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Abstract. In this paper, a fractional-order model for the spread of pests in tea plants is presented. This model consists of three components: tea plant, pest, and predator. The stability of the boundary and positive fixed points is studied. The global stability properties of the positive equilibrium point are also investigated. In addition, fractional Hopf bifurcation conditions for the model are proposed. The generalized Adams-Bashforth-Moulton method is used to solve and simulate the system of fractional differential equations.

Abstract. In this paper, a fractional-order model for phytoplankton-toxic phytoplanktonzooplankton system is presented. This model consists of three components: phytoplankton, toxic phytoplankton, and zooplankton. The equilibrium points are computed and stability of the equilibrium points is analyzed. In addition, fractional Hopf bifurcation conditions for the model are proposed. The generalized Adams-Bashforth-Moulton method is used to solve and simulate the system of fractional differential equations.

In this paper, the dynamical behavior of a discrete SIR epidemic model with fractional-order with non-monotonic incidence rate is discussed. The sufficient conditions of the locally asymptotic stability and bifurcation analysis of the equilibrium points are also discussed. The numerical simulations come to illustrate the dynamical behaviors of the model such as flip bifurcation, Hopf bifurcation and chaos phenomenon. The results of numerical simulation verify our theoretical results.

Throughout this article, symbolic computation will be used in order to construct a more general exact solutions of the nonlinear evolution equations through a new method called the double auxiliary equations' method, the method represent the study focus of this article. The method has proven applicable and practical through its applications to the generalized regularized long wave (RLW) equation and nonlinear Schrodinger equation.