, since all variation in selection gradients must come from
environmental changes affecting βt .
When the selection episode under investigation is based on
viability rather than fecundity selection, the number of surviving
individuals is best modelled as a binomially distributed variable,
or a betabinomial to account for overdispersion. The survival
probability w(z) can then be related to covariates of interest
through a logit link function ln( p/(1 − p)) (logistic regression),
as in Janzen and Stern (1998), or a complementary log log link
(ln(− ln(1 − p))), which arises naturally in survival analysis
with a constant hazard rate (instantaneous mortality risk). Both
link functions produce symmetric stabilizing selection, but the
sigmoid shape of these functions necessarily implies a flattening
of the fitness function and weaker selection for high survival
rates close to one, because high survival implies less opportunity
for selection to occur. This also makes the overall fitness
function non-Gaussian, although both fitness functions are well
approximated by a Gaussian for low survival.
INFERENCE AND MODEL SELECTION
All inferences were made using INLA (Rue et al. 2009), an R
package for Bayesian statistics, which uses integrated nested
Laplace approximation for the rapid computation of posterior
densities for latent Gaussian hierarchical models. The rationale
behind our choice of priors is discussed in details in Appendix S1
(Supporting Information). In brief, we relied on non- or weakly
informative priors for most parameters, except for ω when the latter was used in computations that assume stabilizing selection, in
which case we sometimes used a prior that assumes that βzz < 0
in the case study.
Our approach relies on comparison between models with
different degrees of complexity and number of parameters. Regardless of the actual underlying process, the quality and quantity
of available data should determine whether the best model is one
that includes temporal changes in the optimum, or a simpler one
that considers a constant fitness function (with or without an optimum). In order to compare the fit obtained with different models,
we use the deviance information criterion (hereafter DIC), an
analog to Akaike’s information criterion (AIC) for hierarchical
Bayesian models (Spiegelhalter et al. 2002). Both the AIC and
E S T I M AT I N G F L U C T UAT I N G S E L E C T I O N
1.0
0.8
0.6
0.4
0.0
0.05
0.10
2
0.15
0
0.20
1
2
2
σθ
C
D
3
4
0.8
0.6
0.4
0.2
0.6
0.8
Pr(AR1 | fluct. optimum)
1.0
1.0
S= 1 ( ω + σ z )
0.4
Pr(AR1 optimum)
0.2
Pr(fluctuating optimum)
1.0
0.8
0.6
0.4
0.2
Pr(stabilizing selection)
0.0
0.00
0.0
The ability of the method to identify the best model was tested
using simulations of fluctuating selection. We modeled samples
from a population undergoing stabilizing selection with autocorrelated fluctuations in the optimum, as in equations (1) and (2).
The sample size was drawn each year from a Poisson distribution with mean n = 25 or 50 individuals. We considered 40 time
points, which is a large but still realistic number that compares
to existing long-term studies of natural populations (e.g., Grant
and Grant 2002; Reed et al. 2013a; Vedder et al. 2013). We drew
individual phenotypes from the same normal distribution each
year, that is we neglected responses to selection and genetic drift
for simplicity. Indeed, our approach estimates selection based on
trait-fitness relationships in each time, and does not use information on responses to selection. However, responses to selection
may buffer or amplify deviations of the mean phenotype from the
optimum (Lande and Shannon 1996; Chevin 2013; Tufto 2015),
in effect changing the magnitude of environmental fluctuations,
which may affect the estimation procedure to some extent (see
effect of σθ in Fig. 1B). The phenotypic variance before selection
σ2z was set to 1 without loss of generality, which is equivalent to
focusing on standardized trait values scaled to their phenotypic
standard deviation. For each individual, its fitness was computed
from its phenotype using (1), and its actual number of offspring
was then drawn from a Poisson distribution with mean Wt (z). We
varied the width of the fitness function ω such that the strength
of stabilizing selection S = 1/(ω2 + σ2z ) (e.g., Lande 1976) was
0.025, 0.05, 0.1, or 0.2. For each width of the fitness function,
the standard deviation (hereafter SD) of fluctuations in the optimum was varied from 0 (constant optimum) to 0.5, 1, 2, and
4. This combination of widths of the fitness function and magnitudes of fluctuations in the optimum causes Sσθ , the SD of
directional selection gradients without response to selection and
neglecting genetic drift, to vary from 0 to 0.8. For each combination of these parameters, we varied the autocorrelation of the optimum, α = 0 (no autocorrelation), 0.1, 0.25, 0.5, 0.75, 0.9. We
B
0.2
SIMULATIONS
A
0.0
the DIC are model selection criteria based on information theory
(Burnham and Anderson 2002), which allow ranking the predictive power of different models using their likelihoods, penalizing
models with more parameters. Among the models considered,
the model with the lowest DIC is chosen, and models within
DIC 2 to 3 points larger than the best model are considered to
have nearly as strong support from the data as the best model
(Burnham and Anderson 2002; Spiegelhalter et al. 2002). Using
other newly developed criteria, the Watanabe–Akaike information criterion (Gelman et al. 2014), as well as a cross-validation
score based on conditional predictive ordinate-values (Gneiting
and Raftery 2007; Held et al. 2010) yielded very similar model
rankings (Fig. S3.1, Supporting Information).
