Integrative and Comparative Biology
Integrative and Comparative Biology, volume 52, number 1, pp. 197–212
doi:10.1093/icb/ics055
Society for Integrative and Comparative Biology
SYMPOSIUM
Asymmetric Dispersal Can Maintain Larval Polymorphism: A Model
Motivated by Streblospio benedicti
Christina Zakas1 and David W. Hall
Department of Genetics, University of Georgia, Athens, GA 30606, USA
From the symposium ‘‘Poecilogony as a Window on Larval Evolution: Polymorphism of Developmental Mode within
Marine Invertebrate Species’’ presented at the annual meeting of the Society for Integrative and Comparative Biology,
January 3–7, 2012 at Charleston, South Carolina.
1
E-mail: [email protected]
Synopsis Polymorphism in traits affecting dispersal occurs in a diverse variety of taxa. Typically, the maintenance of a
dispersal polymorphism is attributed to environmental heterogeneity where parental bet-hedging can be favored. There
are, however, examples of dispersal polymorphisms that occur across similar environments. For example, the estuarine
polychaete Streblospio benedicti has a highly heritable offspring dimorphism that affects larval dispersal potential. We use
analytical models of dispersal to determine the conditions necessary for a stable dispersal polymorphism to exist. We
show that in asexual haploids, sexual haploids, and in sexual diploids in the absence of overdominance, asymmetric
dispersal is required in order to maintain a dispersal polymorphism when patches do not vary in intrinsic quality. Our
study adds an additional factor, dispersal asymmetry, to the short list of mechanisms that can maintain polymorphism in
nature. The region of the parameter space in which polymorphism is possible is limited, suggesting why dispersal
polymorphisms within species are rare.
Introduction
Dispersal ability is an intriguing organismal trait because it can have profound effects on a species’ ecology and evolution. Dispersal influences population
dynamics by mitigating differences across local populations and allowing persistence through recolonization following extinction. Selection on life history
characters, especially those that influence reproductive trait optima, are also directly influenced by the
dispersal ability of a species. For sessile species such
as plants and many marine invertebrates, dispersal
can only be achieved through propagule movement,
making these early developmental stages particularly
important for gene flow (Kinlan and Gaines 2003;
Levin et al. 2003). Here, we address conditions for
which genetic differences in propagule dispersal ability across patches can be maintained within a
population.
Life history traits such as propagule size and duration of the propagule stage will influence the extent
to which dispersal among local populations can be
achieved. Seeds that are small or pelagic larvae that
delay metamorphosis are more likely to disperse further. Such traits are often heritable, and seem to be
controlled by multiple loci, making them classic
quantitative traits (Venable and Burquez 1989;
Levin et al. 1991). This observation implies that dispersal rates can evolve in response to such selective
pressures as intraspecific competition, inbreeding
depression,
and
environmental
stochasticity
(Johnson and Gaines 1990; Byers and Pringle 2006,
reviewed in Ronce 2007). For example, temporal variability in environmental quality may lead to selection for high rates of propagule dispersal and thereby
escape from poor conditions and colonization of new
habitats (van Valen 1971; Balkau and Feldman 1973;
Levin et al. 1984) whereas spatially heterogeneous
environments could select for a reduction in dispersal to maintain locally adapted traits (Altenberg
and Feldman 1987).
Polymorphism in dispersal ability has been documented widely among plants (Morse and Schmitt
1985; Olivieri and Berger 1985; Venable 1985;
Venable and Levin 1985, and others; Imbert
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198
et al. 1996) and occasionally in marine invertebrates
(Blake and Kudenov 1981; Levin 1984a; Qian and
Chia 1992; Krug and Zimmer 2004; Krug 2009), insects (Zera and Denno 1997; Langellotto and Denno
2001), and amphibians (Semlitsch et al. 1990). In
many of these cases, the proportion of widely dispersing offspring is variable within and between
clutches and depends on maternal resource availability. This variation may thus represent a bet-hedging
strategy and possibly evolved in response to landscapes where temporal and spatial fluctuations
occur across local populations, and where females
cannot perfectly predict the environmental conditions their offspring will experience (Cohen and
Levin 1991; Roff 1994; Olivieri et al. 1995; Mathias
et al. 2001; Crean and Marshall 2009). However, for
species where offspring type is highly heritable, such
that a female’s clutch shows no plasticity with respect
to dispersal ability, environmental differences in resource availability are unlikely to be a prominent
factor in maintaining a dispersal polymorphism.
For these taxa, there may be other environmental
factors that are more important.
Understanding the processes responsible for maintaining polymorphism in genes that have direct
effects on fitness is a critical question in evolutionary
biology (Stearns 1992). A balance between local adaptation and dispersal can maintain genetic polymorphism across heterogeneous habitats (Levene 1953;
Bulmer 1972; Smith and Hoekstra 1980; Holt 1996;
Hedrick 1998; Spichtig and Kawecki 2004). Recent
work has shown that in heterogeneous habitats limited dispersal can lead to maladaptation of specialist
types allowing generalists to also be maintained, and
resulting in more than n types being maintained in
n different environments at equilibrium (Débarre
and Lenormand 2011). Some theoretical work has
specifically addressed the maintenance of a dispersal
polymorphism (McPeek and Holt 1992; Doebeli
1995; Olivieri et al. 1995; Holt and McPeek 1996;
Doebeli and Ruxton 1997; Mathias et al. 2001;
Parvinen 2002; Bonte et al. 2010; Fronhofer et al.
2011; Poethke et al. 2011). The majority of these
models focus on the role of environmental heterogeneity, population structure, and source-sink
dynamics in maintaining dimorphic strategies for
dispersal.
While previous models have resulted in numerous
insights into the maintenance of genetic polymorphism, they generally exclude species where dispersal
polymorphisms occur despite no obvious spatial
C. Zakas and D. W. Hall
heterogeneity in habitat quality (but see Massol
et al. 2011 for a situation in which patches differ
in size). Here, we develop a two-patch model
where environmental quality does not vary between
patches. Rather, dispersal ability is determined by
two factors. The first is an intrinsic property of the
propagule: small propagules are better able to disperse because their longer development time increases the probability that they will reach distant,
but suitable habitat. This is the opposite of many
dispersal models, which assume dispersal ability increases with offspring size. The second dispersal
determinant is associated with the spatial position
of a patch. This occurs with vector-mediated dispersal, such as wind or ocean currents, when some
patches are predominantly ‘‘upstream’’ of others in
the population. In this situation, the cost of dispersal
will be greater for individuals that are at the furthest
point ‘‘downstream’’ in a population because their
offspring are less likely to disperse throughout the
full range of the habitat (Pringle and Wares 2007).
Abiotic dispersal vectors are of particularly interest
because unlike intrinsic propagule-specific dispersal
ability, these vectors cannot evolve, and therefore
dispersal due to abiotic vectors may not be adaptive.
Abiotic vectors are probably particularly important
for marine species, where long-dispersing offspring
are likely the consequence of small development
size rather than an adaptation for dispersal per se.
Some studies have specifically addressed the effects of
asymmetrical propagule dispersal due to abiotic vectors on patterns of genetic variation in the range of
sessile species (Bertness et al. 1996; Kawecki and Holt
2002; Véliz et al. 2006; Wares and Pringle 2008).
However, to our knowledge, no models have
addressed whether asymmetrical dispersal due to abiotic vectors alone can maintain a dispersal polymorphism within a population.
Recent models are beginning to address the occurrence of dispersal polymorphisms in relatively stable
environments, where patch heterogeneity is not driving selection for offspring dimorphism. Massol et al.
