Why is adaptation prevented at ecological margins?
New insights from individual-based simulations
Jon R. Bridle,1,2*†Jitka
Polechova ´,3,4Masakado Kawata5
and Roger K. Butlin6
1Institute of Zoology, Zoological
Society of London, Regent?s
Park, London NW1 4RY, UK
2School of Biological Sciences,
University of Bristol, Bristol BS8
3Biomathematics & Statistics
Scotland, JCMB, King?s
Buildings, Edinburgh EH9 3JZ,
4IST Austria (Institute of Science
and Technology Austria), Am
Campus 1, Klosterneuburg
5Division of Ecology and
Evolutionary Biology, Graduate
School of Life Sciences, Tohoku
University, Sendai 980-8578,
6Department of Animal and
Plant Sciences, University of
Sheffield, Sheffield S10 2TN, UK
†All authors contributed equally
to this publication.
All species are restricted in their distribution. Currently, ecological models can only
explain such limits if patches vary in quality, leading to asymmetrical dispersal, or if
genetic variation is too low at the margins for adaptation. However, population genetic
models suggest that the increase in genetic variance resulting from dispersal should allow
adaptation to almost any ecological gradient. Clearly therefore, these models miss
something that prevents evolution in natural populations. We developed an individual-
based simulation to explore stochastic effects in these models. At high carrying
capacities, our simulations largely agree with deterministic predictions. However, when
carrying capacity is low, the population fails to establish for a wide range of parameter
values where adaptation was expected from previous models. Stochastic or transient
effects appear critical around the boundaries in parameter space between simulation
behaviours. Dispersal, gradient steepness, and population density emerge as key factors
determining adaptation on an ecological gradient.
Adaptation, dispersal load, ecological margin, habitat patch, individual-based simulation,
population ecology, population genetics.
Ecology Letters (2010) 13: 485–494
Geographical variation in fitness-related traits shows that
populations can adapt to ecological change within a species?
range (e.g. Hoffmann & Willi 2008). Nevertheless, many
species? distributions end abruptly, either in space or in time,
often without the presence of obvious ecological or physical
barriers. Currently, ecological models can only explain such
distributional limits if there is an increased chance of
extinction at margins or if habitats become scarce and
colonization more difficult (e.g. Holt & Keitt 2005), if there
are large asymmetries in carrying capacities in the environ-
ment (Kawecki & Holt 2002), or if genetic variance is too
low to allow perfect adaptation (see Kawecki & Ebert 2004;
Kawecki 2008). Recently, models of ecological margins have
made important advances by coupling population genetics
with population ecology in that the match of a genetically
variable trait to the optimum determines absolute fitness
(see Lenormand 2002; Bridle & Vines 2007; Bridle et al.
2009a). These models typically pertain to ecological or patch
margins, rather than species? margins, because they consider
a continuous spatial area that is not many times greater than
the dispersal distance, whereas species? ranges are larger and
typically consist of discrete patches of habitat, especially
near their margins (see Gaston 2003).
Dispersal along spatial ecological gradients generates a
fitness cost (termed ?dispersal load?). When the trait mean
matches the local optimum, this cost is generated by
Ecology Letters, (2010) 13: 485–494doi: 10.1111/j.1461-0248.2010.01442.x
? 2010 Blackwell Publishing Ltd/CNRS
dispersal to positions on the gradient where the optimum is
different (the load equals the drop in fitness due to dispersal
of one standard deviation). ?Standing load? arises due to the
genetic variation around this optimum. Where the popula-
tion fails to match the local optimum, there is an additional
?maladaptation load?, which increases with the mismatch
between the actual trait mean and its local optimum.
Limited adaptation (LA) occurs when genetic variance in
the trait (which determines the response to selection) is low
relative to the rate of change of the optimum. Three regimes
therefore emerge from Kirkpatrick & Barton (1997):
?unlimited adaptation? (where the trait evolves to match
the spatially changing selective optimum everywhere), ?LA?
(where the population is well-adapted to the local optimum
only in the centre of the species? range), and ?extinction?
(where the population cannot be sustained at any point on
the gradient). ?LA? behaviour is characterized by asymmet-
rical dispersal from the well-adapted central region to the
poorly adapted margins.
