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Keywords:

  • cline;
  • ecotype;
  • gene flow;
  • geographic variation;
  • maladaptation;
  • source–sink dynamics;
  • species’ range

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. The models
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

In an island population receiving immigrants from a larger continental population, gene flow causes maladaptation, decreasing mean fitness and producing continued directional selection to restore the local mean phenotype to its optimum. We show that this causes higher plasticity to evolve on the island than on the continent at migration-selection equilibrium, assuming genetic variation of reaction norms is such that phenotypic variance is higher on the island, where phenotypes are not canalized. For a species distributed continuously in space along an environmental gradient, higher plasticity evolves at the edges of the geographic range, and in environments where phenotypes are not canalized. Constant or evolving partially adaptive plasticity also alleviates maladaptation owing to gene flow in a heterogeneous environment and produces higher mean fitness and larger population size in marginal populations, preventing them from becoming sinks and facilitating invasion of new habitats. Our results shed light on the widely observed involvement of partially adaptive plasticity in phenotypic clines, and on the mechanisms causing geographic variation in plasticity.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. The models
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

Species with large geographic ranges commonly exhibit substantial phenotypic differences between local populations or races, often following predictable patterns along spatial and environmental gradients. In animals, ecogeographical rules (Mayr, 1956) have been described for over 150 years. For instance, Bergmann’s (1847) rule that across a species’ geographic range individual body mass increases with latitude (putatively because reduced surface/volume ratio minimizes energy loss in colder environments) has been confirmed in birds (Ashton, 2002), mammals (Meiri & Dayan, 2003) and insects (Huey et al., 2000). Plants have been extensively studied regarding morphological and phenological differences between altitudinal and geographic races or ecotypes (Bonnier, 1895; Turesson, 1922; Clausen et al., 1940; Knight, 1973; Gurevitch, 1992; Linhart & Grant, 1996; Etterson, 2004).

The origin of these patterns has long been of interest to evolutionists and ecologists. Correlation between phenotypes and environmental variables may be caused either by genetic differences between populations or by phenotypic plasticity, the direct influence of the environment on development of individual phenotypes. Genetic differences often evolve by local adaptation owing to natural selection in a heterogeneous environment. Random genetic drift may also cause genetic differentiation of spatially distributed populations, but this will correlate with environmental variables only in special cases (Cavalli-Sforza et al., 1993; Francois et al., 2010).

In plants, the basis of geographic variation within species has been investigated using common garden experiments and reciprocal transplantation, sometimes combined with controlled crosses (Hall, 1932; Clausen et al., 1940, 1947; Linhart & Grant, 1996). A classic example is the long-term study of Clausen, Keck and Hiesey, who placed ecomorphs of Potentilla glandulosa at various altitudes to investigate differences in height, branching and phenology (Clausen et al., 1940). In animals, the contributions of genetic differentiation and plasticity to biogeographic patterns has been studied in the laboratory and in the wild (Stillwell, 2010). Laboratory experiments on ectotherms focused on the ‘temperature-size rule’ whereby individuals reared at higher temperature develop smaller adult size (Atkinson, 1994), as a possible explanation of Bergmann’s rule (vanVoorhies, 1996; Partridge & Coyne, 1997). Studies of natural populations used statistical methods to test whether the observed phenotype–environment correlation exceeds that expected from plasticity alone (Phillimore et al., 2010).

A major finding common to these studies is that phenotypic differentiation among geographic races or ecotypes is generally caused by a combination of partially adaptive phenotypic plasticity and genetic differences (Clausen et al., 1940; James et al., 1997; Schlichting & Pigliucci, 1998; Gilchrist & Huey, 2004; Kingsolver & Huey, 2007; Wilczek et al., 2009; Phillimore et al., 2010). For example, James et al. (1997) measured the contribution of phenotypic plasticity to latitudinal clines in body size traits of Drosophila melanogaster in Australia, by comparing the phenotypes of individuals collected in the wild to those of lineages from the same locations bred in a laboratory environment. They showed that latitudinal clines were always steeper in the wild than in the laboratory environment, indicating a contribution from plasticity. Table 1 shows that plasticity accounts for a substantial proportion (56% on average) of the slopes of the clines they observed in nature. Similarly, for the classic experiment on altitudinal ecotypes of Potentilla glandusa in California (Clausen et al., 1940, tables 3, 5 and 7), we estimated that 54% of variance in plant size was attributable to plasticity. This was measured as the variance of the mean reaction norm (averaged over races) across environments, divided by the total variance of population means across races and environments (assigning size 0 to races that do not grow in a given environment). Plasticity is thus an important component of observed geographic variation.

Table 1.   Proportional contribution of phenotypic plasticity to latitudinal clines in body size traits of Drosophila melanogaster in Australia (from data of James et al., 1997; Fig. 1).
TraitContribution of plasticity (%)*
FemalesMales
  1. *The proportional contribution of phenotypic plasticity was calculated as (swild − slab)/swild, where swild and slab are the estimated slopes of the latitudinal clines measured in the wild and in the laboratory. Linear reaction norms with little genetic differentiation in phenotypic plasticity are assumed (consistent with James et al., 1997; Fig. 3).

Thorax length6766
Wing area4359
Cell area7484
Cell number1341

Phenotypic plasticity itself can vary substantially among populations of the same species (Morin et al., 1999; Gilchrist & Huey, 2004; Kingsolver & Huey, 2007; Lind et al., 2011). However, only a few empirical studies have investigated the reasons for observed levels of plasticity in a spatially heterogeneous environment (Lind et al., 2011), and factors determining the quantitative contribution of plasticity to geographic variation are not well understood. Theoretical models of the evolution of plasticity in the presence of gene flow may help to identify those factors, and to generate testable predictions.

