### Abstract

- Top of page
- Abstract
- Introduction
- Model and analysis
- Results
- Discussion
- Acknowledgments
- 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

- Top of page
- Abstract
- Introduction
- Model and analysis
- Results
- Discussion
- Acknowledgments
- 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

- Top of page
- Abstract
- Introduction
- Model and analysis
- Results
- Discussion
- Acknowledgments
- 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 . 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 *V*_{S}. The expected fitness of an adult with phenotype *z* living at point *x* is

- (1)

where *r*_{max} 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 *V*_{E}. 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

- (2)

where

- (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:

- (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 . 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 *Z *=* *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,

- (5)

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

- (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 ). To find the deterministic equilibrium of the resident population, , 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 δ_{Z} (measured in terms of the rescaled units defined by eqn 4). In the case of an asexual (or haploid) population, δ_{Z} 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*, δ_{Z}) 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 10^{4} times for each set of parameters, which gives a maximum relative error for the survival probabilities of 1.0%.

### Discussion

- Top of page
- Abstract
- Introduction
- Model and analysis
- Results
- Discussion
- Acknowledgments
- 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 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 δ_{Z} = 0.5 seen in Fig. 4 corresponds to about 1.5 phenotypic standard deviations (assuming *V*_{S} = 10*V*_{E}, 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 σ_{d}, 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 σ_{d}, 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 *F*_{ST} (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 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.