In Kirkpatrick & Barton (1997), the genetic variance is
fixed. Increasing the additive genetic variance allows better
adaptation to the ecological gradient. However, additional
genetic variation is supplied by dispersal between popula-
tions at different optima. Barton (2001) extended the
Kirkpatrick & Barton (1997) model to include this effect.
He showed that, for a range of quantitative genetic models,
the increased evolutionary potential resulting from higher
genetic variance allows adaptation across virtually any
steepness of gradient. However, an absolute limit is reached
where very high levels of variance (despite allowing the trait
mean to closely match the local optimum everywhere)
reduce mean fitness sufficiently (through standing load) to
cause extinction of the entire population.
Models that allow dispersal to increase genetic variance
therefore make it hard to explain why adaptation fails at
ecological margins. What are these models missing that
actually limits evolution at ecological margins? One limita-
tion is that these models do not include stochastic effects
such as genetic drift or increased extinction risk at the low
population densities that may frequently occur in real habitat
patches. Successful occupation of patches in nature also
requires a colonization phase which is not included in these
models. Alleaume-Benharira et al. (2005) incorporated the
effects of drift in a stepping-stone model on an environ-
mental gradient. Their results suggest that an intermediate
migration rate provides the best compromise between
dispersal load and genetic drift, increasing fitness in marginal
populations as well as range-wide mean fitness. A similar
result was also observed by Gomulkiewicz et al. (1999). In
this study, an individual-based simulation model briefly
introduced by Butlin et al. (2003) is developed further, and
its behaviour compared to deterministic predictions. The
basic assumptions of the model closely match Barton?s
(2001) analytical ?two-allele? model, which assumes that the
quantitative trait is determined by n freely recombining bi-
allelic loci with additive effects.
The evolutionary dynamics for the simulated population take
place within a continuous region of maximum extent
32 000 · 1000 units. There is an ecological gradient along
the long (x) axis, which is uniform with slope b. The area is
simulatedasacylinder;theedges ofthesecond,short( y)axis
are joined. Individuals occupy the vertices of a grid and more
follows the fate of a starting population of 500 individuals
that are initially distributed in the central 500 · 1000 units of
the environment. Assuming an allelic effect size of a, their
phenotypes ranged from zopt) 2a to zopt+ 2a where zoptis
the optimum phenotype at the centre of the range.
The phenotype is determined by diploid unlinked bi-allelic
loci with additive effects that mutate symmetrically at rate l
(l = 0.0001perlocuspergenerationunlessotherwisestated).
For most runs, 64 loci were used, with allelic effect a = 1
(maximum phenotypic range = 0–128). In selected runs, the
0.5.Wealsofollowedthefateoftenloci(l = 0.001)thatwere
not subjected to selection. Population growth is logistic,
dependent on the local density of individuals (N ) and local
carrying capacity (K). Females choose mates from the males
available within a finite mating distance (MD), with a
probability proportional to the fitness of each male at its
position on the ecological gradient. This was fixed at
MD = 150 (see Butlin et al. (2003) for a description of the
effect of male dispersal on range expansion). Offspring then
disperse and selection occurs after dispersal through the
the female leaves no offspring.
In order to minimize edge effects as the spatial extent of
the population approaches the maximum x range, observa-
tions of simulation behaviour and calculations of genetic
and demographic parameters were restricted to a central
portion of the observed population, 8000 units wide.
The fitness of both sexes is determined by the same
function: WF= WM= 2 + rF (1 ) N⁄K) ) s(bx ) z)2⁄2
(W ‡ 0). The number of offspring a female leaves is drawn
from a Poisson distribution with mean WF. Females choose
males with probability proportional to WM. In our model,
there is no selective mortality or random effects on death
rates. Generations are non-overlapping and the maximum
rate of increase rm= rF⁄2; rFis set to 1.6. K is the carrying
capacity within a circle of radius 50 around the focal
individual, N (density) is the number of individuals in such a
circle (initially N = 7.85 individuals). Ux= bx is the
486 J. R. Bridle et al.
? 2010 Blackwell Publishing Ltd/CNRS
phenotypic optimum at the point (x) on the gradient
occupied by the female. The parameter s measures the rate
of decline in fitness for phenotypes that depart from the
optimum; the strength of stabilizing selection VSis 1⁄(2s).