(2011) focus on dispersal polymorphisms that occur
due to differences in patch carrying capacity and
include a cost of dispersal. They find that when the
carrying capacity is sufficiently variable across populations, such that there are a few large patches and
several small, fitness across populations will also
be skewed depending on dispersal strategy. This
critical result, that fitness must be variable across
patches for a dispersal polymorphism to occur, is
199
Asymmetric dispersal can maintain larval polymorphism
also demonstrated in our model. A key difference
between the Massol et al. model and the one presented here is that we fix population sizes to be equal
in both patches, and therefore, the only mechanism
that is responsible for maintaining polymorphism is
the cost associated with dispersal under asymmetry.
The empirical system
Our model was motivated by the common estuarine
polychaete Streblospio benedicti. Streblospio benedicti
is a particularly interesting species for exploring the
evolution of dispersal polymorphism because there
are two distinct and highly heritable larval types
that can occur simultaneously in populations in
estuaries along the US East Coast (Levin 1984a;
Levin and Huggett 1990; Levin et al. 1991; Schulze
et al. 2000). In marine systems, this type of developmental dimorphism is termed poecilogony
(Giard 1905) and is relatively rare. Examination of
mtDNA haplotypes demonstrates that the two forms
of S. benedicti are likely a single species (Schulze
et al. 2000).
Both larval types, termed Small and Large, of
S. benedicti co-occur in the native distribution of
populations along the Eastern US. There are no morphological or ecological differences between adults
regardless of the larval type from which they developed. However, an individual female will either only
produce large eggs (40 eggs of 100–200 mm
diameter), or small eggs (150 eggs of 60–95 mm
diameter), throughout her life. Both Large and
Small-producing females will have, on average, the
same number of broods (6) in their lifetime
(Levin and Bridges 1994). Females release pelagic
larvae into the water column where they develop
until they become competent to metamorphose and
return to the sediment. Large larvae are capable of
metamorphosing almost immediately and take no
longer than 8 days. Small larvae, on the other
hand, take approximately 3 weeks before they can
metamorphose into benthic juveniles.
There are potentially large differences in dispersal
capability for the Large and Small larvae of S. benedicti. Offspring that take longer to develop in the
water column will be subject to alongshore currents
and advection away from their source populations
for longer than those offspring that develops quickly
(Kinlan and Gaines 2003; Shanks et al. 2003). For
simplification, larvae in our model are treated as
passive particles that have a probability of dispersal
that is only proportional to development time.
Although there may be potential benefits to individuals that maintain a poecilogonous life history,
the rarity of this developmental strategy suggests it
may be an intermediate, unstable evolutionary strategy. Poecilogony may indicate incipient speciation
or an evolutionary transition between stable development modes. Despite extensive previous research that
has investigated poecilogony, and S. benedicti in particular, the factors that allow for the maintenance of
this polymorphism remain unclear. Here, we demonstrate that evolutionarily stable dispersal polymorphisms do not require differences in environmental
quality between patches. Instead, environmentally
driven asymmetric dispersal between patches is sufficient to maintain poecilogony.
Model and numerical methods
To address whether poecilogony in S. benedicti is an
evolutionarily stable polymorphism, we examine the
trade-off between larval retention and dispersal ability by analyzing haploid clonal, haploid sexual, and
diploid sexual two-patch population genetic models
that incorporate asymmetric dispersal (Fig. 1). The
results we obtained from these three models were
qualitatively similar and for simplicity, we focus on
Fig. 1 Schematic of dispersal for the two-patch haploid dispersal
model. mGii is the probability that a larva produced by a female of
genotype G recruits to its natal patch. vGij is the survival probability of a larva produced by a female of genotype G during dispersal from patch i to j. NGi is the proportion of adult individuals
of genotype G in patch i. Width of arrows indicate relative
recruitment rate of Small versus Large females’ offspring through
that dispersal path. Diploid model is essentially identical except
there are three genotypes (LL, LS, and SS), each with its own
dispersal and survival parameters.
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the haploid clonal model. Analytical results for the
haploid and diploid models can be found in the
Supplementary Material online. Here, we present
the construction of the haploid clonal model as
well as the analytical analyses of the conditions for
polymorphism. We then test the predictions of our
analytical model by assessing the likelihood that
randomly drawn parameter sets result in stable polymorphism. Finally, we address the long-term stability
of a polymorphism maintained in this model by assessing the ability of a third offspring’s strategy to
invade.
Lifecycle
We consider a situation in which a population exists
in two patches or subpopulations, 1 and 2. The life
cycle begins with a census of adults. We assume
that genotype frequencies in males and females are
equal, that there are sufficient numbers of males in a
population to fertilize all eggs, and that mating is
random. Females produce eggs that are internally
fertilized, and the resulting offspring are matured
within the mother until released into the water
column. Once released, some offspring return to
their natal patch, others migrate to the other patch
and the remainder is lost. Dispersal between patches
is allowed to be asymmetric and patch 1 is assumed
to be upstream of patch 2 (Fig. 1). Adults experience
mortality following offspring release. After settlement, offspring immediately become adults and the
cycle repeats. The number of breeding individuals is
constant and equal in both patches and independent
of the number of immigrants and emigrants.
C. Zakas and D. W. Hall
females producing offspring who tend not to disperse
from their natal site. The fecundity of a female of
genotype G is denoted by bG, where G is L or S in
the haploid models and LL, LS, or SS in the diploid
models. The genotypes and phenotypes of females in
a population are shown in Table 1.
Mortality of adults
Following reproduction, all adults experience mortality with the same probability d, regardless of their
age or genotype. If d ¼ 1, generations do not overlap.
The value of the parameter d does not affect the
conclusions of the model regarding the maintenance
of polymorphism.
Dispersal and settlement
The time spent in the water column determines the
probabilities of returning to settle on the natal patch.
Time in the water column is determined by offspring
size at release because ‘‘size’’ at metamorphosis
(settling) is assumed to be relatively constant, such
that settlement occurs once this threshold size is attained. Large larvae spend less time in the water
column than Small larvae, and are thus more likely
to be close to the natal patch when it is time to
settle. The probability that a larva from a female of
genotype G in patch i returns to its native patch is
mGii . If a larva is not close to its natal patch when it
reaches the threshold settlement size, which occurs
with probability ð1 mGii Þ, then it will only survive
if it is able to settle in the other patch. This requires
that it has reached the other patch, which occurs
with probability vGij . Thus, the probability that a
Reproduction
Within the population, different females produce offspring of different sizes. Offspring size is determined
by the genotype of the mother. A female’s phenotype, with respect to offspring size and number, is
controlled by her genotype at a single locus segregating two alleles, S and L. Since there are at most two
alleles segregating in the population; in the haploid
model, females produce either all Large or all
Small offspring. (In the diploid model, homozygous
females produce either Large or Small offspring and
heterozygous females produce either intermediatesized offspring or a mixed brood consisting of both
Large and Small offspring.) Offspring size affects the
total number of offspring that a female can produce
(fewer larger offspring). Thus, females producing
larvae that disperse have higher fecundity than
Table 1 Number of offspring produced by females of different
genotypes and associated dispersal parameters
Dispersal
rates to
nonnatal
patch
Offspring
number
Offspring
size
Dispersal
rates to
natal patch
L
bL
Large
mLii
(1 – mLii) vLij
S
bS
Small
mSii
(1 – mSii) vSij
LL
bLL
Large
mLLii
(1 – mLLii) vLLij
LS
bLS
Intermediate/
mixed
mLSii
(1 – mLSii) vLSij
SS
bSS
Small
mSSii
(1 – mSSii) vSSij
Genotype
Haploid
Diploid
(i, j ¼ 1 or 2)
Asymmetric dispersal can maintain larval polymorphism
larva migrates from its natal to the other patch is
ð1 mGii ÞvGij . Since Large offspring spend less time
in the water column and are thus closer to their natal
habitat when they reach threshold size for settlement,
we assume that mLii > mSii and mLLii > mSSii . This
must also be the case because Small larvae take
longer to achieve settlement size, and therefore,
they will experience greater mortality during the dispersal period even if they eventually return to their
natal patch. Therefore, the fitness of Small returning
larvae is less than that of Large returning larvae to
either patch. Similarly, because Large larvae are less
likely to reach the other patch before reaching settlement size, we assume that vLij < vSij and vLLij < vSSij .
