Research On Branching Models In Random Environments

Branching models in random environments, one of the newest topics in the recent research on stochastic processes at home and abroad, have important theoretical signif-icance, and open up a vast range of prospect for application too. In this paper, we con-sider several classes of specific branching models in random environments, which are branching process in random environment, bisexual branching process with population-size-dependent mating in random environment, age-dependent branching process in random environment, branching random walk in random environment and single-birth chain in Markovian environment.Chapter1, firstly, we briefly introduce the research background and develop-ment history of Galton-Watson branching process, bisexual branching process and age-dependent branching process, then give an introduction of the main content、structure and new ideas of this thesis.Chapter2, branching processes in random environments being Markov chains in random environments, we use the method of studying Markov chains in random envi-ronments to discuss transience and recurrence of branching processes in random envi-ronments. By Markov property of branching processes in random environments, we obtain the relationship between the extinction probability and the transition probability of the model, and give a result similar to Ratio Theorem in classical branching pro-cesses.Chapter3, we introduce the model of bisexual branching processes with population-size-dependent mating in random environments {Zn,n∈N}, prove the Markov prop-erty of the process {Zn, n∈N} and the vector sequence {(Zn-1, Fn, Mn), n∈N+} where Zn and (Fn, Mn) denote the mating units and the number of female and male in nth generation respectively, and discuss the conditions for the dual law to hold and the sufficient conditions for its certain extinction or non-certain extinction, and then discuss the properties of its probability generating function, obtain the super and lower bound of its conditional expectation. Normalized by these two bounds, the conditions of almost sure convergence、L1-convergence and L2-convergence are discussed respec-tively. In an identical and independent environment, normalized by rn with r denoting the expectation of supremum of the conditional mean growth rates per mating unit, the conditions of almost sure convergence、L1-convergence and L2-convergence are discussed, too.Chapter4, in age-dependent branching processes in random environments, let Z(x, t) denote the number of particles alive at moment t whose age do not exceed x. We discuss the properties of the process{Z(x,t):0≤x<t}. We prove that its conditional probability generating function is the unique fixed point of transform0, and can be iterated by any initial function, two specific examples given. We obtain the expressions of its conditional expectation and total expectation, and show that its expectation is the unique non-zero solution of some integral equation.Chapter5, in the model of branching random walks in random environments with locations of particles and the growth of population relying on the environments, Z(n) denotes the point process giving the positions of nth generation particles. Then for arbitrary subset of real numbers, A, Z(n)(A) represents the number of nth generation particles in set A. The Laplace transform of all point processes in each generation, normalized by their own conditional expectation, is a martingale. We consider the sufficient conditions for the martingale convergent uniformly、almost surely and in mean in some domain.Chapter6, by researching the Rθ-chain corresponding to single-birth chains in Markovian environments, the operator Sθ is obtained. By the operator So, we give sufficient conditions for certain extinction and non-certain extinction of the single-birth chain in a Markovian environment where the environment is a Markov chain with two states. At last, an upper bound of its extinction probability and a counter example are shown.