Deterministic Quantities For Non-Gaussian Stochastic Dynamical Systems And Applications

In natural science and applied science,the dynamical systems are used to describe the evolution of complex phenomena.But dynamical systems are always influenced by many random factors in the environment,especially in biophysics,and the small random disturbance may have a great effect on the whole dynamical system.Thus we use stochastic differential equations to describe dynamical systems with noise disturbance.Dynamical systems driven by Gaussian noise are common.But in some complex systems,such as the regulation of gene expression,transcriptions of DNA from genes and translations into proteins and climate change in the oceans,taking place in a bursty,intermittent,unpredictable manner.It is more suitable to use non-Gaussian Lévy noise to express the fluctuations of these systems.Non-Gaussian asymmetrical Lévy process is more general and more representative of Lévy noise.In this thesis,we first consider a gene regulation system driven by Non-Gaussian asymmetrical Lévy noise,which quantitatively describes the dynamical behaviors of the system in the cases of additive noise and multiplicative noise.Then,by the most probable phase portraits,we consider the stochastic pitchfork bifurcation under the multiplicative Lévy noise.This thesis is organized as follows.In Chapter 1,we introduce some research background of stochastic gene regulation system and stochastic bifurcation.In Chapter 2,we review some basic concepts: Non-Gaussian Lévy process,stochastic differential equation,the background of dynamical systems,non-Gaussian stochastic dynamical systems and some methods to quantify stochastic dynamical systems.At last,we briefly present the theoretical concept of stochastic bifurcation.In Chapter 3,we study a gene regulation system for the concentration evolution of transcription factor activator(TF-A).Two cases are considered: the synthesis rate is affected by additive asymmetric Lévy noise and multiplicative asymmetry Lévy noise.We focus on the effects of the skewness parameter of asymmetric Lévy noise on the dynamical behaviors of the system.The numerical results show that under the interaction of non-Gaussian Lévy noise and nonlinearity,the mean first exit time(MFET)and the first escape probability(FEP)reveal a series of interesting phenomena,for ‘regulating' the likelihood for transcription of genetic regulatory system.For example,in the case of additive noise,the mean first exit time(MFET)generate stochastic bifurcation at the non-Gaussianity index value ? = 1.While in the case of multiplicative noise,the mean first exit time(MFET)and the first escape probability(FEP)both generate stochastic bifurcation at the non-Gaussianity index value? = 1.These phenomena do not occur in the symmetric Lévy noise.The effects of noise enhanced stability system is also studied.Hence we are able to select combinations of these parameters,in order to achieve the desired likelihood for transcription.In Chapter 4,we investigate stochastic bifurcation of a deterministic system under multiplicative Lévy noise.We have nonlocal Fokker-Planck equation to obtain most probable phase portraits.Then by judging whether a state is attracting or repelling nearby trajectories,we give the definition of most probable equilibrium state.Through the qualitative change of most probable equilibrium state,we study the bifurcation phenomenon with respect to vector field parameter r and non-Gaussianity parameter ? in terms of most probable equilibrium state.Meanwhile,we have also compared the system with multiplicative Gauss noise case and the deterministic system.In Chapter 5,we summarize the main contents of this thesis and present some future research topics.