Dynamical Analysis Of Fractional Predator-prey Systems

In recent years, fractional dynamical system has been an active research topic be-cause of its great potential applications in biomedical mathematics, sociology, statistical mechanics and so on. As a new research direction of nonlinear science, it is important to study fractional population systems in both theory and applications. Now, the main concerns on fractional population systems are the single specie or two species systems whose stability and numerical solutions are researched. However, the analysis of frac-tional three species systems is few. In this paper,3-dimensional fractional predator-prey systems are considered, four kinds of fractional predator-prey systems are proposed and carefully studied.On the one hand, two kinds of fractional three species predator-prey systems are formulated, and the stability and bifurcation properties are investigated, respectively. For one fractional predator-prey system with interspecific competition, the asymptotic stability of the equilibria is analyzed by providing some sufficient conditions, and the numerical simulation is presented to demonstrate the validity and feasibility of the pro-posed conditions. For another fractional predator-prey system with intraspecific compe-tition, bifurcation behavior is studied by choosing the fractional order as the bifurcation parameter, and the critical value of the fractional order is identified. Particularly, Hopf bifurcation phenomenon is obtained in numerical simulation.On the other hand, two kinds of fractional stage-structured predator-prey systems with delay are provided, and the stability is analyzed. Sufficient conditions are derived for the asymptotic stability of the equilibria based on the stability theorem of fraction-al systems with delay, and the effectiveness of the proposed conditions is verified by the numerical results. Moreover, the numerical simulation is illustrated to discuss the impact of delay on the convergence rate of the system.