Natural selection on fecundity variance
in subdivided populations:
kin selection meets bet-hedging
Laurent Lehmann
∗†Fran¸cois Balloux
‡February 4, 2007
∗Department of Genetics, University of Cambridge, CB2 3EH Cambridge, UK, ll316@cam.ac.uk †Corresponding author, phone +44-1223-332584, fax +44-1223-333992
‡Department of Genetics, University of Cambridge, CB2 3EH Cambridge, UK, fb255@mole.bio.cam.ac.uk
Abstract
In a series of seminal papers, Gillespie (1974, 1975, 1977) challenged the notion that the “fittest” individuals are those that produce on average the highest number of offspring. He showed that in small populations, the variance in fecundity can deter-mine fitness as much as mean fecundity. One likely reason why Gillespie’s concept of within-generation bet-hedging has been largely ignored is the general consensus that natural populations are of large size. As a consequence, essentially no work has inves-tigated the role of the fecundity variance on the evolutionary stable state of life-history strategies. While typically large, natural populations also tend to be subdivided in local demes connected by migration. Here, we integrate Gillespie’s measure of selection for within-generation bet-hedging into the inclusive fitness and game theoretic mea-sure of selection for structured populations. The resulting framework demonstrates that selection against high variance in offspring number is a potent force in large, but structured populations. More generally, the results highlight that variance in offspring number will directly affect various life-history strategies, especially those involving kin interaction. The selective pressure on three key traits are directly investigated here, namely within-generation bet-hedging, helping behaviors and the evolutionary stable dispersal rate. The evolutionary dynamics of all three traits is markedly affected by variance in offspring number, though to a different extent and under different demo-graphic conditions.
Keywords: within-generation bet-hedging, variance in offspring number, helping, dispersal, kin selection, game theory, structured population, population genetics, Poisson distribution.
Introduction
Predicting the fate of an allele newly arisen through mutation is possibly the oldest and
1
most recurrent problem in population genetics. It has been known for long that when the
2
fecundity of individuals follows a Poisson distribution, the survival probability of a mutant
3
gene lineage increases with an increase in the mean fecundity of individuals (Haldane, 1927;
4
Fisher, 1930). It has been later recognized that when the variance in offspring number differs
5
from the mean, the survival probability of a mutant gene lineage decreases with an increase
6
in the variance in offspring number (Bartlett, 1955; Ewens, 1969). This can be understood
by realizing that a random line of genes that failed to transmit itself at any generation in the
1
past is lost forever, and this independently of the mean fecundity of individuals. Intuitively,
2
we thus expect that between two competing strategies with equal mean fecundity, the one
3
with the lower variance will out-propagate its alternative.
4
In a series of insightful papers, Gillespie (1974, 1975, 1977) elucidated the conditions
5
under which natural selection will decrease the variance in offspring number in panmictic
6
populations of constant size. His key result is that the intensity of selection against the
7
variance is proportional to the inverse of population size. The selective pressure against the
8
variance thus decreases with increasing population size and vanishes for very large
popula-9
tions. This result can be understood by noting that in a population subject to regulation,
10
the expected number of recruited offspring of an individual (i.e., its fitness) depends on its
11
total random number of offspring produced relative to the total random number of offspring
12
produced by its competitors. When population size becomes large, the law of large numbers
13
informs us that the total number of offspring produced by competitors converges to a fixed
14
value, which is given by the mean fecundity and is devoid of any variance. In this situation,
15
a mutant allele has only to produce a mean number of offspring exceeding the mean of the
16
resident allele in order to invade the population. Only when the population is small enough
17
can the variance in offspring number play a role as important as the mean in determining the
18
fitness of individuals. While most natural populations are arguably large, they also tend to
19
be discontinuously distributed and typically consist of local demes connected by migration.
20
Such a spatial structure implies that demes can be of very small size while the population as
21
a whole remains large. This feature is renowned to affect the evolutionary process because it
22
alters the forces shaping the changes in gene frequencies such as selection or random genetic
23
drift (Whitlock, 2003; Cherry and Wakeley, 2003; Roze and Rousset, 2003). We thus expect
24
that the effect of the variance in fecundity on the average number of recruited offspring of
25
an individual will also depend on population structure.
26
The nature of selection on the variance in offspring number in subdivided populations
27
has been studied by Shpak (2005), who concluded that the conditions under which
high-28
variance strategies are outperformed by low-variance strategies depends on the timing of
29
the regulation of the size of the population. When density-dependent competition occurs
before dispersal (i.e., soft selection), the variance in the number of offspring competing in
1
a deme is determined by the number of individuals reproducing locally. In this case, the
2
intensity of selection against the variance is proportional to the inverse of the size of the
3
pool of individuals contributing to this variance, that is, deme size. However, in natural
4
populations, density-dependent competition and competition between individuals can also
5
occur after dispersal. Under this process (called hard selection), individuals in each deme
6
send juveniles that will compete in other demes. Then, the individuals contributing to
7
the variance in the number of juveniles competing in a focal deme can be separated into
8
two pools: one small “resident” pool of individuals breeding in the focal deme and one large
9
“foreign” pool, consisting of all those individuals breeding in different demes and which affect
10
the demography of the focal deme through the dispersal of their progeny. This suggest that
11
the contribution to the variance in offspring number in a focal deme should mainly come
12
from the “resident” pool because the “foreign” pool, being very large, will contribute an
13
essentially fixed number of individuals to each deme. Thereby, demes affect each other
14
in a deterministic way when deme number is large (Chesson, 1981), a result that follows
15
again from the law of large numbers. The nature of selection on the variance in offspring
16
number under hard selection was recently investigated by Shpak and Proulx (2006) and their
17
formalization suggests that selection on the variance in fecundity decreases as migration rate
18
increases (i.e., the “foreign” pool of individuals becomes larger).
