| Species dispersal is one of the most ubiquitous prevalent phenomena in nature, it is important for us to know the ecosystem and nature, there are a large number of mathematical models focus on the dynamical behaviors of the system with di?usion, but the research of random population di?usion with white noise is not much. This motivated us to learn the ecosystem more deeply. Based on these study, we also give some intensive research for some population dispersal problem: A class of single-species di?usion in two patches with white noise and time delay and analysis for a delayed predator-prey system with symmetric dispersal between two patches.The main contents in this paper can be summarized as follows:In section 1, we introduce the ecological background of our study. Then, we introduce some current research situation and some research results about the di?usion model. We give the main research system ?nally.In section 2, we propose a class of single-species di?usion with white noise and time delay between two patches. By constructing a suitable Liapunov functional, making full use of the It?o formula and Chebyshev’s inequality we have the existence, uniqueness and global positive of the solution of the system, we also have the bounded, the average integral is bounded and the persistence and extinction. Finally, numerical simulations are presented to illustrate our theoretical results.In section 3, we analysis a predator-prey system with symmetric dispersal between two patches, by constructing a suitable Liapunov functional, the It?o formula and making full use of Chebyshev’s inequality and some analysis skills, we have the positive and bounded of the prey-predator system. Using the comparison theorem of stochastic di?erential equation, the It?o formula and Chebyshev’s inequality we have the persistence and extinction of the prey-predator system. Finally, numerical simulations are presented to illustrate our theoretical results.In section 4, we have a summary. |
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