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Keywords:

  • adaptation;
  • branching process;
  • survival probability

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model and analysis
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References

A species’ range can be limited when there is no genetic variation for a trait that allows for adaptation to more extreme environments. We study how range expansion occurs by the establishment of a new mutation that affects a quantitative trait in a spatially continuous population. The optimal phenotype for the trait varies linearly in space. The survival probabilities of new mutations affecting the trait are found by simulation. Shallow environmental gradients favour mutations that arise nearer to the range margin and that have smaller phenotypic effects than do steep gradients. Mutations that become established in shallow environmental gradients typically result in proportionally larger range expansions than those that establish in steep gradients. Mutations that become established in populations with high maximum growth rates tend to originate nearer to the range edge and to cause relatively smaller range expansion than mutations that establish in populations with low maximum growth rates. Under plausible parameter values, mutations that allow for range expansion tend to have large phenotypic effects (more than one phenotypic standard deviation) and cause substantial range expansions (15% or more). Sexual reproduction allows for larger range expansions and adaptation to more extreme environments than asexual reproduction.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model and analysis
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References

The limit to the range of some species corresponds to an obvious and abrupt change in the environment. Other range limits, however, occur where the environment changes gradually. In these cases, it is unclear why species fail to adapt at their range margins and expand their ranges outwards (Gaston, 2003, 2009; Bridle & Vines, 2007; Kawecki, 2008; Sexton et al., 2009).

Three hypotheses that include evolutionary dynamics have been proposed to explain the occurrence of stable range boundaries where environmental change is gradual. The first hypothesis is that there is no genetic variation for the traits that limit the range (Hoffmann & Blows, 1994; Blows & Hoffmann, 2005; Blows, 2007; Eckert et al., 2008). In the second hypothesis, gene flow from the centre of the species’ range prevents adaptation at the periphery (Haldane, 1956; Garcia-Ramos & Kirkpatrick, 1997; Kirkpatrick & Barton, 1997). In the third hypothesis, species ranges are constrained by biotic interactions, despite genetic variation for the trait that influences their range (Case & Taper, 2000; Case et al., 2005; Price & Kirkpatrick, 2009).

The first hypothesis, lack of genetic variation, might seem unlikely because most individual quantitative traits show a substantial amount of standing genetic variation (Houle, 1992). Consequently, existing genetic models for range limits are virtually all based on a quantitative genetic framework that presumes standing variation (Sexton et al., 2009). It appears, however, that trade-offs or constraints may often cause genetic variation for combinations of traits to be limiting (Hansen & Houle, 2008; Kirkpatrick, 2009). Empirical studies have revealed ecologically important traits that do not respond to strong selection or for which there is extremely low genetic variation (Hoffmann et al., 2003; Blows & Hoffmann, 2005; Kellermann et al., 2006, 2009; Angert et al., 2008; van Heerwaarden et al., 2009). Thus, insufficient genetic variation may prevent range expansion for many more species than is commonly recognized. In this case, the survival of a beneficial mutation is necessary for adaptation and range expansion.

Our goal in this paper is to model the expansion of species ranges that result from locally advantageous mutations. We begin our analysis by finding the survival probability of a mutation that occurs at a given location and with a given phenotypic effect. Those results form a foundation that we then use to study three basic questions about range expansions. First, where in a species’ range do mutations that cause range expansion occur? One might expect mutations that allow for adaptation to arise in the range centre where population sizes are larger, and as a result, there is more mutational input. On the other hand, mutations that arise at the range edge experience strong selection and consequently have a higher chance for survival. Resolving this question is of applied as well as fundamental interest; populations that are important for adaptation may be targets of conservation efforts. Second, what are the phenotypic effects of new mutations that cause range expansion? An ongoing debate in evolutionary biology is whether the basis of adaptation is typically many mutations of small effect or a few mutations of large effect (Orr, 2005). Here, we consider this issue in the context of mutations that cause range expansion. Third, how far will a species range typically expand following the establishment of a single beneficial mutation? One would like to determine whether a range expands mainly by gradual increases or by large jumps.

