The Long Time Behavior Of Stochastic Population Biological Model(With State-dependent Switching)

In this thesis,we first study the almost sure exponentially stability and almost sure exponentially instability of the Logistic population growth model with Brownian motion and state-dependent switching.By means of introducing auxiliary Markov chains and constructing order-preserving coupling,the upper and lower "stability envelopes" are constructed,which lead to systems with "upper and lower" approximating Markov chains.Using the "upper and lower" approximation of the Markov chain system,the Logistic population growth model with Brownian motion and state-dependent switching is obtained as a relatively easy to verify sufficient condition for almost sure exponentially stability and instability.Then the existence of the global positive solution and ergodicity of the single-population biological model with Markov switching are studied.Firstly,it proves that the global positive solution of the model exists,and then by demonstrating the positive recurrent of the process(X(t),γ(t)),the unique stationary distribution is obtained,that is,the process is ergodicity.This thesis is divided into five chapters,the specific contents are as follows:The first chapter mainly introduces the background and significance of the model studied in this thesis,as well as the main ideas,contents and innovations are given.Finally,the working arrangements of the thesis are also introduced.In the second chapter,we first give the preliminary knowledge needed in this thesis,then,we give the structure of order-preserving coupling used in the following research.In the third chapter,the almost sure exponentially stability and almost sure exponentially instability of the Logistic population bio-growth model with Brownian motion and state-dependent switching are studied.By using the generalized Ito formula,the strong majority theorem and stochastic analysis techniques,we obtain the proof of the almost sure exponentially stability and almost sure exponentially instability theorems at equilibrium point,and numerical examples and numerical simulation are given,the correctness of the theoretical results is verified.In the fourth chapter,the existence of the global positive solution and ergodicity of the single-population biological model with Markov switching are studied.By time segmentation,it is proved that the global positive solution of the model exists.Then the Ito formula and stochastic analysis techniques are used to prove the positive recurrent about the process(X(t),γ(t)),Thus,the unique stationary distribution is obtained,that is,the process has ergodicity.The fifth chapter is the conclusion and outlook,summarizes the conclusions of this thesis,and gives some questions worthy of further study.