Dynamical Analysis Of Fractional Order Population Models

In recent years, fractional order dynamical system has attracted many reseqarchers' attention and been one of the top research topics because of its wide potential applications in many fields such as ecology, economices, immunology, sociology and so on. As one of the research diection of fractional order dynamical systems, fractional order population systems have important theoretical and practical significance. Now, the main research of fractional odrder population system focus on the stability of equilibrium points and numerical simulations, but the other dynamical behaviors' is rarely analyzed and the stability of fractional order population systems with time delays needs to be futhur analyzed. In this thesis, based on the analysis and summary of research status of fractional order population models, employing the stability and bifurcation of fractional order systems, the author studies the dynamical behaviors of population models. The paper is organized as follows.The first Chapter elaborates fractional order population systems, as well as analyzing the status and progress, then introduces the relevant basic contents of fractional order calculus and expounds the main contents.The seconed Chapter establishes a fractional order predator-prey system with harvesting. Firstly, the nonnegativity and boundedness of solutions for the system is proved. Secondly, based on the stability theory of fractional order system, some conditions for the locally asymptotic stability of the system at the equilibrium point are provided. Finally, according to the LaSalle invariable principle of fractional order system, a sufificient condition for the globally asymptotic stability of the system at the positive equilibrium point is presented, and numerical simulations are provided to verify the correctness of theoretical analysis.The third Chapter studies fractional order food chain model. Fisrstly, the global existence and uniqueness of solution for the system is proved and the boundedness of solution for the system is analyzed. Secondly, using the stability theory of fractional order system, some conditions for the locally asymptotic stability of the equilibrium point are provides. Finally, the bifurcation of the system is discussed by selecting fractional order as a parameter.The forth Chapter talks about fractional order delayed predator-prey system with harvesting. the stability of the system is discussed by analyzing the corresponding characteristic equation and the effect of harvesting on the dynamical behaviors of the system is studied.The fifth Chapter summarizes this dissertation and makes the future research direction.