Table 1.

Definition of parameters

Parameter	Dimension†	Definition
B	$Z/D$	Gradient in the environmental optimum
$V_s$	$Z^{2}T$	Variance of stabilizing selection
σ	$D/\sqrt{T}$	Dispersal per generation
K	$T/D$	Carrying capacity for a well-adapted phenotype
$r_m$	$1/T$	Maximum exponential growth rate
α	Z	Allelic effect
μ‡	$1/T$	Mutation rate
s	$1/T$	Selection per locus: $s\equiv {\alpha }^2/\left(2V_s\right)$
$r*$	$1/T$	Rate of return to equilibrium population size: $r*\equiv {-N\hspace{0.17em}\partial \overline{r}/\partial N|}_{N\to \widehat{N}}=r_m-V_G/\left(2V_s\right)$
B	—	Effective environmental gradient,
		$B=b\sigma /\left(r*\sqrt{2V_s}\right)$
$N\sigma \sqrt{s}$	—	Efficacy of selection relative to strength of genetic drift
$N\sigma \sqrt{r*}$	—	Efficacy of population regulation relative to demographic stochasticity

Three scale-free parameters, B, $N\sigma \sqrt{s}$, and $N\sigma \sqrt{r*}$ (bottom), describe the system; the top section gives seven parameters of the model before rescaling and the middle section defines two important composite parameters.

^†Dimensions: T stands for time, D stands for distance, and Z stands for trait. Note that, with a Poisson number of offspring, the effective population size $N_e$ (which measures rate of genetic drift/coalescence) is identical to the N that regulates population growth due to crowding: hence, both carrying capacity K and population density N have units of $T/D$.

^‡Mutation rate μ is set to be small, with minimal contribution to the dynamics, and, hence, $\mu /r*$ is neglected in the rescaled parameterization.