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Is the sky the limit? On the expansion threshold of a species’ range

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.

doi: https://doi.org/10.1371/journal.pbio.2005372.g005