When a species’ range is constrained by genetic variation, expansion can occur if a new beneficial mutation arises and avoids stochastic loss when rare. Most advantageous mutations are lost by chance shortly after they appear, and this could be a key limitation for range expansion. The survival probability of a beneficial mutation in a single population was studied by Fisher (1922, 1930) and Haldane (1927) using discrete branching processes and by Kimura (1962) using a diffusion approximation [see Patwa & Wahl (2008) for a review of more recent developments]. The survival of a mutation in discrete subdivided populations has also been studied (Pollak, 1966; Maruyama, 1974; Gavrilets & Gibson, 2002; Whitlock, 2003; Whitlock & Gomulkiewicz, 2005). Results for a population living in a continuous habitat with selection that varies in space will be developed in this paper.

We begin by developing a model for a species living in a continuous environment where the optimum value of a quantitative trait (or combination of traits) changes linearly in space. Initially, the population is at a demographic equilibrium and has no genetic variation for the trait. First, we use simulations to determine how the survival probability of a new mutation varies as a function of where it originates and the size of its phenotypic effect. This probability is then used to answer our three questions.

Model and analysis

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model and analysis
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References

The model is motivated by considering a quantitative trait whose optimum varies along an ecological gradient in space. This trait might be, for example, the optimum temperature to which individuals are adapted. We then ask about the properties of mutations that affect the trait and that succeed in becoming established. The assumptions are described in terms of a single quantitative trait. The model also applies to a combination of correlated traits for which there is no genetic variation to adapt to the environmental gradient. Most generally, our model applies to any mutation whose intrinsic rate of increase varies quadratically in space.

The population lives in a spatially continuous habitat. Generations are nonoverlapping. Following birth, juveniles disperse according to a Gaussian kernel with mean 0 and variance inline image. Individuals reproduce at a rate that is determined by their fitness, which in turn depends on two factors. The first is the local population density. We assume there is a carrying capacity K that is equal at all points in space. The second is the degree to which the individual’s phenotype matches the local trait optimum, which is denoted θ(x) at point x. Fitness declines as a Gaussian function of the deviation of the individual’s phenotype from the optimum; the width (variance) of the fitness function is VS. The expected fitness of an adult with phenotype z living at point x is

  • image(1)

where rmax is the logarithm of the maximum possible fitness and n(x) is the total population density at point x. The phenotypic distribution of a genotype is assumed to be normal with environmental variance VE. The number of offspring left by an adult is Poisson-distributed with the expectation given by eqn (1). In the cases studied here, the optimum for the trait θ(x) is assumed to change linearly in space: θ(x) = bx. Therefore, the parameter b measures the steepness of the environmental gradient in space. These assumptions about density dependence and selection are essentially the same as those of Kirkpatrick & Barton (1997), adapted to discrete time.

The fitness of an individual depends on the population density at location x. We consider cases in which a new mutation invades a resident population that is at a demographic equilibrium. We use two approaches to determine n(x) for the residents. The first approach, which we term the ‘deterministic approximation’, assumes that the resident population densities are sufficiently high that demographic stochasticity can be ignored. The population densities in the following generation are then given by

  • image(2)

where

  • image(3)

is the mean fitness at location X. Equations (2) and (3) have been simplified using rescaled measures of spatial location, mean phenotype and population density:

  • image(4)

Space is now measured in units of the standard deviation of dispersal, and the trait is measured in units defined by the width of the fitness function. The rescaled measure for density is more difficult to interpret, but it simplifies in the biologically plausible case that selection is weak inline image. Then, N(X) is approximately twice the population density n(X) measured relative to the carrying capacity K. Without loss of generality, we define the mean phenotype of the resident population to be = 0.

The advantage of this rescaling is that it reduces the number of independent parameters from six to two. These are a rescaled measure of the steepness of the environmental gradient,

  • image(5)

and a rescaled measure of the maximum intrinsic rate of increase,

  • image(6)

Equation (2) shows that the density of individuals in the next generation at point X depends on how many adults there are following dispersal (represented by the integral) and their average fitness at that point in space (represented by inline image). To find the deterministic equilibrium of the resident population, inline image, we iterated eqn (2) numerically until equilibrium was reached.

