Boundary Behaviors For A Continuous-state Nonlinear Neveu’s Branching Process

In recently,branching process has been one of the hot topics in probability theory,which can often be used to describe the final size of population,which makes the study of branching process has theoretical significance and practical application value.Discrete state branching process is suitable for describing the evolutionary behavior of small population systems and continuous-state branching process is suitable for describing the evolutionary behavior of complex population systems.In real life,when the model system with interaction is generated,the continuous-state nonlinear branching process is proposed.When the stable index ? = 1,it is a continuous-state nonlinear Neveu’s branching process.In this thesis,the boundary behavior of continuous-state nonlinear Neveu’s branching process is studied.Firstly,a class of continuous-state nonlinear Neveu’s branching processes is constructed based on the solutions of stochastic integral equations driven by the Poisson stochastic measure.The discriminant criteria of the uniqueness problem of Markov jumps by Academical Mufa Chen are extended,and the discriminant criteria of non-extinction,extinction,non-explosion,explosion and staying infinity are given.Secondly,using the extended Chen criterion,the necessary and sufficient conditions for extinction,explosion and coming down from infinity of continuous nonlinear Neveu’s branching process under power function are established,and the logarithmic-logarithmic test function and power function test function are selected to find out the violent influence of different forms of large jumps and small jumps on boundary behavior.Finally,the boundary behavior of the continuous-state nonlinear Neveu’s branching process in the case of power function is extended to the boundary behavior of the continuous-state nonlinear Neveu’s branching process in the general case,and the corresponding relationship between them is given.