Island-island migration

Both islands move

In the continent-island model only the island changes. Here two populations exchange migrants, so both change. The Island-Island module follows one locus with alleles A and a in two infinitely large populations. Drift, selection and mutation are absent. Each generation a fraction m₁ of island 1 is replaced by migrants from island 2, and a fraction m₂ of island 2 is replaced by migrants from island 1.

The updates are p₁(t + 1) = (1 − m₁)p₁(t) + m₁p₂(t) and p₂(t + 1) = (1 − m₂)p₂(t) + m₂p₁(t). The defaults are p₁ = 0.8, p₂ = 0.2 and m₁ = m₂ = 0.05.

The difference shrinks, the weighted mean stays

Subtract the two updates. The difference p₁ − p₂ is multiplied by λ = 1 − m₁ − m₂ every generation. With the defaults λ = 0.9, so the difference falls from 0.6 to 0.54 in generation 1, and it halves every ln(0.5) / ln(0.9) = 6.6 generations. One-way flow at m = 0.05 needs 13.5 generations for the same halving. Two-way flow is about twice as fast because both islands move toward each other.

The meeting point comes from a quantity that migration never changes. It is m₂p₁ + m₁p₂, the two frequencies weighted by the opposite migration rates. Setting both islands equal to p_eq in that quantity gives p_eq = (m₂p₁ + m₁p₂) / (m₁ + m₂). With equal rates this is the plain average of the two starting frequencies, (0.8 + 0.2) / 2 = 0.5.

The two results separate speed from destination. The speed depends on the sum m₁ + m₂. The destination depends on the ratio of the rates. The runs m₁ = m₂ = 0.05 and m₁ = 0.01 with m₂ = 0.09 both have a sum of 0.10 and λ = 0.9, so they close the gap at the same pace. They end at 0.5 and 0.74.

Special cases and overshoot

Set m₁ = 0 and island 1 never receives migrants, so it stays at 0.8. It is a continent. Island 2 with m₂ = 0.05 then passes 0.5 in generation 14 (0.5074) and reaches 0.7964 in generation 100, the same numbers as the continent-island page.

Set m₁ = m₂ = 0.5 and λ = 0. Both islands read 0.5000 from generation 1, because each generation replaces half of each gene pool with the other half and the mixture is already even.

Set m₁ + m₂ above 1 and λ turns negative. With m₁ = 0.8 and m₂ = 0.6, λ = −0.4. Island 1 reads 0.8, 0.3200, 0.5120, 0.4352 and 0.4659 in generations 0 to 4. Each generation it overshoots the meeting point p_eq = 0.4571 and then returns, and the swings shrink by 0.4 each time. Migration rates that high replace most of an island each generation, which real populations rarely do.