Here, VS is set to 4 and b (the spatial gradient in the
optimum) is set to 0.004. Note that when drift and the
effects of the margins are negligible, increasing dispersal
with constant gradient is equivalent to increasing the
gradient with constant dispersal (by dispersal of a distance
r, fitness decreases byb2r2
The growth rate of a particular phenotype is
r½z;N? ¼ rm
z ? Ux
the growth rate of the population is the average over all
h i ¼ rm
z ? Ux
Throughout, we assume addictive genetic variation, and
no environmental variation in phenotype. (see Kirkpatrick &
Barton 1997; Barton 2001; Polechova ´ et al. 2009).
Total dispersal (TD) is determined by the combined
effects of the two phases of dispersal, by males or their
gametes (MD) and by offspring (D). The offspring of each
female disperse to new positions in the habitat with a
Gaussian distribution of dispersal distances, mean 0 and
standard deviation D, in uniformly distributed random
directions. As mating is a form of dispersal by males (or
their gametes), the standard deviation of TD is given by
expected distance between mating partners when a female
chooses from a circle with radius MD, hence SM = (1⁄?2
MD). The expected distance r along the x-axis is only in
one dimension, hence r = TD⁄?2.
For an infinite population, the population dynamics
should approximately match the continuous time model
described by eqn 7 in Kirkpatrick & Barton (1997) and if no
linkage disequilibria (LD) are generated, the evolution of
phenotype should follow the 2-allele n loci model of Barton
(2001). In our model, population regulation occurs over
discrete generations, and populations are finite in size,
therefore allowing stochastic effects on demography and
allele frequency, and the generation of lags. The program
was written in C++, developed from that introduced by
Kawata (2002), and is available on request from MK
(email@example.com). Output from the simula-
tions for a given generation was analysed using a Genstat
v10.1 (VSN International; http://www.vsni.co.uk) program
(see Crawford 1984), where SM is the
(see Supplementary Information). This program is available
on request from RKB (firstname.lastname@example.org). Numer-
ical predictions, based on analytical models, were calculated
using Mathematica v7.0.0 (Wolfram Research, Champaign,
Simulations were typically run for 3000 generations,
although by 1000 generations, one of three outcomes was
usually achieved: (1) range expansion to fill the entire
ecological gradient (UA); (2) extinction throughout the
range (E); or (3) LA, where a cline in trait mean formed
which was shallower than the optimum, generating a finite
range (Fig. 1). At the boundary between UA and LA, a
region of ?slow adaptation? (SA) was observed, where the
population filled the available habitat, but took more than
3000 generations. Long runs (up to 10 000 generations)
were used at the boundaries between these behaviours to
explore the temporal stability of different outcomes, the rate
at which allele-frequency clines formed (and a stable
gradient in trait mean was reached), and how quickly
different components of genetic variance (segregating
variance and variance due to LD) became stable (Table S1).
how it matches the predictions of the population genetic
models of Kirkpatrick & Barton (1997) and Barton (2001).
Unlimited adaptation and deterministic extinction
For many parameter combinations (the dark blue region in
Fig. 1, shown for a typical run in Fig. 2a), the populations
spread to fill the available area, albeit with reduced density as
standing load increased (Fig. 3a). This fits the predictions of
the bi-allelic model of Barton (2001), where genetic variance
is allowed to evolve but linkage equilibrium (LE) is assumed.
Such behaviour was characteristic of runs with high carrying
capacity unless the standing load due to dispersal was very
high, leading to extinction over the whole range, also as
predicted (see below).