19
More generally, we also expect that the variance in fecundity will not only affect the
20
evolution of bet-hedging strategies but also other life-history traits. In particular, traits
21
involving kin interactions such as sex-ratio, dispersal or altruism depend on the local
relat-22
edness arising through population structure, which is itself a dynamical variable shaped by
23
the variance in fecundity. The effect of the variance in offspring number on the evolution of
24
such behaviors has been overlooked so far. Apart from sex-ratio homeostasis (Verner, 1965;
25
Taylor and Sauer, 1980) in panmicitic populations and sex-allocation (Proulx, 2000, 2004),
26
we are aware of no model investigating the role of the fecundity variance on the evolutionary
27
stable state of various life-history strategies.
28
The objective of the present paper is twofold. First, we integrate the classical measure
29
of selection for within-generation bet-hedging (Gillespie, 1974, 1975, 1977) into the
sive fitness and game theoretic measure of selection for structured populations (e.g., Taylor,
1
1990; Taylor and Frank, 1996; Frank, 1998; Rousset and Billiard, 2000; Rousset, 2004, 2006).
2
This allows us to set up a framework for studying the effect of the mean and the variance in
3
fecundity for selection on various life-history traits and under different life-cycle assumptions
4
(e.g., timing of regulation, population structure). Second, we apply the resulting framework
5
to investigate the selective pressure on three life-history traits in subdivided populations
un-6
der soft and hard selections regime. These traits are within generation bet-hedging, helping
7
between deme-mates and dispersal. Our results for selection on the fecundity variance are
8
qualitatively consistent with those of Shpak (2005) and Shpak and Proulx (2006) but differ
9
quantitatively because the approach endorsed in this paper allows us to take the variance
10
in gene frequency between demes explicitly into account. Our results also indicate that the
11
magnitude of selection on both helping behaviors and dispersal rates is markedly affected
12
by the fecundity variance, and can even lead to outcomes qualitatively different from the
13
classical predictions considering mean fecundity only.
14
Model
15Life cycle and change in gene frequency
16
Consider that the population consists of haploid organisms occupying an infinite number of
17
demes of sizeN that are connected by migration (finite deme number is considered in the
18
Appendix). The life cycle is punctuated by the following events. (1) Each adult produces
19
a large number of independently distributed juveniles. The mean and the variance of the
20
fecundity distribution of an adult depend on its own phenotype and on the phenotype
21
of all other deme-mates. (2) Each juvenile disperses independently from each other with
22
probabilitymto another deme. (3) Density-dependent ceiling regulation occurs so that only
23
N juveniles reach adulthood in each deme. The size of each deme is thus assumed to be
24
constant over time.
25
The evolution of a phenotypic traitzunder selection in this population can be described
26
by a gradual, step-by-step transformation caused by the successive invasion of mutant alleles
27
resulting in different phenotypic effects than resident alleles fixed in the population. The
invasion of a mutant allele can in turn be ascertained by the change in its frequencypover
1
one generation, which for a mutant with small phenotypic effect δ(weak selection) can be
2
written as
3
∆p=δSp(1−p) +O(δ2). (1)
4
The term O(δ2) is a remainder and S is Hamilton’s (1964) inclusive fitness effect, which
5
measures the direction of selection on the mutant and also provides the first order
pheno-6
typic effect of the mutant on its probability of fixation when the population is of finite size
7
(Rousset, 2004, p. 99, pp. 108–109 and pp. 206–207; Roze and Rousset, 2003, 2004; Rousset,
8
2006). The inclusive fitness effect (S) can be computed by the direct fitness method (Taylor
9
and Frank, 1996; Rousset and Billiard, 2000), as
10 S≡ ∂w ∂z• + ∂w ∂zD 0 FST, (2) 11
wherew ≡w(z•, z0D) is a fitness function giving the expected number of adult offspring of
12
a focal individual (FI) and FST is the relatedness between the FI and another individual
13
randomly sampled from its deme (here Wright’s measure of population structure). The
14
partial derivatives ofware the effects of actors (i.e., individuals bearing the mutant allele)
15
on the fitness of the FI. The actors being the FI itself with phenotype denoted z•, and 16
the other individuals from the focal deme with average phenotype denotedz0D. The partial
17
derivatives can be evaluated at a candidate evolutionary stable strategyz?. This phenotype
18
towards which selection leads through small mutational steps is obtained by solvingS = 0 at
19
z•=z0D=z
?. In the next section we provide expressions for both the fitness functionwand
20
the relatednessFST in terms of the means and the variances of the fecundity distributions
21
of the individuals in the population.
22
Fitness function
23
The direct fitness function of a FI can be decomposed into two terms:
24
w=wp+wd. (3)
25
This formulation emphasizes that fitness depends on the expected number of FI’s offspring
26
recruited in the focal deme (wp) and on the ones reaching adulthood in a foreign deme after
dispersal (wd). In the Appendix we show that when deme size is large, each component of
1
fitness can be expressed as a function of only the mean and the variance of the fecundity
2
distribution of the different individuals in the population affecting the fitness of the FI.
3
Neglecting terms of order 1/N2 and higher order, the philopatric component of fitness can
4 be written as 5 wp= (1−m)f• fp 1 + σ 2 p N f2 p ! − σ 2 •,p N f2 p (4) 6
where f• is the mean fecundity of the FI and fp= (1−m)f0+mf1 is the average of the
7
mean number of juveniles coming in competition in the focal deme (eq. 38 of the Appendix).
8
This number depends on the average mean fecundityf0 of individuals breeding in the focal
9
deme and on the average mean fecundityf1of individuals breeding in different demes. The
10
fitness functionwpalso involves the variance in the number of offspring of the FI coming in
11
competition in the focal deme
12
σ2•,p= (1−m)mf•+ (1−m)2σ2•, (5)
13
where σ•2 is the variance of the fecundity distribution of the FI (here, the variance of the
14
number of offspring produced before the dispersal stage). The first term (1−m)mf• in this 15
equation can be interpreted as the variance in offspring number entering in competition in
16
the focal deme and caused by dispersal being a random event. The second term in eq. 5 is
17
the additional variance in offspring number entering in competition in the focal deme and
18
resulting from fecundity being a random variable. Accordingly, when fecundity is a fixed
19
number and migration is random, we are left only with the first term while if fecundity is a
20
random number but a fixed proportion of juvenile migrate, we are left only with the second
21
term. The fitness function also depends on the average variance of the number of juveniles
22
entering in competition in the focal deme, which reads
23
σ2p= (1−m)mf0+ (1−m)2σ20+mf1, (6)
24
whereσ2
0 is the average variance in fecundity of individuals breeding in the focal deme.
