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This paper finds approximate solutions for Feller-like neutral multi-type branching diffusions in the limit of small mutation rates by solving approximations to the Laplace transformed forward Kolmogorov equation via integration along characteristics. The super-critical diffusion collapses onto a line density aligned with the stationary left eigenvector of the rate matrix, with a rapid change in behavior at a critical time $t_c$ that logarithmically depends on the mutation rate scale $\theta$. First-order approximations to the density and moments are also derived, corresponding to allowing at most one mutation in the coalescent tree.
Neutral multi-type branching diffusions with small mutation rates exhibit a sudden collapse to a stationary density, with critical time depending logarithmically on mutation scale.
Approximate solutions are found for Feller-like neutral multi-type branching diffusions $\big{(}\mathbf{X}(t)\big{)}_{t \in \mathbb{R}_{\ge 0}}$ 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 $\theta$ 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 $t_{\rm c}$, which has a weak logarithmic dependence on $\theta$. First order in $\theta$ approximations to the density and moments of $\mathbf{X}(t)$ 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