Multi-Type Branching Process

Firstly, in this paper some elementary theories about GW branching processes, some primary conclusions about branching processes in random environments and two-type branching processes are introduced. On these foundations, through using the elementary theories and method for example as limit, derivative, Taylor formula and so on in the higher mathematics, and some knowledge of non-negative matrix, the decision theorem about that a two-type branching process is certainly extinct or not is proved. And the relation between the extinction probabilities and the expectation matrix is given, when the matnx is symmetrical.In addition, a model about mixing or migration of branching processes in random environments is introduced. On a basis of the model, a two-type independent identically distributed branching process in independent identically distributed random environments with 2 states whose reproduction probability generating function derivative is an undetermined constant is given. When this branching process in random environments is subcritical, it is a certainty that its two copy processes can be migrated a supercritical branching process in random environments through giving an probability generating function expectation. However, its two supercritical copy processes can not always be migrated a subcritical branching process in random environments.In the end, the multi-type branching process and the multi-type branching process in random environments are introduced briefly in this paper.