Migration-selection balance
Whereas genetic variation is constantly removed each generation via
purifying selection, it is continually renewed by mutation and
migration. A multi-niche polymorphism describes how genetic variation
can be maintained in a population though spatially-variable selection,
where low-fitness alleles persist in a population given gene flow
between niches that favour different optimal phenotypes (Maynard Smith
1970, Bulmer 1972). Having obtained estimates of environmental variance
(VE) and additive genetic variance (VA)
for traits in addition the genetic variance in fitness (γ), we can
estimate the rate of migration (m) necessary to sustain these observed
levels of variation given a range in the selection difference among
niches (Bulmer 1985). If θ1 is the optimal phenotype in
niche 1, and θ2 is the optimal phenotype in niche 2,
then we can solve for \(m\), the proportion of the population that must
migrate between niches each generation to maintain the polymorphism
(Equation 2, from Bulmer 1985, Eq 10.65, pg. 181):
\(\left[{V_{A}+2m\left(V_{E}+\gamma\right)}^{2}\right]=m\left(1-m\right)\left(\theta_{1}-\theta_{2}\right)^{2}\left(V_{A}+V_{E}+\gamma\right)\) (2)
We calculated the rate of dispersal necessary to maintain the observed
variation in frond area (Fig. 4A) and root length (Fig. 4B), over a
range of selection differences (\(\theta_{1}-\theta_{2}\)), in the
absence of mutation. The hyperbolic function indicates that, in the
absence of mutation, there rate of dispersal of about 1% that is
sufficient to sustain the observed diversity given a selection gradient
of 7mm2 for frond area, and 15mm for root length). In
a study on the genetic structure of L. minor populations in
central Minnesota, Cole and Voskuil (1996) estimated much lower rates of
gene flow, Nm =0.3, which suggests that mutation must play a critical
role in maintaining the genetic variation we observed.