Theory on the evolution of plasticity in a heterogeneous environment has concentrated on constraints preventing the evolution of perfect plasticity which, if it occurred, would preclude local genetic adaptation. Via & Lande (1985) predicted that with no cost or constraint, perfect plasticity (where the mean phenotype matches the optimum phenotype in all environments) should evolve, regardless of the frequency of each environment, or whether selection is hard or soft. With a cost of plasticity such that more plastic genotypes have lower fitness, van Tienderen (1991) showed that under hard selection, specialists and/or generalists may evolve, depending on initial conditions. Even without a cost, the equilibrium plasticity is limited by the predictability of the environment when development and selection occur in different environments owing to temporal changes in the environment or individual dispersal (Gavrilets & Scheiner, 1993; de Jong, 1999). Imperfect environmental cues produce a similar result (Tufto, 2000a).

Few theoretical studies have investigated the evolution of spatial differentiation in plasticity itself. de Jong and others showed that in a spatially heterogeneous environment, the combination of a life cycle with unpredictable selection and spatial variation in the strength of density dependence after selection could cause the evolution of a polymorphic reaction norm at the level of the metapopulation (de Jong, 1999, 2005; Sasaki & de Jong, 1999; de Jong & Behera, 2002). However, these studies did not explicitly quantify geographic variation in plasticity.

Here, we investigate (i) how plasticity and its evolution modify the impact of gene flow on local adaptation and demography; (ii) how plasticity differentiates across discrete localities and continuous spatial environments. We incorporated plasticity (and its evolution) into an analytical model of the interaction of genetic and demographic processes, with dispersal between local populations differing in abundance, mean phenotype and environment. This was also recently done using individual-based simulations by Thibert-Plante & Hendry (2010) in the context of ecological speciation, but these authors did not specifically investigate how plasticity differentiates across space. They used a model where the plastic response depends on the distance of the mean phenotype from the local optimum, and plasticity cannot exceed a predetermined threshold. The empirical justifications for such a model are unclear, so it is worth re-examining these subjects using more conventional models of plasticity and its evolution, based on empirically well-established norms of reaction describing how the average phenotype of a genotype changes with the environment of development (Gavrilets & Scheiner, 1993; Scheiner, 1993).

Local adaptation and gene flow in spatially heterogeneous environments have been intensively analysed theoretically because of their close connection to evolution of a species’ ecological niche (Holt & Gaines, 1992) and geographic range (Kirkpatrick & Barton, 1997; Holt, 2003). With local genetic adaptation, gene flow from a differentiated population causes local maladaptation, lowering the mean fitness of the recipient population (Lenormand, 2002). This may reduce local population size if the maladaptive influence of immigrants on population growth rate exceeds their direct demographic contribution to population abundance. Furthermore, gene flow increases genetic variance, accelerating adaptive evolution, but also increasing the variance load from stabilizing selection. With gene flow between populations having different optimum phenotypes but otherwise equivalent, Ronce & Kirkpatrick (2001) showed that a small initial difference in deme size can produce a highly asymmetric outcome, where one population becomes a source that is self-sustaining in the absence of immigration and the other population a sink that is not self-sustaining (Pulliam, 1988). This results from a feedback loop (termed migrational meltdown), whereby the larger deme sends more migrants, thus causing more maladaptation in the smaller deme and augmenting differences in population size. A similar process can restrict a species’ range in continuous space over a steep environmental gradient (Haldane, 1956; Kirkpatrick & Barton, 1997; Polechova et al., 2009). These studies emphasized that the impact of gene flow on local adaptation and its interaction with demography should be most pronounced for populations at the edge of a species’ range, or in marginal habitats that receive more migrants than they produce, resulting in net immigration (Kawecki, 2004, 2008). The limiting case is a ‘black-hole sink’ with immigration but no emigration (Gomulkiewicz et al., 1999; Tufto, 2001), a particular case of the ‘continent–island’ model of dispersal where the island is not self-sustaining.

Few models investigated how plasticity and its evolution affect the phenotypic differentiation of populations and the demographic consequences of gene flow (but see Thibert-Plante & Hendry, 2010). To address this, we first consider a population occupying a marginal habitat (the ‘island’) receiving immigrants from a much larger population in the core habitat of the species (the ‘continent’). The environment differs between the core and marginal habitats, with different local optimum phenotypes for a quantitative trait. We investigate the consequences of constant or evolving plasticity on local adaptation and demography on the island, and the basis of phenotypic differentiation between the core and marginal habitats. We then turn to a spatially explicit model of gene flow along a continuous environmental gradient to analyse how evolving plasticity affects geographic variation in plasticity and a species’ geographic range at equilibrium. We show that, provided they correspond to environments of decanalization, marginal and edge populations are characterized by increased plasticity, which generally improves local adaptation and increases population size.

The models

  1. Top of page
  2. Abstract
  3. Introduction
  4. The models
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

Phenotypic plasticity is often modelled using reaction norms describing the expected phenotype of each genotype as a function of the environment (Scheiner, 1993). For simplicity, we assume linear reaction norms, such that the phenotype of an individual developing in environment ɛ is z = a +  + e. The reaction norm parameters a (elevation) and b (slope) are properties of the genotype with polygenic inheritance, whereas e is a normally distributed residual component of variation, with mean 0 and variance inline image. The elevation a is the breeding value (expected phenotype of a given genotype) measured in a reference environment defined below. The slope b (or plasticity) describes how the breeding value changes with deviation ɛ of the environment of development from this reference environment. We may also write in vector notation z = cTx + e (where superscript T denotes transposition), with xT = (a,b) and cT = (1,ɛ). Denoting inline image as the vector of mean reaction norm parameters, and

  • image

as their covariance matrix, the mean phenotype is inline image, and the phenotypic variance is inline image.

With linear reaction norms, there is necessarily an environment where reaction norm slope and elevation are uncorrelated (Gab = 0; illustrated in (Lande, 2009), Fig. 1), and in this environment, the trait is ‘canalized’, that is, its phenotypic variance is minimized. We will use this as the reference environment to define elevation, without loss of generality. Although this form of canalization (caused by the crossing of reaction norms) is a natural consequence of assuming linear reaction norms, it should also hold for other kinds of monotonic smooth reaction norms; however, for nonlinear reaction norms, the description of plasticity is more complex.

image

Figure 1.  Equilibria with constant plasticity. Scaled genetic differentiation between source and sink inline image (upper row) and scaled population size in the sink N/K (lower row), both at equilibrium, are plotted against scaled number of immigrants NiD/K, under constant partially adaptive plasticity. The black lines illustrate development after migration, and the grey lines development before migration. Lines were plotted with α = 0.2, and the black line is unchanged for other values of α. Strength of divergent selection between source and sink is (1 − α)/ω = 0.2 in A, 0.7 in B and 1.2 in C; other parameters are = 2, ω = 7, inline image, R0 = 1.15, = 5000.

