Study On The Related Properties Of A Class Of Birth And Death Processes With Geometric Catastrophes

The birth and death process with catastrophes is further developed on the basis of birth and death processes.It has important practical significance in the reproduction and extinction of biological species.It is a kind of Markov processes with great research value.In this paper,we focus on the specific model of a class of birth and death processes with geometric catastrophes.The extinction probability,extinction time,decay parameter,quasi-stationary distributions and limiting condition distributions are discussed.In the first chapter,we mainly introduce the background knowledge of the birth and death process with catastrophes.The main conclusions of this paper are also maintioned.In the second chapter,firstly,we introduce some basic theories,including continuous-time Markov chains,birth and death processes,single birth processes and quasi-stationary distributions.Then we introduce a specific model of geomet-ric catastrophes,which is studied in this paper.In the third chapter,we mainly investigate the extinction probability and extinction time of model maintioned above.We give the necessary and sufficient conditions of the eventual extinction by two kinds of methods,and the extinction probability is calculated by another method.At the same time,we also study the necessary and sufficient condition for EiT<?.In the fourth chapter,by using the stochastic monotonicity of q-matrix,we provide an upper and lower bound of decay parameter.In the five chapter,we consider quasi-stationary distributions and limiting condition distributions in the special case of d = 0.We prove that there exists a family of quasi-stationary distributions when absorption of the process is cer-tain and the expression of the quasi-stationary distribution generating function is given.In addition,when the absorption is uncertain,the process has limiting condition distributions,and the expression of the generating function is also given.