Fig 1.
Assuming that genetic variance is fixed, deterministic theory [24] predicts that there are two modes of adaptation to an environmental gradient. When the effective environmental gradient B is steep relative to the genetic potential for adaptation A, clinal adaptation fails, and the population only matches the optimum at the very centre of its range (limited adaptation). These parameters can be understood as fitness loads scaled relative to the strength of density dependence r* (see [24, 30] and [31, Appendix D]). A is a measure of standing load due to genetic variance Ar*, and B is a measure of dispersal load B2r*2—the maladaptation incurred by dispersal across heterogeneous environment. Thus, conversely, when the standing load is large relative to the dispersal load A>B2/2, a population adapts continuously, gradually expanding its range (uniform adaptation). Black dashed lines depict the trait optimum; blue lines depict the trait mean. Population density is shown in grey: it has a sharp and stable margin for limited adaptation, but it is steadily expanding under uniform adaptation. Two subpopulations (or perhaps species) are given for illustration of limited adaptation—depending on further parameters and initial conditions (discussed in this study), a wide species’ range with uniform adaptation can collapse to a single population or fragment to multiple subpopulations.
Fig 2.
Two dimensionless parameters—the neighbourhood size 𝒩 and the effective environmental gradient B —give a clear prediction whether a species’ range can expand.
The red line shows the fitted boundary between expanding populations (in blue) and collapsing ranges (red hues): populations expand above the expansion threshold when 𝒩 ⪆ 6.3B + 0.56. The grey region gives 95% bootstrap confidence intervals, whilst the dashed lines depict the predicted expansion threshold for weak selection, s/r*<0.005 (− −), and for strong selection, s/r*>0.005 (— —). Stagnant populations, changing by less than 5 demes per 1,000 generations, are shown in grey. Solid (blue, grey) dots depict populations with uniform adaptation (illustrated by Fig 3). Open circles denote populations in which continuous adaptation has collapsed and the population consists of many discrete phenotypes adapted to a single optimum each (limited adaptation, Fig 4), whilst local genetic variance is very small. (Specifically, these are defined by mean heterozygosity smaller than 10% of the predicted value in the absence of genetic drift.) Simulations were run for 5,000 generations, starting from a population adapted to a linearly changing optimum in the central part of the available habitat. Populations that went extinct are marked with a black dot. Note that both axes are on a log scale. The top corner legend gives the colour-coding for the rate of range collapse and expansion in units of demes per generation; rates of collapse are capped at −1. The expansion threshold is fitted as a step function changing linearly along B: all blue dots are assigned a value of 1; all red dots and open circles are assigned a value of 0. The expansion threshold has a coefficient of determination R2 = 0.94, calculated from 589 simulations (all but well-adapted stagnant populations). Data for this figure—and all subsequent ones—are deposited at Dryad Digital Repository, https://doi.org/10.5061/dryad.5vv37 [36].
Fig 3.
Uniform adaptation: Above the expansion threshold, the population expands gradually through the available habitat.
(a) Trait (in blue) closely matches the environmental gradient (grey) along the x-axis. (b) Population steadily expands, whilst population density stays continuous across space, with (mean ± standard deviation). The prediction at
is shown by the blue contours; darker shading represents higher density. (c) Adaptation to the environmental gradient is maintained by a series of staggered clines: as each allele frequency changes from 0 to 1, the trait value increases by α. Population starts from the centre (blue hues reflect initial cline position relative to the centre of the range), and as it expands, new clines arising from loci previously fixed to 0 or 1 contribute to the adaptation (in red hues). At each location, multiple clines contribute to the trait (and variance); clines are shown at Y = 25. (d) Genetic variance changes continuously across space with mean
and stays slightly lower than is the deterministic prediction (green contours, VG = 0.045; higher variance is illustrated by darker shading). Deterministic predictions are based on [13] and are explained in the Methods section, along with the specification of the unscaled parameters. The population evolves for 2,000 generations, starting from a population adapted to the central habitat. The predicted neighbourhood size is
; effective environmental gradient is B = 0.48.
Fig 4.