This deterministic approximation for the resident population densities assumes that demographic stochasticity can be ignored. The appeal of this approach is that it allows results to be computed quickly. A potential drawback is that this approximation is expected to be particularly poor near the range edge, where densities are low. To determine whether the results are sensitive to this approximation, our second approach, referred to as the ‘full stochastic model’, explicitly accounts for demographic stochasticity. We used an individual-based simulation that tracks the movement and reproduction of each individual in the resident population. The carrying capacity per unit space, X, was set to 10 000 individuals. The spatial limits of the simulation were plus and minus four standard deviations of the resident density distribution calculated from the deterministic approximation. Density regulation (see eqn 3) was enforced by dividing the range into 241 equally spaced intervals and using the number of individuals within each interval to determine the local density. Before a mutation was introduced, simulations were run until a stochastic equilibrium was reached (500 generations). We verified that the results are insensitive to the carrying capacity and number of spatial intervals by doubling and halving each of those parameter values and verifying that the results did not change significantly.

We begin the analysis by considering the fate of a mutation that arises at location X with mean phenotypic effect δZ (measured in terms of the rescaled units defined by eqn 4). In the case of an asexual (or haploid) population, δZ is the mean effect of the mutation on all individuals that carry it, whereas for a sexual diploid population, it is the difference in mean phenotype between mutant heterozygotes and the resident homozygotes. We use p(X, δZ) to denote the survival probability of such a mutation. That probability was determined using individual-based simulations. We assume that a mutation either is lost when it is initially rare or that it rises to a sufficiently high frequency that it persists indefinitely. If it does so, we say the mutation is ‘established’. A mutation never becomes fixed throughout the range because the resident genotype always has higher fitness in part of the range and so persists there.

The probability of establishment was determined using both approaches to find the resident population density. With the deterministic approximation, we assume that the mutation’s density is sufficiently small relative to the carrying capacity that it has a negligible effect on its own fitness during the critical time its fate is decided. With the full stochastic model, we allow the number of mutant individuals to contribute to the local density and so affect fitness. In both approaches, a single mutation is introduced to the resident population and the fate of its descendants is followed by an individual-based simulation. The mutation is assumed to be established if the number of copies exceeded 500. We verified that increasing this value has a negligible effect on the results. Simulations were run 104 times for each set of parameters, which gives a maximum relative error for the survival probabilities of 1.0%.

Two additional approximations for the probability of establishment are possible when considering weak selection (inline image  1 << 1) and the limit of no dispersal. The first approximation, referred to as the ‘density-dependent’ probability of establishment, was calculated numerically by inline image, where inline image is the mean fitness at location X (given by eqn 3) when resident population density is at its deterministic equilibrium N(X) = inline image. The second approximation, referred to as the ‘density-independent’ probability of establishment, was calculated analytically by inline image, where inline image is the mean fitness at location X when the resident population density is zero everywhere N(X) = 0. These approximations are based on Haldane’s (1927) approximation for the fixation probability of a single mutation under weak selection.

Results

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model and analysis
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References

We begin this section by finding the probability of establishment of a mutation that arises in the population with a given phenotypic effect and at a given spatial location. Next, we ask where mutations that successfully establish tend to originate, taking into account the variation in space of mutational input caused by differences in the densities of the residents. We then study the distribution of phenotypic effects of mutations that become established. Last, we look at what affects the size of a range expansion caused by the establishment of a new mutation.

Establishment of a new mutation

Figure 1 shows the probability of establishment p(X, δZ) of a mutant allele that originates at location X and has phenotypic effect δZ. For a given combination of parameters, there is a spatial origin where the probability of establishment is maximized. That point depends on two factors: where the mutation has highest fitness in the absence of density effects and the density effects of the residents. In the absence of density effects, a mutation with phenotypic effect δZ has highest fitness (or intrinsic rate of increase) at X* = δZ/B. We refer to this point in space as the mutation’s ‘ecological optimum’, and it is indicated by the arrows in Fig. 1. The origin that maximizes the probability of establishment of a mutation is displaced from X* because of competition from the residents. The effect of competition is to decrease the establishment of mutations that originate close to the centre of the species’ range near the mutation’s ecological optimum. The probability of establishment calculated from the deterministic approximation (Fig. 1, curves) does not differ significantly from results based on the full stochastic model (Fig. 1, closed circles).

