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. 2012 May 2;52(1):197–212. doi: 10.1093/icb/ics055

Asymmetric Dispersal Can Maintain Larval Polymorphism: A Model Motivated by Streblospio benedicti

Christina Zakas 1,1, David W Hall 1
PMCID: PMC4817586  PMID: 22576818

Abstract

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 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 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 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 µm diameter), or small eggs (∼150 eggs of 60–95 µm 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 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.

Fig. 1.

Fig. 1

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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.

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.

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 intermediate-sized 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 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.

Table 1.

Number of offspring produced by females of different genotypes and associated dispersal parameters

Genotype Offspring number Offspring size Dispersal rates to natal patch Dispersal rates to nonnatal patch
Haploid
    L bL Large mLii (1 – mLii) vLij
    S bS Small mSii (1 – mSii) vSij
Diploid
    LL bLL Large mLLii (1 – mLLii) vLLij
    LS bLS Intermediate/ mixed mLSii (1 – mLSii) vLSij
    SS bSS Small mSSii (1 – mSSii) vSSij
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(i, j = 1 or 2)

Mortality of adults

Following reproduction, all adults experience mortality with the same probability d, regardless of their age or genotype. If d = 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 Inline graphic. If a larva is not close to its natal patch when it reaches the threshold settlement size, which occurs with probability Inline graphic, 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 Inline graphic. Thus, the probability that a larva migrates from its natal to the other patch is Inline graphic. 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 Inline graphic and Inline graphic. 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 Inline graphic and Inline graphic. The effect of asymmetric dispersal is modeled by assuming that Inline graphic.

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) = 1 − NSi(t). With the assumptions given above, recursion equations for the proportion of Small and Large individuals in patch i are:

graphic file with name ics055m1.jpg (1a)
graphic file with name ics055m2.jpg (1b)

where Ti is the sum of the right hand sides of the equations such that,

graphic file with name ics055m3.jpg (1c)

The first term on the right hand side of Equations (1a) and (1b) represents adults from the previous generation who survive. In Equations (1a) and (1b), respectively, the second term represents recruitment of S or L offspring into patch i from offspring produced in patch i by S or L females, and the third term is an equivalent term for recruitment into patch 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:

graphic file with name ics055m4.jpg (2)
graphic file with name ics055m5.jpg (3)

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). In the alternative situation, Large recruits better in patch 2 and Small does better in patch 1. These necessary requirements immediately clarify why asymmetric dispersal is required for protected polymorphism. Without asymmetric dispersal, both patches are equivalent, so that if one type does better than the other in one patch, it will also do better in the other patch.

Given our assumptions about the biology of the system, namely that natal recruitment does not vary across patches (mG11 = mG22) and downstream dispersal is greater (vG12 > vG21), it is more likely for Condition (3) to hold, and (2) not to hold, than vice-versa. In this case, we expect the frequency of the S allele should be higher in patch 2 than in patch 1 at an internal equilibrium.

Analysis of the haploid sexual model gives identical conditions for protected polymorphism. The diploid sexual model also gives essentially identical conditions for protected polymorphism for the case 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., Inline graphic). In the second scenario, the degree of asymmetry was the same for both genotypes (i.e., Inline graphic). In the third scenario, dispersal was symmetric (Inline graphic). In all scenarios, we required that the Large type produced fewer offspring than Small (bL < bS), had more offspring return to its natal patch (Inline graphic), and fewer offspring migrate to the other patch [Inline graphic] 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 <10−11, 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. 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 ∼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).

Table 2.

Equilibria obtained in the numerical analyses for the clonal haploid model

Dispersal asymmetry Large fixation Small fixation Polymorphism
Different 4009 863 128
Constant 4019 857 124
None 3731 1269 0
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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.

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.

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 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 (Inline graphic) or nonnatal recruitment (Inline graphic) of the new strategy because this in biologically impossible].

Fig. 2.

Fig. 2

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The relationship between the total recruitment to a patch [Inline graphic] 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)].

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. = 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.

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.

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).

Fig. 3.

Fig. 3

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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).

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 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. 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 10−11, 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 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).

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 Polymorphism
Same maxima Monomorphism 7461 0 0
Different maxima Polymorphism possible 7 2033 499
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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 = 40, maximum offspring number = 150, natal dispersal minimum and maximum fall in (0, 1), minimum and maximum nonnatal dispersal viability fall in (0, 0.2), and asymmetry (= 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.

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 = 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 (Inline graphic), 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 Inline graphic 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 (Inline graphic). 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 symmetric dispersal will generally not be sufficient to maintain a polymorphism. Fixations of the Large strategy are also increased with increasing asymmetry.

Fig. 4.

Fig. 4

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Total number of stable pure Small, pure Large or polymorphic equilibria that occur in the simulated data under different levels of upstream migration (a = degree of asymmetry, vGji/vGij) where large values indicate less asymmetry (asymmetry = 1 is symmetric dispersal and asymmetry = 0 is one-directional dispersal). The distribution is for the clonal model with the same asymmetry exhibited by both Small and Large genotypes (Inline graphic).

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 regulation of population size occurs before genotype-dependent 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 frequency-dependent 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 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 over-dominance 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, negative-frequency-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 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 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 well-studied 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 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 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 (IOS-1157279), The Company of Biologists Ltd., the American Microscopical Society, and the Society for Integrative and Comparative Biology, including SICB divisions DEDB, DEE, and DIZ.

Supplementary Material

Supplementary Data
supp_52_1_197__index.html (1KB, html)

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