Stochastic Stability And Stochastic Hopf Bifurcation In A Stage-Structured Singular Biological Economical Model With Stochastic Fluctuations

Biomathematics is an cross discipline between mathematics and life sciences.Many scholars established a number of biological mathematical models from the perspective of mathematics and mechanics and achieved some valuable results.However,some researchers also have begun to realize the impact of biological ecosystem development is multifaceted,these external factors often form a force such noise so that biological variables exhibit randomness and uncertainty.In this case,the determined and generalized model of biological systems has been unable to reflect the real movement of biological and ecological systems.Because of this,discussions of singular biological economical model with stochastic fluctuations have gradually become the frontier of biological systems,and rapid progress has been made.The presented thesis mainly discusses the modeling for a stage-structure singular stochastic bio-economic system,which is described by differential-algebraic equations because of economic factors and random excitation and the stability and bifurcation problems on the basis of stochastic averaging method,nonlinear stochastic dynamic system theory,singular system theory and other related control theory.The main works are summarized as follows:Firstly,the formation,development and application of singular system theory and biological mathematics are introduced.Then a briefly introduction of basic knowledge is given as a foundation of the research.Secondly,the local stability and bifurcations of a staged-structure singula bio-economic system are discussed in the case of model without stochastic fluctuations.Furthermore controller is designed to eliminate the singularity induced branching and pulse phenomena.Thirdly,simplifying the former model with stochastic factor through a stochastic averaging method,a two-dimensional diffusion process of averaged amplitude and phase can be obtained.Stochastic stability and stochastic Hopf bifurcation can be analytically determined based on the singular boundary theory of diffusion process,the Maximal Lyapunov exponent and the invariant measure theory.The critical value of the stochastic Hopf bifurcations parameter was obtained and the conclusion that the position of Hopf bifurcation drifting with the parameter increase is presented as a result.Numerical simulation results are presented to verify the effectiveness of the conclusions.