Supplementary Text: Speed of invasion of an expanding population by a horizontally-transmitted trait

The three relevant solutions of these equations are (i) (N T , N ∗ A) = (0, 0): this is the trivial case where both populations are zero. (ii) (N T , N ∗ A) = (K, 0): this solution describes a domain full with individuals that do not carry the trait; no trait-carrying individuals are present. (iii) (N T , N ∗ A) = (K,K − β/γ): this solution is the most interesting one for our purposes. It describes a domain which contains coexisting subpopulations of trait-carrying and non-trait carrying individuals. This solution is only valid if γ ≥ β/K (such that N A > 0) i.e. if the rate of horizontal transmission is high enough that the trait can be sustained in the population. We note that mathematically there is also another solution (N T , N ∗ A) = (0, (Kα−β)/γ); however this is not physically relevant, since it implies that N A = −N ∗ B i.e. one of the subpopulations has a negative density.