Abstract. The idea of fractional derivatives was raised first by L'Hospital in 1695. The fractional calculus and its mathematical consequences attracted many mathematicians such as Fourier, Euler, Laplace. Various definitions of non-integer order integral or derivative was given by many mathematicians. In this paper, firstly, we discuss the positive integer higher order derivative of a function, and obtain general formula of high order derivative. Secondly, we introduce the definitions of Gr\"unwald-Letnikov, Riemann-Liouville and Caputo. Finally, we point out the relationship between these definitions. Caputo's integral definition and Gr\"unwald-Letnikov integral definition are consistent with the Riemann-Liouville integral definition. When f has 1  m order continuous derivative and, Gr\"unwald-Letnikov positive non-integer order derivative definition is consistent with the Riemann-Liouville positive non-integer order derivative definition.

In order to improve delivery network efficiency and to solve consumer satisfaction problems, parcel locker location optimisation scheme is proposed based on the delivery demand under the e-commerce environment. In this paper, a bilevel programming (BLP) model is established to identify the optimal location for parcel lockers by considering benefits of consumers and logistics planning departments. The upper-level model is to determine the optimal location by minimising the planners’ cost, and the lower one gives an equilibrium demand distribution by minimising the consumers’ pick-up cost. On the special form of constraints, a bilevel genetic algorithm is proposed based on GIS data and a genetic algorithm. Finally, a numerical example is employed to demonstrate the application of the method, which indicates that the model can solve the problem of parcel locker location.

In this paper, the stability of equilibrium points of two species competitive model are studied. On the base of logistic model, the competitive model of two species living in the same natural environment is considered. Firstly, the four equilibrium points of the two species competition model are given, and then stability of each equilibrium point is judged by ordinary differential equation stability theory. Secondly, according to the characteristics of the model, the phase plane is divided into several regions. Finally, matlab programs are wrote, and the numerical solutions of two species competition model are obtained for several different initial conditions, and the graphics of corresponding phase trajectory are drew, and the stability of equilibrium points are analyzed.

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