Title:

Interacting Markov branching processes

In engineering, biology and physics, in many systems, the particles or members give birth and die through time. These systems can be modeled by continuoustime Markov Chains and Markov Processes. Applications of Markov Processes are investigated by many scientists, Jagers [1975] for example . In ordinary Markov branching processes, each particles or members are assumed to be identical and independent. However, in some cases, each two members of the species may interact/collide together to give new birth. In considering these cases, we need to have some more general processes. We may use collision branching processes to model such systems. Then, in order to consider an even more general model, i.e. each particles can have branching and collision effect. In this case the branching component and collision component will have an interaction effect. We consider this model as interacting branching collision processes. In this thesis, in Chapter 1, we firstly look at some background, basic concepts of continuoustime Markov Chains and ordinary Markov branching processes. After revising some basic concepts and models, we look into more complicated models, collision branching processes and interacting branching collision processes. In Chapter 2, for collision branching processes, we investigate the basic properties, criteria of uniqueness, and explicit expressions for the extinction probability and the expected/mean extinction time and expected/mean explosion time. In Chapter 3, for interacting branching collision processes, similar to the structure in last chapter, we investigate the basic properties, criteria of uniqueness. Because of the more complicated model settings, a lot more details are required in considering the extinction probability. We will divide this section into several parts and consider the extinction probability under different cases and assumptions. After considering the extinction probability for the interacting branching processes, we notice that the explicit form of the extinction probability may be too complicated. In the last part of Chapter 3, we discuss the asymptotic behavior for the extinction probability of the interacting branching collision processes. In Chapter 4, we look at a related but still important branching model, Markov branching processes with immigration, emigration and resurrection. We investigate the basic properties, criteria of uniqueness. The most interesting part is that we investigate the extinction probability with our technique/methods using in Chapter 4. This can also be served as a good example of the methods introducing in Chapter 3. In Chapter 5, we look at two interacting branching models, One is interacting collision process with immigration, emigration and resurrection. The other one is interacting branching collision processes with immigration, emigration and resurrection. we investigate the basic properties, criteria of uniqueness and extinction probability. My original material starts from Chapter 4. The model used in chapter 4 were introduced by Li and Liu [2011]. In Li and Liu [2011], some calculation in cases of extinction probability evaluation were not strictly defined. My contribution focuses on the extinction probability evaluation and discussing the asymptotic behavior for the extinction probability in Chapter 4. A paper for this model will be submitted in this year. While two interacting branching models are discussed in Chapter 5. Some important properties for the two models are studied in detail.
