Using the ergodic theory of nonnegative matrices, conditions are obtained for the $L^2$ and almost sure convergence of a supercritical multitype branching process with varying environment, normed by its mean. We also give conditions for the extinction probability of the limit to equal that of the process.
The theory developed allows for different types to grow at different rates, and an example of this is given, taken from the construction of a spatially inhomogeneous diffusion on the Sierpinski gasket.
"On the convergence of multitype branching processes with varying environments." Ann. Appl. Probab. 7 (3) 772 - 801, August 1997. https://doi.org/10.1214/aoap/1034801253