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O R I G I NA L A RT I C L E
doi:10.1111/j.1558-5646.2011.01318.x
COMPARING ENVIRONMENTAL AND GENETIC
VARIANCE AS ADAPTIVE RESPONSE
TO FLUCTUATING SELECTION
Hannes Svardal,1,2 Claus Rueffler,1,3 and Joachim Hermisson1,4
1
Mathematics and Biosciences Group, Department of Mathematics, University of Vienna, Nordbergstrasse 15,
1090 Vienna, Austria
2
E-mail: [email protected]
3
E-mail: [email protected]
4
E-mail: [email protected]
Received November 28, 2010
Accepted March 28, 2011
Phenotypic variation within populations has two sources: genetic variation and environmental variation. Here, we investigate the
coevolution of these two components under fluctuating selection. Our analysis is based on the lottery model in which genetic
polymorphism can be maintained by negative frequency-dependent selection, whereas environmental variation can be favored
due to bet-hedging. In our model, phenotypes are characterized by a quantitative trait under stabilizing selection with the optimal
phenotype fluctuating in time. Genotypes are characterized by their phenotypic offspring distribution, which is assumed to be
Gaussian with heritable variation for its mean and variance. Polymorphism in the mean corresponds to genetic variance while
the width of the offspring distribution corresponds to environmental variance. We show that increased environmental variance is
favored whenever fluctuations in the selective optima are sufficiently strong. Given the environmental variance has evolved to its
optimum, genetic polymorphism can still emerge if the distribution of selective optima is sufficiently asymmetric or leptokurtic.
Polymorphism evolves in a diagonal direction in trait space: one type becomes a canalized specialist for the more common ecological
conditions and the other type a de-canalized bet-hedger thriving on the less-common conditions. All results are based on analytical
approximations, complemented by individual-based simulations.
KEY WORDS:
Bet-hedging, developmental noise, evolutionary branching, frequency dependence, genetic polymorphism, lottery
model.
Explaining the amount of phenotypic variation observed in nature
is one of the defining problems of evolutionary theory. The classification of ecological conditions that favor phenotypic variation
is a primary issue in this debate. Another part is the question how
much phenotypic variation can be attributed to genetic variation
and how much to environmental variation. One mechanism that
can result in increased levels of phenotypic variation is fluctuating
selection, but the conditions for the evolution and maintenance of
variation are different for the genetic and the environmental component.
1
C 2011 The Author(s).
Evolution
Several models showed that genetic variation erodes under
density- and frequency-independent fluctuating selection (e.g.,
Cohen 1966; Bull 1987; Seger and Brockmann 1987; Frank and
Slatkin 1990). Under such conditions genetic variation can only be
maintained by genetic constraints or recurrent mutation, similar to
the case of constant selection. In particular, heterozygote advantage with respect to geometric mean fitness facilitates the maintenance of genetic variation (Gillespie 1973; Karlin and Liberman
1975). The effect of fluctuating selection on the amount of genetic variation maintained at mutation–selection balance depends
H A N N E S S VA R DA L E T A L .
on the nature of the fluctuations. Randomly fluctuating selection
does not, or only slightly, increase genetic variation at mutation–
selection balance (Lande 1977; Turelli 1988) whereas periodically
changing selection can significantly increase genetic variation if
the selection cycle is sufficiently long and the mutation rate is
sufficiently high (Bürger and Gimelfarb 2002).
In an ecological model for species coexistence by Chesson
and Warner (1981), which became known as the “lottery model,”
it was pointed out by Seger and Brockmann (1987) and elaborated by Ellner and co-workers (Ellner and Hairston 1994; Sasaki
and Ellner 1995; Ellner 1996; Ellner and Sasaki 1996; Sasaki and
Ellner 1997) that fluctuating selection can lead to the maintenance
of genetic polymorphism. The crucial feature of the lottery model
with generation overlap is that fluctuating phenotype-dependent
viability selection only acts on juvenile survival whereas longlived adults (or a persistent dormant stage) are not affected by
environmental fluctuations. Together these ingredients result in
negative frequency dependence and allow for the coexistence of
genotypes that are specialized on different sets of the stochastically occurring environmental conditions. Ellner and Hairston
(1994) showed for this model that under certain conditions genetic
dimorphisms are not only protected from extinction on the ecological time scale but also protected on the evolutionary time scale
in the sense that no single phenotypically monomorphic genotype
exists that could invade and replace both genotypes present in
the dimorphism. Although Ellner and Hairston did not phrase it
this way, they essentially proved the existence of an evolutionary
branching point in the sense of Metz et al. (1996) and Geritz et al.
(1998).
Environmental variation, the fact that a single genotype produces different phenotypes, is favored under fluctuating selection
by two mechanisms. First, with adaptive phenotypic plasticity a
single genotype can produce different phenotypes, such that the
realized phenotype is adapted to the realized environment (e.g.,
Schlichting and Pigliucci 1998; West-Eberhard 2003). Adaptive
phenotypic plasticity relies on the existence of cues that reliably predict the future selective environment, a sensory machinery that allows a developing organism to perceive the cue and a
developmental switch, such that the appropriate phenotype can
be produced. The second mechanism is known as bet-hedging
(Slatkin 1974; Seger and Brockmann 1987; Philippi and Seger
1989). In unpredictably fluctuating environments a genotype can
increase its long-term growth rate by simultaneously producing
offspring with a range of different phenotypes because in this way
a genotype increases the chance that at least some of its offspring
will enjoy high reproductive success under future environmental conditions. In this article, we will be interested in the second
mechanism but not in the first one.
Classical models of bet-hedging were developed to explain
within-genotype variation in germination strategies (Cohen 1966;
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Brown and Venable 1986; Evans and Dennehy 2005), pupations
date (Hopper 1999), and diapause length (Hopper 1999; Menu
et al. 2000). More recently this idea received attention in the
context of alternative phenotypes in bacteria (Kussel et al. 2005;
King and Masel 2007; Malik and Smith 2008; Veening et al.
2008) and viruses (Stumpf et al. 2002). In all these models, the
distribution of offspring phenotypes is assumed to vary in a discrete manner with few alternative realizations. And indeed, for
the lottery model Sasaki and Ellner (1995) showed that a discrete distribution of offspring phenotypes is usually the optimal
bet-hedging strategy, even if the distribution of environmental
conditions experienced by the population is continuous. Just like
in the case of phenotypic plasticity, however, discrete phenotype
distributions require the existence of complicated developmental
switches. Several recent studies report such switches in bacteria (reviewed in Dubnau and Losick 2006; Veening et al. 2008)
where they might be more common than previously thought. In
multicellular organisms, there are only few well-supported examples for discrete phenotypic distributions due to bet-hedging and
these examples almost exclusively deal with occurrence of some
type of resting stage (Hopper 1999; Evans and Dennehy 2005).
It therefore seems that the distributions predicted by Sasaki and
Ellner (1995), despite of their theoretical optimality, are usually difficult to realize. However, there is continuous variation
in the expression of any quantitative trait and if environmental
variance effects many steps in a developmental pathway the natural expectation is that it is of Gaussian shape. The Gaussian
shape of environmental variance is also a standard assumption of
quantitative genetics. For fluctuating selection, also such a continuous shape of environmental variation can increase long-term
fitness due to bet-hedging, as has been pointed out by Bull (1987),
Haccou and Iwasa (1995), and Simons and Johnston (1997).
The question about the relative importance of genetic and environmental variance is particularly interesting under conditions
where these are favored simultaneously because this raises questions about their relative advantage and their interaction. A model
that allows to investigate these questions is the above-mentioned
lottery model with generation overlap (Chesson and Warner 1981;
Warner and Chesson 1985). Both Seger and Brockmann (1987)
and Ellner and Hairston (1994) showed for this model that a
genetic dimorphism can always be invaded and replaced by a
single bet-hedging genotype capable of producing the mixture of
the phenotypes present in the genetic polymorphism. Sasaki and
Ellner (1995) generalized these findings and showed that any
genetic polymorphism can be invaded and replaced by a single
genotype producing the optimal distribution of phenotypic variance. Thus, at least under the conditions of the lottery model, the
optimal polymorphic bet-hedging genotype (if it exists) outperforms any protected polymorphism of phenotypically monomorphic genotypes. Leimar (2005) showed that in a certain sense
C O M PA R I N G E N V I RO N M E N TA L A N D G E N E T I C VA R I A N C E
also the converse is true: in the lottery model the strength of selection for a mutation that introduces bet-hedging by changing
a phenotypically monomorphic genotype into a phenotypically
dimorphic genotype always exceeds the strength of disruptive selection favoring a genetic polymorphism. Thus, in the presence of
a complex optimal bet-hedger, genetic variance in the model cannot be maintained beyond mutation–selection balance. This leads
us to the question treated in this paper: How do environmental
and genetic variation evolve if we do not allow for elaborate genetic switches, that is, if phenotypic diversity can only evolve due
to changes in the genetic composition of the population and due
to changes in the width of the Gaussian phenotype distribution
produced by a single genotype.
The structure of this article is as follows. In the next section, we introduce the lottery model and adapt it for our purpose.
Specifically, we assume that at the phenotypic level individuals are
characterized by a quantitative character that is under stabilizing
selection with respect to an optimal value that fluctuates in time.
At the genotypic level individuals are characterized by their Gaussian offspring distribution with heritable genetic variation for the
mean and variance. An analytical treatment of the full mutation–
selection dynamics with a large number of co-segregating alleles
is not possible. To gain insights into the long-term evolutionary
dynamics, we will take a double approach. First, we study the
model in the adaptive dynamics framework, which relies on invasion analysis in the limit of rare mutations with small effect. In a
second step, the analytical predictions and insights are tested and
complemented by individual-based simulations of the full model.
We analyze two versions of the model. First we take the perspective that the amount of environmental variance is not evolvable
but results from a fixed constraint. Here we ask how a given level
of environmental variance affects the existence of evolutionary
branching points and thus the potential for the evolution of genetic polymorphism and increased genetic variance. We show that
increased environmental variance changes the condition for evolutionary branching to become more stringent. In the second version
of the model, we assume that both the mean and the variance of
the offspring distribution are subject to heritable genetic variation
and ask how the potential for genetic polymorphism is affected if
the environmental variance evolves toward its optimal value. We
show that genetic polymorphism can still evolve if the distribution
of selective optima is sufficiently different from a Gaussian distribution, either because of skewness or because it is leptokurtic.
The Model
ECOLOGICAL MODEL: LIFE CYCLE, PHENOTYPES,
AND FLUCTUATING SELECTION
Our model closely follows the lottery model introduced by
Chesson and Warner (1981). Consider a population of organisms
Figure 1.
Life-cycle graph. The adult stage can be entered in two
ways: First, by adults surviving with probability γ from one time
step to the next. Second, by newborns that first undergo a phase
of phenotype-dependent selection where survival depends on the
match between phenotype and ecological condition and second a
phase of phenotype-independent density regulation.
with a two-stage life cycle: a short-lived “juvenile” stage and a
long-lived “adult” stage (Fig. 1 ). Time is measured in discrete
steps, where one time step (or season) corresponds to the duration
of the juvenile stage, that is, the time for a newborn to reach its reproductive age. Census takes place just before reproduction when
the population entirely consists of adults. We assume that the population dynamics has settled at a stable nontrivial equilibrium of N̂
individuals. At this equilibrium, adults survive from one season to
the next with probability γ and thus persist on average for 1/(1 − γ)
seasons. We assume that all adults are identical with respect to
survival, and that γ (at the equilibrium) is constant in time. There
is thus no selection at the adult state. Each season, k = (1 − γ) N̂
newborns are recruited to the adult population. Recruitment takes
place in two steps in analogy to models of soft selection (Levene
1953; Wallace 1975). First, phenotype-dependent viability selection determines the frequency of phenotypes in the offspring pool.
Second, phenotype-independent density-dependence reduces the
number of newborns in the offspring pool to k. (It is this “lottery”
with k prizes that earned this model family its name.)
