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