Barton (2001) derived the equilibrium solution for an
additive trait with n loci and two alleles. Assuming LE, the
b⁄(2a) clines per unit distance to match the gradient b. This
implies variance VG¼ br
approached by our simulations when the population adapted
over the whole range, unless selection due to dispersal across
the spatial gradient was strong (Fig. 3b,c,d). In our simula-
tions, the critical limit for extinction (independently of K )
occurred at the expected steepness of gradient, though the
genetic variance was higher than predicted (Fig. 1, and see
. This prediction was
LetterLimits to adaptation at ecological margins 487
? 2010 Blackwell Publishing Ltd/CNRS
below for derivation). This was probably because the
assumption of a Gaussian distribution of phenotypes was
violated at high values of dispersal, where variance due to LD
was high (Table S1). Kurtosis in these simulations was above
zero, which would reduce the fitness cost of genetic variance
(e.g. UA: TD400, K50, median kurtosis = 0.35; UA: TD900
K50: median kurtosis = 0.55; LA: TD1000 K50; median
kurtosis = )0.31).
Analysis of cline widths and positions for each locus
(Table S1;Fig. 2a)showedtheexpectedevenspacingofcline
centres. The trait mean therefore closely matched the
optimum and population density remained uniform through-
widths and spacing were close to those expected from
analytical predictions (Table S1, Figs 2a, 3). The increase in
total and segregating variance with dispersal was unaffected
by carrying capacity (all the values for variance lie along the
the gradient), demonstrating that the effects of gene flow
rather than neighbourhood population size dominated
genetic variation throughout the range. As a result, neutral
genetic variance was not reduced at the margins (Fig. 2).
As dispersal increased, segregating variance fell below
analytical predictions, although the total genetic variance
exceeded predictions due to a contribution from LD
(Fig. 3b-d, Tables S1, S2). Note that total variance is simply
the sum of segregating variance and variance due to LD
(Fig. 3d). The LD can be predicted from a balance between
recombination and the mixing effect of dispersal (Barton
1986, p. 418). The pair-wise LD depends on the gradients in
allele frequencies at two different loci whose clines overlap,
loci i and j, set to ½ here). LD variance is then 2P
(for 64 loci, the allelic effects a are all set to 1; for 128 loci, a
is set to 0.5). We did not estimate, however, what the cline
shape is with LD. In the Fig. 3, we use the expected clines
for uniform adaptation (UA) and LE, hence we necessarily
overestimate the LD variance component. If LD variance is
calculated from the observed clines, we get a significantly
better fit (as expected, see Tables S1 and S2). Observed
mean cline width increased with dispersal, as expected
(Fig. 3c, Tables S1 and S2). However, observed mean width
was consistently lower than predicted, because LD increases
the effective selection on each locus. This effect increased as
standing load increased, until populations entered the region
of parameter space where adaptation was limited (LA).
Clines were then present for only a few loci (see below), and
these clines were typically very wide (Tables S1 and S2).
Even at very high values of K, unlimited adaptation
was prevented with high TD, as predicted by determinis-
N ¼ Kð1 ?
cost of dispersal across the gradient is
should therefore occur when the effect of phenotypic
variance causes the population growth rate to become zero,
regardless of population size, i.e. when b>2rmffiffiffiffi
2001). As b is fixed to 0.004, rm= 0.8 and VS= 4,
extinction should be observed for TD greater than
TD = r⁄?2 = 1130. In our simulations, the behaviour
@x(where rijis the recombination rate between
p Þ (Barton 2001). The
2rmVSÞ ¼ Kð1 ?
Figure 1 Effect of carrying capacity, K, on
simulation outcomes after 3000 generations,
or when expansion first reached the margins
of the simulated area, for 10 runs per
parameter combination as follows: blue
(unlimited adaptation); light blue (slow
spread); purple (limited adaptation, some
clinal divergence); white (extinction). For
parameters that apply to all simulations, see
text. The solid line at TD = 1130, is the
analytical prediction for the limit of popu-
lation persistence assuming a Gaussian
distribution of phenotypes.
488 J. R. Bridle et al.
? 2010 Blackwell Publishing Ltd/CNRS
changed to LA before this value, even when carrying
capacity was high (Fig. 1). Behaviour varied among runs but
only extinction occurred at a marginally higher TD than
predicted. This value of TD and b therefore represents the
absolute limit where the genetic variance required to track
the optimum cannot be sustained demographically given
this maximum reproductive rate.
Extinction at low carrying capacity
At the opposite end of parameter space, very low dispersal
at low carrying capacities also leads to extinction (Fig. 1).