All selection in the model is thus due to differential viability
of juveniles. Newborn individuals are characterized by a quantitative trait z. In any given season, this trait is under Gaussian
stabilizing selection according to
(θt − z)2
,
w(z; θt ) = exp −
2σs2
where 1/σs2 determines the strength of selection and θt denotes the
trait value that maximizes juvenile survival in season t. Variation
in the selective optimum θt among seasons represents the fluctuating environment in the model. We assume that the random process
generating the selective optima is ergodic and converges toward a
stationary distribution with probability density function f (θ; y),
where the vector y contains the necessary parameters to describe
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H A N N E S S VA R DA L E T A L .
the distribution. Note that the requirements on ergodicity are not
very restrictive and that our analytical results are independent of
possible temporal autocorrelations in the occurrence of selective
optima given that the autocorrelations decay sufficiently fast over
time (Tuljapurkar 1990).
Our main results are independent of the exact distribution
f (θ; y). All that is needed are the first four central moments of
f (θ; y), or, more precisely, the mean μθ , variance σθ2 , skewness
gθ1 , and kurtosis gθ2 . To illustrate our results we will use four
example distributions f (θ; y). (1) A Bernoulli distribution f (θ; p)
with only two selective optima at θ1 = 1 and θ2 = 0, which
occur with frequency p and 1 − p; (2) a Gaussian distribution
f (θ; μθ , σθ2 ) with mean μθ and variance σθ2 ; (3) a Poisson distribution f (θ; λ) with rate parameter λ; and (4) a leptokurtic
Laplace distribution f (θ; μθ , b) = exp ( − |μθ − θt |/b)/2b, where
large deviations from the mean are overrepresented relative to a
Gaussian. The characteristics of the four distributions are summarized in Table 1.
where Vg and Ve denote the genetic and environmental variation,
respectively. In our notation Vg = Var(μz ) and Ve = σ2z .
For each reproducing individual with genotype x, the expected number of juveniles surviving viability selection in season
t is given by
∞
C
(z − μz )2
exp −
r (x, θt ) = w(z, θt ) dz
2σ2z
2πσ2z ∞
C
(μz − θt )2
= exp − 2
,
2(σz + σs2 )
σ2z + σs2
where C is the number of juveniles before selection (fecundity).
The expected contribution Yti of an individual with genotype x i
to the k juveniles that are recruited in each season is proportional
to r (x i , θt ). With n different genotypes in the population,
Yti (x 1 , . . . , x n , Nt1 , . . . , Ntn , θt ) = k
r (x i , θt )
,
n
j
j, θ )
r
(x
t
t
(1)
j=1
j
GENETIC MODEL
We assume that the quantitative trait z decomposes additively into
a heritable (genetic) and a nonheritable (environmental) component,
z = μz + e,
where the genotypic value μz can take any value in the real numbers and the environmental component e follows a Gaussian distribution with mean zero and variance σ2z . Note that e does not
depend on the external environment that determines θt , but rather
expresses the developmental (or micro-environmental) noise that
differs among individuals. The inverse of the width of the distribution, σ−1
z , measures the faithfulness of the developmental program
while exposed to this noise. It is thus a measure of environmental
robustness or canalization, which is itself a quantitative trait under
genetic control and can take any value in the positive real numbers. Each individual is thus characterized by a two-dimensional
vector x = (μz , σz ) and produces offspring whose phenotypes z
are drawn from a Gaussian distribution with mean μz and variance
σ2z . Although x does not characterize the phenotype of a single
individual but the distribution of phenotypes produced by a single
individual, we will in this article, refer to x as “trait vector” and
to the space spanned by μz ∈ R and σz ∈ [0, ∞) as “trait space.”
For most part of our work, we assume clonal inheritance where,
except for mutations, genotypes x are faithfully inherited from
a single parent to its offspring. Finally, both genotypic components are subject to mutation (see section Simulation Methods).
Population-wide phenotypic variation Vz can be decomposed in
the usual way,
Vz = Vg + Ve ,
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where Nt denotes the abundance of the jth genotype at time
step t. The dynamics of genotype x i is then given by
i
Nt+1
= Nti (Yti + γ).
(2)
In the lottery model, fluctuations in the selective optimum in combination with overlapping generations lead to negative frequency-dependent selection that can promote the coexistence of several species or genotypes. This phenomenon
was called “storage effect of generation overlap” by Chesson
(1983) and Warner and Chesson (1985). Qualitatively, the negative frequency-dependent selection can be understood as follows:
Individuals of a rare genotype have a relative advantage compared to if their own type was frequent in years favorable to them
because the majority of the population is unfit and thus their offspring faces less competition. Conversely, individuals of a rare
genotype have a relative disadvantage in years unfavorable to
them compared to if their own type was frequent because most of
the other individuals have many more surviving offspring. Without overlapping generations these two effects cancel over time.
With overlapping generations the offspring of favorable years are
“stored” in the persistent stage and this storage effect favors rare
over common types. The resulting negative frequency dependence
causes elevated long-term average growth rates of rare genotypes
compared to common genotypes and therefore makes protected
polymorphism possible.
ALTERNATIVE INTERPRETATIONS OF THE MODEL
In our presentation of the lottery model, we equated the shortlived life stage with juveniles that are recruited to the adult population, which represent the long-lived life stage. This is close to
the original design of the model by Chesson and Warner (1981)
C O M PA R I N G E N V I RO N M E N TA L A N D G E N E T I C VA R I A N C E
Four example distributions f (θ; y) of the selective optimum and their mean (μθ ), variance (σθ2 ), skewness (gθ1 = μθ3 /σθ3 ), and
excess kurtosis (gθ2 = μθ4 /σθ4 − 3). Positive (negative) values of gθ1 indicate distributions with a long right (left) tail. The excess kurtosis
is a measure of the peakedness of the distribution. Leptokurtic distributions with g2θ > 0 have a sharper peak and longer, fatter tails
Table 1.
than the normal distribution (e.g., Poisson or Laplace). Distributions with g2θ < 0 are called platykurtic. They have a more rounded peak
and shorter, thinner tails than the normal distribution (e.g., Bernoulli with p = 1/2). The critical generation overlap γ crit can be calculated
from these characteristics using equation (9).
f (θ)
μθ
σθ2
gθ1
gθ2
γcrit
Bernoulli
Gauss
Poisson
Laplace
p
p(1 − p)
1 − 2p
√
(1 − p) p
μθ
σθ2
λ
λ
1
√
λ
1
λ
μθ
2b2
4λ
√
1 + 4λ + 1 + 8λ
2
5
6 p2 − 6 p + 1
p(1 − p)
2 p(1 − p)
1 − 2 p(1 − p)
0
0
1
and Warner and Chesson (1985). As pointed out by Ellner and
Hairston (1994) and Sasaki and Ellner (1995), an alternative interpretation equates the short-lived stage with adult individuals
that reproduce once at the end of a time step and then die. Its
offspring can either immediately develop into a reproducing adult
or persist with probability γ in a resting stage. Prime examples for
this interpretation are annual plants with seed bank and Daphnia
resting eggs. Note that in the context of life cycles with resting stage it is usually the variable duration of this stage due
to γ > 0 that is considered a bet-hedging strategy. In contrast,
here we consider bet-hedging as achieved by producing variable
phenotypes in the trait under selection, as detailed in the next
paragraph.
Two alternative interpretations also exist with respect to selection. Above we assumed that fluctuating stabilizing selection
acts on a quantitative trait z that is expressed in the juveniles and
affects their viability. In this case environmental variation affects
the juvenile production of a single adult. However, as already
noted by Warner and Chesson (1985), the model only requires
that fluctuating selection affects seasonal recruitment (rather than
adult viability). In particular, the mathematical model remains
unaltered if we interpret z as an adult trait value determining its
fecundity with an optimum that fluctuates from season to season.
In this interpretation, environmental variation affects an adult trait:
a single adult contributes adults to subsequent generations with a
range of different phenotypes. We note that there is a slight difference in the way how mutation should be implemented in these two
interpretations. In the case of juvenile viability, selection should
0
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act on the newly mutated genotypes (before density regulation)
whereas in case of adult fertility selection, mutation will only
affect selection in the next season (after density regulation). Because the latter mechanism is computationally somewhat simpler
(i.e., only k recruits undergo potential mutation each season), we
have used this variant in our simulation model. (The difference in
the results should however be negligible.)
Under certain conditions the model presented here is also
equivalent to a consumer-resource model (H. Svardal et al., unpubl. data). This equivalence holds if we interpret the traits μz
and σz as the mean and variance of the Gaussian resource utilization function of a consumer, the parameter σs2 as the variance of
a normally distributed continuous resource whose abundance is
not affected by the presence of the consumer, and the value of
the selective optimum θt as the mean of the resource distribution.
Thus, in this interpretation fluctuating selection follows from a
Gaussian resource spectrum that shifts in time.
We conclude this section with a remark on the population dynamics. Here, we merely stated that the dynamics of the number of
recruits per season reaches a nontrivial fixed point. Many implementations of density dependence will do this job, ranging from
competition for a limited amount of space, as assumed in the original formulation of the lottery model (Chesson and Warner 1981),
the saturating yield model due to Levin et al. (1984) and used by
Ellner and Hairston (1994) to logistic dynamics (Chesson 1984).
Importantly, even the assumption of a fixed point is by no means
decisive for the existence of the storage effect but rather a requirement for mathematical tractability. Chesson (1984) showed that
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the storage effect continues to exist if juvenile recruitment is described by Lotka–Volterra dynamics, indicating that qualitatively
our results should also hold for a much larger family of population
dynamical models.
Analytical Methods: Invasion
Analysis
We are interested in two main aspects of the model: (1) the longterm evolution of enhanced levels of environmental variance, and
(2) the potential for the evolution and maintenance of genetic variation due to balancing selective forces (i.e., even in the absence of
recurrent mutation) in the presence of elevated levels of environmental variance. Both these aspects depend primarily on properties of the (variable) fitness landscape of the model. This fitness
landscape can be explored effectively by assuming a simplified
mutational process where rare mutations change the genotypic
trait values from x to x + δx. Following the adaptive dynamics approach (Dieckmann and Law 1996; Metz et al. 1996; Geritz
et al. 1998), we will also assume that mutational effects δx (which
can occur in all direction of the two-dimensional trait space) are
small. If we assume that the population is initially monomorphic
for a “resident” genotype x, the evolutionary dynamics can then be
determined by following a series of mutation-substitution events.
The fundamental tool to predict this dynamics is the so-called
invasion fitness ρ(x , x), which is defined as the logarithm of the
expected long-term growth rate of an infinitesimally small mutant
subpopulation with genotype x in the resident population (Metz
et al. 1992; Metz 2008). The long-term growth rate of a population in a time-varying environment is given by the product of its
seasonal growth rates, and the average growth rate corresponds to
the geometric mean. Using that the geometric mean is turned into
an arithmetic mean on the log scale, and that time averages can be
replaced by averages over the distribution of selective optima (ensemble averages) under our ergodicity assumptions on the time
series of the selective optimum θt , we can derive the invasion
fitness ρ(x , x) for our model from equation (2) as (Appendix A),
ρ(x , x) =
r (x , θt )
ln (1 − γ)
+ γ f (θ; y)dθt .
r (x, θt )
(3)
In sufficiently large populations, a mutant has a positive probability to invade if ρ(x , x) > 0 and is doomed to extinction if
ρ(x , x) < 0. Furthermore, if mutations have small phenotypic
effect, successful invaders will go to fixation and replace the resident, unless the mutant (as new resident) can itself be invaded by
the former resident (Dercole et al. 2002; Geritz et al. 2002; Geritz
2005; Dercole and Rinaldi 2008).
Points of special interest for the evolutionary dynamics as
it results from a trait substitution sequence are trait vectors x ∗
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where the selection gradient S with entries
∂ρ(x , x) for xi ∈ {μz , σz }
Si (x) =
∂ xi x =x
(4)
equals zero, Si (x ∗ ) = 0, because they are equilibria of the
monomorphic evolutionary dynamics. Such points are called
“candidate ESSs” (e.g., Ellner and Hairston 1994), “potential
ESSs” (e.g., Otto and Day 2007) or “evolutionarily singular
points” (Metz et al. 1996; Geritz et al. 1998). Singular points
can be classified according to two independent properties: “invadability” and “convergence stability” and we will now discuss
these two properties in more detail.