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Continent–island model

We first focus on a model of one-way migration from a continent to an island population, where the continent is unaffected by demography and evolution on the island. This will guide our understanding of the influence of gene flow on the evolution of plasticity with immigration into a single deme. In cases where the population on the island has a negative intrinsic rate of increase, this population may be described as a ‘black-hole sink’ (Gomulkiewicz et al., 1999), so we will sometimes refer to it as the sink.

Generations are assumed to be discrete and nonoverlapping. In each generation, Ni immigrants from the continent move to the island, either before or after development of the plastic phenotype, but before selection. We assume that the continent is at demographic and evolutionary equilibrium, such that the distribution of reaction norms of immigrants before selection does not evolve. Subscript i denotes variables among immigrants arriving on the island in the current generation, and subscript r denotes variables among residents already there before migration.

The environment of the island is δ. The reaction norm genetic parameters a and b are assumed to have a bivariate normal distribution with mean vector inline image and covariance matrix Gr on the island before migration and selection, and similarly on the continent with mean vector inline image and covariance matrix Gi. Immigrants may develop and express the plastic phenotypic response on the continent population before migration, as occurs for animals that migrate after development. In this case ci = (1, ɛc)T with ɛc the environment on the continent. Alternatively, they may develop and express the plastic response after immigration, like plants that disperse as seeds, in which case ci = cr = (1, δ)T. We assume that ɛc = 0, such that phenotypes are canalized in the core environment on the continent. This would be expected if the mean environment on the continent had remained stable over a long period, enabling the evolution of genetic mechanisms decreasing genetic variance in this environment (Kawecki, 2000).

Selection is assumed to be density independent, although the population is density regulated. The absolute (Wrightian) fitness of individuals with phenotype z is

  • image(1a)

where Nr and Ni are the numbers of residents and immigrants before viability selection. The phenotype-dependent component of fitness W(z) represents viability selection. We assume Gaussian stabilizing selection for an optimum phenotype that is a linear function of the environment with proportionality constant B (the environmental sensitivity of selection, Chevin et al., 2010),

  • image(1b)

where ω is the width of the fitness function. The second factor in eqn (1a) is a density-dependent (but phenotype-independent) component of survival and reproduction. inline image is the net reproductive rate, including offspring survival up to migration in the following generation, of an individual with optimum phenotype, and K is the carrying capacity of a population composed of such individuals. Density regulation follows the Ricker model (discrete-time equivalent to logistic population growth) and depends on the number of individuals on the island after migration but before viability selection.

The phenotype-dependent component of mean fitness among residents is obtained by integrating (1b) over the phenotype distribution, yielding

  • image(2a)

where inline image is the strength of stabilizing selection on the mean phenotype of residents. The phenotype-dependent component of mean fitness among immigrants is

  • image(2b)

with inline image.

We use a star * to denote variables after migration and selection, but before reproduction in the current generation. The change in mean reaction norm parameters caused by selection in each group (residents and immigrants) can be found using the bivariate selection gradient inline image, with the gradient operator defined as inline image (Lande, 1976, 2009). This yields for residents

  • image(3a)

with

  • image(3b)

Selection also affects the covariance matrix of reaction norm parameters. The change in (co)variance among residents caused by selection within a generation is inline image, where Γr is the matrix of quadratic selection gradients (Lande & Arnold, 1983). The term in parenthesis also equals inline image, where the operator inline image is the Hessian matrix of second-order partial derivates relative to inline image and inline image. Here, inline image for residents, and because inline image, the covariance matrix after selection is

  • image(3c)

Equations similar to (eqns 3a–c) also apply to immigrants, replacing all subscripts r by i.

Where mentioned in the Results, we also include a cost of plasticity, such that more plastic genotypes have lower fitness, regardless of the phenotype they express (van Tienderen, 1991; Dewitt et al., 1998). This is modelled as a separate episode of selection occurring before development of the plastic phenotype (Chevin & Lande, 2010). We assume Gaussian stabilizing selection towards lower absolute plasticity, inline image, with ωb the width of the fitness function on plasticity. The strength of stabilizing selection on mean plasticity owing to its cost is inline image, where Gbb is the genetic variance in plasticity. By analogy with eqns (3a–c), the cost of plasticity changes the mean reaction norm of migrants and residents by inline image, and their genetic covariance matrix by −γbGLG in each generation before phenotypic selection, where inline image.

Then, in the entire island population after selection but before reproduction

  • image(4a)
  • image(4b)

where inline image, the proportion of immigrants after viability selection in the current generation, measures gene flow. Note that m is a dynamical variable that depends on the total island population size as well as the relative fitnesses of migrants and residents. The last term in eqn (4b) gives the change in variances and covariance of reaction norm parameters due to mixing groups with different mean reaction norms. Hence, our model accounts for the increase in genetic variance caused by the build-up of linkage disequilibrium owing to gene flow in a heterogeneous environment (Bulmer, 1971, 1985; Tufto, 2000b). Although this can be neglected when the environment changes smoothly in space (Slatkin, 1978; Lande, 1982; Kirkpatrick & Barton, 1997), it may be important for highly differentiated continent and island populations as modelled here (Tufto, 2000b), but omitted by Ronce & Kirkpatrick (2001). From eqn (4b), mixing populations with different mean reaction norms also generates covariance between elevation and slope on the island, even when there is no covariance at linkage equilibrium.