This extinction behaviour was not observed when females
were allowed to mate with the nearest available male.
Extinction at these parameter combinations is therefore due
Figure 2 Behaviour of typical simulation runs leading to: (a) unlimited adaptation (K = 50 TD = 400); (b) limited adaptation (K = 50;
TD = 975). Upper panels show individual clines in allele frequencies and the prediction (dashed line) for cline shape of a locus under
selection assuming LE. The bold, solid line shows heterozygosity at 10 neutral loci. The central panels show segregating variance (continuous
line), prediction for variance under LE (dashed line, from Barton 2001) and genetic variance due to LD (dots), with the prediction for LD
variance (dotted line, from Barton 1986). Lower panels show density (continuous line), predicted density for uniform adaptation (a only,
dashed line), actual trait mean (dots, value divided by 10) and gradient in the trait optimum (solid grey line).
LetterLimits to adaptation at ecological margins 489
? 2010 Blackwell Publishing Ltd/CNRS
to Allee effects (the failure of females to find mates within
their prescribed MD).
As dispersal load increased, mean density decreased, close
to the rate predicted by Barton (2001), although actual
densities were slightly lower than expected because total
genetic variance was consistently higher than predicted
(Tables S1 and S2 and Fig. 3a–d). However, with carrying
capacity below K = 300, extinction occurred at dispersal
values significantly lower than the absolute predicted limit of
TD=1130 (Fig. 1). Typically, populations at low values of K
became extinct very quickly (within 100 generations), before
any cline was formed, or any population growth could occur.
The proportion of runs that survived this early extinction
increased with higher K. For runs that survived rapid
extinction, transient ?LA? behaviour was observed, where a
shallow trait mean gradient formed and some population
growth occurred. However, extinction occurred within 3000
generations (and usually within 1000 generations), for
fell below approximately 2 (Fig. 3a). A simple deterministic,
0 2004006008001000 1200
Mean density at centre
Mean segregating variance
K50 (128 loci)
Exp var (total)
Mean variance due to LD
K50 128 Loci
Exp LD var (64 loci)
Exp LD var (128 loci)
0200 400600800 10001200
Mean total genetic variance
K50 (128 loci)
Exp (total) var
Total dispersal (TD)
Mean cline width
64 loci K25
64 loci K50
128 Loci K50
Exp (64 loci)
Exp (128 loci)
Figure 3 Effects of dispersal and carrying capacity on: (a) population density; (b) segregating variance; (c) variance due to LD; (d) total
variance and (e) cline width. All runs used 64 loci except for the K50, 128 loci runs indicated by open diamonds. Error bars are standard
deviations, based on five repeats, or total range for population density. Note that for high values of TD these values may be summed over
runs with variable outcomes (see Fig. 1). Analytical predictions (dashed lines) are based on Barton (1986, 2001).
490 J. R. Bridle et al.
? 2010 Blackwell Publishing Ltd/CNRS
discrete-time series model (with population dynamics as in
our simulation and observed neighbourhood size) revealed
that the population growth rate is too small to cause simple
demographic extinction due to delayed feedback.
As TD increased, a narrow region of LA was observed, with
some parameter combinations generating either UA or LA
behaviour in different runs. This region became wider as
carrying capacity increased, and dispersal load drove
populations deterministically to low densities (i.e. at high
dispersal and high carrying capacity; Figs 1, 2b). Here, the
trait mean only matched the optimum towards the centre of
the range, and density declined as the mismatch increased
further away from this point, resulting in a population that
remained bounded in space. These LA populations persisted
for at least 10000 generations (Table S1). However, vari-
ances and trait slopes varied during these runs, with
resulting fluctuations in population size (and spatial extent)
over short timescales.
In the LA region of parameter space, the gradient in trait
mean was caused by clines in fewer loci and clines were
wider on average than in the region of UA. Segregating
variance was also lower than predicted for UA (Fig. 3b).
Given the segregating variance, the gradient of the trait
mean was close to that expected for LA under the
phenotypic model with fixed variance of Kirkpatrick &
Barton (1997) (Figure S3a, Table S1). If genetic variation
rose above the threshold where LA exists for a given
standing load vs. dispersal load, ?SA? rather than LA was
observed (Figure S3b).