As by definition at a singular point the selection gradient
equals zero, the fitness landscape locally around a singular point
is described by the Hessian matrix H = [h kl ] of invasion fitness
with entries
∂ 2 ρ(x , x) for xk , xl ∈ {μz , σz }.
(5)
h kl =
∂ xk ∂ xl x =x=x ∗
If the Hessian matrix is negative definite, then the singular point is
a local maximum and therefore uninvadable by nearby mutants.
Conversely, if the Hessian matrix is positive definite, then the
singular point is a local minimum of the fitness landscape and
can be invaded by all nearby mutants. Of particular importance to
us will be the case where the Hessian matrix is indefinite. Then
the singular point is a saddle point of the fitness landscape and
can be invaded by some nearby mutants but not by others. Figure 2 shows an example of a contour plot of the fitness landscape
where the Hessian matrix is indefinite. At a saddle point in a twodimensional space, two directions exist in which the height of the
fitness landscape does not change. These directions are given by
the vectors (1, v 1 ) and (1, v 2 ) with (1, vi )H(1, vi )T = 0 and delimit regions in (μz , σz )-space of mutants with positive invasion
fitness from regions of mutants with negative invasion fitness.
Convergence stability indicates whether a singular trait vector is an attractor of the gradual evolutionary dynamics (Eshel
1983; Abrams et al. 1993; Metz et al. 1996; Geritz et al. 1998). In
a two-dimensional trait space convergence stability is determined
by the Jacobian matrix J = [ jkl ] of the selection gradient with
components
∂ Sk (x) (6)
jkl =
∂ xl x=x ∗
(Leimar 2009). A singular point is convergence stable independently of possible genetic correlations (“strongly convergence
stable,” Leimar (2009)) if J is negative definite. If J is indefinite
the point is stable if genetic variation satisfies certain conditions,
and otherwise it is not. Finally, if J is positive definite then the
point is an evolutionary repellor. A more rigorous account on this
can be found in appendix B.
C O M PA R I N G E N V I RO N M E N TA L A N D G E N E T I C VA R I A N C E
Figure 2. Contour plot of the fitness landscape locally around a
singular point (μ∗z , σ z∗ ) as it emerges if the resident type is located
at the singular point. In this example the Hessian matrix of inva-
sion fitness is indefinite and thus the fitness landscape is a saddle
point. The coordinate axes give the difference between mutant
genotypes (μz, σ z ) and the singular strategy located at the origin. Mutants in the gray region have positive invasion fitness and
mutants in the white region have negative invasion fitness. The
hatched line gives the direction of strongest disruptive selection,
corresponding to the dominant eigenvector of the Hessian matrix. α d denotes the angle between the μz -axis and the dominant
eigenvector and α denotes the angle of the sector with positive
invasion fitness. The symmetry in the fitness landscape reflects
the fact that the Hessian matrix is symmetric.
A singular point that is both convergence stable and uninvadable is called a “continuously stable strategy” or CSS (Eshel
1983). It is a final stop of evolution. Singular points that are
convergence stable but invadable by nearby mutants are called
“evolutionary branching points” (Metz et al. 1996; Geritz et al.
1998). Selection initially acts in the direction toward such points,
but once the trait value of the population is sufficiently close to
the singular point selection turns disruptive and favors an increase
in phenotypic variance (Rueffler et al. 2006). In the case of clonal
organisms this increase can be realized by the emergence of two
independent lineages and it is this scenario that earned branching
points their name.
Simulation Methods
Extensive individual-based simulations are performed for three
reasons. First, all analytical results are tested with respect to their
robustness if the assumptions of the adaptive dynamics approximation are violated. Second, the polymorphic evolutionary dynamics after evolutionary branching cannot be investigated analytically and are therefore explored by means of simulations.
Third, simulations are employed to investigate evolution in sexu-
ally reproducing diploid populations. Simulations are performed
using Matlab R2009b (Mathworks 2009). More information on
the computational implementation is given in the Supporting information.
Mutations affect μz and σz separately and occur with percapita probability u for each component. Each mutation adds an
increment ±δ to the genotypic value. The mutational parameters
u and δ vary between 10−1 and 10−3 and population size N varies
between 1000 and 10, 000. The exact choice of parameters is
given in the figure legends.
To investigate the evolutionary dynamics of diploid sexually
reproducing populations, we use an infinite alleles model and
assume that the traits μz and σz are each coded by one additive locus. Adults produce haploid juvenile offspring (gametes) through
meiosis with recombination rate r ∈ [0, 0.5]. Selection and density regulation act on haploid juveniles. Random pair formation
of two surviving haploids results in diploid adult individuals.
Results: Fixed Environmental
Variance
In this section, we investigate the scenario that the environmental variance does not evolve. Thus, we consider σz as a
fixed parameter. For this one-dimensional model we show in
Appendix D that μ∗z = μθ is a singular point. This singular point
is always convergence stable irrespective of the value of σz . It is
invadable if and only if
γ > γcrit =
σs2 + σ2z
.
σθ2
(7)
If condition (7) is fulfilled, μ∗z = μθ is an evolutionary branching point. Here selection is disruptive allowing different alleles to
coexist in a protected polymorphism. If the inequality sign in condition (7) is reversed, then μ∗z = μθ is uninvadable and therefore a
CSS. Condition (7) is a straightforward generalization of a result
by Ellner and Hairston (1994) who showed that a polymorphism
of phenotypically monomorphic genotypes (σz = 0) is evolutionary stable if γ > σs2 /σθ2 . Thus, genetic diversification is favored
by a large generation overlap and strong selection against phenotypes deviating from the selective optimum (small σs2 ) relative
to the variance in the distribution of selective optima. Increasing environmental variation σ2z reduces the potential for genetic
diversification by effectively weakening within-time step stabilizing selection on a genotype in the expected way (e.g., Bürger
2000, p. 160).
We investigate the evolutionary dynamics once genetic variation has been build up at a branching point by means of individualbased simulations. Figure 3 shows representative runs for the
case that f (θ; μθ , σθ2 ) is a Gaussian distribution for different combinations of σs and σz . The simulations show that not only the
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H A N N E S S VA R DA L E T A L .
where σθ2 > σs2 is required for x ∗2 . Figure 4 shows the position of the singular points in the two-dimensional trait space.
The following conditions for convergence stability and uninvadability can be derived using the criteria described above (see
Appendix E for derivations):
•
•
Figure 3.
Branching for a fixed amount of environmental vari-
ance for the case of normally distributed selective optima. Time
in generations (where a generation lasts 1/(1 − γ) time steps) is
plotted on the x-axis and black dots indicate the distribution of
individuals with genotype μz (on the y-axis) in the population. (A)
The evolutionary dynamics for σ s = 0.1 and σ z fixed to zero. The
population branches into two temporally stable sub-populations
that branch again into further unstable sub-populations. (B) Increasing environmental variance to σ z = 0.2 lets the secondary
branching disappear while the pattern of primary branching remains qualitatively similar. (C) A combination of σ z = 0 (like in (A))
and weaker selection σs2 = 0.12 + 0.22 gives similar results as in
(B). We conclude that the main effect of positive environmental
variance is to weaken selection. The set of selective optima are
Gaussian pseudo-random numbers drawn once and then applied
to all of three runs to rule out stochastic differences other than
due to drift. Parameters: γ = 0.5, N = 10, 000, u = 0.01, δ = 0.01.
branching condition but also the potential for secondary branching depends on the sum σs2 + σ2z . In the terminology of competition
models, increased environmental variation increases the effective
niche width and thereby decreases the number of possibly coexisting phenotypic cluster (cf. Bolnick 2006). Qualitatively the
same simulation results can be found if f (θ; y) is a Poisson distribution.
Results: Evolvable Environmental
Variance
For the two-dimensional model, where both the mean and the
standard deviation of the phenotypic offspring distribution, μz
and σz , can evolve, we find that two different singular strategies
can exist (Appendix C),
x ∗1 = (μ∗z1 , σ∗z1 ) = (μθ , 0)
(8a)
x ∗2 = (μ∗z2 , σ∗z2 ) = (μθ , σθ2 − σs2 ),
(8b)
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The singular point x ∗1 , corresponding to a fully canalized
genotype, is convergence stable and uninvadable, and thus
a CSS, if σθ2 < σs2 . For σθ2 > σs2 , x ∗1 is not strongly convergence stable, but a saddle point of the evolutionary dynamics. Then the invasion of mutants with increased σz -values
is possible and the evolutionary dynamics approach x ∗2
(Fig. 4).
The singular point x ∗2 is convergence stable whenever it
exists, thus whenever σθ2 > σs2 . It is invadable and thus an
evolutionary branching point if
γ > γcrit =
4 + gθ2 +
4
2
2
8gθ1
+ gθ2
.
(9)
Then x ∗2 it is a saddle point of the fitness landscape and allows for evolutionary branching in at least some directions
of the trait space, and thus for the emergence and maintenance of genetic variation. The critical generation overlap
γcrit can be calculated explicitly for any distribution f (θ; y)
and is given for the four distributions introduced in Section
The Model in Table 1. If condition (9) is not fulfilled, x ∗2
is uninvadable and thus a CSS.
We note that the “optimal” environmental variance σ∗2
z as it is
found at the convergence stable equilibrium coincides with the optimal environmental variation reported by Bull (1987) for a model
that is similar to the lottery model with γ = 0. However, because
there is no generation overlap in Bull’s model, genetic variation
cannot be maintained. From equation (9) several conclusions can
be drawn.
1. Adaptive genetic diversification is favored by a large generation
overlap γ.
2. Adaptive genetic diversification is favored by a small value of
the critical generation overlap γcrit , and thus by large values
of |gθ1 | and gθ2 , measuring the asymmetry and leptokurtosis
of the distribution f (θ; y) of the optimal phenotype across
seasons.
3. For a symmetric distribution (gθ1 = 0), we obtain γcrit = 4/(4 +
gθ2 + |gθ2 |) and γcrit < 1 if and only if f (θ; y) is leptokurtic (gθ2 > 0). In particular, for the Gaussian distribution with
gθ2 = 0, condition (9) becomes γ > γcrit = 1 which can never
be fulfilled.
4. For asymmetric distributions (gθ1 = 0), however, γcrit < 1 regardless of the kurtosis. Then genetic diversification is possible
if the generation overlap is sufficiently large.
C O M PA R I N G E N V I RO N M E N TA L A N D G E N E T I C VA R I A N C E
Location of singular points x∗ = (μ∗z , σ z∗ ) and direction of the monomorphic evolutionary dynamics as a function of σθ2 and σs2 .
< σs2 the only singular point is x∗ = (μθ , 0) which is an attractor of the evolutionary dynamics. (B) For σθ2 > σs2 two singular
(A) For
points exist: x1∗ = (μθ , 0) and x2∗ = (μθ , σθ2 − σs2 ) with x1∗ being a saddle point and x2∗ being an attractor of the monomorphic evolutionary
Figure 4.
σθ2
dynamics.
5. In the limit γ = 1, we have γ ≥ γcrit and genetic diversification
is favored regardless of the details of the distribution f (θ; y).
These results allow for the following interpretation. Genetic polymorphism can evolve if the Gaussian distribution of
phenotypes produced by a single genotype deviates from the
distribution of selective optima, either because the latter is asymmetric (producing an excess of extreme conditions in one direction), or because it is leptokurtic (where extreme conditions
in both directions are more frequent than under a normal distribution). The extent of this deviation determines the minimum
amount of negative frequency dependence (and thus the minimal
generation overlap) that is needed for the emergence of genetic
diversity.
Let us compare condition (7) for uninvadability in the model
derived condition (9). By
with fixed variance σz with the newly
inserting the singular value σ∗z = σθ2 − σs2 into condition (7), we
find γ > 1, which is never fulfilled. In other words, condition
(7) suggests that at x ∗2 evolutionary branching is never possible.