Viability selection is followed by random mating and reproduction. The mean phenotype is unaffected by reproduction, so denoting with primes values in the next generation,

  • image(5a)

In contrast, genetic (co)variances are affected by recombination reducing linkage disequilibrium, and by segregation. Assuming the infinitesimal model, such that phenotypes are controlled by many unlinked polymorphic loci (Fisher, 1918; Falconer & Mackay, 1996), the within-family distribution of breeding values is normal, with covariance matrix G0/2, where G0 is the (co)variance at linkage equilibrium. The genetic covariance matrix on the island in the following generation is then

  • image(5b)

(Bulmer, 1985; Falconer & Mackay, 1996). The within-family covariance between reaction norm elevation and slope is due to pleiotropic effects of alleles on both traits. Because we assume reaction norm elevation a is measured in a reference environment where it is uncorrelated to reaction norm slope in the absence of gene flow, G0 is a diagonal matrix (Lande, 2009).

Mixing migrants and residents could also cause the distribution of phenotypic and breeding values to depart from normality, altering the response to selection in eqns (3a–c). The strongest effect of mixing occurs for = 1/2, where migrants and residents are in equal numbers before reproduction. Even in this case, the distribution of breeding values in the sink before reproduction is unimodal if inline image, assuming the same covariance matrix G in the source and the sink. For m ≠ 1/2, gene flow is less likely to cause bimodality in the distribution of breeding values, but may generate skewness (Yeaman & Guillaume, 2009). However, under the infinitesimal model, recombination and segregation during reproduction tend to restore unimodality and normality in the distribution in breeding values in the next generation. Turelli & Barton (1994) showed that, in the limit of the infinitesimal model (many loci with small effects), the genetic evolution of the mean phenotype in response to selection is well predicted by assuming normality, even under strong disruptive selection.

Finally, the population size on the island before migration in the next generation is

  • image(6)

which completes the dynamical system. There is no analytical solution to eqns (3–6), because they include sums of different exponential terms (transcendental equations). We therefore found the equilibria by iterating numerical recursions of the dynamical system until the relative change in every variable was <10−6. In all cases, the initial plasticity was set to inline image on the continent and the island, with nonzero α on the continent assumed to be caused by other unspecified selection on plasticity, such as environmental fluctuations (Gavrilets & Scheiner, 1993; de Jong, 1999; Lande, 2009).

We considered two scenarios of particular interest. In the ‘colonization’ scenario, migrants from the continent enter a previously empty patch, which becomes a black-hole sink. In the ‘sudden immigration’ scenario, a previously well-adapted population at demographic equilibrium (at carrying capacity) on the island suddenly becomes exposed to one-way immigration from a much larger continent population with a different mean phenotype.

Cline model

We also studied a spatially explicit model of a cline with local gene flow. We modelled 101 demes evenly spaced along a single spatial dimension characterized by a linear environmental gradient, such that the environment is 0 at the centre of the range and changes from −δ to δ between the edges of the range. The per capita migration rate between adjacent demes is μ in each generation, and migrants move to the nearest demes on both sides with equal probability (as in Goldberg & Lande, 2006). Denoting as d the distance between adjacent demes, this approximates dispersal in continuous (but finite) space, where the most important parameter is mean-squared dispersal distance, σ2 = μd2 (Malécot, 1948; Slatkin, 1978; Lande, 1982; Kirkpatrick & Barton, 1997). We used reflecting boundaries at edges of the geographic space, such that the half of the emigrants from the first and last deme that would move over the edge instead remained in place, and only a proportion μ/2 of the local population moved away from the edge demes. The assumption of reflecting boundaries should have little influence on the results provided that the population density is naturally near zero at the edges, as in all of our numerical examples. We assume that because of intrinsic physiological limitations, the carrying capacity of a well-adapted population is larger in intermediate environments at the centre of the range than in extreme environments at the edges. This is described by making the carrying capacity a Gaussian function of space, with maximum Kmax = 100 at the centre of the range (where ɛ = 0) and width ωK. Stabilizing selection in each deme is modelled by a Gaussian fitness function as in eqn (1b), except that the deviation of the local environment of selection from the environment of canalization is now a continuous function of space, and hence the mean phenotype may change smoothly in space. We neglect changes in genetic variance, which should be limited in this model as we assume polygenic inheritance and the environment (and hence the local mean breeding value) changes smoothly over space (Slatkin, 1978; Lande, 1982; Kirkpatrick & Barton, 1997), as opposed to the continent–island model above.

We investigated two types of initial conditions, analogous to those for the continent–island model. In the sudden migration scenario, the mean phenotype initially is everywhere at its local optimum, forming a cline that perfectly matches the cline in the optimum, and local population are at carrying capacity everywhere, as would occur without gene flow. In the invasion scenario, the population size initially has a narrow Gaussian distribution with mode at the centre of the range, width equal to the distance between adjacent demes, and mean phenotype 0 everywhere, such that only the local mean phenotype at the centre of the range is at its optimum.

Results

  1. Top of page
  2. Abstract
  3. Introduction
  4. The models
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

Equilibria in the continent–island model

We first wish to know how (evolving) plasticity affects the demography of an island population receiving immigrants from a large continent population with different environment and mean phenotype.

Equilibria with constant plasticity

We start by analysing constant (or possibly no) plasticity . In this case, genetic differentiation between the continent and the island (measured in a common environment) equals the difference in mean reaction norm elevation inline image, which is at most (1 − α), whereas the plastic component of phenotypic differentiation is Bδα. If development occurs after migration, the evolutionary dynamics and equilibrium of genetic differentiation (scaled to its maximum), the fitness of immigrants, and the equilibrium population size on the island depend only on (1 − α)/ω. In Fig. 1, the black lines (corresponding to development after migration) are unchanged when modifying α. This occurs because constant plasticity reduces the strength of divergent selection by a factor 1 − α, as for directional selection in a closed population (Lande, 2007; Chevin & Lande, 2010). Note that total phenotypic differentiation is larger than the genetic differentiation in Fig. 1 (first row), because it also includes a phenotypic plastic response. The relative contribution of genetic change to phenotypic differentiation is inline image and cannot exceed 1 − α.