0.0
0.2
0.4
α
0.6
0.8
0.0
0.2
0.4
α
0.6
0.8
Figure 1. Test of the approach with simulations. The proportion of simulations of fluctuating selection (out of 200 repeats)
where the model that best fits the data by at least two DIC
points includes a specified feature of selection (below, where
this feature is “detected”), is shown against the actual parameters used in simulations. (A) Detection of stabilizing selection
(quadratic term in (3)), against the strength of stabilizing selection S = 1/(ω2 + σ z2 ). Values for S = 0 correspond to simulations
with Wt (z) = Wmax,t exp {β t z}, which include no stabilizing selection. (B) Detection of any form of random fluctuation in an optimum phenotype, against the actual standard deviation σθ of
simulated fluctuations in the optimum. (C) Detection of an autoregressive (AR1) optimum, against the actual autocorrelation
α of simulated fluctuations in the optimum. (D) Detection of an
AR1 optimum, conditional on detecting a fluctuating optimum.
The gray scale and line width represents the magnitude of fluctuations in (A), (C), and (D), with darker and thicker lines indicating larger σθ , and the strength of stabilizing selection in (B). The
lighter and thiner lines in (B) correspond to the exponential fitness function (with no optimum), while only simulations with a
fluctuating optimum were used in (C) and (D). Simulations were
run for 40 time points, with the mean sample size per time point
being n = 50, the phenotypic variance σ z2 = 1, the width of the
fitness function such that S = 0.025, 0.05, 0.1, 0.2. Fluctuations in
the optimum are determined by their standard deviation σθ = 0
(constant optimum), 0.5, 1, 2, 4, and autocorrelation α = 0 (white
noise), 0.1, 0.25, 0.5, 0.75, 0.9.
also allowed temporal changes in fitness beyond those caused by
fluctuating selection, by drawing the maximum fitness Wmax,t
from a log normal distribution with mean 5 and SD 0.2 on
the log scale. The mean number of offspring across years was
thus 5.1 for hypothetical individuals with the optimum phenotype. We also ran simulations with an exponential fitness function
Wt (z) = Wmax,t exp {βt z}, which corresponds to pure directional
EVOLUTION SEPTEMBER 2015
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LUIS-MIGUEL CHEVIN ET AL.
selection (no stabilizing selection) at each time, with gradient βt .
In those simulations, we drew βt for each time point from a normal distribution with mean 0 and SD ranging from 0 to 0.8 (by
increments of 0.2), thus covering the same range of variation in
directional selection gradients as in simulations with a moving
optimum. We used the same values for the autocorrelation of βt
as those for θt .
We then ran the estimation procedure under a choice of models. Because the number of parameters of the complete model in
equation (4) is relatively large, we did not consider all possible
models but instead focused on a subset of biologically meaningful
models to test particular questions, as recommended for model selection (Burnham and Anderson 2002). Specifically, we fitted (i)
the AR1 model in equation (4), with no covariate x; (ii) a similar
model but with uncorrelated fluctuations (white noise), that is random Poisson regression with independent random effects on the
slope; (iii) a model with autoregressive slope but no quadratic term
(no optimum); (iv) a model with a constant optimum (quadratic
term and no random effect on the slope), implying no fluctuating
selection. For each parameter set, we ran 200 simulations, and we
computed the proportion of simulations where a given feature of
the models was selected based on DIC, as a way to quantify the
support for this feature. For instance, the proportion of simulations where any model with a quadratic term has the lowest DIC
by at least two points measured support for stabilizing selection,
and the proportion of simulations where any of the models with
a random effect and a quadratic term was chosen based on DIC
measured support for a fluctuating optimum.
CASE STUDY ON GREAT TITS
We then applied our method to a dataset of fecundity selection
on breeding date in the Hoge Veluwe population of great tits
(Parus major), in the Netherlands. We used years from 1973 to
2013, and removed all clutches that had been manipulated experimentally, following Reed et al. (2013b). The remaining dataset
included 3782 clutches. The trait z we investigated was breeding
date (date of first egg laying), and we measured the fecundity
component of fitness W as the number of individuals surviving to
become fledglings. We could have used number of recruits surviving to the next breeding season, as in Reed et al. (2013a), but
this has much smaller mean and hence larger coefficient of variation due to demographic stochasticity, causing more uncertainty
in estimates of all parameters and reducing statistical power. A
somewhat high probability of clutch failure (around 32%, see
below) caused the distribution of clutch sizes to be manifestly
zero-inflated (Fig. S3.2, Supporting Information), so we modeled
a zero-inflated Poisson instead of a Poisson distribution as above,
unless otherwise stated. To account for possible nonindependence
in the data caused by some females reproducing repeatedly in different years, we also investigated models with female identity
2324
EVOLUTION SEPTEMBER 2015
included as a random effect in the analysis. To account for few
replicates for a large proportion of females we used a correction described in Ferkingstad and Rue (2015). Our dataset also
included environmental variables that we used as covariates to
explain movements of the optimum: the mean temperature over
a critical time window during which it influences the date of the
peak in food biomass and, when available, the date of the food
peak itself (Visser et al. 2006).