image

Figure 1.  The probability of establishment p(X, δZ) of a mutation as a function of where it originates and its mean phenotypic effect found using the deterministic approximation (dark curve). The circles show the survival probability including the effect of demographic stochasticity of the resident population. The grey curve represents the deterministic equilibrium of the resident population density. The arrow shows the location where the mutation is optimally adapted. The left column shows the effect of changing the mean effect of the mutation (δZ) for = 0.5 and r* = 1. The middle column shows the effect of changing the slope of the environmental gradient (B) for r* = 1 and δZ = 0.6. The right column shows the effect of changing the maximum growth rate (r*) for B = 0.2 and δZ = 0.6.

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A mutation with a small phenotypic effect δZ has an ecological optimum close to the range centre, where the resident population density is high. As a result, it will suffer greater competition from residents than will a mutation of large effect. Consequently, the maximum probability of establishment increases with δZ and the spatial origin with maximum survival shifts closer to the mutation’s ecological optimum (Fig. 1, left column).

As the environmental gradient B steepens, the maximum probability of establishment decreases and the location that maximizes survival moves closer to the ecological optimum (Fig. 1, centre column). These patterns result from an interaction of three factors. First, we see that the range size (width) and maximum density of the resident population decreases. Second, the ecological optimum for a mutation shifts closer to the range centre where competition from the residents is higher. Third, fitness declines more rapidly as mutants move away from their ecological optimum.

Increasing the population’s maximum growth rate r* affects the probability a mutant will establish in two ways (Fig. 1, right column). First, the probability of establishment increases for a mutation that originates at a given spatial location. This is because as the maximum growth rate increases, more mutant offspring are produced, which decreases the probability that all copies of the mutation will be lost by chance. Second, the spatial origin at which a mutant’s survival probability is maximized moves farther from its ecological optimum. That pattern is a consequence of increased competition from residents around X = 0.

Figure 2 compares results from individual-based simulation (solid curve) with the simplified density-dependent (dashed curve) and density-independent (dotted curve) approximations for the limit of no dispersal. The accuracy of these two approximations depends on the parameters δZ, B and r*. The density-dependent approximation performs best when δZ is small, and consequently, the ecological optimum of the mutation is located where the resident density is high (Fig. 2, left column). The density-independent approximation becomes more accurate as the phenotypic effect δZ of the mutation increases. Both approximations become more accurate as the population’s maximum growth rate r* and the slope of the environmental gradient B decrease (Fig. 2, centre and right column).

image

Figure 2.  The survival probability p(X, δZ) of a mutation calculated by individual-based simulations (solid curve) compared to the density-dependent (dashed curve) and density-independent (dotted curve) approximations for the survival probability as a function of where the mutation originates and its mean phenotypic effect. The grey curve represents the deterministic equilibrium of the resident population density. The left column shows the effect of changing the mean phenotypic effect of the mutation (δZ) for = 0.05 and r* = 0.1. The middle column shows the effect of changing the slope of the environmental gradient (B) for r* = 0.1 and δZ = 0.2. The right column shows the effect of changing the maximum growth rate (r*) for = 0.05 and δZ = 0.2.

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A major source of error in these two approximations comes from dispersal. The effect is most easily seen at the point in space where a mutant’s local fitness drops below 1. The approximations say that a mutant has zero probability of establishment wherever the average fitness is less than 1. With even weak migration, however, there is some chance that the mutation will diffuse into a region where its intrinsic rate of increase is positive and so have the opportunity to become established.