By comparison, if development occurs before migration, immigrant phenotypes depend on the environment in the core rather than the marginal habitat, so the mean phenotype of immigrants before selection deviates more from the optimum on the island. This has two consequences. First, the fitness of immigrants on the island inline image is lower, which reduces gene flow m. Noting that inline image, gene flow is unchanged if the number of immigrants is multiplied by the inverse of mean fitness decrease, inline image, when development occurs before migration, and = 1 otherwise. Second, directional selection on immigrants is stronger (and the response to selection larger), so their mean reaction norm elevation after selection on the island is closer to the optimum elevation (1 − α), by an amount γiBδGaaα. Therefore, with constant plasticity, gene flow from the continent causes less maladaptation on the island than when development occurs after migration. In Fig. 1, the grey line shows development before migration. After rescaling the number of immigrants by D, the equilibria are also almost identical to those of the black line, except with high immigration and strong divergent selection (Fig. 1c), where genetic differentiation is lower as expected from above. In what follows, we assume that development occurs after migration. We also ran simulations with development before migration, but this had negligible impact on the equilibria.

Apart from the rescaling caused by constant plasticity, the equilibria are qualitatively similar to those described by Tufto (2001). With weak divergent selection, (1 − α)/ω < < 1, there is a single stable equilibrium at all migration rates (Fig. 1a, lower row), regardless of initial conditions. Gene flow from the continent has little effect on the island (Fig. 1a), which is almost at carrying capacity at low immigration, and above the carrying capacity at high immigration (Fig. 1a, lower row), where the positive demographic contribution of immigration to population size overcomes the negative demographic impact of maladaptation. At high immigration, there is almost no genetic differentiation between the continent and the island because of swamping by gene flow (Fig. 1a, upper row), but the deviation of the mean phenotype on the island from its local optimum is small relative to the width of the fitness function, so there is little maladaptation.

With strong divergent selection between the continent and the island (1 − α)/ω > 1, the existence of one or two stable equilibria depends on the number of immigrants (Fig. 1c). With low immigration, two stable equilibria exist, depending on initial conditions. The adapted equilibrium has mean phenotype near the optimum and large population size. It arises in the sudden immigration scenario, when immigrants are negligible in number relative to residents and strongly counter-selected on the island, therefore causing little gene flow. The maladapted equilibrium has small population size and mean phenotype far from the optimum. It occurs in the colonization scenario, where the low fitness of migrants does not prevent them from overwhelming the small number of residents, so gene flow is high and strongly affects demography. In the maladapted equilibrium, gene flow from the continent forces the island into remaining a sink at equilibrium. When immigration exceeds a threshold, the two stable equilibria merge into a single stable equilibrium similar to the maladapted one.

At some intermediate strength of divergent selection, the existence of two alternative stable equilibria is confined to a small range of immigration rates (Fig. 1b). In this case, gene flow does not cause sufficient maladaptation to restrict the population size in the colonization scenario at low immigration, but causes too much maladaptation for the population size to remain large in the sudden immigration scenario at high immigration.

Equilibria with evolving plasticity

With linear reaction norms, evolution of the mean phenotype (and mean fitness) may occur much more rapidly than that of the mean reaction norm elevation and slope (Lande, 2009; Chevin & Lande, 2010). When evolution influences demography but selection is density independent as we have assumed, if the strength of density dependence in population growth is comparable to the intensity of natural selection, the dynamics of population size and mean fitness may occur on similar time scales. In Fig. S1 in Supporting Information, the mean phenotype and the population size reach their equilibria around generation 250, whereas reaction norm slope and elevation equilibrate around generation 2000. To facilitate comparison with the situation without plasticity, we first define equilibria for inline image and Nr without considering reaction norm parameters. We later describe the evolution of plasticity itself.

Evolving plasticity changes the conditions for existence of two stable equilibria. Figure 2 shows the parameter range allowing two stable equilibria, with constant or evolving plasticity. In all cases, the upper line represents the largest number of immigrants at which the adapted equilibrium is stable (Fig 1b,c), for each strength of divergent selection (1 − α)/ω. This threshold is lowest for an intermediate strength of divergent selection, at which immigrants have sufficiently high fitness to cause important gene flow leading to substantial maladaptation. Under moderate divergent selection, evolving plasticity moves this threshold towards higher migration rates (Fig. 2, black and dark grey lines above the light grey line). Hence, evolving plasticity increases the stability of the adapted equilibrium under moderate divergent selection.

image

Figure 2.  Condition for existence of two stable equilibria. The range of strengths of divergent selection (1 − α)/ω and immigration rates Ni allowing two stable equilibria is shown for constant plasticity (light grey line) and evolving plasticity (dark grey and black lines). The environmental sensitivity of selection is = 2 (black line) or = 4 (dark grey line), and δ increases when B decreases such that (1 − α)/ω remains constant. Variance in plasticity is Gbb = 0.02; other parameters as in Fig. 1.

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Under strong divergent selection, the threshold number of immigrants Ni for the stability of the adapted equilibrium is lower than with constant plasticity (Fig. 2, right). This occurs because increased phenotypic variance due to variance in plasticity in extreme environments produces lower mean fitness and smaller equilibrium population size on the island (Chevin & Lande, 2010), increasing gene flow measured by m.

In Fig. 2, the environmental sensitivity of selection B is smaller for the black line than for the dark grey line. Lower B causes the threshold immigration rate to be higher under moderate divergent selection, but lower under strong divergent selection (Fig. 2, dark grey vs. black lines). Indeed, for a given strength of divergent selection (1 − α)/ω, smaller B implies larger δ, causing stronger selection on plasticity and faster phenotypic evolution, but also larger variance load in extreme environments (Lande, 2009).

Even when the evolution of plasticity has little influence on the equilibria, it can substantially enlarge the domain of attraction of the adapted equilibrium (the range of initial inline image and N leading to this equilibrium: see Fig. S2). This happens because the evolution of plasticity accelerates phenotypic adaptation (Lande, 2009), allowing the population on the island to decrease slower and to stay larger at all times than with constant plasticity (Chevin & Lande, 2010), thus preventing swamping by gene flow from turning the island into a sink.