Results
EVALUATING THE METHOD WITH SIMULATIONS
We first tested our approach on simulations of an optimum phenotype undergoing an autoregressive process. Our first aim was
to assess whether our method is efficient in detecting stabilizing
selection (assuming constant ω), regardless of patterns of fluctuations. Figure 1A shows that the proportion of simulations where
the best model (by at least two DIC points) includes a quadratic
term increases rapidly with the actual simulated strength of stabilizing selection, and approaches 1 for S as small as 0.05. This
corresponds to weak stabilizing selection (Lande 1975), which
would typically be nonsignificant if estimated from a single time
point with moderate sample size as here (see Fig. 8 in Kingsolver
et al. (2001), which reports estimates of the standardized quadratic
gradient γ = S + β2 ≈ S under weak directional selection). Here,
stabilizing selection is detected more efficiently by pooling data
from several time points.
Figure 1B shows that the proportion of simulations where
the best model includes fluctuations in the optimum phenotype
increases with the strength of stabilizing selection (darker lines
indicate larger S = 1/(ω2 + σ2z )). In particular, simulations with
fluctuating selection but no optimum (lighter lines) essentially do
not lend support to any model with a moving optimum. For a given
S, the probability to detect a fluctuating optimum increases with
the magnitude of actual fluctuations in this optimum, except in the
case of large fluctuations, under strong selection (rightmost point
in darkest lines). In this case, fluctuating selection is strong enough
to substantially reduce the expected number of offspring of most
individuals in most generations, reducing the power to detect
variation in selection as compared to more moderate fluctuations.
The probability to detect fluctuating selection when the optimum
is actually constant (values for σθ = 0 in Fig. 1B), which has some
connection to type I error, was always low in our simulations.
The probability to detect an autocorrelated optimum (AR1
process) increases with the actual autocorrelation of fluctuations,
but also with their magnitude as indicated by the gray scale (Fig.
1C). AR1 is detected in at most 20% of simulations for α < 0.4,
and even strongly autocorrelated fluctuations are not detected as
AR1 if their magnitude is small (lighter lines in (1c), for which
σθ = 0.25, 0.5). However, conditional on detecting a fluctuating
E S T I M AT I N G F L U C T UAT I N G S E L E C T I O N
0.0
0.0
α
0.4
CI( α)
1.0
0.8
2.0
credible interval includes the true optimum at all time points. Reducing the mean sample size to 25 widens the credible interval
and increases the error of the posterior mean, making it closer
to the long-term mean, but the major tendencies of the optimum
(such as the sign of the deviation from the mean optimum) are
still well captured in this example.
0.0
0.4
α
0.8
A Estimated α
Figure 2.
0
1
2
σθ
3
4
B Credible interval for α
Inferred autocorrelation of optimum in simulations. (A)
The posterior mean for the autocorrelation of movements of the
optimum phenotype are plotted against the actual autocorrelation α used in simulations. Darker and thicker lines indicate larger
fluctuations in the optimum (as quantified by σθ ), and the dashed
line plots y = x. (B) The width of the 95% credible interval for α
is plotted against the magnitude of fluctuations in the optimum,
quantified by σθ . Darker and thicker lines indicate stronger stabilizing selection (larger S). Parameters as in Figure 1A.
optimum (either white noise or autoregressive), the proportion
of simulations that do detect autocorrelation becomes substantial
and increases with increasing α (Fig. 1D).
The autocorrelation of the optimum is estimated more accurately under larger fluctuations (larger σθ ): the posterior mean of
α is closer to its actual value for dark as compared to light lines
in Figure 2A, and the 95% credible interval becomes narrower
toward the right in Figure 2B. The credible intervals on α are very
sensitive to the strength of stabilizing selection, being reduced
by a factor of 3 as S changes from small to large (light gray to
dark lines in Fig. 2B). But in all cases the credible intervals in
Figure 2B remain quite large, showing that our simulations with
40 time points do not provide sufficient information to estimate
the autocorrelation of the optimum with high accuracy, even in
cases where an autocorrelated optimum may be detected in Figure
1C. Similarly, the posterior mean α is slightly biased toward 0,
the mean of the prior distribution, even under large fluctuations
(Fig. 2A), as expected generally when limited sample size does
not allow the likelihood to completely dominate the prior.
Our hierarchical Bayesian method estimates the joint posterior distribution for all parameters, from which the marginal
posterior distribution for each time point can be recovered. The
primary parameters of our biological model depend nonlinearly
on parameters in the GLMM (see eq. 4). However, their posterior marginal distribution can be found by drawing samples from
the joint posterior distribution of parameters of the GLMM (as
allowed by packages such INLA), and applying the transformation (4) to each sample. An example of estimated fluctuations of
an optimum phenotype is illustrated in Figure 3. When the mean
sample size is n = 50, the movements of the optimum are inferred
with good accuracy, including outside of the current phenotypic
range (individual phenotypes appear as gray dots), and the 95%
CASE STUDY: PHENOLOGY OF GREAT TITS
To illustrate the usefulness of our approach, we next apply it to
a well-known example of climate-change induced phenotypic selection: breeding time in the Great tit population of Hoge Veluwe
in the Netherlands (Visser et al. 1998; Reed et al. 2013a,b). In
this species, there is strong evidence that the date of egg laying is
under selection for an optimum set by the peak in the biomass of
caterpillars that are the main food resource for hatchlings (Visser
et al. 2006). But movements of the optimum phenotype have not
been estimated directly from the phenotypes and fitnesses of all
individuals in this population.