?Slow adaptation? behaviour, where the population took
more than 3000 generations to fill the range (Table S1;
Figure S1a), was observed at values of dispersal typically just
below those producing LA behaviour. For some parameter
combinations, either LA or SA was observed for different
runs. During the establishment of these populations, a
sigmoid trait mean gradient was observed, with the mean
departing from the optimum at the range edges. These
simulations also showed a gradual increase in the number of
loci with clines that contributed to adaptation, whereas the
LA runs fluctuated around a low number of clines
(Table S1; Figure S1b).
The robustness of our results was tested by varying: (1)
mutation rate from 0.0001 to 0.001; (2) the number of
initially polymorphic loci from 2 to 20; (3) starting
population size from 250 to 500; and (4) the number of
loci from 64 loci (each with a of 1) to 128 loci (each with a
of 0.5) (Tables S1b and S2b; Figure S2). In all cases,
although there were small changes in where the transitions
between behaviours occurred, the qualitative behaviour of
the model was unchanged.
Our results show that the evolutionary fate of a population
adapting to an ecological gradient depends on local carrying
capacity (K) as well as dispersal relative to gradient
steepness. In particular, we observe extinction at lower
levels of dispersal⁄steepness of gradient than predicted by
Kirkpatrick & Barton (1997) and Barton (2001), whose
models do not include the stochastic effects associated with
finite population sizes. In our model, these stochastic effects
are variation in the numbers of offspring left by a females or
fathered by males and dispersal directions and distances
(and mutation). These lead to spatial and temporal variation
in densities and allele frequencies. Excluding these effects is
an important limitation because populations at the margins
of ranges are expected to be small.
begin with complete adaptation to an ecological gradient, and
ignore the effects of colonization and of stochasticity in the
supply of mutations and the establishment of clines. By
adapted to the centre of a gradient can spread. This initial
population spreads widely across the environmental gradient,
increasing initial load due to the mismatch between individ-
uals? phenotypes and the local optimum. This causes very
rapid extinction, within a wide region of parameter space
where unlimited adaptation was predicted by Barton (2001).
The lower the carrying capacity, the lower is the threshold
dispersal for this rapid extinction. Occasionally, runs around
this area of parameter space escape early extinction and form
briefly enter the LA behaviour described below, before
stochastic effects cause extinction, usually within 1000
generations. Given that neighbourhood sizes are initially
large, these stochastic effects are likely to be due to dispersal
affecting local allele frequencies, so taking the phenotypic
mean away from the local optimum, rather than simple
demographic stochasticity. At higher carrying capacities,
although population density is kept low due to dispersal load,
LA behaviour becomes increasingly frequent (Fig. 1), and is
observed over a wider region of parameter space. Increasing
genetic variation in the initial population does not allow
spread at higher dispersal because this variation also brings
with it an increased standing load. Similarly, increasing initial
population size does not aid adaptation in this region of
parameter space because fitness is then affected by density-
dependence, making selection for adaptation less effective.
Letter Limits to adaptation at ecological margins 491
? 2010 Blackwell Publishing Ltd/CNRS
Our results therefore suggest that a species? occupancy of a
habitat patch is mainly limited by the cost of dispersal along
the patch?s ecological gradient, which prevents the establish-
ment of a population soon after colonization.
At high carrying capacity, our simulations behave largely
as predicted by analytical models which assume LE, an
approach that is valid under weak selection. Stochastic
effects, either on genotype frequencies or population sizes,
have little effect in this region of parameter space where UA
is achieved. However, linkage disequilibrium is generated by
overlapping clines in allele frequency. This increases the
standing load so that population density decreases slightly
more than predicted under LE (Figs 1, 3a). The observed
LD is predictable from the dispersal and the observed
shapes of clines using the moderate selection approxima-
tions of Barton (1986). This fit is surprisingly good
considering that the selection generated in these simulations
is strong where dispersal is high (equivalent to a steep
gradient in the environmental optimum).
