Qualitative Research On Solutions Of Several Stochastic Ecological Mathematical Models

There always exist stochastic perturbations in the real ecological systems. In order to more accurately describe the systems, and precisely reveal their variation, it is necessary to take environmental perturbations fully into account during modeling systems, such as white noise, color noise, impulsive phenomena, etc.. This dissertation is mainly concerned the qualitative behavior of solutions for several classes of stochastic ecological mathematical models.This dissertation are summarized as follows:1. The research background, significance and status are reviewed.2. Some preliminaries are briefly introduced, such as probability theory, stochastic processes, stochastic calculus and stochastic differential equations, etc..3. A stochastic ratio-dependent predator-prey model with varible coefficients is investigated. By comparison theorem of stochastic equations and Ito formula, the global existence of a unique positive solution of the ratio-dependent model is obtained. Besides, some results are established such as the stochastically ultimate boundedness and stochastic permanence for this model.4. Stabilization of impulsive stochastic functional differential equation is dis-cussed. Based on the Razumikhin techniques and methods of Lyapunov functions, several criteria on pth moment and almost sure exponential stability are established. Our results show that, solutions of stochastic functional differential systems which are unstable can be exponentially stabilized by impulses.5. Qualitative behavior of the delay logistical model under regime switching dif-fusion in random environment is investigated. By using generalized Ito formula, Gron-wall inequality and Young’s inequality, some sufficient conditions for existence of global positive solutions and stochastically ultimate boundedness are obtained, respectively. Also, the relationships between the stochastic permanence and stationary distribution of the Markovian chain, and extinction and stationary distribution of the Markovian chain are revealed. Moreover, the asymptotic estimations of solutions are investigated by virtue of M-matrix method, Borel-Cantelli Lemma and Chebyshev’s inequality.6. A delay Lotka-Volterra model under regime switching diffusion in random en-vironment is investigated. By using generalized Ito formula, Gronwall inequality and Young’s inequality, some sufficient conditions for existence of global positive solutions, stochastically ultimate boundedness and extinction are obtained, respectively. Mean-while, permanence and asymptotic estimations of solutions are investigated by virtue of M-matrix method, Borel-Cantelli Lemma and Chebyshev’s inequality.