25
The component of the fitness of the FI resulting from dispersing progeny reads
26 wd= mf• f1 1 + σ 2 d N f2 1 − σ 2 •,d N f2 1 , (7) 27
where
1
σ2•,d=mf• (8)
2
is the variance in the number of offspring of the FI entering in competition in other demes
3
(eq. 42 of the Appendix). This variance depends only on dispersal being a random event
4
and not on the variance in fecundity. This is so because the contribution of the variance in
5
fecundity to the variance in the number of competing offspring in a given deme depends on
6
the square of the migration rate to that deme (eq. 36 of the Appendix). Since in the infinite
7
island model two offspring descending from the same parent have a probability of zero of
8
emigrating to the same deme, the effect of the fecundity variance on the variance in the
9
number of competing offspring in a non-focal deme vanishes. Finally, we need the average
10
variance of the number of juveniles entering in competition in different demes, this is
11
σ2d= (1−(1−m)2)f1+ (1−m)2σ21 (9)
12
whereσ2
1 is the average variance in fecundity of individuals breeding in different demes.
13
Equations 4 and 7 highlight that both the means (f) and the variances (σ2) of the
14
fecundity distributions of individuals in the population affect the fitness of a FI. This result is
15
an approximation that follows from the assumption that demes are of large size. Conversely,
16
if demes are small, the fitness function w will also depend on the kurtosis, skewness and
17
other measures of the dispersion of the fecundity distributions (see eq. 26 of the Appendix).
18
By contrast, when fecundity follows a Poisson distribution, fitness depends only on mean
19
fecundity. This is an exact result (see section Poisson distributed fecundity of the Appendix),
20
holding whatever the sizes of demes, which is usually invoked in applications of inclusive
21
fitness theory (e.g., Comins et al., 1980; Rousset and Ronce, 2004) or population genetics
22
(e.g., Karlin and McGregor, 1968; Ewens, 1969; Gillespie, 1975; Ewens, 2004), because it
23
greatly simplifies the calculations. In order to investigate how a distribution of fecundity
24
with a variance in excess of a Poisson distributed fecundity might affect selection, we consider
25
here the fate of mutant alleles affecting three different life-history traits:
26
(a) The mutant results in both a reduction in the mean (byC) and the variance (byD)
27
of its random number of juveniles produced before dispersal.
28
(b) The mutant causes an action that reduces its mean fecundity by some cost C but
which increases the mean fecundity of each individual in the deme (excluding the actor) by
1
a benefitB/(N−1).
2
(c) The mutant expresses a decrease (or increase) in the dispersal rate set by the resident
3
allele.
4
Assumptions for the variance in fecundity
5
According to our assumption that the deme sizes remain constant over time, the mean
6
fecundityf of individuals in a monomorphic population has to be large in order to cancel
7
out the fluctuations of deme sizes induced by demographic stochasticity. If the variance in
8
fecundity is of the same order of magnitude as the mean, all terms involving a variance in the
9
fitness functions (eq. 4 and eq. 7) will vanish because they are divided by the mean squared
10
(i.e.,σ2/f2→0 whenf becomes large). We are then left with a situation where the fitness
11
of a FI depends only on its mean fecundity of that of the other actors in the population.
12
However, when the variance in fecundity is of higher order of magnitude than the mean, the
13
coefficient of variation defined as
14
σv≡
σ
f (10)
15
(e.g., Lynch and Walsh, 1998, p. 23) remains positive under large mean fecundity (i.e.,
16
σv >0 for f large). For the rest of this paper, we assume that the fecundity distribution
17
has a variance of larger order of magnitude than the mean and that the mean fecundity is
18
finite but large enough to prevent demographic stochasticity.
19
Several distributions closely related to the Poisson distribution have a variance of larger
20
order of magnitude than the mean. Such distributions can be conveniently obtained by
mix-21
ing the rate of occurrence of birth eventsf of the Poisson distribution with other distribution
22
(i.e., mixed Poisson distributions, Willmot, 1986). For instance, the Negative-Binomial
dis-23
tribution, which is advocated to be the basis of the fertility in humans and other species
24
(Caswell, 2000), can be obtained by mixing the birth ratef by a Gamma distribution. This
25
distribution has meanf and variance f+f2/α, where the parameterαallows tuning this
26
variance away from that of a Poisson distribution (Fig. 1). When the mean becomes large,
27
the coefficient of variation of this distribution remains positive (σv →1 +f /α). Similarly,
when clutch size follows a Poisson distribution and the whole clutch has a probability (1−s)
1
of complete failure, the resulting distribution has meansf and variancef s+f2s(1−s).
2
Relatedness
3
The variance in fecundity will also affect the dynamics of relatedness between two individuals
4
sampled in the same deme after dispersal. At equilibrium, relatedness satisfies the recursion
5
6
FST= (1−m)2(Pr(C) + [1−Pr(C)]FST), (11)
7
where (1−m)2is the probability that the two individuals are of philopatric origin and Pr(C)
8
is the probability that they descend from the same parent.
9
Using eq. 10 and eq. 51 from the Appendix, assuming that both deme size and mean
10
fecundity are large, and keeping only terms of leading order we find that the coalescence
11 probability is given by 12 Pr(C) =1 +σ 2 v N , (12) 13 whereσ2
v is called the effective variance by Gillespie (1975, p. 407).
14
Using eq. 4, eq. 7 and eq. 12 we investigate in the next section the selective pressure on
15
the three life-history traits discussed above.
