Analysis On Properties Of Three Stochastic Dynamical Systems Driven By White Noise

During the process of analysis and research of the practical problems, in order to describe the change rules of objective things more accurately, we need to take several stochastic factors into consideration, so the stochastic differential equation is arising as an important branch of mathematics. With the deep exploration of scientists, stochastic model has been widely used in the finance, physics, biology, control, etc. It is well known that noise has the significant meaning to the deterministic stochastic systems. Therefore, it becomes necessary for us to study some interference effects on the stochastic systems.Noise can stabilize an unstable differential system, or affect the related properties of the deterministic differential model, such as stability and boundness, etc. It is exactly based on this consideration that this paper mainly probes into the existence and uniqueness of positive solution, boundedness and stability, and many other properties of three types of stochastic models, by applying the Lyapunov analysis methods, ?Ito formula, and the basic theories of stochastic differential equations, it analyzes the interference effects that white noise has on the several properties of the stochastic systems.Firstly, the paper analyzes the interference effects that the white noise described by Brown motion has on the stochastic Gilpin-Ayala model. Through constructing different Lyapunov functions, stopping times and other stochastic differential equation theories, it discusses the existence and uniqueness of the global positive solution, extinction, some kinds of persistences and stability of the equilibrium points, etc. and it also proves that the noise interference is not conducive to the survival of the population, fatherly, the paper verifies the theoretic results obtained above through a series of numerical simulations.Secondly, the nonlinear stochastic model with two independent Brown noises is studied. Using the stochastic techniques, the stochastic inhibitory and stability of the stochastic is analyzed. As a result, it points out that the polynomial Brown noise can inhibit the explosion of the system solution, it also demonstrates that the linear Brown noise can make system be stable by the general decay rate. At the same time, we give out two examples to verify the theoretic results.Finally, we consider the two-dimensional stochastic predator-prey model with white noise. Applying ?Ito formula, the comparison principle and the theories of stochastic differential equation, this paper investigates the questions about the existence and uniqueness of the global positive solution, and also takes the extinction, boundness and other related properties into consideration. the results shows that the noise interference is not conducive to the survival of the population, lastly but not the least, it analyzes the boundness, the orbit estimation of the solution, etc.