The effect of asymmetric dispersal is modeled by assuming that vGij 6¼ vGji .
Haploid, clonal model
In this section, we present the analytical results for
the haploid, clonal model, and a set of simulations
that we performed to examine the likelihood of obtaining different outcomes. The analysis of the haploid clonal version of the model illustrates the
general framework in this simplest case. Let the proportion of Small individuals in patch i at census time
t be NSi(t), and the proportion of Large be NLi(t) ¼ 1
NSi(t). With the assumptions given above, recursion equations for the proportion of Small and Large
individuals in patch i are:
201
i from offspring produced in patch j from S or L
females. While Mathematica (Wolfram 2009;
http://www.wolfram.com/mathematica/) can be used
to determine the equilibria of the system in the general case, these solutions are too complicated to be
useful. However, our interest is primarily in elucidating the conditions under which a polymorphism for
the larval types is maintained (i.e., an internal stable
equilibrium exists). To determine those conditions,
we have thus focused on determining the conditions
for protected polymorphism (Prout 1968). Protected
polymorphism requires that the two fixation equilibria are simultaneously unstable. A stability analysis at
each of the two fixation equilibria gives the conditions for protected polymorphism (Supplementary
Appendix A). These conditions indicate that for
both equilibria to be simultaneously unstable, a necessary condition is for one of the following two conditions to be met and the other violated:
ðbS mSii þ bS ð1mSjj ÞvSji ÞðbL mLii þbL ð1mLjj ÞvLji Þ > 0;
ð2Þ
ðbS mSjj þbS ð1mSjj ÞvSij ÞðbL mLjj þ bL ð1mLii ÞvLij Þ > 0:
ð3Þ
When Condition (3) holds and (2) does not, then
the parameters of recruitment for Large in patch 1
(upstream) are greater than Small, while Small
recruits better than Large in patch 2 (downstream).
Ti NSi ðt þ 1Þ ¼ ð1 dÞNSi ðtÞ þ mSii bS NSi ðtÞ
ð1aÞ In the alternative situation, Large recruits better in
þ ð1 mSjj ÞvSji bS NSj ðtÞ,
patch 2 and Small does better in patch 1. These
necessary requirements immediately clarify why
asymmetric dispersal is required for protected polyTi NLi ðt þ 1Þ ¼ ð1 dÞNLi ðtÞ þ mLii bL NLi ðtÞ
morphism. Without asymmetric dispersal, both
ð1bÞ
patches are equivalent, so that if one type does
þ ð1 mLjj ÞvLji bL NLj ðtÞ,
better than the other in one patch, it will also do
where Ti is the sum of the right hand sides of the better in the other patch.
equations such that,
Given our assumptions about the biology of the
system,
namely that natal recruitment does not vary
Ti ¼ ð1 dÞNSi ðtÞ þ mSii bS NSi ðtÞ þ ð1 mSjj ÞvSji bS NSj ðtÞ
across
patches
(mG11 ¼ mG22) and downstream disþ ð1 dÞNLi ðtÞ þ mLii bL NLi ðtÞ þ ð1 mLjj ÞvLji bL NLj ðtÞ:
persal is greater (vG12 > vG21), it is more likely for
ð1cÞ
Condition (3) to hold, and (2) not to hold, than
The first term on the right hand side of Equations vice-versa. In this case, we expect the frequency of
(1a) and (1b) represents adults from the previous the S allele should be higher in patch 2 than in patch
generation who survive. In Equations (1a) and 1 at an internal equilibrium.
Analysis of the haploid sexual model gives identi(1b), respectively, the second term represents recruitment of S or L offspring into patch i from offspring cal conditions for protected polymorphism. The dipproduced in patch i by S or L females, and the third loid sexual model also gives essentially identical
term is an equivalent term for recruitment into patch conditions for protected polymorphism for the case
202
where the heterozygote is ‘‘consistently intermediate,’’ meaning it produces a mixture of Small and
Large offspring, the ratio of which is determined
by the dominance relationships (h) of the two alleles,
where h is the proportion of the brood that is Small
and (1 h) is the proportion that is Large with
0 h 1. In the diploid model, if the heterozygote
is not consistently intermediate, then it can have
higher recruitment to each patch than the average
of the homozygotes (overdominance), which also
gives protected polymorphism. Analyses for the haploid and diploid sexual cases are presented in the
Supplementary Material online.
Numerical simulations of the haploid clonal model
We performed simulations to gain insight into the
relative likelihoods of achieving a polymorphic life
history by iterating Equation (1) using Mathematica
(Wolfram 2009) under three scenarios. Our primary
interest was in how frequently the conditions for a
stable polymorphism were met when parameters are
randomly drawn from uniform distributions. The
three scenarios we examined reflect increasing constraints on the degree of asymmetry. In the first scenario, the degree of asymmetry was allowed to differ
for each genotype (i.e., 0 < vSji =vSij 6¼ vLji =vLij < 1).
In the second scenario, the degree of asymmetry
was the same for both genotypes (i.e.,
0 < vSji =vSij ¼ vLji =vLij < 1). In the third scenario,
dispersal was symmetric (vSji =vSij ¼ vLji =vLij ¼ 1). In
all scenarios, we required that the Large type produced fewer offspring than Small (bL < bS), had more
offspring return to its natal patch (bS mSii < bL mLii ),
and fewer offspring migrate to the other patch
[bS ð1 mSii ÞvSij > bL ð1 mLii ÞvLij ] than the Small
type, for both patches.
For each scenario, we randomly drew 5000 parameter sets under the set of constraints on asymmetry
outlined above and then iterated the equations until
the change in allele frequency in one generation was
<1011, at which point we assumed that an equilibrium had been reached. We specified ranges for each
parameter (see below) and then randomly chose
values from this range assuming every value was
equally likely (a uniform distribution). This assumption allows us to determine what proportion of the
parameter space will result in coexistence of Large
and Small. This does not necessarily tell us the biological likelihood of an outcome because it is not
necessarily the case that all regions of the parameter
space are equally likely from a biological standpoint.
C. Zakas and D. W. Hall
For instance, if there are genetic trade-offs among
parameters, some regions of parameter space may
be unattainable and so assuming a uniform distribution biases the perceived likelihood of polymorphism. This potential problem, in which randomly
drawn parameters may be subject to different constraints, may result in a biased view of the likelihood
of obtaining a particular outcome (polymorphism in
our model) and is an example of Bertrand’s paradox
(e.g., Calcagno et al. 2006). We return to this issue
later in the methods and in the discussion.
Genotypic fecundities were randomly drawn from
within a range of 40–150, based on estimates from
S. benedicti, though using other ranges only had a
minor impact on the quantitative results (data not
shown). For the results presented here, we assumed
that the maximum viability for an offspring migrating to the other patch was 20%, which implies that
the largest viability parameter, vS12 is 0.2.
Broadening the possible range for this viability
parameter increased the likelihood that the Small
type would win, but did not appreciably affect the
probability of polymorphism (data not shown). All
other parameters were allowed to take values within
their full range within the constraints described
above. For each parameter set, we started the simulations at three different frequencies of the L allele:
0.01, 0.5, and 0.99. We recorded the equilibrium that
was reached and, if all three starting frequencies
ended at the same equilibrium, we assumed that
the equilibrium was globally stable. We also confirmed that polymorphism only occurred under the
conditions our analysis predicted.
Results of the numerical analyses are summarized
in Table 2 for the haploid clonal model. In the
absence of asymmetry, polymorphism never occurs.