Table 1 shows a number of candidate models for the Hoge
Veluwe great tit data. We first fitted models without any environmental covariates. Given this model form, there was weak
evidence for a nonzero autocorrelation between the optima in different years t (DIC reduced by 3.29 for model 2 vs. 1, with ζt AR1
or iid, respectively). We investigated whether the data showed evidence for nonstationarity, by fitting an alternative model where
the optimum follows a random walk (discrete time equivalent of
Brownian motion as modeled by Estes and Arnold (2007) and
Hansen et al. (2008)) instead of AR1, but this increased the DIC
by 1.4 (model 3 vs. model 2). Fitting an alternative model with a
linear trend in the optimum, equivalent to an environmental covariate xt = t, did not reduce the DIC either (models 4 and 5, with
ζt iid or AR1, respectively).
Selection on laying date in great tits (and other passerines)
is known to depend on the timing of the peak in the biomass of
caterpillars which they use as food (Visser et al. 2006). This in
turn partly depends on the phenology of trees, which is largely
determined by spring temperature. Having estimates of the mean
temperature over a critical time window known to influence the
food peak (Visser et al. 2006), we changed the model to include
this as a covariate xt instead of a linear trend in the optimum,
but still allowing for random fluctuations beyond those caused
by spring temperature. This improved the model, causing DIC to
decrease by 4.21 for models with AR1 residual variance (model 7
vs. 2). With this environmental covariate included, there was still
evidence for nonzero autocorrelation in the remaining variation
in the optimum (t AR1 instead of iid) as shown by the decrease
in DIC by 4.9 when comparing model 7 to model 6.
We then compared the model with the spring temperature and
autocorrelation included (model 7) to a number of alternatives,
including a model without random fluctuations in the optimum
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LUIS-MIGUEL CHEVIN ET AL.
0
10
20
30
0
40
10
20
A
30
40
t
t
B
E(n)=50
E(n)=25
Estimates of fluctuating optimum in an individual simulated sample. The simulated movements of an optimum phenotype
(thick full line) are represented along time in a particular simulation repeat. The posterior mean for the estimated optimum is also
Figure 3.
represented in dashed, together with the 0.95 credible interval (shading). The dark gray dots are the individual phenotypes. The same
simulated pattern of phenotypic selection is estimated with mean sample size per time point n = 50 (left) or 25 (right). The phenotypic
variance is σ z2 = 1, the width of the fitness function is ω = 3, the optimum phenotype fluctuates with σθ = 2 and α = 0.5, and all other
parameters are as in Figure 1A.
Table 1. Model selection for temporal variation in selection on laying date in great tits. Linear predictors are shown for different
candidate models of selection on breeding time, together with the model assumed for the random effect (ζ t being either independent
and identically normally distributed, iid, or following an autoregressive order 1 process, ar1). Also shown are the associated differences
in DIC (DIC) relative to the best model (model 7). Further details are given in the main text. The number of fledglings is obtained
from the linear predictor as a zero-inflated Poisson variable, except in models 13 and 14 (Poisson and zero-inflated negative Binomial,
respectively). Mother identity was included as a random effect in model 15, treating some cases where the mother was unknown as
different mothers.
Model
Linear predictor: μt + βz z+
ζt
DIC
1) No covar.
2) No covar.
3) Random walk
4) Linear trend
5) Linear trend
6) Temper.
7) Temper.
8) No random slope
9) Temper. + trend
10) No stab. sel.
11) Free lin. slopes
12) Free lin. and quadr.
13) No. zero. infl.
14) Zero. infl. neg. bin.
15) Fem. id. rand. eff.
βzz z + ζt z
βzz z 2 + ζt z
βzz z 2 + ζt z
βzz z 2 + βzt zt + ζt z
βzz z 2 + βzt zt + ζt z
βzz z 2 + βzx zxt + ζt z
βzz z 2 + βzx zxt + ζt z
βzz z 2 + βzx zxt
βzz z 2 + βzx zxt + βzt zt + ζt z
βzx zxt + ζt z
βzz z 2 + βzt z
βzz z 2 + βzt z + βzzt z 2
βzz z 2 + βzx zxt + ζt z
βzz z 2 + βzx zxt + ζt z
βzz z 2 + βzx zxt + ζt z
iid
ar1
rw1
iid
ar1
iid
ar1
7.5
4.21
5.61
7.61
4.43
4.9
0
32.4
0.24
25.18
14.55
14.19
6129.08
90.96
8.2
2
beyond those generated by spring temperature (model 8). Removing the random fluctuations led to an increase in DIC of
32.4, strongly suggesting that additional environmental variables
other than spring temperature itself influence the optimum. A
model including both a linear trend in the optimum and effects
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EVOLUTION SEPTEMBER 2015
ar1
ar1
ar1
ar1
ar1
of the spring temperature was also considered (model 9) but this
model did not improve DIC. Comparing model 7 with a model
without the quadratic term βzz z 2 , that is, an exponential fitness
function with random fluctuations in the slope of log fitness on
the trait (model 10), strongly favored our selected model based on
E S T I M AT I N G F L U C T UAT I N G S E L E C T I O N
Estimate of the width ω of the Gaussian fitness function,
the intercept A and slope B in the regression of the optimum θ t on
Table 2.
