Limits to adaptation along environmental gradients

Abstract

Why do species not adapt to ever-wider ranges of conditions, gradually expanding their ecological niche and geographic range? Gene flow across environments has two conflicting effects: although it increases genetic variation, which is a prerequisite for adaptation, gene flow may swamp adaptation to local conditions. In 1956, Haldane proposed that, when the environment varies across space, "swamping" by gene flow creates a positive feedback between low population size and maladaptation, leading to a sharp range margin. However, current deterministic theory shows that, when variance can evolve, there is no such limit. Using simple analytical tools and simulations, we show that genetic drift can generate a sharp margin to a species' range, by reducing genetic variance below the level needed for adaptation to spatially variable conditions. Aided by separation of ecological and evolutionary timescales, the identified effective dimensionless parameters reveal a simple threshold that predicts when adaptation at the range margin fails. Two observable parameters determine the threshold: (i) the effective environmental gradient, which can be measured by the loss of fitness due to dispersal to a different environment; and (ii) the efficacy of selection relative to genetic drift. The theory predicts sharp range margins even in the absence of abrupt changes in the environment. Furthermore, it implies that gradual worsening of conditions across a species' habitat may lead to a sudden range fragmentation, when adaptation to a wide span of conditions within a single species becomes impossible.

Keywords: genetic drift; genetic variation; heterogeneous environment; range margin; species’ range.

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 (, 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.

Fig. 2.

Species’ range starts to contract when the effective environmental gradient is steep compared with the efficacy of selection relative to genetic drift: $B\gtrsim 0.15N\sigma \sqrt{s}$, where $B=b\sigma /\left(r^{*}\sqrt{2V_s}\right)$. This threshold is shown by a dashed line. The rate of expansion increases from light to dark blue and rate of range contraction increases from orange to red. Gray dots denote populations for which neither expansion nor collapse is significant at $\alpha =2\text{\%}$. Open dots indicate fragmented species’ ranges (illustrated by Fig. S3). The ranges of the underlying (unscaled) parameters are in the following intervals: $b=\left[0.01,1.99\right]$, $\sigma =\left[0.5,4.8\right]$, $V_s=\left[0.006,8.4\right]$, $K=\left[\mathrm{4,185}\right]$, $r_m=\left[0.27,2\right]$ and $\alpha =\left[0.01,0.39\right]$, $\mu =\left[{10}^{-8},8\cdot {10}^{-5}\right]$; the number of genes is between 7 and 3,971. The selection coefficient per locus is hence in the interval of $s=\left[3\cdot {10}^{-4},0.66\right]$, with median of 0.007. Parameter distributions are shown in Fig. S1.

Fig. 3.

With a steepening environmental gradient, a stable range margin forms when $B\gtrsim 0.15N\sigma \sqrt{s}$ (red dots). (A) The gradient in trait mean follows the environmental optimum (dashed line) until the gradients steepens so that $B\gtrsim 0.15N\sigma \sqrt{s}$, where expansion stops. (B) Population density drops off sharply when the predicted threshold (red dots) is reached. Dashed line gives the predicted population size assuming variation is not eroded by genetic drift. (C) Three representative clines are shown in black, and other clines form the gray background. (D) Adaptation fails when genetic variance fails to increase fast enough to match the steepening environmental gradient (total variance $V_G$ in black; linkage equilibrium component $V_{LE}$ in blue). For all subfigures, the dashed lines give deterministic predictions (Fig. 1 and ref. 11). Parameters are as follows: central gradient $b_0=0.12$, ${\sigma }^2=0.5$, $V_s=0.5$, $r_m=1.06$, $K=53$, $\mu =2\cdot {10}^{-7}$. Time = 100,000 generations; expansion stops after 40,000 generations; A also shows the initial stage in light blue.

Fig. 4.

The phenotypic model predicts a second sharp transition in the dynamics ($B_c$, dashed curve). As the effective environmental gradient $B\equiv b\sigma /\left(r*\sqrt{2V_s}\right)$ increases, the scaled variance $A\equiv V_G/\left(r*V_s\right)$ increasingly deviates from the deterministic prediction with evolvable variance (11) (gray dotted line). The variance decreases due to the combined forces of genetic drift (Fig. S2A) and selection on small transient deviations of the trait mean from the optimum. Once $B_c\gtrsim \sqrt{2A}$ (dashed curve), the population collapses abruptly. Furthermore, no adaptation is maintained beyond the solid line $B_e=\left(2+A\right)/\sqrt{2}$, where the phenotypic model (9) predicts extinction. Open dots (Lower Right) denote fragmented species’ range (Fig. S3). The colors are as in Fig. 2; the threshold when range starts to contract, $B^{*}\approx 0.15N\sigma \sqrt{s}$, is illustrated by the short steep dashed line. Parameters are as follows: b increases from 0.025 to 1.25, ${\sigma }^2=1/2$, $V_s=1/2$, $K=50$, $\alpha =1/10$, $r_m=1.06$, 5,000 generations. Ten replicates are shown for each B.