Stability In Distribution Of GOMPERTZ Equation With Markovian Switching

The mathematical model has come to play an important role in describing the growthof the population. In recent years, many experts has received lots of achievements in themodels that describing the growth of population. This means that the importance of themathematical models in the real world. Gompertz equation is one of the most importantmodels to describe the growth of single population. It has been almost universally usedto describe the growth of microorganisms and the innovation diffusion such as digitalcellular telephones.In order to let the model obey the real world better, we consider Gompertz equationwith environmental noise that described by Markovian switching. In this paper, we mainlystudied properties and stabilities of the hybrid system with Markov chain.In Chapter 1, it surveys the recent developments of the study achievements andapplications from the point of view of the Gompertz equation and Markovian switching.In Chapter 2, we present some essential theorems about Markov chain and stochasticdifferential equations with Markov chain related to this paper. Including the properties ofMarkov process, the existence , boundedness and stabilities of solution of SDEs withMarkov chain. What's more, we also present the definition related to this paper.In Chapter 3, we present the explicit expression of the unique solution of the modeland prove it is bounded. Meantime, we prove other useful properties of the solution.Specially, we prove it is uniform permanence. This lays the groundwork for provingasymptotically stable in distribution.In Chapter 4, on the basic of properties of solution we get in chapter 3, we prove themain conclusion of this paper: asymptotically stable in distribution. Moreover, we alsogive some examples and give the figures by Matlab.What's more, based on the main conclusion, we make further discussion on theGompertz equation with Markov chain and forecast the long-term behavior of the species.