Age-dependent Branching Processes With Random Environments

The model of Bellman and Harris is a very important model in continuous branching process and is widely used in biology,demology,chymistry and physics, which has had a lot of results by far. In this model, each particle from different generations has the same life distribution and the same probability mechanism to give birth to the next generation particles. This temporal homogeneous assumption is binifited for us to simplify the mathematical theory and actual application. However, in most reproduction in nature, when a particle gives birth to new particles, it will be affected by other ones. As a result, we meet with certain restrictions in application. Hence Professor Li improves Bellman-Harris model and puts forward a new model: age-dependent branching process with random environments. Now Bellman-Harris model has became a special case of the new model. The advent of new mode makes up the Bellman-Harris model in some sense, which is restricted in application because of its homogeneous assumptions. Professor li has extended many conclusions of the Bellman-Harris model to the new model. Based on the reserch of professor li, I investigate the {Z ( x , t )}t≥0 process and the explosive problem of the {Z (t )}t≥0process,( Z (t )is defined for the total number of survival particles at the moment t; Z ( x , t ) is defined for the total number of survival particles at the moment t,whose age is less than x)I also extend the relevant conclusions to the new model.