With asymmetry, the frequency of polymorphism is
Table 2 Equilibria obtained in the numerical analyses for the
clonal haploid model
Dispersal
asymmetry
Large fixation
Different
4009
Constant
None
Small fixation
Polymorphism
863
128
4019
857
124
3731
1269
0
All fixations were globally stable. Random dispersal asymmetry
implies that Small and Large strategies have a different ratio of
upstream to downstream dispersal. Constant asymmetry implies the
ratio is the same for both Small and Large. No asymmetry implies the
ratio is 1 for both Small and Large.
Asymmetric dispersal can maintain larval polymorphism
2.5%, regardless of whether both genotypes have
the same level of asymmetry. Identical patterns
were seen for the haploid sexual model (data not
shown). For diploids, the probability of polymorphism was much higher, around 26%, when the heterozygote was unconstrained (i.e., not necessarily a
consistent intermediate of the homozygotes), because
of overdominance (which also allowed polymorphism to occur in the absence of asymmetric dispersal). Constraining the heterozygote to be
consistently intermediate and produce a mixture of
Large and Small offspring, determined by the dominance of the alleles (h) such that heterozygote advantage was not possible, resulted in polymorphism
in 3% of cases with asymmetry and never in the
absence of asymmetry, similar to the haploid clonal
model (see Supplementary Appendix C for data for
diploids).
Long-term stability of a polymorphic equilibrium
Once we determined the conditions under which a
polymorphism is possible, we asked whether such a
polymorphic equilibrium is stable to invasion by
other strategies. We examined this question for two
cases. Case 1 is when a third strategy is randomly
drawn but consistent with the current strategies
(defined below), such that a linear relationship between offspring size and total recruitment to a patch
is assumed. Case 2 is when the relationships between
the parameters of the model (bG, mG., and vG.) and
offspring size are explicitly specified. In this second
case, we initially assume linear relationships between
offspring size and each of the parameters, noting that
these relationships can make the relationship between offspring size and recruitment to a patch
nonlinear.
203
intermediate strategy, which is one that falls in
between Large and Small types, and produces offspring that are equally intermediate with respect to
recruitment to the natal and to the nonnatal sites.
The second consistent strategy we term ‘‘consistently
extreme’’ and is a strategy that produces either larger
offspring than the resident Large type, or smaller
offspring than the resident Small type (Fig. 2). To
model a consistently extreme strategy, we can utilize
the same method as before, substituting a value of c
that is either negative or >1. Here, c is constrained to
values that maintain nonnegative recruitment values
of that strategy to each patch [i.e., c cannot be
large enough to generate negative natal
(¼ cbS mSii þ ð1 cÞbL mLii ) or nonnatal recruitment
(¼ cbS ð1 mSii ÞvSji þ ð1 cÞbL ð1 mLii ÞvLji ) of the
new strategy because this in biologically impossible].
We examined the invasion of a consistent strategy
in two situations. First, we analytically evaluated stability conditions for the special situation of no dispersal of Large offspring to an alternate patch,
vL. ¼ 0, to gain insight into the more general case.
Second, we numerically examined invasion of a
consistent strategy for more general cases. An overview of the methods we used are presented in
Supplementary Appendix B.
Case 1: invasion by a consistent strategy
Here, we introduced a new, consistent strategy into a
population that is already polymorphic for a Large
and Small strategy. We use the term ‘‘consistent’’ to
mean the third strategy will produce a mixture of
Small and Large offspring, such that a proportion c
of the brood is Small and a proportion (1 c) is
Large with 0 c 1. Graphically, this is equivalent
to drawing lines between the total recruitment for
Large and Small for each patch and assuming that
the third type represents a particular point on the
x-axis with recruitment values given by the two lines
(Fig. 2). We define two types of strategies that represent consistent strategies. The first is a consistently
Fig. 2 The relationship between the total recruitment to a patch
[¼ bG ðmGii þ ð1 mGjj ÞvGji Þ] for the two resident strategies,
Large and Small (black dots), at a polymorphic equilibrium and a
newly introduced strategy (gray dots) that is either consistently
intermediate (C.I.) or consistently extreme (C.E.) with respect to
the resident strategies. Total recruitment to both patch 1 and
patch 2 are shown. Negative slope for patch 2 and positive slope
for patch 1 of dotted lines connecting strategies in each patch is
guaranteed by the necessary conditions for polymorphisms
[Equation (3)].
204
With no Large dispersal to an alternate patch,
we were able to demonstrate analytically that a polymorphic equilibrium cannot be invaded by any
individuals with a consistently intermediate strategy,
but will be invaded by individuals with a consistently
extreme strategy (Supplementary Appendix B). In the
numerical simulations, we introduced a third consistent strategy into a population that contained a
polymorphic resident strategy. The polymorphic
populations we used were those obtained from
the clonal haploid simulations presented above
(Table 2). We found that for 250 of the 252 polymorphic equilibria, successful invasion by individuals
with a consistently intermediate strategy did not
occur, instead individuals with a third strategy were
lost and invasion by individuals with consistently
extreme strategies were successful, with the third
strategy displacing the resident strategy most similar
to it (Supplementary Table B1). In two cases, the
opposite results were found which was unexpected.
The parameter values for these two cases are not
obviously different than for the other cases and
therefore the reason for these results is unclear. It
is notable, however, that we never observed a case
where a polymorphic equilibrium was displaced by a
monomorphic one, which implies that polymorphism is expected to be long-lived for biological systems that meet the assumptions of our model.
Case 2: specified offspring size relationships
In the models considered above, we examined the
evolution of different strategies without specifying
the underlying relationships between offspring size
and parameters of the model. That is, we assumed
that strategies could have any parameter values
within the constraints imposed by the biology of
the system. In this section, our goal is to expand
our analysis to predict long-term evolution of
offspring size by including explicit relationships
between the parameters (bG, mG., and vG.) and offspring size. Specifying these relationships also allows
us to gain insight into whether our previous assumption of uniform distributions from which to draw
model parameters (bG, mG., and vG.) has biased
our understanding of the likelihood of polymorphism (Bertrand’s paradox). Now that we specify
these relationships, we can determine whether the
likelihood of polymorphism changes. If it does not
change very much then any bias caused by assuming
a uniform distribution of parameter values might
well be small.
C. Zakas and D. W. Hall
To specify the relationships between parameters
and offspring size, we begin by reconsidering the
assumptions of our system. We first consider offspring size constraints and assume that there is an
upper threshold size necessary for metamorphosis to
the juvenile stage to occur. Producing offspring
above this size has no benefit in further reducing
dispersal given the assumptions of our models, and
is thus a maximum offspring size. Second, there is a
minimum offspring size below which an offspring is
nonviable. These two assumptions constrain the
range of possible offspring sizes. Our third assumption is that there is a trade-off between offspring size
(with respect to volume) and the number of offspring that can be produced by a female. Fourth,
the time spent in the water column before metamorphosis affects the likelihood of recruiting to the natal
site (longer time lowers this probability) and of
reaching the other settlement site (longer time
increases this probability). Fifth, external flow
dynamics make some populations easier to reach
via dispersal than others (downstream sites are
easier to reach than upstream sites).
In the simplest case, we assumed that the relationships between offspring size and offspring number,
natal recruitment, and viability during dispersal were
all linear, with negative slope for offspring number
and viability during dispersal and positive slope for
recruitment rate. For any particular combination of
minimum and maximum parameter values, we can
calculate recruitment to each patch as a function of
offspring size. The total recruitment curves can then
be calculated and used to predict the evolution of
offspring size. The two total recruitment curves obtained can exhibit one of three relationships (Fig. 3):
(1) maximum recruitment for each patch occurs at
the minimum offspring size, (2) maximum recruitment for each patch occurs at the maximum offspring size, and (3) maximum recruitment occurs
at a different size in each patch, with the maximum
in patch 2 occurring at a smaller offspring size than
in patch 1 (again, due to downstream dispersal).
