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In this paper, we deduce a condition for a strategy S 1 to be more abundant on average at equilibrium under weak selection than another strategy S 2 in a population structured into a finite number of colonies of fixed proportions as the population size tends to infinity. It is assumed that one individual reproduces at a time with some probability depending on the payoff received in pairwise interactions within colonies and between colonies and that the offspring replaces one individual chosen at random in the colony into which the offspring migrates. It is shown that an expected weighted average equilibrium frequency of S 1 under weak symmetric strategy mutation between S 1 and S 2 is increased by weak selection if an expected weighted payoff of S 1 near neutrality exceeds the corresponding expected weighted payoff of S 2 . The weights are given in terms of reproductive values of individuals in the different colonies in the neutral model. This condition for S 1 to be favoured by weak selection is obtained from a strong migration limit of the genealogical process under neutrality for a sample of individuals, which is proven using a two-time scale argument. The condition is applied to games between individuals in colonies with linear or cyclic dominance and between individuals belonging to groups represented by subsets of a given set.

Let 𝒜 be an alphabet of size n ≥ 2. Our goal in this paper is to give a complete description of primitive words p≠q over 𝒜 such that pq is non-primitive. As an application, we will count the cardinality of the set ℰ(l,𝒜) of all couples (p, q) of distinct primitive words such that |p| = |q| = l and pq is non-primitive, where l is a positive integer. Then we give a combinatorial formula for the cardinality ε(n, l) of this set. The density in {(p, q) : p, q are distinct primitive words and |p| = |q| = l} of the set ℰ(l,𝒜) is also discussed.

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