The spatial origin of successful mutations

In the previous section, we determined the probability of establishment of a mutation that originated at a specific spatial location with a given phenotypic effect. In this section, we take into account how changes in resident population density in space affect where mutations arise. The higher the resident population density, the more frequently mutations appear. To find the frequency F(X) that mutations arising at point X become successfully established, we integrate over all possible mutant phenotypic effects δZ:

  • image(7)

where μ(δZ) is the mutation rate for mutations with mean phenotypic effect δZ. To evaluate (7), we assume that μ(δZ) is proportional to a normal distribution with mean 0 and variance inline image. A review of the literature on mutation rate (Drake et al., 1998), mutational heritability (Houle et al., 1996) and strength of stabilizing selection (Kingsolver et al., 2001) suggests that inline image may typically take values between 0.1 and 1. For simplicity, we fixed the mutational variance at inline image = 1 [or 1/Vs after rescaling according to eqn (4)]. The diploid genomic mutation rate was set to 1; our results for the establishment frequency can be applied to any other mutation rate by simply multiplying our values of F(X) by that rate. F(X) was calculated by integrating eqn (7) numerically.

Figure 3 shows the results. As the slope of the environmental gradient B increases, the frequency of establishment decreases and the location that maximizes survival moves closer to the centre of the species’ range (Fig. 3, left column). As the maximum growth rate r* increases, the frequency of establishment increases and the location with the highest frequency of establishment shifts towards the range edge (Fig. 3, right column). These patterns are partly a result of where more mutations occur in the population owing to high density. When the mutational variance is high (inline image = 1), the frequency of establishment is higher and the location that maximizes establishment is closer to the range edge than when the variance is low (inline image = 0.1) (Fig. 3, dashed vs. solid black curves).

image

Figure 3.  The frequency of establishment F(X) of mutations as a function of where they originate found using the deterministic approximation (solid and dashed curve). The mutation variance is inline image = 0.1 (solid curve) and inline image = 1 (dashed curve). The circles show the survival probability including the effect of demographic stochasticity (solid circles, inline image = 0.1, and open circles, inline image = 1). The grey curve represents the deterministic equilibrium of the resident population density. The left column shows the effect of changing the slope of the environmental gradient (B) for r* = 1. The right column shows the effect of changing the maximum growth rate (r*) for B = 0.2. In all cases, we assume selection is weak (VE << 4VSr * + 2VSVE) and so changes in B and r* do not affect the scaling of N and hence the rate of mutations entering the population.

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Once again, the deterministic approximation for the resident density gives results that are virtually identical to the full stochastic model (Fig. 3, points vs. curves). Although the effects of demographic stochasticity in the residents are pronounced at the edge of the range, that part of the population contributes very little to the patterns that we are studying. Because the deterministic approximation is much more rapid to compute, we will present results based only on that method in the rest of the paper.

Phenotypic effects of successful mutations

We now consider the phenotypic effects of mutations that successfully establish. The frequency that mutations with mean phenotypic effect δZ become established is found by integrating over all spatial locations:

  • image(8)

We evaluated (8) by numerical integration.

The results are shown in Fig. 4. The phenotypic values of new mutations that allow for range expansion are large. For the parameter values analysed, the modes of the mean phenotypic effects are approximately 0.5 or greater. That value corresponds to about 1.5 phenotypic standard deviations (assuming width of the fitness function is 10 times the environmental variance, VS = 10VE, Johnson & Barton, 2005). The mode increases as the slope of the environmental gradient increases. Mutations with small phenotypic effects have little chance of becoming established. That is because they are nearly selectively neutral with respect to the residents. Our model assumes that the resident population size is effectively infinite; therefore, a neutral mutation has zero probability of surviving. The frequency of establishment is maximized by mutations of intermediate effect. Mutations with large effect are rare and have an ecological optimum that is far in space from where they typically originate, and consequently, they have a very low chance of establishing.

image

Figure 4.  The frequency of establishment PZ) of mutations as a function of their mean phenotypic effect. The grey curve represents the rate that new mutations with mean phenotypic effect δZ arise. Top panels: the mutational variance is inline image = 0.1; bottom panels: inline image = 1.0; left panels: maximum growth rate is r* = 1; right panels: slope of the environmental gradient is B = 0.2.

