We consider a multitype branching process in which particles move according to probability laws that depend on the type of a particle. The composition of the population, namely the proportions of numbers of particles of different types, varies over the region of motion. In this paper, we study the growth rate and population composition of the process throughout its domain of supercriticality.
M. Bramson. P. Ney. J. Tao. "The Population Composition of a Multitype Branching Random Walk." Ann. Appl. Probab. 2 (3) 575 - 596, August, 1992. https://doi.org/10.1214/aoap/1177005650