Based on our analytical and numerical results
from the general clonal haploid model, we predict
that offspring size will evolve to an extreme when
the maximum recruitment in each patch occurs
at the same extreme offspring size (situations 1 and
2). In situation 3, we predict that the population
will evolve to an offspring size that lies between
the maxima, and may be polymorphic (Fig. 3).
If there is polymorphism, we expect the offspring
Asymmetric dispersal can maintain larval polymorphism
sizes may be as divergent as the difference between
the maxima.
Numerical simulations of offspring size when the
relationships are specified
To test our predictions, we performed numerical
simulations. Our intention was to determine the
equilibrium state for a set of randomly drawn strategies, and then iteratively introduce additional strategies until a final, evolutionarily stable strategy was
met. We first assigned the linear relationships
between offspring size and each parameter by choosing minimum and maximum parameter values.
Fig. 3 Recruitment to each patches as a function of offspring size,
and expected evolution (gray arrows/bar). (A) Maximal recruitment in both patches occurs at the minimal offspring size (arbitrarily set to 1). (B) Maximal recruitment in each patch occurs at
different offspring sizes. Between the two diamonds, a polymorphic equilibrium is possible but will depend on the specific slopes
of the two curves. (C) Maximal recruitment in both patches
occurs at the maximum offspring size (arbitrarily set to 5).
205
For offspring number, we assumed a minimum of
40 and maximum of 150. For settlement rate,
we assumed that the minimum and maximum were
randomly drawn from a (0, 1) uniform distribution.
For viability, we assumed that the minimum and
maximum were randomly drawn from within
(0, 0.2) as explained above. For asymmetry, we
assumed that Large and Small genotypes had the
same value (i.e., constant asymmetry), which was
randomly drawn from a (0, 1) uniform distribution.
These parameter ranges are similar to our previous
simulations. Once the linear relationships were
assigned for each parameter, we calculated total
recruitment curves and determined the maximum
values of offspring size. We then randomly chose
three strategies, each corresponding to a particular
offspring size, and used the linear relationships to
calculate their particular parameter values. We iterated the recursions [Equation (1)] until the allele
frequency change was less than 1011, and recorded
the winning strategy(ies). For the initial set of strategies, we included one offspring size strategy at the
maximum total recruitment value for either patch i
or j [when they were different (situation 3) we chose
both the maximum values for patch i and j as two of
the initial strategies]. We ran a total of 10,000 simulated relationship sets.
Because we are interested in the long-term stability
of a polymorphic equilibrium in particular, once we
found the winning strategy then we introduced one
(for a polymorphic equilibrium) or two (for a
monomorphic equilibrium) randomly drawn new
strategies. We calculated parameter values and
again iterated to equilibrium. We repeated this process 50 times for each set of parameter relationships
and recorded the final equilibrium obtained. To prevent large increases in time to equilibrium and substantial slowdown in the simulations, we assumed
that randomly drawn strategies could not be too
similar to strategies currently in the population.
A newly drawn strategy that was within 2% of a
current strategy was redrawn. Of the 10,000 simulated relationship sets, we found a polymorphism in
499 of them. The percentage of randomly drawn
parameter sets giving polymorphism (5%) when constraints among parameters are explicit is reassuringly
similar to our previous simulations where parameters
were chosen from uniform distributions (2.5%). This
suggests that our conclusion that polymorphism is
constrained to a small region of the parameter
space is perhaps not too biased by our ignorance
206
C. Zakas and D. W. Hall
of the actual biological distribution of possible parameter values, as discussed above (Bertrand’s
paradox).
The simulation results (Table 3) indicate that our
predictions were robust: polymorphism never
occurred when the maximum recruitment values in
both patches did not differ. Interestingly, when a
polymorphic equilibrium occurred, the two strategies
differed by <10% in offspring size, even though the
size of offspring at the recruitment maxima in the
two patches differed up to two-fold (data not
shown). Since offspring size in nature can differ
five-fold (based on S. benedicti egg-size measurements), this suggests that the underlying linear relationships we assumed are not an accurate reflection
of what occurs in nature. Changing this underlying
assumption to more realistic relationships might increase the applicability of this model to biological
systems. For example, we changed the relationship
of viability during dispersal versus offspring size
from linear to cubic and re-ran simulations with
all else remaining the same. In these 5000 simulations, we found that at polymorphic equilibria, the
two strategies could differ by up to five-fold (data
not shown).
We also performed an additional set of simulations to determine whether our primary finding,
that asymmetric dispersal was sufficient to maintain
polymorphism, was peculiar to the two-patch system.
We simulated a three-patch system for the haploid
clonal case. Simulations were set up in the same
manner as for the two-patch system with identical
asymmetries, such that asymmetries for Small and
Large genotypes were the same between any two
patches (e.g., vS13/vS31 ¼ vL13/vL31). We simulated
5000 data sets with specified offspring size relationships and found that 4.4% resulted in stable polymorphism, which is similar to the 5% found in the
equivalent two-patch simulations (Table 3). We have
also examined a three-patch stepping-stone model.
We allowed exchange of migrants only between
neighboring patches and assumed that the migration
parameters between neighboring patches were the
same (i.e., between patches 1 and 2 and between
patches 2 and 3). In 6000 simulations, using parameters chosen at random from the same ranges as
previous simulations, we found polymorphism in
413 of the 6000 simulations (7%). When present,
polymorphism was usually present in all three
patches. In seven cases, polymorphisms were
restricted to only two patches and in five cases to
only one patch. In all cases where polymorphism was
in one or two patches only, downstream patches
were fixed for the ‘‘smaller’’ of the two strategies
(lower mii).
Degree of asymmetry
From the above analyses, it is clear the degree of
dispersal asymmetry (vG21 =vG12 ), affects the occurrence of polymorphism in our model. It is important
to note that asymmetry is only due to environmental
differences for each patch (abiotic vectors). This is
because vG12 and vG21 are based, in part, on the
intrinsic dispersal potentials of the larvae, which
are identical in each patch, and in part on water
currents, which are asymmetric. To determine the
general range of asymmetric dispersal necessary to
maintain a stable polymorphism, we calculated
vG21 =vG12 for our simulated data sets and determined
the number of cases where a pure Large, pure Small,
or polymorphic equilibrium was stable (Fig. 4).
We used the scenario where both phenotypes are
subject to the same asymmetric dispersal
(vS21 =vS12 ¼ vL21 =vL12 ). From these simulations, it is
clear that polymorphism becomes more frequent
when the degree of asymmetric dispersal is large
(dispersal is predominantly one-directional). This
suggests that a small deviation from purely
Table 3 Simulation results when the linear relationships between parameter values and offspring size are explicit
Recruitment to
patch 1 and patch 2
Expected outcome
Monomorphism at an
extreme offspring size
Monomorphism at
intermediate offspring size
Same maxima
Monomorphism
7461
0
0
Different maxima
Polymorphism possible
7
2033
499
Polymorphism
Monomorphism at extreme offspring size indicates that the winning strategy was either at the minimum or the maximum possible offspring size.
For all simulations minimum offspring number ¼ 40, maximum offspring number ¼ 150, natal dispersal minimum and maximum fall in (0, 1),
minimum and maximum nonnatal dispersal viability fall in (0, 0.2), and asymmetry (¼ v.ij/v.ji) is the same for Large and Small genotypes and falls in
(0, 1). For each of the 10,000 simulated relationships, three strategies were introduced, allowed to reach equilibrium, and lost strategies were
replaced with randomly sampled new strategies.
Asymmetric dispersal can maintain larval polymorphism
symmetric dispersal will generally not be sufficient
to maintain a polymorphism. Fixations of the
Large strategy are also increased with increasing
asymmetry.
