The Properties Of Interacting Branching Collision Processes With Immigration

In this paper,we consider Interacting Branching Collision Processes with Im-migration (BCPI),which consists an ordinary Markov branching process,a collision branching process and a state independent immigration process.It is necessary to research the related properties for the importance in biology and physics field-s.The monotonicity and extinction probabilities are primarily presented in this paper.In the first chapter, we introduce the background,significance,history,current situation and primarily conclusion about Branching process,Collision branching process,Interacting branching collision process with immigration.The second chapter is mainly talk about the basic definition and properties of Continuous-time Markov chains,Collision branching process and Interacting branching collision Process.The third chapter introduces the model,regularity,existence and uniqueness, monotonicity and duality,the condition of extinction by probability l,the upper bound of extinction probability, which expand the properties of BCPI.In the fourth chapter,we talk about the Interacting branching collision pro-cesses with immigration and resurrection, It adds to the resurrection when the system is empty.It has no absorption,so we research the recurrence and positive recurrence of the process.The fifth chapter is summary and expectation,a second order differential e-quation is constructed to obtain the extinction probability and expected extinction time,but I can’t solve it,so I expected solve it perfectly by all the wisdom.