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As the slope of the environmental gradient increases, the frequency of establishment decreases and the phenotypic effect that maximizes the frequency of establishment increases (Fig. 4, left column). This pattern is the result of increased competition for mutations optimally adapted near the range centre, and larger fitness decreases as individuals move away from their ecological optimum. For the parameters we studied, mutations with phenotypic values < 0.65 do not allow for adaptation to steep environmental gradients (= 1). This threshold results because mutations with small effects are optimally adapted close to the range centre where they suffer strong competition from the resident population. As the maximum growth rate increases, the frequency of establishment of all phenotypic values increases and the phenotypic value that maximizes survival changes little (Fig. 4, right column). As the mutational variance increases from 0.1 to 1, the range of phenotypic values that survive increases and the phenotypic value that maximizes establishment increases (Fig. 4, top row vs. bottom row).

The size of range expansions

Next, we ask how much a species’ range expands as the result of a mutation successfully establishing. Our measure is R, the relative range expansion, measured as the proportional change in the range size of the residents. We defined range size as the distance between the leftmost and rightmost points at which the population density is half of its maximum value in the resident population.

We considered asexual and sexual reproduction. In the case of asexuals, the population is composed of resident and mutant genotypes. The mean number of offspring for each genotype is determined by eqn (3), with a mean phenotype = 0 for residents and = δZ for mutants, and N(X) equals to the sum of the densities of both genotypes at X. In the case of sexual reproduction, we assumed mutations have additive effects (no dominance). The population now consists of three genotypes: resident homozygotes, mutant heterozygotes and mutant homozygotes. Mates are found within a spatial neighbourhood defined by a Gaussian mating kernel with mean 0 and variance 1. (The following results are also robust to mating kernels with variance < 1.) The mean phenotypes are = 0 for resident homozygotes, = δZ for mutant heterozygotes and = 2 δZ for mutant homozygotes. We determined the new demographic equilibrium for the entire population by iterating eqn (2) for each genotype simultaneously forward in time until equilibrium was reached. At the new equilibrium, the density distribution in space remains unimodal when the phenotypic effect is small, but it becomes bimodal for asexuals and trimodal for sexuals when the mutation has a large effect.

We are interested in calculating the probability density function for the relative range expansion. This calculation depends on two functions: the function relating the relative range expansion to the mean phenotypic effect of a mutation and the density function for the phenotypic effects of new mutations that become established. Both of these functions were evaluated numerically, and then the method of transformations (Mendenhall et al., 1981) was used to calculate the density function for the proportion of range expansion. The results for low (inline image = 0.1) and high (inline image = 1) mutation variance reveal similar trends; therefore, only the results for low mutational variance are presented here.

Figure 5 shows how the relative range expansion is affected by the environmental gradient, the maximum growth rate and the reproductive mode. Individual mutations that successfully establish often lead to substantial increases in the range. Under the parameter values that we used, the modes for the relative range expansion fall near 15%. Variation around the mode increases as the environmental gradient decreases, and with = 0.2, it is not rare for mutations in sexual populations to cause the range to increase by 50% or more.

image

Figure 5.  The frequencies of the relative range expansion caused by new mutations. The top panels are asexual reproduction, and the bottom panels are for sexual reproduction. Left panels: maximum growth rate is r* = 1; right panels: the slope of the environmental gradient is B = 0.2. For all panels, the mutation variance is inline image = 0.1.

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A striking pattern is that sexual reproduction leads to larger range expansions than asexual reproduction (Fig. 5, top vs. bottom rows). The reason for this effect is that the establishment of a mutation in sexual populations depends on its heterozygous effects. (That is because survival is determined when the mutation is still very rare, and virtually all its copies appear in heterozygotes.) Those mutations that do establish increase in frequency, and when they do so, homozygotes are generated. Homozygotes have more extreme phenotypes and therefore are adapted to regions further from the range centre, allowing range expansion into that region of space. In asexual populations, however, only one new genotype is created by a mutation. If the mutation establishes, it is only its initial phenotypic effect that allows the species to expand in space.

Other effects of the parameters can be seen in Fig. 5. As the environmental gradient B decreases, the average size of relative range expansions increases (Fig. 5, left column). This pattern is the result of an increase in the spatial origin that maximizes survival as the slope of the environmental gradient increases (shown in Fig. 4, left column). As the maximum growth rate r* increases, the average relative range expansion decreases, but this effect is small (Fig. 5, right column).

Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model and analysis
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References

Populations can adapt to new environments in two ways: selection on standing genetic variation or selection on new mutations. Although there is empirical evidence for both processes (e.g. Houle (1992) and Grant & Grant (1995) for standing variation; Mongold et al. (1999), Ferris et al. (2007), and Sabeti et al. (2007) for new mutations), their relative importance remains unknown. Range limits in some natural populations appear to result from insufficient genetic variation for a trait or genetic correlations of traits opposing the direction of selection (Jenkins & Hoffmann, 1999; Etterson & Shaw, 2001; Hoffmann et al., 2003; Griffith & Watson, 2006; Kellermann et al., 2006, 2009; Angert et al., 2008; Angert, 2009; van Heerwaarden et al., 2009). Range expansion in these cases depends on the establishment of new mutations.

Our model is the first to study species range expansion by mutations in a spatially continuous population with no genetic variation. Previous models have assumed there is standing genetic variation available for adaptation to the environment (Holt & Gomulkiewicz, 1997; Kirkpatrick & Barton, 1997; Gomulkiewicz et al., 1999; Case & Taper, 2000; Barton, 2001; Holt, 2003; Filin et al., 2008; Polechova et al., 2009; Price & Kirkpatrick, 2009). The results in this paper provide a new set of hypotheses for how range limits evolve.

Our results are consistent with previous results for discrete demes. When discrete demes (or metapopulations) experience heterogeneous selection, low migration rates favour establishment of a new mutation (Nagylaki 1980; Tachida & Iizuka, 1991; Gavrilets & Gibson, 2002; Whitlock & Gomulkiewicz, 2005; Vuilleumier et al., 2008). In our model, low migration (or, equivalently, a shallow environmental gradient) likewise favours establishment provided that the resident population has not already filled the entire habitat. In the case of a single deme, increasing the population growth rate increases the establishment probability of an advantageous mutation (Otto & Whitlock, 1997). This pattern is also seen in our model, despite the fact that larger values of r* also increase the density of residents and the amount of competition that new mutants face.

We used the results on mutant survival probability to investigate three questions. The first is where in a species’ range successful mutations tend to originate when we account for the fact that regions of high population density experience a greater input of mutations. In steep environmental gradients, mutations that establish tend to arise closer to the range centre. With a high maximum growth rate, successful mutations tend to originate near the range edge. These observations show the relative importance of central and peripheral populations to long-term adaptation (and perhaps survival) of a species depends on demographic and ecological variables. Consequently, it does not seem possible to make generalizations that might be helpful, for example, in the context of conservation planning.

The second issue we considered is the size of phenotypic effects of new mutations that cause range expansion. For the parameter values analysed, most range expansions resulted from mutations with large phenotypic effects. A typical effect size of δZ = 0.5 seen in Fig. 4 corresponds to about 1.5 phenotypic standard deviations (assuming VS = 10VE, Johnson & Barton, 2005). Adaptation to steeper environmental gradients results from mutations of relatively larger phenotypic effects than adaptation to shallow gradients. The third topic is the size of the range expansion following the establishment of a single mutation. With plausible parameter values, a successful mutation might typically have a substantial effect, increasing the range by 15% or more. Shallow environmental gradients lead to proportionally larger range expansions than do steep gradients. Further, the reproductive mode has an effect; when mutations have additive effects, sexual reproduction leads to larger range increases than does asexual reproduction. This difference disappears, however, if mutations are dominant or individuals are haploid.

Here, we study cases in which the resident population density is at equilibrium when a single mutation arises. More generally, our model applies to consecutive mutations provided a new equilibrium is reached before each mutation. Previous models of single populations show that mutations that arise in growing populations have a higher chance of establishment than those appearing in populations at equilibrium (Otto & Whitlock, 1997). We expect a similar qualitative effect applies in spatially continuous settings. This would be relevant, for example, when a species is invading a new habitat. We expect that details of how population growth affects the quantities that we have studied (the sizes of mutant phenotypic effects, the spatial locations where successful mutations tend to originate, etc.) depend on the details of how the population grows in space and time.