Increased plasticity on the island

As described earlier (Fig. S1), after the joint phenotypic and demographic equilibrium is nearly achieved, the reaction norm keeps evolving (sometimes much slower) until it also reaches equilibrium. This occurs because near or at phenotypic equilibrium, gene flow still displaces the local mean phenotype on the island from the optimum before selection in any generation. This causes directional selection, which acts more strongly on reaction norm slope (plasticity) than on elevation if the environment of development substantially differs from that where phenotypes are canalized (Lande, 2009). When this phase occurs on different timescales from those of phenotypic and demographic dynamics, it is characterized by a joint change in reaction norm elevation and slope, whereas the mean phenotype remains nearly unchanged. From eqns (3a,b) and (4a), the recursion for the reaction norm can thus be written

  • image(7a)

where both gene flow meq and the response to directional selection inline image are constant when the mean phenotype and the population size are at equilibrium, and inline image is constant by definition. The constancy of the mean reaction norm of immigrants inline image influences the equilibrium of the mean reaction norm on the island. The equilibrium reaction norm is obtained by setting inline image and solving for inline image, yielding

  • image(7b)

The second term on the right in eqn (7b) shows that at migration-selection equilibrium, and assuming that inline image (the mean phenotype does not overshoot of the optimum), plasticity is always higher on the island than on the continent. Equilibrium plasticity on the island increases with decreasing gene flow at demographic and phenotypic equilibrium meq, increasing response to directional selection at phenotypic equilibrium (inline image), and increasing deviation of the environment of development from that where phenotypes are canalized (inline image large). The equilibrium reaction norm on the island can be found from the equilibrium population size and mean phenotype, which may be reached earlier.

Cost of plasticity

The mean reaction norm can converge to equilibrium slowly (Fig. S1), suggesting that this equilibrium may be sensitive to other evolutionary pressures such as a small cost of plasticity (van Tienderen, 1991; Dewitt et al., 1998; van Buskirk & Steiner, 2009). Figure 3 shows the equilibrium plasticity on the island with and without a cost of plasticity, under weak or strong divergent selection. The mean phenotype and population size resemble those in Fig. 1. Under weak divergent selection, the equilibrium plasticity is highest at intermediate immigration, but not much higher than plasticity on the continent (which is 0 in Fig. 3). In contrast, under strong divergent selection, very high plasticity can be reached in the adapted equilibrium, whereas in the maladapted equilibrium, mean plasticity is nearly the same as on the continent. Higher cost causes lower plasticity at low immigration (Fig. 3). This is because with negligible gene flow, directional selection is very weak on the island (low deviation of the mean phenotype from the optimum at equilibrium, second row in Fig. 3), so the equilibrium plasticity is almost the same as for a single panmictic population in a stable environment, which is 0 with a cost of plasticity. With high gene flow, the mean plasticity resembles that on the continent. Note that if local environmental heterogeneity produces nonzero plasticity on the continent (but not on the island), plasticity on the island could be lower than on the continent at very low migration, but it would still be nearly the same as on the continent at high migration because of swamping by gene flow. The cost of plasticity also affects the threshold number of immigrants that allows two stable equilibria to exist (Fig. 3).

image

Figure 3.  Equilibrium plasticity in the sink. The mean plasticity in the sink population at equilibrium is shown against the immigration rate Ni, under weak (left panel, (1 − α)/ω = 0.5) or strong (right panel, (1 − α)/ω = 1.5) divergent selection, with different costs of plasticity: inline image (no cost, light grey), inline image (dark grey) and inline image (black). The right panel shows both the adapted (continuous line) and the maladapted equilibria (dashed line). The scaled deviation of the mean phenotype from the optimum inline image is also represented at equilibrium. There is initially no plasticity (α = 0). Other parameters as in Fig. 2.

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Equilibria in the cline model

So far, we have focused on an island receiving one-way gene flow from a much larger continent population with a different environment and mean phenotype, such that the continent is not affected by evolution and demography on the island. In this case, the island population reaches an equilibrium that is determined by the assumed constancy of parameters on the continent. The outcome may be qualitatively different if local dispersal and gene flow cause neighbouring populations to affect each other’s demography and evolution. We now investigate this using a model of a geographic cline in the optimum phenotype along a continuous environmental gradient.

In the absence of any constraint or cost, perfect plasticity (inline image) evolves, sometimes very slowly, and the optimum phenotype is reached everywhere without any genetic differentiation in reaction norms (not shown). This accords with earlier results for the evolution of plasticity with two-way gene flow between demes in a heterogeneous environment (Via & Lande, 1985), but differs with the results of the model above, where plasticity on the island is constrained by that on the continent.

With a cost of plasticity, there may be two stable equilibria. For a shallow cline in the optimum phenotypes across space (small B), the population converges to an adapted equilibrium where the cline in mean phenotype matches the cline in optimum phenotype closely (Fig. 4, left), and the population size approaches carrying capacity everywhere. With steep cline of the optimum phenotype across space (large B), the population converges to a maladapted equilibrium, where the mean phenotype away from the centre of the range deviates strongly from the local optimum, therefore restricting the geographic range of the population (Fig. 4, right). This resembles results of earlier models of evolution on an environmental gradient without plasticity, where edge populations become sinks because of maladaptation (Kirkpatrick & Barton, 1997; Polechova et al., 2009), except that we here allow carrying capacity to be a function of the environment, which partly constrains local adaptation and further restricts the species’ range.

image

Figure 4.  Evolving plasticity and species’ range over a continuous environmental gradient. Equilibrium spatial pattern of the population size N, the mean phenotype inline image, its scaled deviation from the local optimum inline image, the mean reaction norm elevation inline image and mean scaled plasticity inline image are shown for a shallow (left, = 2) or steep (right, = 6) cline of the optimum phenotype. The environment ε changes from –δ in location 0 to δ in location 101, with δ = 10. First row: dashed line shows the carrying capacity across space for a perfectly adapted population with no plasticity. Second and fourth row: dashed line represents the optimum phenotype. There is initially no plasticity. Per capita migration rate to adjacent demes μ = 0.2, maximum local carrying capacity Kmax = 100, genetic variance in plasticity Gbb = 0.04. Other parameters as in Fig. 1.

