Derivation of Migration / Selection equilibrium

Consider an island adjacent to a mainland, with unidirectional migration to the island.
The fitness values of the AA, AB, and BB genotypes differ in the two environments:
B has high fitness on mainland, and low fitness on island.
Therefore, B is nearly fixed on the mainland, and is rare on the island.

[For this model, allele A is semi-dominant to allele B, so we use t for selection coefficient]

 AA AB BB W0 W1 W2 qinitial Island 1 1-t 1-2t qi  0 Mainland 0 0 1 qm  1

What is the equilibrium frequency f(B) on the island?
change in f(B) from migration: qi = m(qm - qi)
change in f(B) from selection: qi  = -tqi(1 - qi)) / (1 - 2tqi)
-tqi(1 - qi)                     [if tqi << 1]
Then, combined change           qi = m(qm - qi) - tqi(1 - qi)
= mqm - mqi - tqi + tqi2)
= tqi2 - (m + t)qi + mqm

For qi = 0   this can be solved as a quadratic equation for several special cases:
if    t:      qi    qm              migration behaves like mutation
if m >> t:      qi    qm                 mainland alleles 'swamp' island population
if m << t:      qi    (m / t)qm       some equilibrium is achieved, if m is constant

Text material © 2004 by Steven M. Carr