16
Examples
17Within-generation bet-hedging
18
In this section, we investigate the conditions under which an organism is selected to decrease
19
its mean fecundity in order to reduce its variance in fecundity. The setting is the same as
20
the one investigated by Shpak and Proulx (2006). The fecundity means and variances of
21
the various classes of actors affecting the fitness of a FI bearing a mutant bet-hedging allele
22
are evaluated as is usually done by the direct fitness method (e.g., Frank, 1998; Rousset and
23
Billiard, 2000; Rousset, 2004) and are presented in Table 1. Substituting these functionals
into the direct fitness function (eq. 3) allows us to calculate explicitly the selective pressure
1
on the mutant allele (eq. 2). Evaluating this selective pressure at the phenotypic value of
2
the resident allele (z•=z0D=z1= 0) and taking into account only terms of leading order
3
(i.e., neglecting terms of order 1/N2, 1/f2,1/(N f) and beyond), we find that the inclusive
4
fitness effect becomes
5 S=−C 1 +σ 2 v(1−m) 2 N ! +Dσ 2 v(1−m) 2 N +C(1−m) 2 FSTR, (13) 6 where 7 FSTR = 1 N + N−1 N FST (14) 8
is the relatedness between the FI and an individual randomly sampled with replacement
9
from its deme (thus including the FI with probability 1/N) and
10 FST= 1 +σ2v (1−m)2 N(2−m)m . (15) 11
The inclusive fitness effectSis composed of three terms. First, there is a direct cost to the
12
FI stemming from the reduction in its (mean) fecundity. Second, there is a direct benefit to
13
the FI stemming from the reduction in its variance in fecundity, which is proportional to the
14
probability (1−m)2/Nthat two randomly sampled offspring in the focal deme descend from
15
the FI and on the coefficient of variation squared (i.e., effective fecundity). Third, we have to
16
account for an indirect benefit resulting from the reduction in competition in the focal deme
17
faced by the FI’s offspring, which results from its own and its relativesı reduced fecundity.
18
This decrease in kin competition is proportional to the probability (1−m)2that two offspring
19
produced in the focal deme compete against each other. Since we assumed large deme size,
20
there are no effects on fitness resulting from relatives lowering their fecundity variance as such
21
effects are of order 1/N2. However, the variance in fecundity increases relatedness (eq. 15)
22
and thus has an impact on the intensity of kin competition. Rearranging the inclusive fitness
23
effect, the low-variance mutant spreads when the threshold cost to benefit ratio satisfies
24 C D < σ2 v(1−m) 2 (2−m)m (N+ 1) (2−m)m−σ2 v(1−m) 2 (1−2 (2−m)m)−1. (16) 25
This threshold decreases with an increase in migration rate and deme size and with a decrease
26
in the variance in fecundity. The inequality cannot be satisfied when the coefficient of
27
variation is vanishingly small (σ2
v= 0) or when migration is complete (m= 1).
While the intensity of selection on the variance in fecundity is proportional to the inverse
1
of population size in panmictic populations (Gillespie, 1974, 1975, 1977; Demetrius and
2
Gundlach, 2000; Demetrius, 2001), we see from eq. 13 that the intensity of selection on this
3
variance is proportional to (1−m)2/Nin subdivided populations when regulation occurs after
4
dispersal. Thus, selection on the variance decreases as the migration rate and/or deme size
5
increases. This is so because the contribution of the variance in fecundity of a focal individual
6
to the variance in the number of offspring competing in a given deme after dispersal depends
7
on the square of the migration rate to that deme. Under infinite island model assumptions,
8
this contribution vanishes for all demes except for the focal deme (compare eq. 5 and eq. 8).
9
The finding that selection on the variance is of intensity (1−m)2/Ncorroborates the results of
10
Shpak and Proulx (2006). Indeed, the two first term of the inclusive fitness effect is consistent
11
with eq. 20 of Shpak and Proulx (2006). However, we note that the approach endorsed here
12
takes into account the effect of the variance in gene frequency between demes on the expected
13
change of allele frequency at the level of the population given by eq. 1. Indeed, the third
14
term of the inclusive fitness effect, which involves kin competition, depends on the variance
15
in gene frequency among demes (i.e. FST) and is not considered by Shpak and Proulx (2006)
16
but its presence here is consistent with previous population genetic derivation of selection
17
on the mean fecundity in subdivided populations (e.g. Roze and Rousset, 2003; Rousset,
18
2004; Roze and Rousset, 2004). In the Appendix, we complement our analyses by presenting
19
the condition under which an organism is selected to decrease its mean fecundity in order
20
to reduce its variance in fecundity when regulation occurs before dispersal (soft selection).
21
Selection on the variance in fecundity then scales as 1/N as was already established by
22
Shpak (2005).
23
Helping behaviors
24
Here, we investigate the effect of the variance in fecundity on the condition under which an
25
organism should be prepared to sacrifice a fraction of its resources to help its deme-mates.
26
The setting is exactly the same as Taylor (1992a), except the assumptions regarding the
27
distribution of fecundity. While Taylor considers an infinite fecundity, we assume here as
28
in the previous section that the fecundity is finite but large and that its distribution has a
variance in excess of a Poisson distributed fecundity. The means and the variances of the
1
fecundities of the various classes of individuals affecting the fitness of a FI are given in Table
2
1. Evaluating the selective pressure on the helping allele at the phenotypic value of the
3
resident allele (z•=z0D=z1= 0) and taking into account only terms to leading order, the
4
selective pressure on helping becomes
5 S=−C 1 +σ 2 v(1−m) 2 N ! +BFST−(1−m)2(B−C)FSTR, (17) 6 where FR
ST and FST are given by eq. 14 and eq. 15, respectively. This inclusive fitness
7
effect S is also composed of three terms: First, the direct cost of helping; second, the
8
benefit received by the FI from its neighbors; third, the cost of the increase in competition
9
in the focal deme faced by the FI’s offspring caused by him and its neighbors expressing
10
the helping trait. In Taylor’s (1992a) original setting we have FST = (1−m)2FSTR, which
11
inserted into eq. 17 cancels out all the benefit terms. However, in the present situation we
12
have FST > (1−m)2FSTR (see eq. 12) because relatedness is inflated due to the increased
13
frequency of coalescence events stemming from the higher variance in fecundity. This results
14
in a situation where the fecundity benefitB is no longer cancelled out by the concomitant
15
increase in kin competition. Rearranging this selective pressure, we find that helping spreads
16 when 17 C B < σ2v(1−m) 2 (2−m)m (N+ 1) (2−m)m−σ2 v(1−m) 2 (1−2 (2−m)m)−1 (18) 18
is satisfied. As in the preceding section, the threshold cost to benefit ratio decreases with an
19
increase in the migration rate and deme size and with a decrease in the variance in fecundity.
