Fig. 1.

Illustration of the individual-based model at the limit of weak genetic drift, when the species’ range keeps expanding as predicted by the deterministic model (11). (A) Trait mean $\overline{z}$ matches the optimum $\theta =bx$ (dashed line), shown for the starting population (light blue) and after 5,000 generations (dark blue). The spread of the trait values z for all individuals is shown with dots. (B) Local population size is close to the deterministic prediction (dashed line) $\widehat{N}=K{r}^{*}/r_m=K\left[1-\sigma b/\left(2\sqrt{V_s}{r}_m\right)\right]$, where K gives the carrying capacity for a well-adapted phenotype. (C) Clines for allele frequencies are shown by thin black lines; the predicted clines (dashed) have widths $w_s=4\sigma /\sqrt{2s}$ and are spaced $\alpha /b$ apart. (D) Total genetic variance is shown in blue, and the linkage equilibrium component is shown in black; the dashed line gives the prediction $V_G=V_{LE}=b\sigma \sqrt{V_s}$: each cline contributes genetic variance $V_{G,i}={\alpha }_i\sigma \sqrt{V_s}$, and per unit distance, there must be $b/\alpha$ clines if the trait mean matches into the optimum (11, p. 378). Parameters, defined in Table 1, are as follows: $b=0.1$, ${\sigma }^2=1/2$, $V_s=2$, $r_m=1.025$, $K=300$, $\alpha =1/\sqrt{20}$, $\mu ={10}^{-6}$, 5,000 generations.