Two kinds of three-dimensional fractional Lotka-Volterra systems are discussed. For one system, the asymptotic stability of the equilibria is analyzed by providing some sufficient conditions. And bifurcation property is investigated by choosing the fractional order as the bifurcation parameter for the other system. In particular, the critical value of the fractional order is identified at which the Hopf bifurcation may occur. Furthermore, the numerical results are presented to verify the theoretical analysis.

In recent years, fractional calculus has attracted much attention of researchers. It has been pointed out that fractional calculus plays an outstanding role in modelling and simulation of systems, such as viscoelastic systems, dielectric polarization, electromagnetic waves, heat conduction, robotics, and biological systems. In fact, fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes in comparison with the classical integer-order counterparts. Therefore, it may be more important and useful to investigate the fractional systems.

Traditionally, the fractional differential equation defined by mathematicians is a Riemann-Liouville fractional derivative [

As is well known, in the field of mathematical biology, the traditional Lotka-Volterra systems are very important mathematical models which describe multispecies population dynamics in a nonautonomous environment. Many important and interesting results on the dynamical behaviors for the Lotka-Volterra systems have been found in [

Many important results regarding stability of fractional systems have been obtained. For instance, the stability, existence, uniqueness, and numerical solution of the fractional logistic equation are investigated in [

To the best of our knowledge, some papers have concentrated on the dynamic investigation of the fractional population systems [

Motivated by the above discussions, some dynamical properties of two kinds of three-dimensional fractional Lotka-Volterra systems are investigated in this paper. Existence and uniqueness of solutions are considered. Some sufficient conditions are provided for the asymptotic stability of equilibria. Specifically, bifurcation behaviors are analyzed by formulating the critical values of the fractional order at which Hopf’s bifurcations may take place.

The rest of this paper is organized as follows. In Section

Consider a three-dimensional fractional Lotka-Volterra system

In the following, existence and uniqueness of solutions for system (

Here, the fractional Lotka-Volterra system (

For

System (

Let

The linear autonomous system

Let

The proof follows from Theorem

In the following, the stability of system (

Because of the fact that all constant coefficients of system (

For the three-dimensional fractional Lotka-Volterra system (

For

For

For

Similarly, it can be readily derived that the equilibrium

For

For the further dynamic investigation of the fractional population systems, the other fractional Lotka-Volterra systems will be considered in the following section. Particularly, bifurcation properties for the system will be studied in detail.

Consider a three-dimensional fractional Lotka-Volterra system:

On the basis of Theorem

It is clear that there are eight equilibria for system (

In the following, by choosing the fractional order

The positive equilibrium

For the characteristic equation (

In addition, by analyzing the condition (iii) of Proposition

With respect to system (

The conclusions (a) and (c) are obvious. For the statement (b), due to

It is apparent that the critical value satisfies

Under the situation of statement (b), a bifurcation phenomenon must happen at the critical value

According to Proposition

With respect to system (

According to the condition (i), the equilibrium

According to the statement of Theorem

The analysis of periodic solutions in fractional dynamical systems is a very recent and promising research topic. As a consequence, the nonexistence of exact periodic solutions in time invariant fractional systems is obtained [

Even though exact periodic solutions do not exist in autonomous fractional systems [

In this paper, an Adams-type predictor-corrector method is used for the numerical solutions of fractional differential equations. This method has been introduced in [

For system (

The trajectory of system (

For system (

The solution of system (

When

When

In this paper, two kinds of three-dimensional fractional Lotka-Volterra systems have been studied. The main results are divided into two parts. On the one hand, for system (

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Nature Science Foundation of China under Grant no. 11371049 and the Science Foundation of Beijing Jiaotong University under Grant 2011JBM130.