Linkage disequilibrium is expected to be generated in
natural populations wherever multiple loci contribute to local
adaptation and selection is moderately strong. Evolutionary
responses to ecological gradients are typically due to multiple
loci, for example in heavy-metal tolerance in grasses (see
Macnair 1993), and in insecticide resistance in Culex mosqui-
toes (Labbe et al. 2005, 2007). In Anthoxanthum grasses, LD
appears elevated at population margins, with concomitant
levels of LD may not always be the case. For example, allele-
frequency variation at only a few QTL loci (located within a
size along latitudinal clines in Drosophila melanogaster in
Australia (Rako et al. 2006; Kennington et al. 2007).
In our simulations, the selection strength generated by
dispersal along a fixed ecological gradient reaches its absolute
limitatTD = 1130.Howdoesthiscomparetolikelylevelsof
dispersal load in natural populations? Burt (1995, 2000)
reviewed available data on TD load, giving a median of about
correspond to TD = 316 (see Polechova ´ et al. 2009). As
variance is maintained by dispersal (and VG= VP), TD =
316 generates VP⁄VS= b TD⁄(2 VS)1 ⁄ 2= 0.45. This is
moderately strong selection (estimates from Kingsolver et al.
2001 suggestthatthemedianVP⁄VSinnatural populationsis
about 0.2; but see also Hereford et al. 2004).
For a narrow range of parameter values, we observe LA
behaviour, where the trait mean is shallower than the
optimum. This generates a population with a spatial extent
that is bounded by maladaptation, as may be observed in
sticklebacks in North America, where dispersal from large
lake populations to smaller stream populations may limit
adaptation to their local ecological optimum (Moore &
Hendry, 2009). However, because LA behaviour occurs only
in a restricted portion of parameter space in our model we
consider it unlikely to be a common explanation for limited
state for thousands of generations; at lower values of K, this
behaviour may only be a transient state en route to extinction.
between simulation runs and must have a chance element.
What appears to happen in LA runs is that genetic variance
initially increases due to the formation of clines in additional
loci until the population reaches a locally stable state. The
population is then subjected to temporal fluctuations in trait
gradient and population density, but fails to achieve the
optimum gradient (and hence unlimited adaptation). Where
this occurs in our simulations, the trait means and variances
fall within the region that also gives a finite range in the
models by Kirkpatrick & Barton (1997) and Polechova ´ et al.
(2009) (Figure S3a), suggesting that the swamping effect of
asymmetrical dispersal is stronger in these runs than its
is required to predict the conditions at which the population
evolves into this state. If the genetic variance becomes higher
than the Kirkpatrick & Barton (1997) threshold for UA, the
population can move from the LA regime into the SA regime
the margin are probably due to stochastic effects, so we
cannot predict when the variance will cross this threshold, or
when a specific parameter combination will generate a
particular behaviour. Similar ?SA? is also observed in some
source-sink models (e.g. Holt et al. 2003).
Empirical data for these variables are difficult to obtain,
but do suggest that the amount of genetic variance
determines rates of adaptation in real populations. For
example, Willi & Hoffmann (2009) used data from
experimental populations of Drosophila birchii subjected to
heat-knockdown selection, and suggest that large popula-
tions persist due to reduced stochasticity in growth rate,
higher reproductive output, and more additive genetic
variation in heat resistance. Similarly, species? ranges in
Australian Drosophila appear to be limited by low levels of
additive genetic variation in stress resistant traits (Kellerman
et al. 2006; Hoffmann & Willi 2008; Bridle et al. 2009b;
Kellermann et al. 2009).
Relevance to population and species? margins in nature
The population genetic models explored here represent
patches of habitat within species? ranges, within which
persistence depends on evolving to match the ecological
gradients observed along their length. Such patches may vary
substantially in their quality (carrying capacity), even when
populations within them match the local optima. Our results
show that, where the carrying capacity of a patch is large,
492 J. R. Bridle et al.
? 2010 Blackwell Publishing Ltd/CNRS
a population can usually persist and evolve to match the
optimum across the entire patch. Such patches might be
typical of the core of the species? range. In more marginal
patches, with lower K or steeper gradients, populations may
persist but undergo only limited expansion, generating a cline
in trait mean that is shallower than that demanded by
the ecological gradient is steep (either inherently, or due to
expand due to maladaptation load, even if initial population
size is large and the population is adapted to some part of the
patch. Such effects may be particularly important given that
colonizers tend to show elevated levels of dispersal (Hughes
et al. 2007; Duckworth 2008; Anderson et al. 2009).