A metapopulation can form when the population is below the expansion threshold throughout its range.
The population fragments rapidly (within tens of generations) to small patches of tens to a few hundred individuals whilst losing local adaptive variation. In two-dimensional habitats, such a metapopulation with limited adaptation can persist for a long time. Nevertheless, the population very slowly contracts, eventually forming a narrow band adapted to a single optimum. (a) The distribution of phenotypes across space is fragmented. (b) The subpopulations are transient, although they are stabilised by dispersal across space, especially along the neutral direction with no change in the optimum (Y). Locally, the population density may be higher than under uniform adaptation; blue contours depict the deterministic prediction for population density under uniform adaptation, N = 3. The realised density is about (standard deviation); darker shading represents higher density. (c) The adaptive genetic variance is low on average (
)—about an order of magnitude lower than would be maintained by gene flow under uniform adaptation (shown in green contours, VG = 0.23). Typically, only a few clines in allele frequencies contribute to the genetic variance within a subpopulation. The parameterisation and predictions are detailed in the Individual-based simulations section of the Methods; predicted neighbourhood size is
, effective environmental gradient is B = 0.48. Shown here after 5000 generations—the population collapses to a narrow band (at X = 45) after a further 20,000 generations and then appears persistent.
Fig 5.
On a steepening environmental gradient, a sharp and stable range margin forms near the expansion threshold.
This illustrative run shows that as the effective environmental gradient steepens away from the central location, adaptive genetic variance must increase correspondingly in order to maintain uniform adaptation. (a) Median population density stays fairly constant across the range (blue dots), following the deterministic prediction (, blue dashed line). Genetic variance (black dots) increases due to gene flow across the phenotypic gradient—the deterministic expectation is given by the grey dashed line (see Model section of Methods for details). Yet, as the environmental gradient steepens, genetic variance fails to increase fast enough, and near the expansion threshold, adaptation fails. The dotted line gives the corresponding critical genetic variance, below which only limited adaptation is expected in a phenotypic model with a fixed genetic variance (
, in which
is the standing genetic load; [24]). (b) As the environmental gradient steepens, the frequency of limited adaptation within the metapopulation increases (black and grey), and hence neutral variation decreases (blue). The black line gives the proportion of demes with limited adaptation after 50,000 generations, when the range margin appears stable; grey gives the proportion after 40,000 generations (depicted is an average over a sliding window of 15 demes). The median is given over the neutral spatial axis Y (with constant optimum); the trait mean, the population trait mean, variance, and population density in two-dimensional space is shown in S3 Fig, which also lists all the parameters.
Fig 6.
Dispersal aids adaptation in small populations because the neighbourhood size 𝒩 increases with the square of generational dispersal, whereas the effective environmental gradient B increases only linearly.
This chart shows a set of simulated populations, with dispersal increasing from left to right and bottom to top. The hue of the dots indicates the rate of expansion (light to dark blue and purple) or collapse (orange to red). The rates of expansion and collapse are shown in dependency on B and 𝒩. Open circles indicate limited adaptation, in which a species’ range is fragmented and each subpopulation is only matching a single optimum, whilst its genetic variance is very small. As dispersal increases, population characteristics get above the expansion threshold (dashed line), and hence, uniform adaptation becomes stable throughout the species’ range. Local population density stays fairly constant, around N = 3.5, whilst total population size increases abruptly above the expansion threshold as the population maintains a wide range (not shown). Parameters for these simulations are given in the Individual-based simulations section of the Methods; the scaling of 𝒩 and B with dispersal σ is clear from the Methods, section Parameters. The rate of range change is not significantly different from zero for the first three simulations above the expansion threshold; black centre (bottom left) indicates extinction.
Table 1.
Three scale-free parameters: B, 𝒩, and s/r* (top) describe the system.
T Middle section gives informative derived parameters. The bottom section gives seven parameters of the model before rescaling, in which the seventh parameter, mutation rate μ, can be neglected because variance maintained by mutation-selection balance, VG, μ/s = 2μVsnl, is typically much smaller than variance generated by gene flow across environments, The middle column gives the dimensions of the parameters.