This apparent contradiction is resolved by noting that in the onedimensional model genetic diversification is restricted to the μz direction of the two-dimensional trait space. Because x ∗2 is a
saddle point of the fitness landscape for γ > γcrit (rather than
a minimum), branching is only possible in some, but not all,
directions of the trait space. In particular, branching only in the μz direction will never be possible. In terms of Figure 2, the gray area
where mutants have positive invasion fitness never extends in the
region horizontally to the right and left of x ∗2 . Instead, evolutionary
branching can occur in a compound direction, simultaneously
changing the mean and variance of the offspring distribution. In
such a dimorphism, one type becomes a canalized specialist for
the more common ecological conditions and the other type takes
the role of a de-canalized bet-hedger thriving on the less-common
ecological conditions. In Appendix F, we describe in more detail
how the shape of the fitness landscape at x ∗2 depends on the details
of the model given that condition (9) is fulfilled.
SIMULATION RESULTS
The analytical results suggest that genetic diversification evolves
if γ > γcrit where γcrit is given by the right-hand side of inequality
(9). In computer simulations we find that this prediction is highly
accurate for small mutation rates and small mutational step sizes.
However, branching occurs more easily if either of them is large
(Fig. 5). Then it can occur already for γ < γcrit and at points in
trait space that are far away from the singular point. Thus, our
analytical results are conservative concerning the potential for
the emergence of genetic diversity. The deviations result from
the fact that adaptive dynamics is a local analysis, applying to
monomorphic resident populations and small mutational steps.
More detailed explanations for the deviations can be found in
Appendix G.
Due to the stochastic occurrence of selective optima in the
simulations, frequently one of the branches that emerged at an
evolutionary branching point dies out. We therefore discuss the
long-term evolution after branching in two steps, first, for those
simulation runs where extinction did not occur and then for runs
with extinction.
The endpoint of the polymorphic evolutionary dynamics
depends on the distribution of selective optima f (θ; y). For
Bernoulli-distributed selective optima, the two subpopulations
emerging at a branching point evolve the genotypes x 1 = (0, 0)
and x 2 = (1, 0), but do so via different routes (Fig. 6A–C).
Branching occurs in the direction of the dominant eigenvector
of the Hessian matrix of invasion fitness (cf. Fig. 6A). Thus,
the branch that is closer to the more frequently occurring selective optimum evolves a phenotype that matches this selective
optimum while simultaneously canalization increases, that is, environmental variance decreases. The branch that is closer to the
rare selective optimum also evolves a phenotype that matches this
selective optimum but simultaneously becomes a de-canalized
bet-hedger by increasing its environmental variance. The transient increase in environmental variance creates a path from
the singular point to the phenotype matching the less-frequent
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H A N N E S S VA R DA L E T A L .
Condition for evolutionary branching at the singular point x2∗ for the case of Bernoulli-distributed selective optima as a
function of the frequency of selective optimum 1, p, and the generation overlap, γ. Based on the adaptive dynamics approximation
Figure 5.
(cf. Table 1) branching occurs for combinations of p and γ corresponding to points above the white hatched line. Gray scales indicate
whether branching occurred in a simulation run within one million time steps for different values of the mutational step size δ (A) and
the mutation rate u (B). In panel (A) black areas correspond to combinations of p and γ where branching occurs for δ = 0.005 and u =
0.01. Increasingly lighter shades of gray correspond to combinations of p and γ where branching occurs for δ = 0.0075, 0.01, 0.025, 0.05,
respectively, but not for smaller values. In panel (B) black areas correspond to combinations of p and γ where branching occurs for u =
0.001 and δ = 0.01. Increasingly lighter shades of gray correspond to combination of p and γ where branching occurs for u = 0.005, 0.01,
0.05, 0.1, respectively, but not for smaller values. White areas correspond to combinations of p and γ where branching did not occur for
any for the tested mutation parameters. The adaptive dynamics approximation predicts the onset of branching well for small mutational
step size and low mutation rates. Larger mutational step sizes and higher mutation rates result in branching occurring for combinations
of lower p- and γ-values than predicted by the adaptive dynamics approximation. Parameters: N = 10, 000, σ s = 0.1.
selective environment that bypasses the fitness valley that blocks
the direct path.
Also with Poisson-distributed selective optima branching
takes place in a diagonal direction (Fig. 6D,G) such that the
branch that matches the more common selective optima, the one
with the lower value of μz , decreases its environmental variance
whereas the branch evolving higher values of μz increases its
environmental variance. Following branching, two main types of
long-term evolutionary dynamics are observed. First, for larger
values of λ, resulting in more symmetric distributions, only one
branching event occurs and the two emerging branches reach endpoints where they both maintain positive environmental variance
(Fig. 6D–F). For smaller values of λ, resulting in more asymmetric distributions, more than one branching event can be observed
(Fig. 6G–I). The subpopulations become completely canalized
(σz = 0) and evolve phenotypes that match the most common
environmental conditions (μz = 0, 1, 2, . . .). This is true for all
subpopulations except for the subpopulation that is characterized
by the largest μz -value, which maintains positive environmental
variance (Fig. 6D–F).
For Laplace-distributed selective optima, branching occurs
only in σz . Simulations with one or two branching events have been
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EVOLUTION 2011
observed, with larger generation overlap favoring two branching
events. In the case of a single branching event, dimorphic evolution proceeds to a coalition x 1 = (μ1 , σ1 ) and x 2 = (μ2 , σ2 ) with
μ1 = μθ = μ2 and 0 < σ1 < σ∗z < σ2 (Fig. 6J–L). In this polymorphism one subpopulation mainly exploits the dense region
around the mean whereas the other can persist by also thriving on the substantial tails of the distribution of selective optima. In the case of two branching events, three subpopulations
evolve, differing only in their level of positive environmental
variation.
In many simulations, we observed that one of the subpopulations went extinct before the equilibrium described above could
be reached (Fig. 7). This leads to a characteristic pattern of transient genetic dimorphism with recurrent branching and extinction. Extinction typically happens after a series of environmental conditions that is unfavorable for the individuals in one of
the branches. Consequently, we observe that with negative autocorrelation in the sequence of selective optima extinction occurs less frequently. Almost always extinction affects the branch
with μz -values matching the less-frequent environments. The reason is that this branch consists on average of fewer individuals and is therefore more susceptible to stochastic fluctuations.
C O M PA R I N G E N V I RO N M E N TA L A N D G E N E T I C VA R I A N C E
Figure 6.
Clonal evolutionary dynamics before and after branching for Bernoulli-, Poisson-, and Laplace-distributed selective optima.
Panels in the left column show the dynamics in (μz , σ z )-space where time proceeds from lighter toward darker gray. Panels in the right
column show identical simulation runs but with time plotted on the x-axis and values for μz and σ z on the y-axis. The cross in the panels
in the left column correspond to the singular point x2∗ . The simulations show that for the chosen parameters the adaptive dynamics
approximation accurately predicts the location of this singular point. (A)–(C) With Bernoulli-distributed selective optima, the population
splits into two discrete clusters, each of which specializes on one selective optimum and becomes fully canalized. (D)–(F) With a Poisson
distribution and larger values of λ, typically only a single branching event takes place. In this case, both branches maintain positive
levels of environmental variation σ z . (G)–(I) With a Poisson distribution and intermediate values of λ, typically more then one branching
event takes place. The resulting subpopulations evolve to become specialists for the most frequently occurring selective optima θ t =
0, 1, . . ., except for the subpopulation characterized by the highest μz -value, which maintains a positive environmental variance σ z2 . (J)–(L)
With Laplace-distributed selective optima, branching occurs only in the component σ z . The resulting subpopulations are thus arranged
vertically above each other in trait space. Parameters: N = 10, 000, σ s = 0.1, u = 0.001; (A)–(C) γ = 0.95, p = 0.8, δ = 0.005; (D)–(F) γ =
√
0.975, λ = 1, δ = 0.01; (G)–(I) γ = 0.95, λ = 1.5, δ = 0.01; (J)–(L) γ = 0.85, b = 0.5, δ = 0.01.
Extinction of branches becomes more likely with decreasing generation overlap γ, increasing skewness gθ1 , stronger selection σs−1
and decreasing population size N. All these changes increase
the stochastic component in the population dynamics. Among
these factors, variation in γ has the strongest effect (cf. Fig. 7
where we show characteristic simulation runs for four different
values of γ). Interestingly, while increasing skewness facilitates
branching (cf. condition 9), such an increase has negative effects on the stability of the emerging branches in the face of
stochasticity.
Many organisms do not reproduce clonally, but sexually, and
it is therefore important to evaluate to what extend our results carry
over to diploid sexually reproducing populations. The dynamics
of the traits μz , σz and of the genetic variance Vg is expected to
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H A N N E S S VA R DA L E T A L .
Figure 7. Polymorphic evolutionary dynamics with extinction for the case of Bernoulli-distributed selective optima for different values of
the generation overlap γ. For each value of γ, two panels are shown, one depicting the dynamics of μz (upper panel) and one depicting
the dynamics of σ z (lower panel). Time is plotted on the x-axis as number of generations, where one generation equals 1/(1 − γ)
time steps. Hatched lines indicate the value of μz and σ z corresponding to x2∗ . In all cases γ > γ crit such that x2∗ is a branching point.
However, the subpopulation evolving to the rare environmental condition frequently goes extinct resulting in characteristic patterns of
transient genetic dimorphism with recurrent branching and extinction. (A) For γ = 0.6, the waiting time until branching is relatively long
and emerging branches are short-lived. (B) Increasing the generation overlap γ shortens the waiting time until branching and branches
on average diverge further before extinction takes place. Note that in both (A) and (B) after extinction the remaining branch evolves
back to the singular point before re-branching takes place. Panel (C) illustrates how stochasticity affects the branching location. For
γ = 0.8 the remaining branch only evolves back to the singular point in some cases while in other cases branching takes place while
the remaining branch is still located at (μz , σ z ) = (0, 0), corresponding to a canalized specialist for the more common environment. (D)
Increasing the generation overlap even further stabilizes the evolutionary dynamics and branches become very long-lived. Even larger
generation overlap results in persistent sub-populations as shown in Figure 6. Parameter values: N = 10, 000, p = 0.8, σ s = 0.1, u = 0.001,
δ = 0.005.
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C O M PA R I N G E N V I RO N M E N TA L A N D G E N E T I C VA R I A N C E
A
B
z
0.4
0.5
0.3
0.3
0
0.2
0.5
z
z
1
0.8
0.5
0.1
0.3
0
0
0
0.2
0.4
0.6
0.8
1
0
10
20
30
40
50
60
3
10 generations
z
Figure 8.
Diploid evolutionary dynamics before and after branching for Bernoulli-distributed selective optima. Panel (A) shows the
dynamics in (μz , σ z )-space. The simulation is initialized at (μz , σ z ) = (0.5, 0) from where the dynamics proceed to the singular point x2∗
(indicated by the “+”). The dynamics after branching is represented by three snapshots in time (squares: generation 21 × 103 , triangles:
24.5 × 103 , dots: 60 × 103 ). Panels (B) and (C) show the identical simulation run but with time plotted on the textit x-axis and values for
μz and σ z on the y-axis. Note that from generation 55 × 103 onwards genetic variation in σ z is lost and that the evolutionary endpoint is
identical to the one predicted by the clonal dynamics (cf. Fig. 6A–C). Panel (B) shows that also individuals heterozygous in μz disappear
when the variation in σ z is lost. This is due to the fact that diploid adults are formed after selection on the gametes and in each step only
one of the specialist genotypes has a notable amount of offspring. Parameter values: recombination rate=0.25, N = 2000, p = 0.8, γ =
0.95, σ s = 0.1, u = 0.001, δ = 0.01.
depend strongly on the details of the genetic architecture. Thus, a
full exploration of these dynamics in sexual populations is beyond
the scope of this article. Here, we analyze the case that each trait
is characterized by a single diploid additive locus with a continuum of alleles. As in the asexual case, allelic values can change
through mutations of small effect in the individual-based simulations (cf. Section Simulation Methods). Populations evolve to the
predicted singular point as long as mutation rates and effects are
sufficiently small. If γ > γcrit , such that x ∗2 is a branching point,
an increase in genetic variation is observed also in the diploid
case. Two qualitatively different outcomes can be distinguished
and their appearance shows a clear correlation with the recombination rate. First, with high recombination rates the population often
remains a single connected cloud in trait space and genetic variation increases only slightly. Second, with low recombination rates
branching occurs more frequently and discrete diverging clusters
appear (Fig. 8). This result can be explained in the following way.