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In both the adapted and maladapted equilibria, the mean plasticity may approach its equilibrium slower than the mean phenotype. At equilibrium, the mean plasticity is only partially adaptive (inline image), and greater at the edge than at the centre of the range (Fig. 4, bottom row, both sides). Consequently, the cline in reaction norm elevation is shallower than that in mean phenotype, even in the adapted equilibrium (Fig. 4, compare second and fourth row on the left). Although the initial plasticity was assumed to be 0 in Fig. 4, we also checked that the equilibrium plasticity across space was not affected by the initial (partially adaptive) plasticity in this model. When the environment of canalization does not coincide with the best environment physiologically (highest K), the minimum plasticity (and maximum population size) occurs in between these two environments, but generally closer to the environment of canalization (not shown).

For a given strength of stabilizing selection γ, genetic covariance matrix G, and per capita migration rate μ, the steepness of the cline in optimum phenotype determines the size of the species’ geographic range (Fig. 5, first row), the genetic differentiation in plasticity across the range (Fig. 5, second row) and the contribution of plasticity to geographic variation in the mean phenotype (Fig. 5, third row). The adapted equilibrium with large range size remains stable for steeper clines in optimum phenotype with evolving plasticity than with no plasticity (black and grey lines in Fig. 5, first row). Geographic variation in plasticity and the contribution of plasticity to phenotypic divergence are both larger in the adapted than in the maladapted equilibrium. Even with parameter values for which the species is restricted to a small geographic range (Fig. 5, first row right side), with little differentiation in mean phenotype or plasticity across the range (Fig. 5, second row right side), the contribution of plasticity to geographic variation in the mean phenotype remains high at equilibrium (30% to over 50% in Fig. 5, third row).

image

Figure 5.  Impact of the steepness of the cline in optimum phenotype. Equilibrium relative range size (top panel), differentiation in plasticity between the centre and edges of the range (middle panel), and relative contribution of phenotypic plasticity to phenotypic differentiation between the centre and the edge of the geographic range (bottom panel), plotted against the environmental sensitivity of selection B. Edges were defined as the locations where the interpolated population size is 5% of its maximum. The relative range size is the distance between edges, divided by the equivalent measure for the carrying capacity K. The geographic differentiation in plasticity is the difference in plasticity between the edge and the centre of the range, scaled to B. The relative contribution of phenotypic plasticity is the difference in mean plastic phenotypic responses to the environment inline image between the edge and the centre of the range, divided by the corresponding difference in mean phenotype. In all cases, the dashed line represents the maladapted equilibrium. Top panel: the grey line is with no plasticity, and the black line is with evolving plasticity. All parameters except B as in Fig. 4.

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Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. The models
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

We studied the influence of phenotypic plasticity and its evolution on adaptation and demography in an island population receiving immigrants from a much larger continent population with a different environment and mean phenotype, and in a cline with local migration along an environmental gradient. Constant plasticity reduces the strength of diversifying selection, thus allowing a stable adapted equilibrium with large population size on the island to exist for larger differences between the optimum on the continent and on the island (Fig. 1). Evolving plasticity further enhances the stability of the adapted equilibrium, except in environments where phenotypic variance due to variation in reaction norm slope substantially reduces mean fitness. The island generally evolves higher plasticity than the continent, assuming that the core habitat on the continent corresponds to an environment where the phenotype is canalized. For a population with a continuous spatial distribution along an environmental gradient, higher plasticity also evolves at the edge of the range, assuming that the cost of plasticity (or another constraint) prevents perfect plasticity from evolving everywhere. In both models, higher plasticity evolves in marginal populations as a consequence of directional selection maintained by maladaptive gene flow in a heterogeneous environment, combined with increased genetic variance in marginal environments because of variance in reaction norm slope.

Below, we compare our main findings with those of previous models of evolution of plasticity in spatially heterogeneous environments. A brief overview of classic and recent empirical evidence then serves to clarify critical data needed to test predictions of the models.

Comparison to previous results

Scheiner (1998) and de Jong (1999) studied the evolution of plasticity in a genetically structured population occupying a heterogeneous environment. Both found that the equilibrium mean plasticity in the metapopulation was lower than that yielding the optimum phenotype in each environment (inline image), such that geographic variation in the mean phenotype was caused by a combination of local adaptation and phenotypic plasticity. de Jong (1999) identified that this was caused by imperfect predictability of the environment of selection, when migration occurs between development and selection. The influence of a source–sink demographic structure caused by varying strength of density regulation on the evolution of reaction norms in a heterogeneous environment was investigated using optimality models (Sasaki & de Jong, 1999; de Jong & Behera, 2002; de Jong, 2005). Those revealed that if the environment of development differs from that of selection, causing unpredictability, the evolutionarily stable (ESS) mean phenotype in each patch is a weighted average of the optima that emigrants from this patch encounter upon selection, weighting each optimum by the strength of density dependence after selection. Under strong stabilizing selection, this ESS reaction norm may become polymorphic, such that the metapopulation contains a mixture of generalist (plastic) and specialist genotypes (Sasaki & de Jong, 1999; de Jong & Behera, 2002). How plasticity differs in space was not the primary subject of these models. de Jong (2005) found lower plasticity at range edges for some parameter values (see her Figs. 7h and 8i), but this appears to be an artefact of allowing large populations at the range edges such that reflecting boundaries dominate the evolutionary dynamics there (whereas we focused on populations that vanish near the edges of the range). Here, we show that higher equilibrium plasticity evolves in marginal populations because they receive more gene flow per capita and are more likely to experience environments where phenotypes are not canalized.

Thibert-Plante & Hendry (2010) found that the evolution of plasticity could facilitate colonization of new environments, reducing the time to successful invasion. This parallels our finding that evolution of plasticity can facilitate local adaptation despite gene flow and expand the domain of attraction of the adapted equilibrium where the marginal population is not a sink. Despite differences between our modelling approaches and assumptions about the form of plasticity, our predictions for colonization are qualitatively consistent with theirs, demonstrating the robustness of these conclusions.