20
When the coefficient of variation is vanishingly small (σ2v →0), we have C/B <0 so that
21
the direction of selection on helping is determined solely by direct fecundity benefits. The
22
behavior is then selected for if−C >0, that is, if the number of juveniles counted before any
23
competition stage, is increased (Taylor, 1992a; Rousset, 2004). Here, helping can be selected
24
for at a direct fecundity cost to the actor wheneverσv2>0. We note that Taylor’s seminal
25
hard selection model and relaxations of its life-cycle assumptions has been investigated by
26
several authors (Taylor and Irwin, 2000; Irwin and Taylor, 2001; Perrin and Lehmann, 2001;
27
Rousset, 2004; Roze and Rousset, 2004; Gardner and West, 2006; Lehmann et al., 2006;
28
Lehmann, 2006).
Dispersal
1
Finally, we examine the effect of the variance in fecundity on the evolutionary stable (ES)
2
dispersal rate. The setting is exactly the same as in Frank (1985, 1998), at the exception
3
that we assume again as in the previous sections that fecundity is finite but large and that its
4
distribution has a variance in excess of a Poisson distributed fecundity. The fitness functions
5
wof a focal parent bearing a mutant genotype affecting the dispersal rate of its offspring is
6
more complicated than those investigated so far because we have to take into account the
7
survival rates of juveniles during dispersal. Let us designate by d the evolving dispersal
8
rate and byd•the average dispersal rate of the offspring of a FI. A proportion 1−d• of the 9
offspring produced by the FI remain in the focal deme and a fractionsd• of these offspring 10
enter in competition in another deme after dispersal. The direct fitness function for the
11
evolution of dispersal can then be obtained by setting (1−m)≡1−d• in the numerator 12
of the first ratio in eq. 4 andm≡sd• in the numerator of the first ratio of eq. 7, while all 13
the remaining terms of these fitness functions are given in Table 2. The fitness function for
14
dispersal can then be written as
15 w= 1−d• (1−dR 0) +sd1 1 + σ 2 p N f2 p ! + sd• (1−d1) +sd1 1 + σ 2 d N f12 − σ 2 •,p N f2 p − σ 2 •,d N f12, (19) 16
which, for a large mean fecundity and vanishing coefficient of variation (σv = 0), reduces
17
to the classical fitness function for dispersal (e.g., Frank, 1998; Gandon and Rousset, 1999;
18
Perrin and Mazalov, 2000). Substituting the fitness function into eq. 2, evaluating the
19
selective pressure at the candidate ES dispersal rate, that is at d• = dR0 = d1 = d, and
20
taking into account only terms to leading order, we find that the selective pressure on
21 dispersal becomes 22 S=− 1−s 1−d(1−s)+ (1−d) (1−d(1−s))2F R ST+ σ2v(1−d)(1−d(1−s) +s) N(1−d(1−s))3 , (20) 23
whereFSTR andFSTare given by eq. 14 and eq. 15 with the backward migration rate expressed
24
as m≡sd/(1−d+sd), which represents the probability that an individual sampled in a
25
deme is an immigrant. This selective pressure on dispersal is decomposed into three terms.
26
First, the classical direct cost of increasing the dispersal rate, which varies directly with
27
the survival probabilitysduring dispersal. Second, the indirect benefit stemming from the
28
reduction in competition in the focal deme faced by the FI’s offspring, which results from his
and its neighbors offspring dispersing (e.g., Hamilton and May, 1977; Taylor, 1988; Frank,
1
1998; Gandon and Michalakis, 1999; Gandon and Rousset, 1999; Perrin and Mazalov, 2000).
2
Third, a benefit resulting from the dispersing progeny of the FI reducing the variance in its
3
number of offspring reaching adulthood in the focal deme. This term is new and is driven
4
by eq. 5. By dispersing, juveniles minimize the variance in the number of offspring of a
5
focal parent entering in competition in the focal deme. Since the variance in the number
6
of offspring entering locally in competition (eq. 5) dominates the variance in the number of
7
offspring entering in competition in different demes by dispersing (eq. 8), dispersal decreases
8
the overall variance in the number of offspring entering in competition. As a consequence,
9
fitness increases through dispersing and the ES dispersal rates are boosted by higher variance
10
in fecundity. Fig. 2 compares the ES dispersal rate obtained from the present model with the
11
classical ES dispersal rate (e.g., Hamilton and May, 1977; Frank, 1998; Perrin and Mazalov,
12
2000; Gandon and Rousset, 1999) obtained under vanishing coefficient of variation (σv2= 0
13
in eq. 20 and eq. 15).
14
Discussion
15In this paper, we integrated the classical measure of selection for within-generation
bet-16
hedging (Gillespie, 1974, 1975, 1977) into the game theoretic and inclusive fitness measure
17
of selection for structured populations (e.g., Taylor, 1990; Taylor and Frank, 1996; Frank,
18
1998; Rousset and Billiard, 2000; Rousset, 2004, 2006). The resulting framework in which
19
fitness depends explicitly on both the means and the variances of the fecundity distribution
20
can be applied to study the evolution of various life-history traits (i.e., sex-ratio, helping,
21
reproductive effort), under soft and hard selection regimes. The selective pressure on three
22
traits where directly investigated here, namely within-generation bet-hedging, helping
be-23
haviors and the evolutionary stable dispersal rate. The results suggest that the evolutionary
24
dynamics of all these traits is affected by the variance in fecundity, though with variable
25
intensity and under different demographic conditions.
Within-generation bet-hedging
1
Natural selection selects against the variance in offspring number when panmictic
pop-2
ulations are not too large (Gillespie, 1974, 1975, 1977; Demetrius and Gundlach, 2000;
3
Demetrius, 2001). Our models suggest that selection against this variance in subdivided
4
populations scales as 1/N (eq. 57) when regulation occurs before dispersal (soft selection)
5
and scales as (1−m)2/N (eq. 13) when regulation occurs after dispersal (hard selection).
