Research On The Dynamic Behavior Of Logistic System Under The Excitation Of Random Noise

Noise has been proven to exist in all aspects of nature.Reasonable noise has a positive effect on nonlinear systems under certain conditions.It has been widely used in various systems in nature and human society.The Logistic system model can describe the growth process of population quantity,predict the environmental carrying capacity or the social possession of a durable commodity,so this essay focuses on the effect caused by the combination of them.This essay mainly studies the non-stationary state of Logistic system under the excitation of non-Gaussian noise and Gaussian white noise.In addition,stationary probability density,stochastic resonance and mean first passage time of a tumor growth model under the excitation of Lévy noise and Gaussian white noise are investigated.The main research contents of this paper are as follows:1.We study the non-stationary state evolution problem of Logistic system under the excitation of non-Gaussian noise.First,the path integration method and the space expansion method are used to perform an equivalent transformation on the system,and then the expansion theory of the Green function is used to linearize the nonlinear dynamic system in the initial region.Eigenvector theory gives approximate expressions of unsteady solutions p(x,t).Next,the effects of non-Gaussian noise correlation time?,noise intensity D and non-Gaussian noise deviation parameter q on p(x,t)and the first moment are analyzed with the Logistic model.The results show that when the yield x is large,p(x,t)has a single peak phenomenon with the increase of time t.In a certain range,p(x,t)decreases monotonously with the increase of D,and increases with the increase of ?.The first moment decreases monotonously with the increase of D and q,monotonically increases with the increase of ?.When the Logistic model is used to describe the product yield growth,the non-steady state solution can better reflect the evolution behavior of the product yield near the unstable point.Furthermore,the stationary probability density of a tumor growth model under the excitation of Lévy noise and Gaussian white noise is investigated.First,a tumor cell growth system stimulated by Lévy noise and Gaussian white noise is given.Then,the characteristic function of Lévy noise is introduced briefly and the Lévy random number is generated by Janick-Weron algorithm.The steady-state probability density function of the system is simulated by the fourth-order Runge-Kutta method.The results show that both Lévy noise and Gaussian white noise can give rise to a noise-induced transition of the system,smaller stability index ? and Lévy noiseintensity D enhance the likelihood of tumor cell death.2.We aims to cope with stochastic resonance and mean first passage time of a tumor growth model under the excitation of Lévy noise and Gaussian white noise.The fourth-order Runge-Kutta method and the Janick-Weron algorithm are used to simulate the signal-to-noise ratio(SNR),and mean first passage time(MFPT)is studied as a function of Lévy noise intensity D and Gaussian white noise intensity Q.We find SR phenomenon and calculate the MFPT.Moreover,we discuss the effects of Lévy noise and Gaussian white noise on tumor cell growth.The results indicate that both noise parameters and system parameters can induce the occurrence of stochastic resonance.Although the Lévy noise intensity has different effects on the transition between the two states,it is not conducive to the extinction of tumor cells.When the number of tumor cells is at a high level,increasing the Gaussian white noise intensity has a positive effect on the reduction of tumor cells.Stability index ? and skewness parameter ? work in the opposite way in tumor growth dynamics.