Kimbrell & Holt (2007) showed that an increase in genetic variance caused by decanalization in an extreme environment can facilitate adaptation in a sink population. In our model, evolving plasticity favours the adapted equilibrium because with linear reaction norms, variance in plasticity increases the genetic variance in extreme environments, augmenting the rate of adaptative phenotypic evolution (Lande, 2009; Chevin & Lande, 2010). Our results again agree qualitatively despite substantial differences in the models. In our model, the increase in phenotypic variance occurs in one generation because of variation in reaction norm slope, whereas in their models, it takes many generations of selection for decanalization to evolve.

Limitations of the models

For simplicity, we used a deterministic quantitative genetic model assuming constant genetic variances in linear reaction norm slope and elevation in the reference environment, but accounting for changes in variances owing to linkage disequilibrium created by migration and selection. Holt et al. (2003) studied the influence of demographic and genetic stochasticity in a ‘black-hole’ sink with one-way migration using individual-based simulations. They showed that stochasticity causes random shifts between two states of the population, corresponding to the two stable equilibria described here. Explicit genetics was included by Barton (2001) in a cline model similar to ours, but without plasticity. Comparing the infinitesimal model of quantitative trait inheritance to models with a Gaussian distribution of alleles at each locus, or with a large number of diallelic loci, he showed that for all but the infinitesimal model the change in genetic variance owing to linkage disequilibrium created by dispersal makes the equilibrium with limited geographic range unstable, regardless of the steepness of the cline in the optimum phenotype. Bridle et al. (2010) showed that when genetic and demographic stochasticity are included in a cline model with many diallelic loci, the equilibrium with limited range is stable only for a narrow range of parameters. Although these complications may be worth investigating in the context of evolving plasticity, they should not qualitatively alter our main conclusions.

We assumed that phenotypic variance is minimized in the core environment of the species, where most of the population occurs. This accords with the idea that largest local population sizes determine the evolution of genetic variation in the long run, with the greatest canalization evolving in the average environment experienced by the species (Kawecki, 2000). From the available empirical evidence, it is not clear whether phenotypic variance generally increases in extreme or rare environments (Holloway et al., 1990; Scharloo, 1991; Hoffmann & Merila, 1999; Charmantier & Garant, 2005; le Rouzic & Carlborg, 2008; Schlichting, 2008). Most such studies are framed in terms of new/stressful environments rather than marginal environments at the edge of a species’ geographic range, which may or may not coincide depending on what restricts the species’ range. Future theoretical work should investigate how the evolution of spatial differentiation in canalization influences geographic variation in plasticity.

Causes of geographic variation of phenotypes

Our results help to elucidate factors governing the contribution of phenotypic plasticity to geographic variation in the mean phenotype, the origin of ecotypes (Turesson, 1922; Clausen et al., 1940, 1947; de Jong, 2005) and the mechanisms underlying ecogeographical rules (Bergmann, 1847; Mayr, 1956; Meiri & Dayan, 2003).

Crispo (2008) discussed how constant plasticity alters the impact of gene flow on local adaptation in a heterogeneous environment. With constant partially adaptive phenotypic plasticity, gene flow homogenizes the genetic composition among demes but does not alter the plastic phenotypic response. Hence, larger gene flow reduces the relative contribution of genetic differentiation to phenotypic clines (Fig. 1). Our model also entails that partially adaptive phenotypic plasticity decreases the strength of divergent selection, thus augmenting gene flow by increasing the fitness of the immigrants. Nevertheless, as stated above, we found that constant or evolving plasticity facilitates local adaptation despite gene flow.

When plasticity is heritable, higher phenotypic plasticity evolves in environments that differ most from the environment of canalization (Lande, 2009), and in demes subject to stronger directional selection at migration-selection equilibrium (eqn 7b). Sink or marginal populations occupying extreme environments at the edge of species’ range (or intermingled with a preferred environment) combine these two features, so our model predicts they should display more phenotypic plasticity. Empirical evidence is currently scarce to test the prediction of higher plasticity in sink populations. A few studies compared plasticity in invasive populations to that in their population of origin (or in another, noninvasive, population of the same species). Although increased plasticity or environmental tolerance in invasive population was sometimes found (Lee et al., 2003; Cano et al., 2008), most studies remained inconclusive (e.g. Richards et al., 2006). Genetic differentiation in phenotypic plasticity has been documented among populations from several parts of the geographic range of a species (Morin et al., 1999; Gilchrist & Huey, 2004; Kingsolver et al., 2007; Lind et al., 2011), but with no reference to potential source or sink status. Lind et al. (2011) showed that island populations of the common frog in Sweden exhibit substantial differentiation in phenotypic plasticity and that plasticity correlates positively with both within- and among-island environmental heterogeneity. Otaki et al. (2010) reported the occurrence of novel colour patterns caused by phenotypic plasticity in a population at the northern margin of the Japanese range of a lycaenid butterfly, but no increase in phenotypic plasticity was shown. The adaptive value of this colour pattern is unknown, but it may be correlated to cold resistance (Otaki et al., 2010).

To test the predictions of our model, experiments should be conducted using genotypes sampled from multiple localities across a species’ range and tested in a comparable range of environments to measure geographic variation in plasticity. Local populations can be classified as sources or sinks by demographic and molecular genetic measurements of immigration and emigration, using estimates of local intrinsic rates of population growth (per capita rate of birth minus death minus emigration [excluding immigration]) and directional selection gradients to assess local adaptation or maladaptation. We expect that such experiments should find higher plasticity in marginal habitats. We also predict higher proportional contribution of plasticity to phenotypic differentiation when comparing populations from the centre and edge of the species’ range (or source and sink populations) than when comparing populations near the centre of the species’ range. These patterns should be more pronounced if the core habitat (near the centre of the range) is also the environment where phenotypic variance is smallest.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. The models
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

We thank two anonymous reviewers for helpful criticisms. LMC is supported by a Newton International Fellowship, and RL by a Research Professorship, both from the Royal Society.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. The models
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. Introduction
  4. The models
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

Figure S1 Evolutionary and demographic dynamics with evolving plasticity.

Figure S2 Domains of attraction.

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