Table 3. Same as Table 2, but for the model without environmental covariates (model 2 in Table 1).
temperature xt and the standard deviation σ2 and autocorrelation
α of additional fluctuations in the optimum, for the selected model
of selection on onset of breeding through the number of fledglings
(model 7 in Table 1). The parameter estimates were computed by
transforming samples from the posterior according to equation
(4). Also reported are the zero-inflation parameter p0 , and the
standard deviation σν of the random effect on maternal identity.
Parameter
ω (days)
σ (days)
Autocorrelation α
Intercept A (April
day)
Slope B (days/◦ C)
p0
Posterior
mean ± S.E.
95% credible
interval
20.55 ± 1.7
6.75 ± 1.66
0.3029 ± 0.2419
19.43 ± 1.95
(17.85, 24.42)
(4.4, 10.53)
(−0.2176, 0.7113)
(14.97, 22.89)
−5.01 ± 1.09
0.32 ± 0.01
(−7.38, −2.93)
(0.31, 0.33)
stabilizing Gaussian selection (difference in DIC equal to 25.18),
where directional selection comes from mismatches between the
mean phenotype and the optimum.
We also considered models with the regression slopes βzt
estimated as free parameters in different years (model 11), rather
than being modeled as random effects. This model was further
extended to also include different quadratic regression coefficient βzzt in different years (model 12). These more flexible
model alternatives, providing a form of goodness-of-fit test of
our model, did not lead to any improvement in DIC. Thus we
conclude that there is no evidence in the data for any deviations in the local slope and curvature of the fitness function beyond what is predicted by our Gaussian model with a fluctuating
optimum (1).
The last three models were used as tests for basic features of
our statistical model. Our choice of residual distribution of fitness
was supported by the strong increase in DIC when replacing the
zero-inflated Poisson by a regular Poisson distribution (model
13), or by a zero-inflated negative binomial (model 14). This
last comparison suggests that there is little overdispersion in the
number of fledglings produced by a given female in a given year,
beyond that caused by the zero-inflation. DIC also increased (by
8.2) if including a random effect associated to female identity,
indicating that there was no significant variation in annual number
of fledglings between females.
Parameter estimates under our selected model (model 7) and
the best model without any environmental covariates (model 2)
are listed in Tables 2 and 3, respectively. Both models estimate
32% zero inflation ( p0 ). The inclusion of temperature as a covariate greatly improved the precision of estimates for several of
our model parameters (smaller S.E. and CI in Table 2 than in
Posterior
mean ± S.E.
Parameter
ω (days)
σ (days)
Autocorrelation α
A (April day)
p0
24.11 ± 2.73
11.3 ± 2.56
0.2472 ± 0.2075
17.66 ± 2.95
0.32 ± 0.01
95%
credible interval
(20.08, 30.78)
(7.5, 17.94)
(−0.1745, 0.626)
(11.3, 23.06)
(0.31, 0.33)
Table 3). This improvement agrees well with our intuition about
how our method extracts information about the different parameters from the data. For instance regarding ω, with some—albeit
incomplete—information about current location of the optimum
provided by the environmental covariate at different time points,
model 7 is better able to infer the strength of stabilizing selection
from the local steepness of the fitness function. In contrast, for
model 2 not utilizing this information, the strength of stabilizing
selection can to a greater extent only be inferred from the local
curvature of the fitness function, making the uncertainty in ω
much greater.
The estimated width of the fitness function in the best
model was ω = 20.55 days. With the average within-year phenotypic standard deviation being σz = 5.35 days, this yields for
the standardized strength of stabilizing selection on the unitless trait measured in the scale of phenotypic standard deviation,
S = σ2z /(ω2 + σ2z ) = 0.063, which can be compared directly to
the S used in simulations (see Fig. 1A). The estimated environmental sensitivity of selection, as quantified by the slope B of
the optimum on the environmental covariate (here temperature),
is such that the optimum laying date advances by 5.01 days/◦ C. In
model 2 that does not include temperature as a covariate, A is simply the mean optimal laying date across years, which is estimated
to be April 18 (Table 3) whereas in model 7 A is the optimal laying date (April 19) at the mean value of spring temperature (since
most of our covariates were mean centered). Finally, note that
the standard deviation of the optimum is estimated to be larger
in model 2 than in model 7 (11.3 vs. 6.75 days, respectively),
as expected because in the latter model part of the fluctuations
are captured by the relationship with temperature, and σ only
describes variation that is left after accounting for this.
