
W_{AA}  W_{AB}  W_{BB}  q_{init} 
Island  1  1  t  1  2t  q_{i } 0 
Mainland  0  0  1  q_{m } 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 q_{i
}= f(B) = 0 on island
Change in f(B)
from migration: q_{i }= m(q_{m}
 q_{i})
Change in f(B)
from selection: q_{i} = tq_{i}(1
 q_{i})) / (1  2tq_{i})
tq_{i}(1  q_{i})
[if tq_{i
}<< 1]
Then,
combined change
q_{i }=
m(q_{m}
 q_{i})  tq_{i}(1  q_{i})
= mq_{m}  mq_{i}  tq_{i} + tq_{i}^{2})
= tq_{i}^{2}
 (m + t)(q_{i})
+ mq_{m}
For q_{i }= 0
solve as quadratic
equation for several special cases:
if m
t: q_{i }q_{m}
migration
behaves
like mutation
[except: alleles introduced at higher
rate, as diploids]
if m >> t: q_{i }q_{m}
mainland B allele 'swamps' island A allele
if m
<< t:
q_{i }
(m / t)(q_{m})
some equilibrium achieved, iff m constant
Intermediate cases can be simulated
in GSM worksheet
Approach to MutationSelection Equilibrium where
q_{i
}= 0
for t = 0.1 & m = 0.01
~ 0.250
For m = 0.250, q_{I} rapidly approaches g_{M}: migration swamps locally favored allele, prevents local adaptation.
For m = 0.01 or 0.05,
q_{I}
reaches equilibrium at (m / t) as predicted:
deleterious mainland allele maintained at relatively high
frequency.
For m = 0.1 = t, q_{I
}approaches q_{M} but very
slowly. The exact solution with m = 0.1* flattens earlier.