6
This is so because the contribution of the variance in fecundity of a focal individual to the
7
variance in its number of offspring competing in a given deme depends on the square of the
8
migration rate to that deme. Under soft selection, all the competition occurs in the focal
9
deme so that migration has no effect on selection. By contrast, under hard selection, the
10
offspring of an individual compete in different demes and, due to our infinite island model
11
assumptions, the contribution of the variance in fecundity to the various demes vanishes for
12
all demes except for the focal deme (compare eq. 5 and eq. 8). These results corroborate the
13
findings of both Shpak (2005) and Shpak and Proulx (2006) and illustrate that they can be
14
derived from the same model.
15
It is sometimes assumed in the literature that selection against the variance in offspring
16
number and its impact on various life-history trait will be of limited importance in natural
17
populations because they are of large size (e.g., Seger and Brockman, 1985). Our results and
18
those of Shpak (2005) and Shpak and Proulx (2006) demonstrate that selection against this
19
variance can perfectly operate in large but subdivided populations in the presence of both
20
soft and hard selection regimes. That selection can target the fecundity variance in large
21
populations may be relevant for the evolution of bet-hedging strategies in general and for the
22
theory of sex-ratio homeostasis (Verner, 1965; Taylor and Sauer, 1980) in particular. Indeed,
23
the results suggest that the demographic parameters of the population will condition whether
24
strategies involving probabilistic sex-ratio determination (i.e. each individual develops into
25
a male or a female with a given probability) will resist invasion by strategies involving a
26
rigid determination of it (i.e. a fixed proportion of offspring develop into either sex).
27
More generally, the fecundity variance might also play a role for the evolution of systems of
28
phenotypic determination. Systems of adaptive polymorphism are expected in the presence
29
of fluctuating environments or resource specialization and involve, among other mechanisms,
the random determination of an individual’s phenotype or its genetic determination (Grafen,
1
1999; Leimar, 2005). Since probabilistic phenotype determination can introduce additional
2
variance in the number of offspring entering in competition, the conditions under which
3
mixed evolutionary stable strategies are favored over pure genetic polymorphism are likely
4
to be dependent on the variance in the number of offspring sent into competition by each
5
strategy.
6
Helping behaviors
7
While the preceding examples of bet-hedging strategies resulted in directional selection
8
against the variance in fecundity, the selective pressure on other strategies can depend on
9
the magnitude of this variance. Helping behaviors involving kin interaction fall into this
cat-10
egory because their evolution depends on the relatedness between interacting individuals,
11
whose dynamic is determined by the probability of common ancestry within deme, itself a
12
function of the fecundity distribution (Ewens, 2004; Rousset, 2004). An increase in
fecun-13
dity variance decreases the number of effective ancestors within demes and thus increases
14
the relatedness between deme-mates possibly leading to an increase in the selective pressure
15
on helping behaviors.
16
Taylor (1992a,b) has demonstrated that when demes are of constant size and when
fe-17
cundity is infinite (i.e., no variance in fecundity), selection on helping is determined solely
18
by direct fecundity benefits. Here, Taylor’s assumption on fecundity was replaced by the
as-19
sumption that fecundity follows a distribution with finite but large mean and with a variance
20
exceeding the mean. Allowing for a non-vanishingly small coefficient of variation increases
re-21
latedness between deme members with the consequence that all kin selected effects, whether
22
positive or negative, are increased. Here, this translates in helping being selected for at a
23
direct fecundity cost to the actor with the intensity of selection on helping decreasing when
24
the variance in fecundity decreases (eq. 18). This result suggests that selection on helping is
25
also bound to depend on the mating system because the variance in the number of mating
26
partners obtained by males will increase relatedness within demes.
27
The finding that the magnitude of the variance in fecundity qualitatively affects the
28
outcome of selection on costly helping might help us to understand why different authors
have repeatedly reached contradictory results when investigating selection on helping under
1
apparently identical life cycles. While several authors have emphasized that unconditional
2
costly helping is counter selected when social interactions occur between semelparous
indi-3
viduals living in structured populations of constant size (Wilson et al., 1992; Taylor, 1992a,b;
4
Rousset, 2004; Lehmann et al., 2006), other authors have suggested that costly helping is
5
selected for in that case (Nowak and May, 1992; Killingback et al., 1999; Hauert and Doebeli,
6
2004). The results by the former group of authors are based on analytical models assuming
7
infinite and/or Poisson distributed fecundities. The results by the latter group are mostly
8
based on simulations. In this case, it remains unclear how the updating rules used by the
9
authors to simulate the transmission of strategies from one generation to the next in the
pop-10
ulation impact on relatedness and may thus result in significant departure from the Poisson
11
distributed fecundity assumption. For instance, copying the fittest neighbor will result in a
12
different dynamics of relatedness than copying neighbors with a probability proportional to
13
their fitness. These specific features might help to explain why some authors observed the
14
emergence of costly helping behaviors while others did not.
15
Dispersal
16
The present paper emphasizes that the variance in the number of offspring sent into
com-17
petition can be as important in determining fitness as the mean of such offspring (eq. 4 and
18
eq. 7). This variance will depend among other factors on the probability that an individual
19
reproduces, its offspring disperse, migrants survive dispersal and successful migrants survive
20
competition. While a near endless chain of stochastic events may affect individuals as they
21
move along their life cycle, we focused here only on the stochasticity induced by reproduction
22
and migration. Migration has a striking and non-intuitive impact on the total variance in
23
the number of offspring entering competition because it results in two distinct effects on this
24
variance. First, migration redistributes the variance in fecundity of an individual over the
25
whole population. Since the variance in the number of offspring competing in a given deme
26
after dispersal depends on the square of the migration rate to that deme, the contribution
27
of the variance in fecundity to the variance in competing offspring in that deme vanishes for
28
all demes except for the focal one, whenever there is a large number of demes (compare eq. 5
and eq. 8). Second, we assumed that migration itself is a probabilistic event, which thus
1
generate a variance in the number of competing offspring even for fixed fecundity schedules
2
(first term in eq. 5 and eq. 8). These two effects of migration on fitness will have two distinct
3
consequences for the evolution of stable dispersal rates.