Having established an effect of the spring temperature xt on
the optimum, we analyzed the pattern of fluctuations in this variable. Direct observations of temperature yields more power to detect autocorrelation, but after removal of the long-term trend in xt
by ordinary linear regression, there was no evidence for autocorrelation in the residuals of xt . The standard deviation σxt around the
long-term trend was estimated to 1.07◦ C with a 95%-confidence
interval of (0.87, 1.37). This gives for the standard deviation of
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LUIS-MIGUEL CHEVIN ET AL.
fluctuations in the optimum, arising from the combined effect of
variation
in spring temperature and the latent random variable t ,
σθ = B 2 σ2xt + σ2 = 8.62 days. A proportion B 2 σ2xt /σθ2 = 38%
of the total variance in θt is thus attributable to fluctuations in
temperature around the long-term trend.
The estimated patterns of fluctuations in the optimum are
represented in Figure 4. The right panel shows how the fitness
function scales in comparison to patterns of phenotypic variation
in the trait, represented by a Gaussian curve (both curves are standardized to the same height). The average phenotype distribution
(represented by the gray curve) is narrower than the fitness function, consistent with the estimated weak stabilizing selection. The
time series of estimates of θt are shown in the left panel (mean
of the posterior marginal for each date in continuous line, 95%
credible interval in dashed lines), using the same scale for the Y
axis as in the right panel. This gives a sense of how much the optimum fluctuates relative to the widths of both the fitness function
and the phenotype distribution. Interestingly, the mean laying date
(represented as dots in Fig. 4) is later than the optimum in most
years (by an average 5.1 days, based on posterior mean estimates
of θt each year), resulting in a biased mismatch with optimum and
directional selection for earlier breeding. Figure 4 also shows the
date of the peak in the biomass of caterpillar food (dotted line).
The latter is either directly available (after 1985), or predicted
from the spring temperature (before 1985), using a linear model
based on years where measurements exist (Visser et al. 2006). It is
striking that the optimum phenotype closely tracks major movements of the food peak, but with some advance. To investigate
this further, we reanalyzed the data using the food peak date as a
covariate in addition to spring temperature, for years where this
date was available (Appendix S2, Supporting Information). Using the food peak date as a covariate and assuming a constant lag
between the optimum and the food peak yielded a better model
than using any other covariate in isolation (temperature, or time
for linear trend, models 8–11 in Table S2.1). However, the best
model included all three covariates (model 1 in Table S2.1). In
models that assume that the optimum laying date occurs a fixed
number of days before the food peak and that no other covariate
affects the optimum (model 10 in Table S2.1), the mean lag is
about 33 days (Table S2.2).
Discussion
IMPLICATIONS OF THE APPROACH AND RESULTS
The variance of phenotypic selection across time or space cannot be estimated from individual estimates of selection gradients
(or differentials), because this conflates the true variance with
sampling variance (as emphasized by Morrissey and Hadfield
2012). Instead, it can be estimated through a random effect (across
years or locations) on the regression slope of relative fitness on
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individual phenotype. Allowing a flexible covariance structure
for this random effect makes it possible to model different forms
of autocorrelation in phenotypic selection. Here, we assumed a
first-order autoregressive process (AR1) for phenotypic selection
(and we also fitted a random walk), but our method can easily
accommodate other structures, such as ARp (p-order autoregressive process), or continuous-time processes such as white noise,
Ornstein-Uhlenbeck, or Brownian motion (as in e.g., Arnold et al.
(2001); Hansen et al. (2008)).
The linear regression method of Lande and Arnold (1983)
could be directly extended to the measurement of fluctuating selection, by including a random effect on the linear (directional)
gradient β, thus measuring the variance and autocorrelation of
directional selection. Linear regression with a random effect on
the slope in the form of a stochastic process is sometimes described as dynamic linear regression (Petris et al. 2009), a class of
state-space models (Shumway and Stoffer 2010). However, our
approach based on GLMM with a log link function proves more
useful for several reasons. First, we directly estimate a single parameter ω that determines the strength of stabilizing selection,
while the quadratic gradient γ does not. Indeed, movements of an
optimum with constant width would produce changes in γ, while
they would not affect the strength of stabilizing selection (see
Methods, and Lande and Arnold (1983); Schluter (1988); Phillips
and Arnold (1989b)). Hence, temporal variation in γ, as reported
in some empirical studies (see e.g. Fig. 2B in Charmantier et al.
2008), can be perfectly consistent with constant stabilizing selection (constant ω in our model), but changing deviation of the mean
phenotype from the optimum. Second, the use of linear regression for analysis of phenotypic selection has been criticized for its
reliance on the nonrealistic assumption of Gaussian distribution
of fitness residuals, which leads to wrong confidence intervals
and hypothesis testing using F- or t-tests, despite providing unbiased estimates of linear and quadratic regression coefficients
(Mitchell-Olds and Shaw 1987; Janzen and Stern 1998; Shaw and
Geyer 2010; Morrissey and Sakrejda 2013). This problem may be
circumvented by using nonparametric resampling methods such
as bootstrapping or jack-knifing, as first suggested by Lande and
Arnold (1983). Here, we chose to use the more realistic statistical framework of generalized linear-mixed models (GLMM),
which in general makes more efficient use of the data than linear
regression.
