We consider a multitype population size dependent branching process in discrete time. A process is considered to be near-supercritical if the mean matrices of offspring distributions approach the mean matrix of a supercritical process as the population size increases. We show that if the convergence of the means to the supercritical mean is fast enough and the second moments of offspring distributions do not grow too fast as the population size increases, then the process grows geometrically fast. Similarly to the classical multitype Galton-Watson process, the process grows at the geometric rate determined by the largest eigenvalue of the limiting matrix in the direction of the corresponding left eigenvector.
"Geometric Growth in Near-Supercritical Population Size Dependent Multitype Galton-Watson Processes." Ann. Probab. 17 (4) 1466 - 1477, October, 1989. https://doi.org/10.1214/aop/1176991167