Discussion
We have presented a simple haploid two-patch
genetic model to examine the maintenance of a dispersal polymorphism, with particular reference to
S. benedicti. We have shown that polymorphism
can be maintained when patches are identical if
there is asymmetric dispersal between patches. This
result holds for clonal haploid, sexual haploid, and
sexual diploid (with no overdominance) cases, none
of which would otherwise give polymorphism in the
absence of intrinsic differences in patch quality. To
the best of our knowledge, this result has not been
demonstrated before. In our numerical simulations,
we show that the likelihood of randomly choosing
parameters resulting in polymorphism is quite restricted (3% of the scenarios in the general haploid
clonal case and 5% of the scenarios where parameter relationships are explicitly defined), suggesting
why such dispersal polymorphisms are rare in
nature.
Our model follows a Ravigné-type selection regime,
where during the life-cycle genotype-independent
207
regulation of population size occurs before genotypedependent selection within a site (Ravigné et al. 2004).
The behavior of this model depends on whether habitat
selection is occurring among propagules. In our model,
the biological constraint of asymmetric ocean current
dispersal imposes such habitat selection. Under a
Ravigné-type model, the greater the habitat selection,
the more likely the model will promote polymorphism
in traits related to local adaptation. Thus, with greater
asymmetry imposed on the model (greater habitat selection), we would expect a greater likelihood for a
stable polymorphism in dispersal ability, which is generally what we observe (Fig. 4). Dispersal ability is a
trait that is dependent on the patch of origin and thus
can be considered a local adaptation in this respect.
Understanding the maintenance of genetic variation has been a long-standing question in population
genetics. Polymorphism is generally assumed to be
transient (Gemmell and Slate 2006; Hedrick 2006),
and explanations of persistent polymorphism often
invoke either some form of balancing selection,
such as overdominance or negative frequencydependent selection, or a balance between directional
selection and mutation, such that variation is
constantly being eroded by selection and generated
by mutation. Neither of these explanations seems
applicable for the dispersal polymorphism in
Fig. 4 Total number of stable pure Small, pure Large or polymorphic equilibria that occur in the simulated data under different levels of
upstream migration (a ¼ degree of asymmetry, vGji/vGij) where large values indicate less asymmetry (asymmetry ¼ 1 is symmetric
dispersal and asymmetry ¼ 0 is one-directional dispersal). The distribution is for the clonal model with the same asymmetry exhibited
by both Small and Large genotypes (vS21 =vS12 ¼ vL21 =vL12 ).
208
S. benedicti. In this species, both the Small and Large
females are common, which argues against mutation–selection balance, and the variation across populations is inconsistent with frequency-dependence.
Adults are morphologically indistinguishable, and
mate freely in lab crosses (Levin et al. 1991;
Schulze et al. 2000), which suggests that there is
not larval type competition at this stage.
In diploid S. benedicti, the dimorphic distribution
of dispersal types suggests that overdominance is unlikely. Experimental data does not indicate that there
are phenotypically intermediate or mixed strategies,
so there is no evidence for heterozygote advantage.
In addition, a heterozygote exhibiting the optimal
strategy could be invaded by a homozygote with
the optimal strategy. One situation in which overdominance might be occurring is if the heterozygote
can show either the Large or Small phenotype depending on the patch in which it resides rather than
exhibiting an intermediate form. However, the
genetic data do not indicate that this is occurring:
pure crosses of Large or Small types do not give
offspring of the other type (Levin et al. 1991).
With high levels of dispersal, which seems likely in
S. benedicti and which is supported by genetic data
(Schulze et al. 2000), polymorphism requires overdominance for harmonic fitness if dispersal is symmetric (Prout 1968). That is, the harmonic fitness
across patches of the heterozygote must be greater
than the harmonic fitness of both of the homozygotes. Our diploid model demonstrates that even in
the absence of mutation–selection balance, negativefrequency-dependent selection, and harmonic overdominance, a stable polymorphism is still possible
if dispersal is asymmetric. Therefore, asymmetric dispersal between patches is an additional mechanism
for the maintenance of stable polymorphisms, and
may be the underlying explanation for the maintenance of dispersal polymorphism in S. benedicti.
Long-term stability of a polymorphic equilibrium
We were able to demonstrate the long-term stability
of a dispersal polymorphism that occurs due to
asymmetric dispersal. From our numerical results,
we predict that, if there is a linear relationship between total recruitment to a patch and egg size, then
egg sizes would evolve to the minimum and maximum of its range in those cases where the slopes of
the relationships allow polymorphism (i.e., satisfy
the conditions for polymorphism given in the
Appendix). Our prediction is definitely true with
C. Zakas and D. W. Hall
no Large dispersal between patches, as we showed
analytically. When Large dispersal does occur, there
is an unexpected and rare situation where a strategy
intermediate to the two polymorphic strategies can
invade (Supplementary Table B1). However, none of
the simulated scenarios resulted in a monomorphic
equilibrium replacing a polymorphic equilibrium,
strongly suggesting that with a linear relationship
between total recruitment to a patch and egg size,
polymorphism is evolutionarily stable.
While it is encouraging that a stable polymorphism can persist when there is a linear relationship
between egg size and total recruitment, it is not likely
to reflect the true underlying relationship in nature.
Rather, the parameters of our model (natal recruitment, dispersal ability, and fecundity) have explicit
relationships to each other that are constrained by
trade-offs in egg size, such that an individual with
large fecundity will have small offspring, and offspring with a high probability of natal recruitment
will have a low probability of successfully dispersing,
etc. By assuming explicit relationships between these
parameters and egg size, we can make more realistic
predictions of the long-term stability of a dispersal
polymorphism. To illustrate this, we again chose the
simplest assumption, where the relationships between
the parameters and egg size are linear. As expected
from our analytic work, a dispersal polymorphism is
only possible when fitness is maximized at different
egg sizes for the two patches (Fig. 3). Again, such a
scenario is only possible due to asymmetric dispersal
between patches. Even when the maxima for the two
patches differ, polymorphism may not result because
this condition is necessary but not sufficient. A stable
polymorphism occurs in 5% of these simulations,
even though 20% of the simulations resulted in
different maximum egg sizes for the two patches
(Table 3). The small proportion of simulations that
result in polymorphism suggests why poecilogony is
rare in this and other marine groups.
We noticed that when we assumed an explicit and
linear relationship between egg size and the parameters of our model, the simulations that resulted in
polymorphism only occurred when the two strategies
were similar (only 10% different in egg size). This is
much smaller then the difference in egg sizes usually
reported for poecilogonous species, which have distinctively different offspring sizes as noted above for
S. benedicti. When the underlying relationship between egg size and the parameter values is adjusted,
a larger range of polymorphic egg sizes becomes
Asymmetric dispersal can maintain larval polymorphism
possible. We demonstrate this by changing the relationship between dispersal probability and egg size
from linear to cubic, and find a five-fold difference
in egg size is possible when polymorphism occurs.
The biological assumption behind a higher power
than linear function is simply that halving egg size
more than doubles development time. As these underlying relationships become more tailored to an
individual system, it becomes possible to generate
predictions that reflect the biology of the system.
Therefore, for systems where these relationships
between egg size and fecundity, dispersal ability,
and natal recruitment are well understood, the
regions of parameter space that allow for polymorphic dispersal can be more accurately predicted.
Streblospio benedicti
Our model of asymmetric dispersal predicts that a
stable polymorphism will be rare, but not absent. It
is possible that the dispersal, recruitment, and fecundity probabilities for S. benedicti are such that at least
part of the species’ range falls within the parameter
space where a persistent polymorphism is stable.
Currently, we have no accurate means to predict
the probability of survival for developing larvae in
this species. Field measurements of larval recruitment
for both larval types are necessary to improve our
parameter estimates so we can accurately predict
where S. benedicti, or other poecilogonous species,
falls within this parameter space. However, if these
parameters were obtained, it would be possible to
determine if asymmetric dispersal is responsible for
poecilogony in this, or any other system.