Instead of inferring the selection gradients per se, we use
a model of stabilizing selection with a moving optimum, thus
providing estimates of a number of parameters appearing in theoretical models of fluctuating selection (Bull 1987; Charlesworth
1993; Lande and Shannon 1996; Bürger and Gimelfarb 2002;
Lande 2009; Chevin 2013; Tufto 2015). This should make results of empirical studies more readily interpretable in terms of
these theoretical models, while the variance and autocorrelation
June 1
May 1
April 1
Laying date, food peak date
E S T I M AT I N G F L U C T UAT I N G S E L E C T I O N
1973
1980
1990
Year
Figure 4.
2000
2010
Fitness w(z) and
mean distribution of z
Estimated movements of the optimum laying date. The time series of estimates for the optimum phenotype θ t (solid line),
based on the selected model for number of fledglings, is shown on the left panel, together with its 95% credible interval (dashed
lines). Also shown is the mean laying date in each year (dots), and the timing of the peak in caterpillar biomass (dotted line). For years
before 1985, the food peak date was not available, so an estimate based on linear regression on spring temperature was used, with a
95%-prediction interval represented by the shaded zone. The right panel shows the estimated Gaussian fitness function in solid curve,
and a normal density representing the mean within-year phenotypic distribution (variance equal to mean empirical variance in onset of
breeding across all years) in dashed curve, both centered on the mean laying date across years.
of selection gradients themselves generally are not sufficient to
infer patterns of fluctuations of an optimum phenotype (Chevin
and Haller 2014). In our approach, the assumption of stabilizing
selection can be removed simply by not including the quadratic
term in equation (3), which also allows comparing models with
and without stabilizing selection (through likelihood-ratio tests
or using information criteria, as we did here). Also in line with
most theory, we modeled stabilizing selection as a Gaussian fitness function, which approximates well any fitness function with
an optimum when sufficiently close to the optimum (Lande 1976)
as in Figure 4. In general, however, the Gaussian fitness function
should not be seen as a realistic representation of the actual fitness
landscape, which may be better estimated through nonparametric approaches such as cubic splines (Schluter 1988; Morrissey
and Sakrejda 2013). Rather, it is an idealized fitness function that
allows quantitative comparison of patterns of fluctuating selection across studies, as long as stabilizing selection with a moving
optimum is biologically meaningful. A perhaps useful analogy is
the Wright–Fisher population, which is an idealization that allows
comparing the intensity of drift across populations with different
complex demographies, through the summary parameter Ne .
Our simulations resulted in rather imprecise estimates of the
autocorrelation in the optimum phenotype, with large confidence
intervals (Fig. 2B). This is because we restricted our attention to
the modest sample sizes and number of time points that are typical of studies on organisms with intermediate to large generation
times (mostly birds and large mammals) in (semi-)natural conditions. The precision for these estimates cannot be increased by
only increasing the sample size within time points, and more time
points are needed instead. For instance, even with perfect knowledge of the optimum at each time point, the estimate of the temporal variance in the optimum (σθ2 , or σ2 in the case without an en√
vironmental covariate) has coefficient of variation (2/(n − 1)),
such that n > 2/u 2 − 1 time points are required in order for the
coefficient of variation to be below a given treshold u. This problem is further amplified by the uncertainty in the estimate of each
optimum. More precise estimation of patterns of fluctuating selection may be obtained for species with shorter generations, such
as those studied in experimental evolution, allowing for measurements over a larger number of time points.
The results from our analysis of the Hoge Veluwe great tit
data were partly in agreement with previous findings, but also
revealed new and intriguing patterns. We showed that the optimum breeding time responds to spring temperature, which could
be anticipated since the latter is a good predictor of the food peak
date. Furthermore, we were able to show that the optimum laying
date undergoes temporally autocorrelated random fluctuations,
beyond those caused by spring temperature. These fluctuations
may be caused for instance by other climatic variables than temperatures, other spring food sources that are less dependent on
the timing of bud burst in trees, or winter conditions. A possible
candidate driver of these fluctuations is the number of sunspots,
which has been shown to influence laying date independently of
mean spring temperature (Visser and Sanz 2009), and may thus
also affect selection on this trait. Another plausible candidate is
the height of the food peak, which determines the total amount of
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LUIS-MIGUEL CHEVIN ET AL.
food available in each year, and which strongly affects the number of fledglings produced. It has also been noted that food peak
height correlates with the number of sunspots (Visser and Sanz
2009). In models that assumed a constant time lag between the
optimum laying date and the food peak, we found that this lag was
about 33 days, which is consistent with (although slightly lower
than) what is known of the biology of this population: egg laying
takes about 9 days, incubation 12 days and hence it is optimal
to have nestlings close to fledging (which is at day 16-18) at the
food peak (Visser et al. 1998, 2006; Gienapp et al. 2013; Reed
et al. 2013b).
POSSIBLE EXTENSIONS
Here, we focused for simplicity on selection on a individual trait.