Real habitat patches are likely to be characterized by
central areas with shallow ecological gradients, with steeper
gradients at the edges, rather than the linear gradients
considered here. Patches beyond the current range margin
may also have core environments that differ from occupied
patches (see Holt et al. 2004). In this case, the genetic
variance maintained along gradients within occupied patches
may aid colonization, as is observed in Holt et al. (2005).
Understanding ecological margins therefore requires inte-
grating the effects of gradients within and between patches,
and the effects of dispersal on colonization as well as on load.
Limits to adaptation in natural populations also depend on
other factors: how ecological gradients vary in time as well as
space (Polechova ´ et al. 2009), biotic as well as abiotic
interactions (Bridle et al. 2009a), and the form of density-
dependence (Filin et al. 2008). It is also unlikely that any
ecological gradient will remain constant over even a few
generations, let alone several thousand. Fluctuating selection
could also elevate genetic variance substantially, without
necessarily incurring high dispersal load. Given the key effect
of dispersal on adaptation, it is important to understand
when mean dispersal distances, or conditional or habitat-
specific dispersal (Holt 2003) can evolve to aid expansion
along otherwise impenetrable ecological gradients (Oliveri
et al. 1995; Dytham 2009). This requires the development of
more complex models as well as the gathering of empirical
data on genetic variation in fitness, levels of dispersal,
colonization rates and the local steepness of ecological
gradients. In particular, future exploration of models where
colonization between patches is combined with evolutionary
responses to within-patch gradients is likely to prove fruitful.
Understanding these models will involve studying the
transient effects in early generations that our current model
suggests are critical for successful colonization.
We are very grateful to Nick Barton, Sergey Gavrilets, and
Mark Kirkpatrick for discussion of our results, and earlier
drafts of this MS. We also thank Robert Holt, and two
anonymous referees for very helpful comments on the MS.
JB was funded by a Zoological Society of London
Postdoctoral Fellowship. JB and RB were both supported
in part by funding from NERC and the Japanese Society for
the Promotion of Science. MK was supported in part by the
Global COE Program ?Centre for ecosystem management
adapting to global change? (J03) of the Ministry of
Education, Culture, Sports, Science and Technology of
Japan. JP has been supported by EPSRC funded NANIA
network, GRT11777 and IST Austria.
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Additional Supporting Information may be found in the
online version of this article:
Figure S1 Plots of slow adaptation behaviour: TD950 K50,
at generation 500, 1000 and 5000, compared to a limited
adaptation run TD975 K50; the sigmoid shape of the trait
slope indicates slow rather than persistently limited spread
(up to 10 000 generations). Note also the steady increase in
number of clines involved in the change in trait mean over
Figure S2 Plots of limited adaptation and full adaptation for
128 loci runs, for comparison with Figs 2a and b.
Figure S3 The relationship between gradients in trait means
(b) and segregating variance (Vg) in runs with limited
adaptation (a; top panel) or slow adaptation (b: lower panel).
Table S1 Temporal stability of runs at different parameter
combinations for UA, SA and LA, showing rate of
recruitment of clines, and match of observed clines widths,
spacings, genetic variance, trait slope and density to
expectations. These are shown for runs where the trait is
controlled for (a) 64 loci and (b) 128 loci.
Table S2 Expected and observed cline widths, cline spacing,
and segregating variance with increased TD, K, and numbers
of loci. These are shown for runs where the trait is
controlled for (a) 64 loci and (b) 128 loci.
As a service to our authors and readers, this journal provides
supporting information supplied by the authors. Such
materials are peer-reviewed and may be re-organized for
online delivery, but are not copy-edited or typeset. Technical
support issues arising from supporting information (other
than missing files) should be addressed to the authors.
Editor, Marcel Holyoak
Manuscript received 5 October 2009
First decision made 9 November 2009
Manuscript accepted 23 December 2009
494 J. R. Bridle et al.
? 2010 Blackwell Publishing Ltd/CNRS