In diploid sexual populations interbreeding and recombination result in up to nine different trait combinations, of which not all lie
in the sector with positive invasion fitness (cf. Fig. 2). At high
recombination rates, individuals with maladapted trait combinations occur at high frequency and selection against them overrides
the disruptive selection acting on individuals with favorable trait
combinations. Disruptive selection acting on some trait combinations then results only in a slightly increased genetic variance
at mutation–selection balance. In contrast, if the frequency of
maladapted trait combinations is sufficiently low, more extreme
phenotypes are favored, such that discrete diverging clusters in
trait space can appear.
The genetic architecture of most quantitative traits is polygenic. What can we expect for such a more realistic genetic architecture? If μz and σz are determined by many loci of similar effect
size, we expect that discrete clusters cannot evolve. However,
two previous studies have shown that under frequency-dependent
disruptive selection a genetic architecture can evolve where allelic effects are concentrated at few (or even single) loci (Kopp
and Hermisson 2006; van Doorn and Dieckmann 2006). Based
on these results, we suggest that the scenario investigated here
might also capture the long-term behavior of traits with more
complicated genetic architectures.
Discussion
Assume that natural selection favors phenotypic diversity within
a population. What, then, are the evolutionary mechanisms that
underlie this variation? In particular, under which conditions is
(part of) this variation heritable, that is, due to the maintenance
of genetic variation? In this article, we investigate the coevolution of genetic and environmental variation under fluctuating
selection. Our analysis is based on the lottery model (Chesson
and Warner 1981; Warner and Chesson 1985), which is characterized by a life cycle with a short-lived and a long-lived stage.
Only the short-lived stage is sensitive to fluctuating environmental
conditions: the optimal phenotype maximizing recruitment to the
long-lived stage varies according to some distribution. In contrast,
individuals in the long-lived stage are not affected by the fluctuating selection pressure and experience a constant mortality rate.
This two-stage life cycle is decisive because under these conditions both, environmental and genetic variation, can be selectively
EVOLUTION 2011
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H A N N E S S VA R DA L E T A L .
favored. Increased environmental variation can be favored because of bet-hedging: genotypes that produce a range of different
phenotypes increase their chance that at least some of their offspring enjoy high reproductive success under each realized environmental condition (Bull 1987; Seger and Brockmann 1987).
Genetic variation can be maintained because of the “storage effect” of the two-stage life cycle: Genotypes can survive adverse
environments in the unaffected long-lived stage, and rare genotypes contribute disproportionally to the recruitment pool in generations with their favored environment. As a result selection becomes negatively frequency-dependent leading to the emergence
and maintenance of genetic polymorphism (Chesson and Warner
1981; Ellner and Hairston 1994).
In nature, the key assumptions of the lottery model apply to
a wide range of organisms. In many perennial plant and animal
species, major selection components affect only a short juvenile
stage, whereas long-lived adults are relatively insensitive. Alternatively, the short-lived stage can be interpreted as reproductively
active individuals, which produce a long-lived persistent stage
such as seeds or resting eggs (see the last part in Section Genetic
Model for a discussion of different model interpretations). An organism with a probability γ to survive as an adult (or seed) from
one generation to the next lives on average 1/(1 − γ) generations.
Organisms surviving, for example, 10 reproductive periods, corresponding to γ = 0.9, are no rarity. Often cited examples include
trees, fish, sessile marine organisms, animals with resting eggs,
and plants with seed banks (Chesson and Warner 1981; Ellner
and Hairston 1994; Sasaki and Ellner 1995; Warner and Chesson
1985).
Our model assumptions concerning the environmental fluctuations are almost completely general: the value of the selective
trait optimum can be picked from an arbitrary discrete or continuous distribution, with or without autocorrelations between generations. The example distributions that we have chosen for a detailed
study represent different plausible types of ecological variation:
(1) The Bernoulli distribution with two discrete selective optima
can be a useful approximation if the strength of selection depends
on whether a particular event (epidemic disease, forest fire, thunderstorm) happens at all during the short-lived phase, rather than
on quantitative details; (2) a Gaussian distribution can be expected
whenever the selective optimum is determined by many independent contributions of small effect; (3) a Poisson distribution with
rate parameter λ expresses the probability for a given number of
events (such as predator attacks) occurring in a fixed period of
time if these events are independent and occur with an average
rate λ. Also the number of independent events in other specified
intervals such as distance, area, or volume are Poisson-distributed
(e.g., amount of rain drops on a square meter); (4) the last example
is chosen more for theoretical reasons: the Laplace distribution
allows for variation in the kurtosis without introducing skew.
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Past research has shown that any genetic polymorphism in
the lottery model can be invaded and displaced by a genotype
producing an optimal distribution of environmental variation and
that the phenotype distribution produced by this “optimal bethedger” is generically discrete rather than continuous (Seger and
Brockmann 1987; Ellner and Hairston 1994; Sasaki and Ellner
1995). The punch line of this research is that under fluctuating selection genetic variation beyond mutation–selection balance can
only be maintained if constraints exist that prevent a single genotype from producing environmental variation of the optimal kind.
This leads to the central assumption of our model. Our starting
point is the observation that relatively few examples of discrete
bet-hedgers are known in nature, at least if compared with the
prevalence of unstructured continuous environmental variation,
which is ubiquitous across traits and taxa. We therefore constrain the environmental variation on the focal trait z to follow
a Gaussian distribution—as it is assumed throughout in models
of quantitative genetics. Bet-hedging can still evolve, but only by
increasing the width σz of this distribution. Modeling the environmental variance as an evolvable trait in this way, relates our
approach to the theory of environmental (de)canalization, where
environmental variance results from incomplete buffering of the
developmental system against the effects of environmental fluctuations (Gavrilets and Hastings 1994; Wagner et al. 1997; de
Visser et al. 2003). We note that our theory also applies to more
general shapes of continuous environmental variation, as long
as this shape can be transformed into a Gaussian by an appropriate choice of scale. The distribution of the selective optima
needs to be transformed accordingly. A shape of the environmental variation that closely matches the initial optimum distribution
will therefore result in an optima distribution that is close to a
Gaussian after the transformation. Whether genetic variation can
evolve then depends on the mismatch between the transformed
distributions.
Our main results are all based on analytical work and can be
summarized as follows. Fluctuating selection in the lottery model
will often favor a mixture of genetic variation and Gaussian bethedging, and both factors combine to produce the observed phenotypic diversity. Depending on the genetic and environmental
details, we find: (1) If the environmental variance σ2z is fixed (no
heritable variation), selection on the trait z turns disruptive, and
genetic variance can evolve if the generation overlap γ exceeds a
critical value γcrit . This critical overlap increases linearly with σ2z
(cf. eq. 7). (2) If σz is free to evolve, it evolves under fluctuating
selection to positive values if and only if σθ2 > σs2 , where σθ2 denotes the variance in the distribution of selective optima and σs−1
denotes the strength of Gaussian stabilizing selection within each
season.
The optimal width of the phenotype distribution equals
σ∗z = σθ2 − σs2 . (3) If the environmental variance is at its optimal level σ∗2
z , genetic diversification due to disruptive selection
C O M PA R I N G E N V I RO N M E N TA L A N D G E N E T I C VA R I A N C E
is still possible for a sufficiently large generation overlap if the
environmental distribution of the selective optima is either asymmetric or leptokurtic (cf. eq. 9 for the precise condition). We find
that genetic diversification occurs never purely in the direction
of the mean trait value μz of a genotype, but generally along a
diagonal direction in the two-dimensional (μz , σz )-trait space: a
genotype with reduced σz that specializes on common environments coexists with a genotype with increased σz , exploiting a
wide variety of rarer environments via bet-hedging.
Finally, we have used computer simulations to test the robustness of the analytical predictions and to study the long-term consequences of disruptive selection and genetic differentiation. With
a sufficiently small mutation rate and mutational step size both the
clonal and the diploid simulations confirm that populations evolve
an optimal environmental variance σ∗2
z . However, in particular increasing mutational step size results in the evolution of genetic
polymorphism under much wider conditions than predicted by
the analytical theory (cf. Fig. 5) and at points in trait space far
away from a branching point. (Thus, by choosing a rather small
mutational step size in most simulations we somewhat constrain
the emergence of genetic polymorphism relative to the emergence
of environmental variance.) Our analysis of the long-term polymorphic evolution focuses on clonal reproduction, where distinct
clusters in phenotype space can be easily be formed. Although
a Bernoulli distribution with only two selective optima favors at
most a single split into two phenotypic clusters, other types of
environmental fluctuations (e.g., Poisson) can result in secondary
or even higher order branching events. On the other hand, we
frequently observe that phenotype clusters die out after a series
of adverse environmental conditions, leading to a dynamics of
recurrent branching and extinction events in phenotype space (cf.
Fig. 7). The prediction of evolutionary equilibria is much more
difficult for diploid sexual populations, where our study gives
only a preliminary account. Here, the formation of heterozygotes
and recombinants can prevent clustering and the long-term fate
strongly depends on constraints on the genetic system. Indeed,
because under disruptive selection intermediate phenotypes have
lower fitness than homozygotes, any mechanism preventing the
production of the intermediate heterozygotes will be selectively
favored (Rueffler et al. 2006) and it has been shown that discrete
phenotypic clusters can emerge, for example, through the evolution of assortative mating (e.g., Dieckmann and Doebeli 1999;
Pennings et al. 2008), through the evolution of sexual dimorphism
(Bolnick and Doebeli 2003; Van Dooren et al. 2004), or through
the evolution of dominance modifiers (e.g., Van Dooren 1999;
Peischl and Schneider 2010).
We can interpret these results in various ways. First, we
confirm predictions by Bull (1987) that Gaussian environmental variance can act as a selectively favored bet-hedging strategy.
As a consequence, elevated levels of environmental variance are
predicted for populations experiencing large variation in the selective optima between reproductive periods. Second, in contrast
to Sasaki and Ellner (1995), our results show that genetic variation can be maintained by fluctuating selection even in the face
of evolving environmental variation. Genetic variation is favored
by two main factors: (1) A large generation overlap γ due to the
long-lived stage in the life cycle which strengthens frequencydependent selection; (2) environmental fluctuations leading to a
non-Gaussian distribution of the selective trait optimum with a
strong asymmetry or large leptokurtosis. This second factor lends
a crucial advantage to genetic variation over environmental variation, because the latter is constrained to a Gaussian shape and as
such less flexible. We therefore predict that long-lived organisms
facing environments with asymmetric or leptokurtic distribution
should have higher heritabilities than short-lived organisms in
more normally distributed environments. Third, stochasticity is
expected to cause fluctuations in the amount of genetic variation. In particular, it can also lead to the temporary extinction of
phenotype clusters. These fluctuations are expected to increase
with decreasing generation overlap and population size, and with
increasing strength of selection and increasing positive autocorrelation in the sequence of environmental optima and increasing
skewness in their distribution. Fourth, if the distribution of selective optima is strongly asymmetric, genetic variation should be
structured such that phenotypes matching more common environmental conditions on average show a higher degree of canalization than phenotypes matching less-common environmental conditions. We note that the phenomenon that disruptive selection
typically exists in a compound direction in trait space (selecting simultaneously for variation in μz and σz ) while selection is
stabilizing in each separate component has also been described
in a recent study by Ravigné et al. (2009) in a model studying
the coevolution of habitat specialization and habitat choice. Even
more recently, based on an analysis of Lotka–Volterra competition models, Doebeli and Ispolatov (2010) proposed as a general
principle that the likelihood of disruptive selection increases with
increasing dimension of the trait space. Our results support this
proposal.
