Random Noise Is Driving The Next Class Of Biological Dynamics Of Nonlinear Systems

In this paper, the logistic model, which has been widely used to describe the biological species with breeding and death processes in nature, is intensively studied to investigate its steady states, transient behavior and kinetic properties based on the stochastic dynamical theory. This dissertation is divided into two parts:the first part describes the research background, the status of the study, the theoretical knowledge, the research methods and the related theories of the stochastic dynamical systems in the presence of noises. In the second part, based on the Logistic biological growth equation, the stationary probability distribution function, the mean first passage time and the population correlations of the insect outbreak model in the insect breeding and death processes driven by Gaussian white noise and Gaussian colored noise are closely studied. The associated relaxation time and the normalized correlation function for a tumor cell growth system driven by colored noises are considered further as well.In the first chapter, the background and current situations about the effect of the noise in the nonlinear stochastic system, the theories of Gaussian white noise and colored noise, the basic theoretical knowledge to obtain approximate Fokker-Planck equation with Fox method, Novikov theory and Hanggi theory are introduced. A detailed introduction to the unified colored-noise approximation which is very important used later in the thesis is also given.In the second chapter, the insect outbreak model driven by Gaussian white noise is studied. Starting with the insect outbreak model and based on its corresponding Fokker-Planck equation, the fluctuations on growth rate and predation rate, the effects of noise correlation are considered, respectively. The corresponding stationary probability distribution functions are obtained; the population distributions and the mean value of the insect density are analyzed. According to numerical analysis, the noise strength D and a can induce a phase transition on species population, and they can even lead the insects go into extinction. But the correlation strength λ have an opposite effect, it can make the insect population increase rapidly, and it is a favorable factor to the growth of insect, indicating a positive coherent effect of the correlation between different noises.In the third chapter, steady states and the transient properties of an insect outbreak model driven by Gaussian colored noise are studied. Based on the Fokker-Planck equation in the unified colored-noise approximation, the distribution versus x for different self-correlation time τ1and τ2are analyzed. The effects of the correlation noises are also investigated in this case. By using the steepest-descent approximation, the analytical expression of the mean first passage time is obtained. The effects of D、α、 τ1、τ2、τ3and λ on the population dynamics are then analyzed. When the colored noises present in the system, the self-correlation time τ1、τ2、and τ3performance an enhancement effect on the insect population divergence. In the contrary, the noise intensity D or α plays a negative role on the insect population growth, but the correlation strength λ plays a positive role on the insect population.In the fourth chapter, the associated relaxation time and the normalized correlation function for a tumor cell growth system driven by colored noise are finally studied. Based on the Logistic model as well and using the Novikov theorem and Fox approach, the corresponding Fokker-Planck equation for tumor cell growth is obtained. First, I calculate the steady probability distribution of the tumor cell growth system, and then get the expression of the associated relaxation time and the correlation function in the framework of projection operator method. By numerical simulation, the effects of the noise parameters on the system are analyzed, and some meaningful phenomena are found. The main conclusions read:(1) the damping time,Tc, is increased when the noise parameters λ、α、τ1are improved and the tumor cell numbers can be recovered by them; The increase of D and τ3can accelerate the damping speed while τ2has no effect on the tumor cell numbers.(2) The effect of noise parameters on C(s) is that λ weakens the related activity between two different states at two different times, and enhance the tumor cell growth in the steady state stability; on the contrary,τ1and τ3can enhance the related activity between two different states at two different time while τ2has no effect on the related activity.