4
First, assuming that each juvenile disperses independently from each other and that the
5
environment is constant, the evolutionary stable dispersal rate depends on a balance between
6
two opposing forces. There is a cost of dispersal resulting from reduced survival during
7
migration opposed by a benefit leading to local decrease in kin competition (e.g., Hamilton
8
and May, 1977; Frank, 1985, 1998; Gandon and Michalakis, 1999; Gandon and Rousset, 1999;
9
Perrin and Mazalov, 2000). Here, this balance is displaced in favor of increased dispersal
10
because the cost of dispersal is reduced as migration reduces the total variance in the number
11
of offspring of an individual sent into competition. By dispersing, the offspring of the focal
12
individual decrease the impact of the variance in fecundity on fitness. A large variance in
13
fecundity can result in a two-fold increase of the evolutionary stable dispersal rate (Fig. 2).
14
That the variance in fecundity can affect the candidate ESS strategy has already been shown
15
for models of sex-allocation (e.g., Proulx, 2000, 2004). Our result that the ES dispersal rate
16
is affected by the variance in fecundity might be of practical importance when evaluating
17
the causes of dispersal in natural populations and it suggests that the fecundity distribution
18
might explain a part of the observed variance in dispersal rates.
19
The second consequence of subdivided populations for the evolution of dispersal rates
20
stem from the fact that if migration is a probabilistic event, it increases the variance in
21
offspring number of an adult coming into competition. One might then wonder whether a
22
strategy resulting in a fixed proportion of individual dispersing would not be favored over a
23
strategy involving probabilistic dispersal, in complete analogy with the theory of sex-ratio
24
homeostasis (Taylor and Sauer, 1980), where the production of a fixed proportion of males
25
and females is favored over probabilistic sex-ratio determination. In the section entitled
26
“Dispersal homeostasis” of the Appendix, we constructed a model precisely along these lines
27
(see eq. 63), which demonstrates that dispersal homeostasis can indeed evolve but that the
28
selective pressure on it is only of order 1/(f N) wheref is the mean fecundity. This result
29
is in line with work by (Taylor and Sauer, 1980), who showed that selection on “sex-ratio
homeostasis” in panmictic populations is proportional to the inverse of the total number
1
f N of offspring born in a group. According to our assumption that both deme size and
2
fecundity is large, terms of such order were generally dismissed in the calculation of the
3
various selective pressures (eq. 13, eq. 17 and eq. 20). However, selection on both dispersal
4
and sex-ratio homeostasis might be effective in subdivided population when both the mean
5
fecundity and deme size are not too large.
6
Limitations and extensions of the model
7
All results derived in this paper are based on five main assumptions that greatly simplified
8
the analyses and that we now discuss. First, we assumed an island model of migration. This
9
assumption was introduced in order to highlight in a simple way the effects of the fecundity
10
variance on fitness in subdivided populations, although the model derived in the Appendix
11
is more general and allows for isolation by distance (e.g., stepping-stone dispersal). In this
12
case we expect selection on the variance in fecundity to be increased because the variance
13
in the number of offspring competing in a given deme after dispersal depends on the square
14
of the migration rate to that deme (eq. 36). This probability will not be vanishingly small
15
for migration at short spatial distances so that selection against the variance will generally
16
exceed the intensity (1−m)2/N established here for infinite island situations. Second,
17
we assumed that demes are of large size (terms of order 1/N2 are neglected, see section
18
Regulation of the Appendix). This assumption was introduced in order to focus primarily
19
on selection on the variance in fecundity and to neglect selection on other measures of
20
dispersion of the fecundity distribution and which are likely to matter when demes are of
21
small size. As can be noted by inspecting eq. 26, natural selection will not only reduce
22
the variance in fecundity but also favor positively skewed fecundity distributions that are
23
platykurtic. That is, distributions that are asymmetric with a fatter right tail and that are
24
less peaked than a normally distributed variable. This is intuitively expected because it
25
results in a shape of the fecundity distribution that reduces the likelihood of producing no
26
offspring at all. Exactly how natural selection will shape the fecundity distribution given
27
the constraints on reproductions remains to be investigated. Third, we assumed that the
28
mean fecundity is finite but large. This assumption was introduced in order to be able to
neglect demographic stochasticity and to obtain analytical expressions as simple as possible.
1
Under ceiling regulation, demographic stochasticity is completely prevented from occurring
2
for demes of large size as soon as individuals produce more than two offspring (Lehmann
3
et al., 2006). However, other forms of population regulation occur in nature, such as
density-4
dependent survival with no specific ceiling effect, which might result in wider population
5
fluctuations. Overall, allowing for low fecundities and small deme sizes should in principle
6
lead to situations where selection on the measures of statistical dispersions (e.g., variance,
7
skewness) is exacerbated. Fourth, we assumed that the variance in fecundity is of larger order
8
of magnitude than the mean so that the coefficient of variation (σv) is not vanishingly small
9
even when the mean fecundity is large. This assumption was introduced to ensure that the
10
variance of the fecundity distribution still impacts on fitness under our hypotheses of large
11
fecundities and demes sizes. A typical example of such a fecundity distribution is given by the
12
Negative-Binomial distribution, which follows from many models of reproduction (Caswell,
13
2000, p. 455), and has been advocated to be characteristic of the fertility in humans and
14
other species (Anscombe, 1949; Cerda-Flores and Davila-Rodriguez, 2000). That organisms
15
have a variance in fecundity in excess to the mean has also been reported in studies of
16
reproductive success (Crow and Kimura, 1970, Table 7.6.4.2.; Clutton-Brock, 1988). Fifth,
17
we assumed an infinite number of demes so that the effect of drift at the level of the total
18
population on gene frequency change is not taken into account. However, our direct fitness
19
derivation presented in the Appendix is framed within a finite population so that the inclusive
20
fitness effectS also allows us to evaluate the first order phenotypic effect of a mutant allele
21
on its probability of fixation (Rousset, 2004, 2006). The evaluation of the probability of
22
fixation itself deserves further formalizations that might be achieved by following along
23
the lines of the direct fitness approach to diffusion equations of Roze and Rousset (2003,
24
2004). We finally mention that we investigated here only three phenotypic traits, whose
25
evolution depends on the magnitude of relatedness (i.e.,FST) that is itself of function of the
26
variance in fecundity. However, the evolution of many other traits including sex-ratio,
mate-27
choice, harming behaviors such as spiteful acts, niche-constructing phenotypes or resource
28
exploitation curves depend on relatedness (e.g., Hamilton, 1971; Boomsma and Grafen,
29
1991; Taylor and Getz, 1994; Gandon and Michalakis, 1999; Ajar, 2003; Lehmann, 2006).