How can we interpret these results in the larger context of
the adaptive maintenance of phenotypic variation? There are three
ecologically important mechanisms under which phenotypic diversity is truly adaptive, that is, not due to genetic constraints
(such as overdominance, where the optimal genotype cannot breed
true) or due to mutation–selection balance. These are (1) negative frequency dependence, (2) temporally fluctuating selection,
and (3) heterogenous selection in a spatially structured population. However, these mechanisms affect the genetic and the environmental component of phenotypic variation in different ways:
Under negative frequency-dependent selection genetic and environmental variation are in principle equivalent. In contrast, under
EVOLUTION 2011
15
H A N N E S S VA R DA L E T A L .
temporally fluctuating selection without frequency dependence
only environmental variation can evolve, and under spatially heterogeneous selection without frequency dependence only genetic
polymorphism can evolve. In the first case, environmental variation serves as a bet-hedging mechanism, whereas in the latter case
genetic variation reflects local adaptation.
The lottery model combines two of these mechanisms: temporally variable selection and negative frequency dependence.
The parameter γ for the generation overlap allows for a continuous transition between two extremes: purely temporal variation
without frequency dependence for γ = 0, and pure negative frequency dependence for γ → 1, where all individuals face the
same distribution of environmental challenges for their offspring
during their infinitely long life times, and temporal fluctuations
are averaged out (cf Appendix A).
If both genetic and environmental variance are unconstrained, selection for them is equally strong for γ = 1. Any
value of γ < 1 introduces a time-dependent selection component,
leading to an advantage of the environmental variance component
due to its bet-hedging capacity as indeed observed by Sasaki and
Ellner (1995). With a Gaussian constraint on the environmental
variance, however, a nontrivial trade-off results, as expressed by
condition (9) for the maintenance of genetic polymorphisms. In
particular, genetic variation is always favored due to this constraint in the limit γ → 1 (unless the distribution of selective trait
optima is exactly Gaussian, results not shown). We expect that
such a trade-off is a generic feature that should be observed also
beyond the limits of our particular model. Like the lottery model
(Sasaki and Ellner 1995), also competition models typically lead
to phenotypic clustering (Doebeli et al. 2007; Pigolotti et al. 2007),
where the “optimal” phenotype distribution that emerges in the
long-term equilibrium is discrete rather than continuous. It is this
complex shape of the phenotype distribution and the limitations
of nonheritable variation to produce this shape that lend the more
flexible genetic component its crucial advantage.
Two previous studies compare the coevolution of environmental and genetic variation under frequency-dependent competition, thus in a scenario similar to γ = 1 in our model. The
results from both studies are fully consistent with our interpretation. For a two-patch Levene model of soft selection, Leimar
(2005) finds that the condition where a genetic polymorphism
can arise is identical to the condition where a mutant producing
two alternative phenotypes can invade a genetically monomorphic
population with zero environmental variance. Moreover, also the
strength of selection in both scenarios is identical, suggesting that
in this model genetic and environmental variation are, in principle, equivalent. Importantly, Leimar imposes no constraints on
the shape of either type of variation: reproduction is clonal with a
continuum of genotypes, and the environmental variation directly
comes in its optimal shape, with the two alternative phenotypes
16
EVOLUTION 2011
each favored in one of the patches. This is different in a second
study by Zhang and Hill (2007). They investigate a classical competition model, where a quantitative trait under Gaussian stabilizing selection also mediates competition among individuals with
similar phenotypes, resulting in frequency-dependent disruptive
selection whenever competition outweighs the stabilizing component of selection. Like in the present study, environmental variance
is constrained to a Gaussian shape with evolvable width. Zhang
and Hill find that Gaussian environmental variation can only be
adaptively maintained if the genotypic values are restricted to
a narrow interval and, hence, the genetic variance is even more
severely constrained than the environmental variance.
The third mechanism that can maintain adaptive phenotypic
diversity, spatially heterogeneous selection, is not part of the lottery model. It is studied by Leimar (2005) who extends his analysis
of the Levene model to restricted migration. Decreasing migration allows for a continuous transition between the pure negative
frequency dependence and pure spatial structure. In accordance
with the above expectation, he shows that decreasing migration
progressively favors the genetic over the environmental variance
component. We note that many scenarios in nature will contain elements of all three mechanisms. Selection frequently varies over
space and time and soft selection with gene flow entails negative
frequency dependence. The scope for the evolution and maintenance of the genetic and environmental variation components
under these conditions, in the presence of constraints, remains a
promising field for future studies.
ACKNOWLEDGMENTS
The authors thank M. Doebeli, O. Leimar, and an anonymous reviewer for
comments on the manuscript and gratefully acknowledge funding from
the Vienna Science and Technology Fund (WWTF).
LITERATURE CITED
Abrams, P. A., H. Matsuda, and Y. Harada. 1993. Evolutionary unstable fitness maxima and stable fitness minima of continuous traits. Evol. Ecol.
7:465–487.
Bolnick, D. I. 2006. Multi-species outcomes in a common model of sympatric
speciation. J. Theor. Biol. 241:734–744.
Bolnick, D. I., and M. Doebeli. 2003. Sexual dimorphism and adaptive speciation: two sides of the same ecological coin. Evolution 57:2433–2449.
Brown, J. S., and D. L. Venable. 1986. Evolutionary ecology of seed bank
annuals in temporally varying environments. Am. Nat. 127:31–47.
Bull, J. J. 1987. Evolution of phenotypic variance. Am. Nat. 41:303–315.
Bürger, R. 2000. The mathematical theory of selection, recombination, and
mutation. Wiley, Chichester.
Bürger, R., and A. Gimelfarb. 2002. Fluctuating environments and the role
of mutation in maintaining quatitative genetic variation. Genet. Res.
80:31–46.
Chesson, P. 1984. The shortage effect in stochastic population models. Lecture
Notes Biomath. 54:76–89.
Chesson, P., and R. Warner. 1981. Environmental variability promotes coexistence in lottery competitive systems. Am. Nat. 117:923–943.
C O M PA R I N G E N V I RO N M E N TA L A N D G E N E T I C VA R I A N C E
Chesson, P. L. 1983. Coexistence of competitors in a stochastic environment:
the storage effect. Pp. 188–198 in H. I. Freedman and C. Strobeck, eds.
Population biology, Lecture Notes in Biomathematics, vol. 52. SpringerVerlag, New York.
Claessen, D., J. Andersson, L. Persson, and A. de Roos. 2007. Delayed evolutionary branching in small populations. Evol. Ecol. Res. 9:51–69.
——. 2008. The effect of population size and recombination on delayed
evolution of polymorphism and speciation in sexual populations. Am.
Nat. 172:E18–E34.
Cohen, D. 1966. Optimizing reproduction in a randomly varying environment.
J. Theor. Biol. 12:119–129.
de Visser, J., J. Hermisson, G. Wagner, L. Meyers, H. Bagheri, J. Blanchard,
L. Chao, J. Cheverud, S. Elena, W. Fontana, et al. 2003. Perspective:
Evolution and detection of genetic robustness. Evolution 57:1959–1972.
Dercole, F., and S. Rinaldi. 2008. Analysis of evolutionary processes: the
adaptive dynamics approach and its applications. Princeton Univ. Press,
Princeton, NJ.
Dercole, F., R. Ferrière, and S. Rinaldi. 2002. Ecological bistability and evolutionary reversals under asymmetrical competition. Evolution 56:1081–
1090.
Dieckmann, U., and R. Law. 1996. The dynamical theory of coevolution: a
derivation from stochastic ecological processes. J. Math. Biol. 34:579–
612.
Dieckmann, U., and M. Doebeli. 1999. On the origin of species by sympatric
speciation. Nature 400:354–357.
Doebeli, M., and I. Ispolatov. 2010. Complexity and diversity. Science
328:494–497.
Doebeli, M., H. J. Blok, O. Leimar, and U. Dieckmann. 2007. Multimodal
pattern formation in phenotype distributions of sexual populations. Proc.
R. Soc. Lond. B 274:347–357.
Dubnau, D., and R. Losick. 2006. Bistability in bacteria. Mol. Microbiol.
61:564–572.
Ellner, S. 1996. Environmental fluctuations and the maintenance of genetic diversity in age or stage-structured populations. Bull. Math. Biol. 58:103–
127.
Ellner, S., and N. G. Hairston. 1994. Role of overlapping generations in
maintaining genetic variation in a fluctuating environment. Am. Nat.
143:403–417.
Ellner, S. P., and A. Sasaki. 1996. Patterns of genetic polymorphism maintained by fluctuating selection with overlapping generations. Theor.
Popul. Biol. 50:31–65.
Eshel, I. 1983. Evolutionary and continuous stability. J. Theor. Biol. 103:99–
111.
Evans, M. E. K., and J. J. Dennehy. 2005. Germ banking: bet-hedging and
variable release from egg and seed dormancy. Q. Rev. Biol. 80:431–451.
Frank, S., and M. Slatkin. 1990. Evolution in variable environments. Am. Nat.
136:244–260.
Gavrilets, S., and A. Hastings. 1994. A quantitative-genetic model for selection
on developmental noise. Evolution 48:1478–1486.
Geritz, S., É. Kisdi, G. Meszéna, and J. Metz. 1998. Evolutionarily singular
strategies and the adaptive growth and branching of the evolutionary
tree. Evol. Ecol. 12:35–57.
Geritz, S. A. H. 2005. Resident-invader dynamics and the coexistence of
similar strategies. J. Math. Biol. 50:67–82.
Geritz, S. A. H., M. Gyllenberg, F. J. A. Jacobs, and K. Parvinen. 2002.
Invasion dynamics and attractor inheritance. J. Math. Biol. 44:548–
560.
Gillespie, J. H. 1973. Polymorphism in random environments. Theor. Popul.
Biol. 4:193–195.
Haccou, P., and Y. Iwasa. 1995. Optimal mixed strategies in stochastic environments. Theor. Popul. Biol. 47:212–243.
Hopper, K. R.1999. Risk-spreading and bet-hedging in insect population biology. Annu. Rev. Entomol. 44:535–560.
Iwasa, Y., A. Pomiankowski, and S. Nee. 1991. The evolution of costly mate
preferences ii. the “handicap” principle. Evolution 45:1431–1442.
Karlin, S., and U. Liberman. 1975. Random temporal variation in selection
intensities: one-locus two-allele model. J. Math. Biol. 2:1–17.
King, O. D., and J. Masel. 2007. The evolution of bet-hedging adaptations to
rare scenarios. Theor. Popul. Biol. 72:560–575.
Kopp, M., and J. Hermisson. 2006. The evolution of genetic architecture under
frequency-dependent disruptive selection. Evolution 60:1537–1550.
Kussel, E., R. Kishony, N. Q. Balaban, and S. Leibler. 2005. Bacterial
persistence: a model of survival in changing environments. Genetics
169:1807–1814.
Lande, R. 1976. Natural selection and random genetic drift in phenotypic
evolution. Evolution 30:314–334.
——. 1977. The influence of the mating system on the maintenance of genetic
variability in polygenic characters. Genetics 86:485–498.
Leimar, O. 2005. The evolution of phenotypic polymorphism: randomized
strategies versus evolutionary branching. Am. Nat. 165:669–681.
——. 2009. Multidimensional convergence stability. Evol. Ecol. Res. 11:191–
208.
Levene, H. 1953. Genetic equilibrium when more than one ecological niche
is available. Am. Nat. 87:331–333.
Levin, S., D. Cohen, and A. Hastings. 1984. Dispersal strategies in patchy
environments. Theor. Popul. Biol. 26:165–191.
Malik, T., and H. L. Smith. 2008. Does dormancy increase fitness of bacterial
populations in time-varying environments? Bull. Math. Biol. 70:1140–
1162.
MathWorks, T. 2009. Matlab r2009b. Natick, MA.
Menu, F., J.-P. Roebuck, and M. Viala. 2000. Bet-hedging diapause strategies
in stochastic environments. Am. Nat. 155:724–734.
Metz, J. 2008. Fitness. Pp. 1599–1612 in S. Jørgensen and B. Fath, eds.
Evolutionary Ecology, vol. 2 of Encyclopedia of Ecology. Elsevier,
Oxford.
Metz, J., R. Nisbet, and S. Geritz. 1992. How should we define ‘fitness’ for
general ecological scenarios? Trends Ecol. Evol. 7:198–202.
