The Qualitative Analysis Of Three Types Of Stochastic Ecosystems

Epidemiology is to predict the development trend of the disease and to prevent the spread of the disease becoming threat to human survival by studying the spread rule of disease and establishing mathematical model.Population dynamics is a science that studies the change of population quantity and the law of development by finding the interaction mechanism both between and within groups and establishing mathematical model,so as to help us to manage the biological population effectively.In recent years,many scholars both here and abroad have described the dynamical behavior of ecosystems by establishing deterministic mathematical model.However,in the real world,all kinds of random disturbances affect every aspect of biological growth in different degrees,so mathematical models with stochastic interference reflect the reality more accurately.Based on this,this article covers the main research contents as follows:A type of SIQS epidemic model with quarantined item and fluctuation around the transmission coefficient is studied in the first part of this thesis.Our community is separated into three compartments:the susceptible,the infective and the isolated individuals.We have shown that there exists a unique positive solution of this stochastic system by using of Lyapunov functional method.Based on the dynamic propertity,we obtain the threshold of this stochastic SIQS model by the method of the deterministic model,It(?)'s formula and constructing Lyapunov functions and analysis of dynamic behavior of the boundary equilibrium of the system,to guarantee the criteria of which the disease will spread or die out.The second part of this thesis considers a nonautonomous SIRS epidemic model with fluctuation around the transmission coefficient.Our SIRS epidemic model admits the existence and uniqueness of a global positive solution by means of It(?)'s formula and constructing Lyapunov functions,which laid a solid foundation for the further study of the long-term trend of disease.These results show that there exists a positive nontrivial periodic solution and the disease will prevail when the parameters are controlled in a suitable range and the sufficient conditions for the extinction of the disease is investigated.One-prey two-predator food chain system is proposed in the third part.The random perturbations is different from the first two chapters It turns out that the stochastic model admits the same positive equilibrium with the corresponding deterministic model.We show this stochastic model admits a global positive solution.By Chebyshev's inequality,It formula,combined with the Lyapunov functional method,we cleverly got a very good property of the global positive solution,that is,stochastically ultimate boundedness.And we further obtain the sufficient condition for the stochastically asymptotic stability of the positive equilibrium.According to what we have observed in three papers,stochastic models retain the properties of the corresponding deterministic models when intensities of white noises are controlled in a suitable range.The original properties of stochastic models are destroyed if intensities of white noises are rather large within a certain range.