There are two tantalizing pieces of evidence that
suggest asymmetric dispersal may be maintaining
this dispersal polymorphism. First, because this species occurs in estuaries throughout the entire East
and West Coast of the US, there are potentially a
large number of habitats where dispersal is predominantly asymmetric. One important example of this
has been documented within Bogue Sound, North
Carolina (Levin and Huggett 1990). Two wellstudied populations have been described: Island
Quay Marsh (IQM) has predominantly Large individuals (74%) while Tar Bay Landing (TBL),
1.5 km west, has been predominantly Small individuals (64%). These populations were monitored
for over 2 years and this pattern persists year
round. In this location, the net direction of the current is predominantly from west to east (Churchill
et al. 1999; Logan et al. 2000) on average placing
209
IQM downstream from TBL, a result consistent
with our predictions. Unfortunately, no other populations of S. benedicti are so well-characterized for
further evaluation of our predictions.
In addition to the direction of asymmetry, the
degree of asymmetric dispersal is a further consideration (Fig. 4). An unusual phenomenon exists in the
global distribution of S. benedicti. This species was
introduced to the US West Coast starting in the early
20th century (as well as a number of other estuarine
locations worldwide [Carlton 1979]). Interestingly,
the larval polymorphism in this species is only
known to occur in the native range, while the introduced populations on the Pacific coast are strictly
Large individuals (Levin 1984a; Schulze et al.
2000). This reduction in dispersal ability is consistent
with the development mode of many West Coast
infaunal species (Levin 1984b). According to the predictions from our model, an alternative scenario may
be that the degree of asymmetry between suitable
habitats on the West Coast is such that a polymorphism cannot persist. For example, when the viabilities of migrants become small (vGij close to zero),
the probability of fixation of Large becomes high.
Estuarine habitats on the West Coast are notably
more dispersed and larvae released from these habitats are subject to strong offshore equatorial currents
(Bertness et al. 1996; Sotka et al. 2004). Due to these
stronger currents, it is possible that much of the
suitable West Coast habitats fall into a region of
parameter space where the necessary asymmetric dispersal cannot be achieved. However, this prediction
warrants further investigation.
The degree of asymmetry needed to maintain a
polymorphism need not be constant. It is, of
course, a gross simplification that flow between natural habitats would be constant over time. A natural
extension of our model would be to account for
stochastic environmental fluctuations in current
movement between patches. Most estuarine habitats
will experience daily fluctuations in flow due to tides,
and events such as storm surges or El Niño can dramatically alter the net water flow for extended periods of time (Engle and Richards 2001; Montagne
and Casien 2001). We have demonstrated the net
amount of asymmetry between patches that is necessary to maintain a dispersal polymorphism, but
allowing for changes in flow regime is necessary for
more accurate predictions in natural populations.
Recognizing the conditions under which this
diversity is retained may suggest why certain broad
210
geographic or temporal patterns in larval type have
persisted. There has long been recognition that
dispersal patterns, and in particular asymmetries in
dispersal, would influence patterns of genetic diversity among populations. While most of this work has
illustrated the tendency of such scenarios to reduce
genetic diversity from panmictic expectations (Wares
and Pringle 2008), we show here that there are conditions under which the asymmetry itself is responsible for maintaining or even promoting diversity.
Our model helps to illustrate the conditions where
diversity in development mode may not only be
possible, but also advantageous, and provides an
explanation for the persistence of rare dispersal
polymorphisms.
Supplementary Data
Supplementary Data are available at ICB online.
Funding
Organization of the symposium was sponsored by
the US National Science Foundation (IOS1157279), The Company of Biologists Ltd., the
American Microscopical Society, and the Society
for Integrative and Comparative Biology, including
SICB divisions DEDB, DEE, and DIZ.
References
Altenberg L, Feldman MW. 1987. Selection, generalized transmission and the evolution of modifier genes. I. The reduction principle. Genetics 117:559–72.
Balkau BJ, Feldman MW. 1973. Selection for migration modification. Genetics 74:171–74.
Bertness MD, Gaines SD, Wahle RA. 1996. Wind-driven settlement patterns in the acorn barnacle Semibalanus balanoides. Mar Ecol Prog Ser 137:103–10.
Blake J, Kudenov J. 1981. Larval development, larval nutrition
and growth for two Boccardia species (Polychaeta:
Spionidae) from Victoria, Australia. Mar Ecol Prog Ser
6:175–82.
Bonte D, Hovestadt T, Poethke H. 2010. Evolution of dispersal polymorphism and local adaptation of dispersal distance in spatially structured landscapes. Oikos 119:560–66.
Bulmer MG. 1972. Multiple niche polymorphism. Am Nat
106:254–57.
Byers J, Pringle JM. 2006. Going against the flow: retention,
range limits and invasions in advective environments. Mar
Ecol Prog Ser 313:27–41.
Calcagno V, Mouquet N, Jarne P, David P. 2006. Rejoinder to
Calcagno et al (2006): which immigration policy for optimal coexistence? Ecol Lett 9:909–11.
C. Zakas and D. W. Hall
Carlton JT. 1979. History, biogeography and ecology of the
introduced marine and estuarine invertebrates of the Pacific
Coast of North America [PhD Thesis]. [Davis (CA)]:
University of California.
Churchill JH, Hench JL, Luettich RA, Blanton JO, Werner FE.
1999. Flood tide circulation near Beaufort Inlet, North
Carolina: implications for larval recruitment. Estuaries
22:1057.
Cohen D, Levin SA. 1991. Dispersal in patchy environments:
the effects of temporal and spatial structure. Theor Popul
Biol 39:63–99.
Crean AJ, Marshall J. 2009. Coping with environmental uncertainty: dynamic bet hedging as a maternal effect. Philos
T R Soc B 364:1087–96.
Débarre F, Lenormand T. 2011. Distance-limited dispersal promotes coexistence at habitat boundaries: reconsidering the competitive exclusion principle. Ecol Lett
14:260–6.
Doebeli M. 1995. Dispersal and dynamics. Theor Popul Biol
47:82–106.
Doebeli M, Ruxton GD. 1997. Evolution of dispersal rates in
metapopulation models: branching and cyclic dynamics in
phenotype space. Evolution 51:1730–41.
Engle JM, Richards DV. 2001. New and unusual marine invertebrates discovered at the California Channel Islands
during the 1997-1998 El Niño. Bull South Calif Acad Sci
100:186–98.
Fronhofer E, Kubisch A, Hovestadt T, Poethke HJ. 2011.
Assortative mating counteracts the evolution of dispersal
polymorphisms. Evolution 65:2461–69.
Gemmell NJ, Slate J. 2006. Heterozygote advantage for fecundity. PloS One 1:e125.
Giard A. 1905. La poecilogony. CR. 6th Congress of
International Zoology, Berne, Switzerland, p. 617–46.
Hedrick PW. 1998. Maintenance of genetic polymorphism:
spatial selection and self fertilization. Am Nat 152:145–50.
Hedrick PW. 2006. Genetic polymorphism in heterogeneous
environments: the age of genomics. Ann Rev Ecol Syst
37:67–93.
Holt RD. 1996. Adaptive evolution in source-sink environments: effects of density-dependence on niche evolution.
Oikos 75:182–92.
Holt RD, McPeek MA. 1996. Chaotic population dynamics
favors the evolution of dispersal. Am Nat 148:709–18.
Imbert E, Escarré J, Lepart J. 1996. Achene dimorphism and
among-population variation in Crepis sancta (Asteracae).
Int J Plant Sci 157:309–15.
Johnson ML, Gaines MS. 1990. Evolution of dispersal: theoretical models and empirical tests using birds and mammals. Ann Rev Ecol System 21:449–480.
Kawecki TJ, Holt RD. 2002. Evolutionary consequences of
asymmetric dispersal rates. Am Nat 160:333–47.
