The Asymptotic Behavior Of The Stochastic Gilpin-Ayala Competition Models

Lotka-Volterra model was the most important one of related eco-dynamical system models. Its research now is prevailing. But most of the research focus only on some properties under certain state. Due to professor Mao's efficient and instructive work, the research on stochastical Lotka-Volterra competition model break through. They discuss their proprieties of Moment boundedness; Pathwise estimation; Stochastically ultimate boundedness. However, Gilpin and Ayala have given a modification for Lotka-Volterra competition model, called Gilpin-Ayala competition model. It is more general and realistic, but the research is just beginning. Gilpin-Ayala competition model is a highly nonlinear system, while Lotka-Volterra model is a quadratic system, many instrument can be used for the latter.We reveal that the environmental noise will not only suppress a potential population explosion in the stochastic delay Gilpin-Ayala competition system but will also make the solutions to be stochastically ultimately bounded. Comparing the classical Lotka-Volterra with Gilpin-Ayala competition system, we find that the latter has better proprieties.First, we summarize the existing research on stochastic Lotka-Volterra model. It is mainly on its properties of non-volatile, moment boundedness, and pathwise estimation. At the same time this prepared for our extension and generalizing in the content and form.Next, the Gilpin-Ayala competition system is so complex that we cannot display the all form of it.so we try our best to discuss some of the special case, we specially showed the stability issue of a stochastic Gilpin-Ayala competition model in the third section, Stability itself is a center problem in the stochastic system.Lastly, we respectively demonstrate the properties of stochastic Gilpin-Ayala model without delay, with delay and with markovian switch in the section 4, 5, 6. It is: Existing condition of global positive solutions. We introduce quadratic transform technique used in the third section. Due to generalization of this kind of models. Our result regulate the stronger limit on parameterθ_i. Our result can take the conclusion of stochastic Lotka-Volterra competition model as its special case. Simultaneity it have more form. Solution's stochastically boundedness, approximate estimation, pathwise estimation also is our research's focus. We can obtain various properties reflected by different equation and different on research methods. If compare these and the properties by stochastic Lotka-Volterra model reflected, we can see our extension take foregone results as specials, at the same time foregone methods be continued and altered.All in all, the thesis present several special stochastic Gilpin-Ayala model's properties of stability, stochastic boundedness, approximate estimation, pathwise estimation, obtain same result. on the one hand, They include analogous results of stochastic Lotka-Volterra model; on the other hand, they initiate a controvert on stochastic Gilpin-Ayala model. It's a long way to do this research.