Metz, J., S. Geritz, G. Meszéna, F. Jacobs, and J. Van Heerwaarden. 1996. Adaptive dynamics: a geometrical study of the consequences of nearly faithful reproduction. Pp. 183–231 in S. van
Strien and S. Verduyn Lunel, eds. Stochastic and spatial structures
of dynamical systems, Proceedings of the Royal Dutch Academy
of Science. North Holland, Dordrecht, Netherlands; available at
http://www.iiasa.ac.at/Research/ADN/Series.html.
Otto, S. P., and T. Day. 2007. A biologist’s guide to mathematical modeling
in ecology and evolution. Princeton Univ. Press, Princeton, NJ.
Peischl, S., and K. A. Schneider. 2010. Evolution of dominance under
frequency-dependent intraspecific competition in an assortatively mating population. Evolution 64:561–582.
Pennings, P. S., M. Kopp, G. Meszéna, U. Dieckmann, and J. Hermisson.
2008. An analytically tractable model for competitive speciation. Am.
Nat. 171:E44–E71.
Philippi, T., and J. Seger. 1989. Hedging one’s evolutionary bets revisited.
Trends Ecol. Evol. 4:41–44.
Pigolotti, S., C. López, and E. Hernández-Garcı́a. 2007. Species clustering in
competitive Lotka-Volterra models. Phys. Rev. Lett. 98:258101.
Ravigné, V., U. Dieckmann, and I. Olivieri. 2009. Live where you thrive: joint
evolution of habitat choice and local adaptation facilitates specialization
and promotes diversity. Am. Nat. 174:E141–E169.
Rueffler, C., T. J. M. Van Dooren, O. Leimar, and P. A. Abrams. 2006.
Disruptive selection and then what? Trends Ecol. Evol. 21:238–
245.
EVOLUTION 2011
17
H A N N E S S VA R DA L E T A L .
Sasaki, A., and S. Ellner. 1995. The evolutionary stable phenotype distribution
in a random environment. Evolution 49:337–350.
Sasaki, A., and S. P. Ellner. 1997. Quantitative genetic variance maintained by
fluctuating selection with overlapping generations: variance components
and covariances. Evolution 51:682–696.
Schlichting, C. D., and M. Pigliucci. 1998. Phenotypic evolution: a reaction
norm perspective. Sinauer Associates Inc., Sunderland, MA.
Seger, J., and H. J. Brockmann. 1987. What is bet-hedging? Oxford Surv.
Evol. Biol. 4:182–211.
Simons, A., and M. Johnston. 1997. Developmental instability as a bet-hedging
strategy. Oikos 80:401–406.
Slatkin, M. 1974. Hedging one’s evolutionary bets. Nature 250:704–705.
Stumpf, M. P. H., Z. Laidlaw, and V. A. A. Jansen. 2002. Herpes viruses hedge
their bets. Proc. Natl. Acad. Sci. USA 99:15234–15237.
Tuljapurkar, S. 1990. Population dynamics in variable environments, vol. 85
of Lecture Notes in Biomathematics. Springer Verlag, Berlin, Germany.
Turelli, M. 1988. Population genetic models for polygenic variation and evolution. Pp. 601–618 in E. J. Eisen, M. M. Goodman, G. Namkoong, and
B. S. Weir, eds. The Second International conference on quantitative
genetics. Sinauer, Boston.
Van Dooren, T. J. M. 1999. The evolutionary ecology of dominance. J. Theor.
Biol. 198:519–532.
Van Dooren, T. J. M., I. Demon, and M. Durinx. 2004. Sexual dimorphism or
evolutionary branching? Evol. Ecol. Res. 6:857–871.
van Doorn, G., and U. Dieckmann. 2006. The long-term evolution of multilocus traits under frequency-dependent disruptive selection. Evolution
60:2226–2238.
Veening, J.-W., W. K. Smits, and O. P. Kuipers. 2008. Bistability, epigenetics,
and bet-hedging in bacteria. Annu. Rev. Microbiol. 62:193–210.
Wagner, G. P., G. Booth, and H. C. Bagheri. 1997. A population genetic theory
of canalization. Evolution 51:329–347.
Wallace, B. 1975. Hard and soft selection revisited. Evolution 29:465–473.
Warner, R., and P. Chesson. 1985. Coexistence mediated by recruitment fluctuations: a field guide to the storage effect. Am. Nat. 125:769–787.
West-Eberhard, M. J. 2003. Developmental plasticity and evolution. Oxford
Univ. Press, New York.
Zhang, X.-S., and W. G. Hill. 2007. Competition can maintain genetic but not
environmental variance in the presence of stabilizing selection. Evolution 61:1532–1545.
APPENDIX A: Invasion Fitness
Invasion fitness ρ(x , x) is given by the logarithm of long-term per
capita growth rate of an infinitesimal small mutant subpopulation
with genotype x in a resident population with genotype x at
population dynamical equilibrium (Metz et al. 1992; Metz 2008).
The growth rate of the mutant subpopulation Nt from one
year to the next is given by
and invasion fitness can be calculated as
⎤
⎡
T T1
,
θ
)
r
(x
t
⎦
ρ(x , x) = ln ⎣ lim
+γ
(1 − γ)
T →∞
r
(x,
θ
)
t
i=1
1
= lim
T →∞ T
r (x , θt )
ln (1 − γ)
+ γ . (A3)
r (x, θt )
i=1
T
Using that time averages can be replaced by ensemble averages
given that the random process generating the selective optimum
is ergodic and converges toward a stationary distribution, ρ(x , x)
can be written as in equation (3).
Based on a Taylor approximation up to first order and in the
limit γ → 1 invasion fitness ρ(x , x), as written in equation (3),
becomes
r (x , θt )
ρ(x , x) ≈ −1 + γ + (1 − γ)
f (θ; y)dθt . (A4)
r (x, θt )
Thus, in this limit invasion is determined by the arithmetic mean
of the year-to-year recruitment, sampled over an infinitely long
life. As a result, in this limit the lottery model becomes equivalent
to a Levene-type soft selection model where invasion fitness is
determined by the arithmetic mean of the recruitment from all
patches and where the per patch recruitment can be written as
r (x , θi )/r (x, θi ), where θi is the optimal phenotype in the ith
patch.
APPENDIX B: Two-dimensional
convergence stability
The mathematical characterization of convergence stability is
straightforward in one-dimensional trait spaces (Geritz et al.
1998), but for two-dimensional trait spaces the situation is more
complicated because then the evolutionary dynamics is not only
governed by the shape of the fitness landscape but also by the
distribution of mutational effects. We follow Leimar (2009) and
use the concept of “strong convergence stability.” This concept
is based on the observation that in the limit of small mutational
effects and large population size the path of the evolutionary dynamics in the two-dimensional trait space can be described by an
equation of the form
dxi
= m(x)
ci j (x)S j (x),
dt
j=1
2
= Nt (Yt + γ) = Nt k
Nt+1
r (x , θt )
N̂r (x, θt )
+γ .
(A1)
Inserting k = N̂ (1 − γ) gives
Nt+1
18
=
Nt
r (x , θt )
+γ
(1 − γ)
r (x, θt )
EVOLUTION 2011
(A2)
(B1)
where m(x) is a positive number related to the production of
new mutants and C = [ci j ] the mutational variance-covariance
matrix (Dieckmann and Law 1996). The last factor S j (x) is the
jth element of the selection gradient S(x) with entries given by
the right-hand side of equation (4). We note that this equation
has the same form as the one derived for the change in the mean
C O M PA R I N G E N V I RO N M E N TA L A N D G E N E T I C VA R I A N C E
phenotype under the weak selection limit of quantitative genetics
(Lande 1976; Iwasa et al. 1991).
A singular point x ∗ is then defined as strongly convergence
stable if the singular point is asymptotically stable under the dynamics described by equation (B1) independent of the mutational
variance–covariance matrix. Strong convergence stability is determined by the Jacobian matrix J = [ jkl ] of the selection gradient
with components
∂ Sk (x) ∂ 2 ρ(x , x) ∂ 2 ρ(x , x) =
+
.
jkl =
∂ xl x=x ∗
∂ xk ∂ xl x =x=x ∗
∂ xk ∂ xl x =x=x ∗
exactly. For S1 (x) we find
S1 (x) = (1 − γ)
μθ − μz
.
σ2z + σs2
(C5)
Solving S1 (x) = 0 for μz , we find that the singular trait value μ∗z
equals μθ . Using this result we get
σθ2
(1 − γ)σz
S2 (x) = 2
−1 .
(C6)
σz + σs2 σ2z + σs2
Solving S2 (x) = 0 for σz , we find the twosingular strategies given
in equation (8) with σ∗z1 = 0 and σ∗z2 = σθ2 − σs2 .
(B2)
Leimar (2009) showed that a singular point x ∗ is strongly convergence stable for any positive definite covariance matrix C, if
the Jacobian matrix is negative definite. If the Jacobian matrix is
positive definite, then x ∗ is not convergence stable but repelling
for any positive definite covariance matrix C. In cases where the
Jacobian matrix is indefinite the singular point is a saddle point of
the evolutionary dynamics in the absence of covariance and also
for most cases with covariance.
APPENDIX C: Location of Singular
Points
The invasion fitness gradient (eq. 4) is given by
∂ρ(x , x) Si (x) =
∂μz x =x=x ∗
r (x , θt )
∂ ln (1 − γ)
+ γ f (θt ; y)dθt r (x, θt )
=
∂ xi
Using the same ideas as in Appendix C, this condition can shown
to equal
−
x =x
(C1)
After exchanging integration and differentiation we get
r (x , θt )
∂ ln (1 − γ)
+ γ r (x, θt )
Si (x) =
f (θt ; y)
dθt
∂ xi
(C2)
for xi ∈ {μz , σz }.
,θt )
+γ
Executing the differentiation and using that (1 − γ) rr(x
(x,θt )
evaluated at x = x equals 1, we find
θt − μz
S1 (x) =
f (θt ; y)(1 − γ) 2
dθt ,
(C3)
σz + σs2
S2 (x) =
γ2 σθ2 + σs2 + σ2z − γ(σθ2 + σs2 + σ2z )
> 0.
(σs2 + σ2z )2
(D2)
Solving the inequality for γ results in condition (7). The singular
point is convergence stable if and only if (Geritz et al. 1998)
∂ 2 ρ(μz , μz ) ∂ 2 ρ(μz , μz ) +
< 0. (D3)
2
∂μz
∂μz ∂μz μz =μz =μ∗z
μz =μz =μ∗z
It can be shown that this condition equals
x =x
(1 − γ)σz
f (θt ; y) 2
σz + σs2
For the version of our model where σz is a fixed parameter and
where only μz can evolve (Section Results: Fixed Environmental
Variance) we find analogously to the two-dimensional case above
that μ∗z = μθ is a unique singular point. This singular point is
invadable if and only if (Geritz et al. 1998)
∂ 2 ρ(μz , μz ) ∂μz2
μz =μz =μ∗z
r (μz , θt )
(D1)
∂ 2 ln (1 − γ)
+ γ f (θt ; y)dθt r (μz , θt )
=
> 0.
∂μz2
∗
μz =μz =μz
for xi ∈ {μz , σz }.
APPENDIX D: Fixed Environmental
Variance
(θt − μz )2
− 1 dθt . (C4)
σ2z + σs2
Using that
f (θt ; y)dθt = 1,
f (θt ; y)θt dθt = μθ , and
f (θt ; y)(θt − μθ )2 dθt = σθ2 , the integrals can be calculated
−
γ2 σθ2 + σs2 + σ2z − γ(σθ2 + σs2 + σ2z ) (1 − γ)γσθ2
− 2
(σs2 + σ2z )2
(σs + σ2z )2
=−
1−γ
< 0,
σs2 + σ2z
(D4)
which is always fulfilled.
APPENDIX E: Evolvable
Environmental Variance
Invadibility of singular points in a trait space with more than one
dimension is determined by the Hessian matrix H = [h kl ] of invasion fitness with entries given by equation (5). After exchanging
EVOLUTION 2011
19
H A N N E S S VA R DA L E T A L .
differentiation and integration, hkl can be written as
r (x , θt )
∂ 2 ln (1 − γ)
+ γ r (x, θt )
f (θt ; y)
h kl =
∂ xk ∂ xl
dθt
x =x=x ∗
with xk , xl ∈ {μz , σz }.