A multivariate statistical model of stabilizing selection on multiple traits (analog to the multiple linear regression in Lande and
Arnold (1983)) is straightforward to formulate in our framework.
After transformation of the GLMM, the fitness function becomes
a multivariate Gaussian, with orientation and shape determined
by a symmetric matrix whose diagonal elements measure stabilizing selection, and off-diagonal elements measure correlational
selection (Lande 1979). In a stochastic environment, stationary
fluctuations in the optimum would then also be multivariate,
characterized by the covariance matrix of the stationary joint
distribution of optima for all traits (rather than the single variance
parameter used here), and a nonsymmetric cross-covariance matrix representing the autocovariance over one time step (rather than
the single autocorrelation parameter used here) (see e.g., appendix
in Chevin (2013)). In practice however, the number of parameters
to estimate is likely to become problematic. For k measured traits,
the three matrices described above require estimation of k(2k + 1)
parameters in total. The slope of the k-dimensional optimum
relative to n environmental variables (environmental sensitivity
of selection) would involve kn additional parameters. Estimation
of these parameters would require not only a large number of
time points, but also a huge sample size at each time point.
We focused on a single selection episode, acting either
through a single fitness component, or through an integrative measurement of lifetime fitness. A natural extension of our method
would be to combine multiple selection episodes, acting through
different fitness components. For instance in our study case, we
could consider clutch failure (causing zero-inflation in the distribution of fledgling number) as a separate episode of selection,
which may be correlated to some extent with the selection episode
we focused on above, based on the nonzero-inflated part of the
distribution. More generally, it would be useful to model selection operating through all possible vital rates (age-specific survivals and fecundities), in an age-structured population. Several
approaches have been proposed to integrate different fitness components into measurements of selection, from combining selection
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EVOLUTION SEPTEMBER 2015
gradients measured in each selection episode (Arnold and Wade
1984), to aster modeling of lifetime reproductive success (Shaw
and Geyer 2010), or explicit age-structured demographical models
(Lande 1982; van Tienderen 2000; Engen et al. 2012). Engen et al.
(2011) showed that when (weak) stabilizing selection operates on
a trait through all vital rates in an age-structured population, the
resulting overall selection on this trait is well approximated by a
single Gaussian function with width and optimum that are averages of those at each age, weighted by the relevant reproductive
values. Engen et al. (2012) further showed how to measure fluctuating selection in an age-structured population, but assuming
no autocorrelation, and focusing on linear gradients. All these approaches could be combined with the present method in order to
measure autocorrelated fluctuations in age-specific optima, and a
resulting single optimum for overall selection. A likely outcome
will be that overall selection through lifetime fitness fluctuates
less than selection at each episode (e.g., one breeding season),
because of a buffering of environmental fluctuations as individuals experience different optima throughout their life. Note that a
serious complication of such an approach would be that optimum
phenotypes (and more generally selection strengths) are likely to
be correlated across ages, especially if they depend on the same
unobserved environmental variables.
Our method can be applied to analyze spatial (rather than
temporal) variation in phenotypic selection, as long as there is
no dispersal during selection. For one-dimensional space (e.g.,
latitudinal or altitudinal transects), the analyses conducted above
are readily transposable, using spatial coordinates instead of time.
The first-order autoregressive process should be replaced by its
continuous equivalent (an Ornstein–Uhlenbeck process), where
the autocorrelation over one step is replaced by a rate of exponential decay of autocorrelation with distance. A tendency with
space would then indicate a spatial cline in the optimum, while the
random process would estimate how the random component of
phenotypic selection varies (and correlates) across space. Spatiotemporal variation in phenotypic selection would also be straightforward to analyze under the assumption that temporal and spatial
effects act additively on the optimum. Such a model could represent permanent differences between optimal egg-laying dates
at different spatial locations, arising for instance from topography or nutrient availability. Beyond this, analysis of spatiotemporal variation in selection (including the spatial scale of synchrony in selection) would be more involved, and requires further
developments.
Conclusion
We introduced an approach that connects empirical measurements
of traits and fitness across time to theoretical predictions for
evolution under environmentally driven fluctuating selection.
E S T I M AT I N G F L U C T UAT I N G S E L E C T I O N
Applications of this method and its future extensions to datasets
from natural populations should provide a more quantitative picture of the eco-evolutionary consequences of randomly changing
environments.
ACKNOWLEDGMENTS
We thank R. G. Shaw, J. Hadfield, and two anonymous reviewers for
constructive comments. L.-M. C. is supported by the grants ContempEvol (ANR-11-PDOC-005-01) and PEPS (ANR-12-ADAP-0006) from
the Agence Nationale de la Recherche, and M. E. V is supported by the
European Research Council (ERC-2013-AdG 339092).
DATA ARCHIVING
http://dx.doi.org/10.5061/dryad.6td18
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Supporting Information
Additional Supporting Information may be found in the online version of this article at the publisher’s website:
Appendix S1: Prior distributions.
Appendix S2: Constant lag behind the optimum.
Appendix S3: Supplementary figures.
2332
EVOLUTION SEPTEMBER 2015
Associate Editor: J. Hadfield
Handling Editor: R. Shaw
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