Generalized Nonlinear Branching Process

The ordinary Markov branching process forms one of the most important classes of stochastic processes. The one-dimensional Markov branching process occupies a major niche in the whole theory of Markov branching processes. The aim of this paper is to consider non-linear Markov branching process with state-dependent immigration and resurrection (i.e. we increase non-linear immigration in non-linear branching processes and rescue a species from extinction). In this paper, we study the basic properties of a special Markov branching process through continuous-time Markov chains.In chapter one, we give a short introduction about Markov branching processes and introduce the model which we discuss in our paper. In chapter two, we give some basic knowledge about continuous-time Markov chains which we can use so that we can discuss branching processes better.In chapter three we discuss regularity and uniqueness of such process, regularity and uniqueness criteria together with the result that immigration does not have any effect on the regularity and uniqueness property are established. In chapter four, we concentrate on discussing the absorbing process for which the most interesting problems are the extinction probability and extinction time. Especially, the explicit expressions of them are presented. In chapter five the explosion time andthe mean global holding time are obtained. In chapter six we discuss theergodicity of such processes and we focus on the recurrent, the positiverecurrent and the equilibrium distribution. In chapter seven the non-linerbranching processes are discussed and the criteria of existence areestablished.