Multi-type branching diffusions with small mutation rates

Approximate solutions are found for Feller-like neutral multi-type branching diffusions X(t) t in R 0 in the limit of small mutation rates. The method employed involves solving approximations to the Laplace transformed forward Kolmogorov equation by integrating along characteristics. To leading order in the scale θ of the overall mutation rate the super-critical diffusion is found to collapse onto a line density aligned with the stationary left eigenvector of the rate matrix following a rapid change of behaviour at a critical time t c, which has a weak logarithmic dependence on θ. First order in θ approximations to the density and moments of X(t) are also determined. The first-order approximation corresponds to allowing at most one mutation in the coalescent tree of a sample and is valid for t < t c in the supercritical case, and more generally for the critical and sub-critical cases.