30
Consequently, the expression of all these traits is likely to be indirectly influenced by the
variance in fecundity.
1
Based on all these comments and the premise that there is as much genetic variation for
2
the variance in fecundity as there is for the mean, we expect that the results presented in this
3
paper are of relevance for natural systems. As a final note, we can thus only re-emphasize
4
Gillespie’s (1977) view that the role of the variance deserves a more prominent place in our
5
thinking about evolutionary processes.
6
Acknowledgements
7
We thank one anonymous reviewer for helpful comments on this manuscript. LL
acknowl-8
edges financial support from from the Swiss NSF.
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Wade, M. J. 1985. Soft selection, hard selection, kin selection, and group selection.American 25
Naturalist 125:61–73.
Whitlock, M. C. 2003. Fixation probability and time in subdivided populations. Genetics 1
164:767–779.
2
Willmot, G. 1986. Mixed compound poisson distributions. Astin Bulletin 16:59–79.
3
Wilson, D., G. Pollock, and L. Dugatkin. 1992. Can altruism evolve in purely viscous
4
populations. Evolutionary Ecology 6:331–341.
Number J of juveniles produced
P
ro
b
ab
ili
ty
5 10 15 20 25 30 0.02 0.04 0.06 0.08 0.1 0.12 0.14Figure 1: Negative-Binomial fecundity probability distribution Pr(J) =
Γ(α+J) Γ(α)J! f α J 1 +αf −(α+J)
for the number J of juveniles produced by an individual. The mean is given byf and the variance by f +f2/α. For all curves we have a mean of f = 10, whileα=∞for the curve with the highest peak (i.e., Poisson distribution), α= 5 for the curve with intermediate peak andα= 1.5 for the curve with the lowest peak.
Patch size
ES
d
is
p
er
sa
l r
at
e
20 40 60 80 100 0.2 0.4 0.6 0.8 1Figure 2: Evolutionary stable dispersal rate graphed as a function of deme size N with survival rate set to s = 0.9. The plain line is the classical ES dispersal rate obtained by assuming that the progeny distribution is Poisson or has an infinite mean. The dotted line (curve in between both other curves) corresponds to the ES dispersal rate whenσ2v = 1/5
while the dashed line corresponds to the ES dispersal rate when σ2
v = 1/1.5. Fecundity
Table 1
1
Symbol Value for within-generation bet-hedging Value for Helping
f• f(1−Cz•) f(1 +Bz0D−Cz•) f0 f(1−CzR0) f(1 + (B−C)zR0) f1 f(1−Cz1) f(1 + (B−C)z1) σ2 • σ2(1−Dz•) σ2 σ2 0 σ2(1−DzR0) σ2 σ21 σ2(1−Dz1) σ2
f Baseline mean fecundity of each individual in a monomorphic population.
σ2 Baseline fecundity variance of each individual
in a monomorphic population. z• Phenotype of a focal individual.
zR
0 Average phenotype of individuals from
the focal deme (zR= 1 Nz•+ N−1 N zD 0). zD
0 Average phenotype of individuals.
from the focal deme but excluding the FI.
z1 Average phenotype of individuals from different demes.
Table 2
1
Symbol Value for dispersal
f• f fp f(1−dR0) +sd1 f1 f[(1−d1) +sd1] σ2 •,p (1−d•)d•f+ (1−d•)2σ2 σ2 •,d sd•f σ2p (1−dR0)d R 0f+ (1−d R 0) 2σ2+sd 1f σ2 d (1−d1)d1f+ (1−d1)2σ2+sd1f
d• Dispersal rate of the offspring of the focal individual.
dR
0 Average dispersal rate of the offspring of individuals
from the focal deme (dR= 1 Nd•+ N−1 N dD 0).
d1 Average dispersal rate of the offspring of individuals
from different demes.
Appendix
1Fitness function
2
In this appendix, we derive an expression for the direct fitness function w (eq. 2), which
3
involves both the mean and the variance of the number of offspring produced by a focal
4
individual and which generalizes the derivation of Gillespie (1975) to subdivided populations
5
(see also Shpak and Proulx, 2006). We assume that the mean fecundity of individuals is
6
finite but sufficiently large to prevent any demographic stochasticity. Accordingly, each
7
deme will be treated as being of constant sizeN. This assumption implies that the model
8
is not exact but provides a balance between accuracy and complexity. An exact model can
9
be obtained by recasting the forthcoming derivation within the framework of the general
10
metapopulation model of Rousset and Ronce (2004) by conditioning the expressions for
11
fitness on demographic states. For completeness, we first consider that the population is
12
constituted of a finite numberndof demes and then take the infinite island limit (nd→ ∞).
13
We now follow the events of the life cycle presented in the main text in reverse order.
14
Regulation
15
The expectation of the random numberNijxof offspring surviving regulation in demeiand
16
descending from individualxbreeding in demej can be written as
17 E[Nijx] = X Ji X Jijx
E[Nijx|Jijx, Ji] Pr(Jijx|Ji) Pr(Ji), (21)
18
where Pr(Jijx | Ji) Pr(Ji) is the joint probability that Jijx offspring of individual xenter
19
in competition in deme iamong a total number ofJi offspring and E[Nijx|Jijx, Ji] is the
20
conditional expectation ofNijx givenJijxand
21 Ji= nd X j=1 N X y=1 Jijy, (22) 22
where each random variableJijy is assumed to be independently distributed.
23
Density-dependent regulation is assumed to affect each individual independently and
24
equally. According to this assumption, the conditional expectation of the number Nijx of