(E1)
Leimar (2009) showed that strong convergence stability of
singular points is determined by definiteness of the Jacobian
matrix J = [ jkl ] of invasion fitness with entries given by equation (B2). Leimar defines the Jacobian matrix as negative definite/positive definite/indefinite if its symmetric part (with elements (jkl + jlk )/2) is negative definite/positive definite/indefinite.
For our model the Jacobian matrix can be shown to be symmetric,
such that its definiteness can be inferred directly.
We denote the second derivative in the expressions hkl as h̃ kl .
These can be explicitly calculated as
⎧
(1 − γ) γ(θt − μθ )2 − σs2
⎪
⎪
for x ∗1
⎪
⎨
σs4
=
h̃ 11
⎪
(1 − γ) γ(θt − μθ )2 − σθ2
⎪
⎪
⎩
for x ∗2 ,
σθ4
⎧
0
for x ∗1
⎪
⎪
⎨
h̃ 12 = h̃ 21 = (1 − γ) (μθ − θt ) σ2 − σ2 (γ + 2)σ2 − γ (θt − μθ ) 2 s
θ
θ
⎪
⎪
⎩
for x ∗2 ,
σθ6
⎧
(1 − γ) (θt − μθ ) 2 − σs2
⎪
⎪
⎪
⎨
σs4
=
h̃ 22
2
2
2
6
2 2
2 4
2
2
2
⎪
σθ − σs2
⎪
⎪ (1 − γ) σθ − 3 (θt − μθ ) σθ − 2σs σθ − 2 (θt − μθ ) σθ + γ (θt − μθ ) − σθ
⎩
σθ8
Using the definitions for the third and fourth central moment of
probability distributions, that is, μθ3 := f (θt ; y)(θt − μθ )3 dθt
and μθ4 := f (θt ; y)(θt − μθ )4 dθt , the integrations can be computed to give
⎧
(1 − γ)(γσθ2 − σs2 )
⎪
⎪
for x ∗1
⎪
⎨
σs4
(E2a)
h 11 =
⎪
(1 − γ)2
⎪
∗
⎪
for x 2 ,
⎩−
σθ2
for x ∗1
for x ∗2 .
The first term on the right-hand side of equation (B2) equals
the entries hkl of the Hessian matrix. The second term on the righthand side, which we will denote qkl , after exchanging integration
and differentiation equals
r (x , θt )
∂ 2 ln (1 − γ)
+ γ r (x, θt )
qkl =
f (θt ; y)
dθt .
∂ xk ∂ xl
∗
x =x=x
(E4)
h 12 = h 21
h 22
⎧
0
for x ∗1
⎪
⎪
⎨
= (1 − γ)γμθ3 σ2 − σ2
s
θ
⎪
⎪
⎩
for x ∗2 ,
6
σθ
⎧
(1 − γ)(σθ2 − σs2 )
⎪
⎪
⎪
⎨
σs4
=
2
4
⎪
⎪ (1 − γ)(σθ − σs2 )(γμθ4 − (γ + 2)σθ )
⎪
⎩
σθ8
(E2b)
for x ∗1
(E2c)
for
x ∗2 .
A two-dimensional matrix A = [akl ] is negative definite if
and only if
2
> 0,
−a11 > 0 and a11 a22 − a12
(E3)
that is, if the principal minors of −A are positive. Inserting
the equalities (E2) into inequality (E3) reveals the invadibility
conditions reported in Section Results: Evolvable Environmental
Variance.
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EVOLUTION 2011
We denote the second derivative in the expressions qkl as q̃kl .
These can be explicitly calculated as
⎧
(γ − 1)γσθ2
⎪
⎪
⎪
for x ∗1
⎨
σs4
q̃11 =
⎪ (γ − 1)γ
⎪
⎪
for x ∗2 ,
⎩
σθ2
⎧
for x ∗1
⎪
⎪0
⎨
q̃12 = q̃21 = (γ − 1)γμθ3 σ2 − σ2
s
θ
⎪
⎪
⎩
for x ∗2 ,
σθ6
⎧
for x ∗1
⎪
⎨0
q̃22 = (γ − 1)γ(σ2 − σ2 )(μθ4 − σ4 )
s
θ
θ
⎪
⎩
for x ∗2 ,
σθ8
where x ∗1 and x ∗2 again denote
the singular strategies (μz , σz ) =
(μθ , 0) and (μz , σz ) = (μθ , σθ2 − σs2 ), respectively. The entries
of the Jacobian matrix of the selection gradient jij = hij + qij
C O M PA R I N G E N V I RO N M E N TA L A N D G E N E T I C VA R I A N C E
calculate to
j11
⎧γ−1
⎪
⎪
⎨ σ2
s
=
⎪
γ
−
1
⎪
⎩ 2
σθ
j12 = j21 =
j22
for x ∗1
(E5a)
for x ∗2 ,
⎧
⎨0
for x ∗1
⎩0
for x ∗2 ,
⎧
(γ − 1)(σθ2 − σs2 )
⎪
⎪
−
⎪
⎨
σs4
=
⎪
⎪ 2(γ − 1)(σθ2 − σs2 )
⎪
⎩
σθ4
(E5b)
for x ∗1
(E5c)
for
x ∗2 .
Inserting the equalities (E5) into inequality (E3) reveals the
conditions for strong convergence stability reported in the resultsection on evolvable environmental variance.
APPENDIX F: Fitness Landscape at
Branching Points
In this appendix, we describe the fitness landscape at the singular
point x ∗2 in more detail. Here we are interested in the case γcrit > γ
where the point x ∗2 is a saddle point of the fitness landscape created
by a population characterized by this trait vector. We analyze
two aspects of these saddle points, the angle of the sector with
positive invasion fitness, α, and the angle between the dominant
eigenvector and the μz -axis, αd (cf. Fig. 2). Figure F1 shows
for Bernoulli and Poisson-distributed selective optima how these
properties depend on the generation overlap γ and the parameters
p and λ, respectively.
Figure F1A,C shows that for both distributions α increases
with increasing asymmetry (increasing deviation of p from 0.5 in
case of the Bernoulli distribution and decreasing λ in case of the
Poisson distribution) and with increasing generation overlap. For
extreme asymmetry α goes back to zero because once p(1 −
p) < σs2 or λ < σs2 , respectively, the singular point x ∗2 ceases to
exist. Branching is not possible for γ < γcrit , corresponding to
a sector with width zero (flat part in Fig. F1A,C). As expected,
the range of γ-values where branching is possible increases with
increasing asymmetry.
The angle between the dominant eigenvector and the μz axis, αd , strongly depends on the parameter p and λ, respectively.
As noted in the section Results: Evolvable Environmental Variance, selection at x ∗2 is never disruptive in the horizontal direction
where only μz is varied. Instead, selection is disruptive in a compound direction, simultaneously changing the mean and variance
of the offspring distribution. Figure F1B,D show that for asymmetric distributions of selective optima the dominant eigenvector
is tilted such that the type that is becoming a specialist for the
environments in the fatter tail of the distribution is also decreasing its environmental variance. Conversely, the type that improves
on the environments in the thinner tail increases its environmental variance. The tilt of the eigenvector increases with increasing
asymmetry of the distribution. This can best be understood by
considering the case of Bernoulli-distributed selective optima.
With increasing (decreasing) p the selective optimum 1 occurs
with increasing (decreasing) frequency and μθ = μ∗z2 decreases
(increases). Thus, μ∗z2 is shifted toward the more common selective optimum. As a consequence, at x ∗2 selection for a genotype
adapted to selective optimum 1 corresponds to a increasingly steep
dominant eigenvector.
For the Laplace distribution the situation is somewhat different. From equation (E2) we can directly deduce that the dominant
eigenvector at x ∗2 equals (0, 1). First note that h11 < 0 and h12 =
4
− 3 = 3 follows μθ4 = 6σθ4 . By inserting
0. From gθ2 = μθ4 /σθ4
this into h22 we see that h22 > 0 ⇔ γ > 2/5 which is exactly the
condition for x ∗2 to be a branching point. This can lead to a genetic
polymorphism purely in the environmental variance σ2z .
APPENDIX G: Accuracy of the
analytical predictions
Figure 5 shows the accuracy of the analytical prediction for evolutionary branching, as given by equation (9), when compared
to individual-based simulations for different values of the mutational step size and the mutation rate. We find that if both are
relatively small, then γcrit predicts the onset of genetic diversification with high accuracy. Substantially increasing the mutational
effect size (Fig. 5A) or the mutation rate (Fig. 5B) facilitates genetic diversification by lowering the critical generation overlap
where a population starts splitting into two. These findings can
be explained as follows. Uninvadability of a singular point as
determined by the Hessian matrix is a local result that need not
hold true for mutants at some distance. Numerical calculations
show that in some directions in trait space an uninvadable singular point is indeed only separated by a narrow fitness valley
from regions where mutants have positive invasion fitness. Simulations show that such invaders coexist with the resident type at
the singular point and that subsequently these two types evolve
further apart. Both increased mutational step size and increased
mutation rate make the appearance of mutants on the other side
of the fitness valley surrounding an uninvadable singular point
more likely. Larger mutational steps allow to cross the fitness
valley with a single mutational step whereas a higher mutation
rate results in larger genetic variance around the singular point
at mutation–selection balance. The waiting time until branching
increases with decreasing values of the mutation rate, mutational
step size, population size and generation overlap γ. Each of these
EVOLUTION 2011
21
H A N N E S S VA R DA L E T A L .
Figure F1.
Characteristics of the fitness landscape at the singular point x2∗ as a function of the generation overlap γ and a shape
parameter of the distribution of selective optima. (A) & (B) Bernoulli distribution with parameter p; (C) & (D) Poisson distribution with
parameter λ. (A) & (C) Angle of the sector with positive invasion fitness, α. (B) & (D) Angle between the dominant eigenvector and
the μz -axis. In case of the Bernoulli distribution results are only shown for 0.5 < p < 1. Graphs for 0 < p < 0.5 are identical if the p-axis
decreases from 0.5 toward 0. In case of the Poisson distribution, for better comparison with the results for the Bernoulli distribution the
axes for λ is reversed such that asymmetry increases further along the λ-axis. Plots are shown for σ s = 0.1.
factors decreases the amount of genetic variation in the neighborhood of a singular point, resulting in smaller fitness differences
and therefore slower evolutionary dynamics. For population size
this phenomenon that has been noted previously (Claessen et al.
2007, 2008), and for γ this phenomenon is illustrated in panels
(A)–(C) of Figure 7.
Simulations show that for small values of the mutation parameters not only the onset of evolutionary branching is predicted
relatively accurately by the analytical results but also the location
in (μz , σz )-space where branching occurs (Fig. 6A,D,G, and J).
However, as with the critical generation overlap allowing for
branching, the accuracy of this prediction is sensitive to the mutation rate and the mutational effect size. If the mutation rate and
in particular the mutational effect size are sufficiently high (but
in a still biologically realistic range), branching can be observed
at points in trait space far away from the singular point x ∗2 . This
phenomenon can be explained in the following way. In some regions of trait space, selection is directional in one direction and
disruptive in another one. Locally, the first-order directional se-
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EVOLUTION 2011
lection exceeds the second-order disruptive selection. However,
with mutants sufficiently far away from the resident type, secondorder terms will dominate the first-order terms, and the effect of
disruptive selection surpasses the effect of directional selection.
This is for instance the case for the line segment connecting x ∗1
and x ∗2 . For simulations that are initialized on this line segment we
frequently observe that branching occurs without the population
evolving to x ∗2 (no figure) for a wide range of mutation parameters.
A similar situation occurs for example with Bernoulli-distributed
environmental optima if one of the optima occurs far more often
than the other. Under this condition, a population that is initially
characterized by (μz , σz ) = (θi , 0), with θi being the more frequent selective optimum, experiences directional selection in μz
toward μθ and disruptive selection in σz . In this situation branching is frequently observed immediately without prior evolution
toward the singular point as illustrated in Figure 7C,D. Last but
not least, whenever parameters are such that no branching occurs,
the evolutionary dynamics settle at the point x ∗2 as predicted by
the analytical theory.