Kinlan BP, Gaines SD. 2003. Propagule dispersal in marine
and terrestrial environments: a community perspective.
Ecology 84:2007–20.
Krug PJ. 2009. Not my "type": larval dispersal dimorphisms
and bet-hedging in opisthobranch life histories. Biol Bull
216:355–72.
Asymmetric dispersal can maintain larval polymorphism
Krug PJ, Zimmer RK. 2004. Developmental dimorphism: consequences for larval behavior and dispersal potential in a
marine gastropod. Biol Bull 207:233–46.
Langellotto GA, Denno RF. 2001. Benefits of dispersal in
patchy environments: mate location by males of a
wing-dimorphic insect. Ecology 82:1870–78.
Levene H. 1953. Genetic equilibrium when more than one
ecological niche is available. Am Nat 87:331–33.
Levin LA. 1984a. Multiple patterns of development in
Streblospio benedicti Webster (Spionidae) from three
coasts of North America. Biol Bull 166:494–508.
Levin LA. 1984b. Life history and dispersal patterns in a dense
infaunal polychaete assemblage: community structure and
response to disturbance. Ecology 65:1185–200.
Levin LA, Bridges S. 1994. Control and consequences of
alternative developmental modesin a poecilogonous polychaete. Am Zool 34:323–32.
Levin LA, Creed EL. 1986. Effect of temperature and food
availability on reproductive responses of Streblospio benedicti (Polychaeta, Spionidae) with planktotrophic or
lecithotrophic development. Mar Biol 92:103–13.
Levin LA, Huggett DV. 1990. Implications of alternative reproductive modes for seasonality and demography in an
estuarine polychaete. Ecology 71:2191–208.
Levin LA, Zhu J, Creed EL. 1991. The genetic basis of
life-history characters in a polychaete exhibiting planktotrophy and lecithotrophy. Evolution 45:380–97.
Levin SA, Cohen D, Hastings A. 1984. Dispersal strategies in
patchy environments. Theor Popul Biol 26:165–91.
Levin SA, Muller-Landau HC, Nathan R, Chave J. 2003. The
ecology and evolution of seed dispersal: a theoretical perspective. Ann Rev Ecol Syst 34:575–604.
Logan DG, Morrison JM, Pietrafesa LJ, Hopkins TS,
Churchill JH. 2000. Physical oceanographic processes affecting inflow/outflow through Beaufort inlet, North Carolina.
J Coastal Res 16:1111–25.
Massol FA, Duputié A, David P, Jarne P. 2011. Asymmetric
patch distribution leads to disruptive selection on dispersal.
Evolution 65:490–500.
Mathias A, Kisdi E, Olivieri I. 2001. Divergent evolution
of dispersal in a heterogeneous landscape. Evolution
55:246–59.
McPeek MA, Holt RD. 1992. The evolution of dispersal in
spatially and temporally varying environments. Am Nat
140:1010–27.
Montagne DE, Casien DB. 2001. Northern range extensions
into the Southern California Bight of ten decapod crustacea
related to the 1991/92 and 1997/98 El Niño events. Bull
South Calif Acad Sci 100:199–211.
Morse DH, Schmitt J. 1985. Propagule size, dispersal ability,
and seedling performance in Asclepias syriaca. Oecologia
67:372–79.
Olivieri I, Berger M. 1985. Seed dimorphism for dispersal:
physiological,
genetic,
and
demographic
aspects.
In: Jacquard P, Heim G, Antonovics J, editors. Genetic
differentiation and dispersal in plants. Berlin: Springer.
p. 413–29.
211
Olivieri I, Michalakis Y, Gouyon P. 1995. Metapopulation
genetics and the evolution ofdispersal. Am Nat 146:202–28.
Otto SP, Day T. 2007. A biologist’s guide to mathematical
modeling in ecology and evolution. Princeton, NJ:
Princeton University Press.
Parvinen K. 2002. Evolutionary branching of dispersal
strategies in structured metapopulations. J Math Biol
45:106–24.
Poethk HJ, Dytham C, Hovestadt T. 2011. A metapopulationparadox: partial improvement of habitat may reduce metapopulation persistence. Am Nat 177:792–99.
Pringle JM, Wares JP. 2007. Going against the flow: maintenance of alongshore variation in allele frequency in a
coastal ocean. Mar Ecol Prog Ser 335:69–84.
Prout T. 1968. Sufficient conditions for multiple niche polymorphism. Am Nat 107:493–96.
Qian P, Chia F. 1992. Effects of diet type on the demographics of Capitella sp. (Annelida: Polychaeta): lecithotrophic development vs. planktotrophic development.
J Mar Biol Ecol 157:159–79.
Ravigné V, Olivieri I, Dieckmann U. 2004. Implications of
habitat choice for protected polymorphisms. Evol Ecol
Res 6:125–45.
Roff DA. 1994. Why is there so much genetic-variation for
wing dimorphism. Popul Ecol 36:145–50.
Ronce O. 2007. How does it feel to be like a rolling stone?
Ten questions about dispersal evolution. Ann Rev Ecol Evol
Syst 38:231–53.
Schulze SR, Rice SA, Simon JL, Karl SA. 2000. Evolution of
poecilogony and the biogeography of North American
populations of the polychaete Streblospio. Evolution
54:1247–59.
Semlitsch RD, Harris RN, Wilbur HM. 1990. Paedomorphosis
in Ambystoma talpoideum: Maintenance of population variation and alternative life-history pathways. Evolution
44:1604–13.
Shanks AL, Grantham BA, Carr MH. 2003. Propagule dispersal distance and the size and spacing of marine reserves.
Ecol Appl 13:S159–69.
Smith JM, Hoekstra R. 1980. Polymorphism in a varied
environment: how robust are the models? Genet Res
35:45–57.
Sotka EE, Wares JP, Barth JA, Grosberg RK, Palumbi SR.
2004. Strong genetic clines and geographical variation in
gene flow in the rocky intertidal barnacle Balanus glandula.
Mol Ecol 13:2143–56.
Spichtig M, Kawecki TJ. 2004. The maintenance (or not) of
polygenic variation by soft selection in heterogeneous environments. Am Nat 164:70–84.
Stearns S. 1992. The Evolution of Life Histories. Oxford:
Oxford University Press.
Van Dooren TJ. 2000. The evolutionary dynamics of direct
phenotypic overdominance: emergence possible, loss probable. Evolution 54:1899–914.
Van Valen L. 1971. Group selection and the evolution of
dispersal. Evolution 25:295.
Venable LD. 1985. The evolutionary ecology of seed heteromorphism. Am Nat 126:577–95.
212
Venable LD, Burquez AM. 1989. Quantitative genetics of size,
shape, life-history, and fruit characteristics of the seedheteromorphic composite Heterosperma pinnatum. I.
Variation within and among populations. Evolution
43:113–24.
Venable LD, Levin DA. 1985. Ecology of achene dimorphism
in Heterotheca latifolia. J Ecol 73:133–45.
Véliz D, Duchesne P, Bourget E, Bernatchez L. 2006. Stable
genetic polymorphism in heterogeneous environments: balance between asymmetrical dispersal and selection in the
acorn barnacle. J Evol Biol 19:589–99.
C. Zakas and D. W. Hall
Wares JP, Hughes AR, Grosberg RK. 2005. Mechanisms
that drive evolutionary change: insights from species introductions and invasions. In: Sax DF, et al. editor.
Species invasions: insights into ecology, evolution and
biogeography. Sunderland, MA: Sinauer Associates.
p. 229–57.
Wares JP, Pringle JM. 2008. Drift by drift: effective population size is limited by advection. BMC Evol Biol 8:235.
Zera AJ, Denno RF. 1997. Physiology and ecology of dispersal polymorphism in insects. Ann Rev Entomol
42:207–30.