Study For Theoretical Properties Of Stochastic Gilpin-Ayala Models Under Regime Switching

The theory of stochastic differential equations has a great significance, boththeoretically and practically, since it takes the environmental noise into account which isinevitably in the real world when it is used to describe the movement of things, so it iswidely applied in autonomous control, disease prevention and control, economics andecology, especially in analyzing the population dynamics in mathematical ecology.This paper mainly makes use of the stochastic differential equations to investigatethe problem of population dynamics. One of the classical biological models is theLotka-Volterra model, where the growth rate of population is assumed to be linear, whichis not always to be in the complicated ecology system. Therefore, a modified version ofLotka-Volterra model was put forward by biologists Gilpin and Ayala, that is, theGilpin-Ayala biological model.Taking both the white noise and color noise into consideration, this paper hasproposed both stochastic differential equation models based on the deterministicsingle-specie Gilpin-Ayala ordinary differential d t dt x t b t a t x t, one isto assume that the noise affects the competition coefficient a mainly; the other is toassume that the noise affects the intrinsic growth rate b. Then the properties of modelsare studied with the Liapunov function and the theory of stochastic differential equations.For the first model, we prove that the model has a unique positive solution, then focuseon the sufficient conditions for stochastic properties of the model, meantime, we obtainthe asymptotic properties such as the nonpersistent in the mean, weakly persistent andthe upper-growth rate of the positive solution; for the second model, we establish theform of solution, mainly investigate the extinction and the nonpersistent in the mean.