Migration (Gene Flow) / Selection equilibrium

Consider an island adjacent to a mainland, with unidirectional migration from the mainland to the island.
   The mainland is uniformly dark, the island uniformly light
phenotypes are light, BB dark, and AB intermediate

Phenotypes are subject to selection according to degree of crypsis
Fitness of AA, AB, & BB genotypes differ between environments,
      so allele frequency q = f(B) differs between mainland (qm) and island (qi)
      B has high fitness on mainland, low fitness on island
For this model, A semi-dominant to B:
allele additive
: WAB = (1 - t)    WBB = (1 - 2t)
[Note: use t so as not to confuse additive vs recessive s]
Island 1 1 - t 1 - 2t q 0
Mainland 0 0 1 q 1

Migration rate m = fraction of island population newly arrived from mainland
                        [m equivalent to fraction of new alleles arriving from mainland]

Migration / Selection equilibria resemble Mutation / Selection equilibria mathematically, except
                Migration introduces alleles two at a time (BB diploid migrants vs AB gametic mutations)
                Migration rates m >> Mutation rates u

Calculate equilibrium frequency where
qi = f(B) = 0 on island
     Change in f(B) from migrationqi = m(qm - qi)
     Change in f(B) from selectionqi  = -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   solve as quadratic equation for several special cases:
       if    t:      qi    qm              migration behaves like mutation
                                                                 [except: alleles introduced at higher rate, as diploids]
       if m >> t:      qi    qm                 mainland B allele 'swamps' island A allele

       if m << t:      qi    (m / t)(qm)     some equilibrium achieved, iff m constant

       Intermediate cases can be simulated in GSM worksheet

Mutation-Selection equilibrium

Approach to Mutation-Selection Equilibrium where  qi = 0
for t = 0.1 & m = 0.01 ~ 0.250

For m = 0.250, qI rapidly approaches gM: migration swamps locally favored allele, prevents local adaptation.

For m = 0.01 or 0.05, qI reaches equilibrium at (m / t) as predicted: deleterious mainland allele maintained at relatively high frequency.

For m = 0.1 = t,
qI approaches qM but very slowly. The exact solution with m = 0.1* flattens earlier.

Text material © 2021 by Steven M. Carr