What effect does dispersal have on the probability that a mutation establishes? The dispersal variance does not explicitly appear in the final parameterization of our model as it has been absorbed into the compound parameters B and X. Increasing the dispersal variance increases B, effectively increasing the change in the environmental optimum per unit movement, and decreases X, effectively causing individuals to disperse further each generation. These effects decrease the mutant survival probability and frequency of establishment.

Our analysis has limitations. We found approximations for the survival probability for the limiting cases of a shallow environmental gradient and weak dispersal and selection. These approximations apply to a very narrow range of parameter values where B << 1 and r* << 1. There are two limiting cases that can be solved analytically. When B is zero (that is, there is no environmental gradient or no dispersal), the resident population will expand to fill the entire landscape. At the other extreme, when B is sufficiently large (a steep gradient or a large dispersal variance), the population goes extinct everywhere. (The conditions leading to these outcomes are described in Kirkpatrick & Barton (1997).) Neither of these cases is informative for our model, because a new mutation cannot become established in either.

Our model assumes the simplest pattern of spatial variation; the trait optimum varies linearly in space. It does not seem possible to do a complete analysis of arbitrary patterns simply because the number of possibilities is infinite. We can, however, anticipate some generalizations. If the optimum varies at very fine spatial scales, one expects that movement of individuals across the habitat will have an averaging effect and result in patterns similar to what we found. By analogy with models of clines (Slatkin, 1973), this argument is expected to apply to spatial variation on a scale that is substantially smaller than σd, the standard deviation of dispersal. At the other spatial extreme, when departures from linearity in the environmental gradient occur on spatial scales much greater than σd, our results should provide good approximations for each segment of the gradient.

How might the role of new mutations in the evolution of species’ ranges be studied empirically? Recently developed methods based on the analysis of neutral DNA polymorphism are able in principle to distinguish between recent adaptation based on standing genetic variation and adaptation resulting from a new favourable allele (Barrett & Schluter, 2008). These techniques are most effective when the genomic region involved has been identified. In the absence of that information, it may be possible to identify candidate regions using a genomic scan to find chromosome segments that are most geographically differentiated, for example, with high FST (Sabeti et al., 2007). The genomic resources for this approach, however, are substantial. Furthermore, once loci associated with local adaptation are identified, additional evidence would be needed to show that they were the cause (rather than an evolutionary consequence) of range expansion.

Another strategy is to focus on situations where the environmental gradient is clear and the trait that responds to it can be readily identified (Bridle et al., 2009). Plants that have locally evolved tolerance to heavy metals in soils (Brady et al., 2005) appear to be a prime candidate for this strategy. Genetic analysis might be coupled with transplants and genetic crosses [see Angert et al. (2008) and Angert (2009)] to test the role of single genes in promoting range expansion.

Several extensions of this model would allow for a better understanding of the processes that control geographical ranges. Our model is of a single quantitative trait whose optimum is static in time and varies linearly in space. One extension would be to consider the interaction of multiple environmental gradients, a situation thought to cause range boundaries for many organisms (Etterson & Shaw, 2001). Second, it would be of interest to study environmental gradients that are not constant in space. In particular, many gradients may be shallow near the range core and steeper towards the range edge. We expect the survival probability of mutations that originate near the range centre to decrease because a larger portion of the range has high population density and the survival probability of mutations that originate near the range edge to decrease as individuals move further away from their ecological optimum faster. Third, one might consider alternatives to eqn (1) to describe selection and population regulation. Under our assumptions, the initial resident density distribution inline image is similar in form to a Gaussian distribution. Other forms of density dependence will produce different density distributions and also affect results for the other quantities we have considered.

Finally, it is evident that environmental conditions change over time. Anthropogenic changes to the global climate are focusing increased attention on the role that adaptation might have (or not have) in the survival of species. In many cases, lack of alternative habitat and demographic challenges may preclude the opportunity for species to avoid extinction via adaptation. The survival of new mutations may play an important role in species evolution and persistence. The model developed in this paper offers the start of a theoretical foundation for studying that process.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model and analysis
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References

We thank Trevor Price, Timothy H. Keitt and two anonymous reviewers for comments on the manuscript. M.K. is grateful for funding from N.S.F. grant DEB-0819901 and for support from the Miller Institute for Basic Research.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model and analysis
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References