Dynamical Analysis Of Fractional-Order Species Models

In recent years, the research on fractional-order dynamical system attracts increas-ing attentions due to its potential applications in biomedical mathematics and statistical mechanics. The fractional-order species models have become a new challenging and interesting problem in nonlinear science. Many researchers pay their attention to differ-ent kinds of fractional-order species systems, such as fractional-order logistic model, fractional-order predator-prey model and so on. However, most of the current results are for the fractional-order single specie or two species models. The analysis of the fractional-order three species models is few. Therefore, our concern in this work is to investigate the dynamical behavior of the fractional-order three species models.Two kinds of the fractional-order three species models are analyzed. One is the fractional-order three species chain structure model. The other is the fractional-order three species non-chain structure model. Moreover, the dynamical behaviors of the fractional-order species models are discussed with and without intraspecific competi-tion. Based on the qualitative theory of the fractional-order differential equations, the existence and uniqueness of the solutions for the fractional-order three species models are proven. And some sufficient conditions are provided for the local asymptotic sta-bility of the corresponding equilibria. Finally, numerical solutions of these models are given by the predictor-corrector method. The simulations results successfully verify the effectiveness of the analysis.