Stability and Bifurcation of Two Kinds of Three-Dimensional Fractional Lotka-Volterra Systems

Abstract

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.

1. Introduction

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 [1]. But this definition is less popular because of the fact that it requires initial conditions to be expressed in terms of fractional integrals and their derivatives. Meanwhile, there is no known physical interpretation for such types of initial conditions. In contrast, the alternative definition of the fractional derivative given by Caputo [2] has the advantage of only requiring the initial conditions given in terms of integer-order derivatives. These initial conditions of integer-order derivatives can be measured accurately and represent well-understood features of a physical situation. In [2], it has been pointed out that Caputo’s derivative is equivalent to the Riemann-Liouville derivative under homogeneous initial conditions and some smoothness conditions. Therefore, Caputo’s definition of fractional derivative is used throughout in this paper.

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 [3–9], such as the existence and uniqueness of solutions, the permanence, extinction, global asymptotic behavior, and bifurcation. Because of the good memory and hereditary properties of fractional derivatives, it is often necessary to study the corresponding fractional systems. Therefore, the dynamical analysis of the fractional Lotka-Volterra systems has attracted a great deal of attention due to its theoretical and practical significance.

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 [10]. The stability and solutions of fractional predator-prey and rabies models are discussed in [11]. In addition, bifurcation properties of fractional systems have been studied in some papers. For example, conditions for the occurrence of Hopf’s bifurcation are explored based on numerical simulations in [12]. The critical values of the fractional order are identified for which Hopf’s bifurcation may occur based on the stability analysis in [13]. Thus, it is significant to study the dynamical behaviors in the fractional population systems.

To the best of our knowledge, some papers have concentrated on the dynamic investigation of the fractional population systems [10, 11]. However, there are few results on bifurcation phenomena of the fractional population systems. Therefore, in the paper, we mainly consider stability and bifurcation in the three-dimensional fractional Lotka-Volterra 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 2, a three-dimensional fractional Lotka-Volterra predator-prey system with interspecific competition is introduced. And the asymptotic stability of the system is studied. In Section 3, a three-dimensional fractional Lotka-Volterra predator-prey system is provided, and bifurcation properties are investigated. The numerical results in Section 4 are given to verify the theoretical findings. Finally, the paper is concluded in Section 5.

2. Stability Analysis of a Three-Dimensional Fractional Lotka-Volterra Predator-Prey System with Interspecific Competition

Consider a three-dimensional fractional Lotka-Volterra system 𝐷^𝑞𝑥₁(𝑡)=𝑥₁(𝑡)𝑏₁−𝑎₁₁𝑥₁(𝑡)−𝑎₁₂𝑥₂(𝑡)−𝑎₁₃𝑥₃(𝑡,𝐷_𝑞)𝑥₂(𝑡)=𝑥₂(𝑡)−𝑏₂+𝑎₂₁𝑥₁(𝑡)−𝑎₂₂𝑥₂(𝑡)−𝑎₂₃𝑥₃(𝑡),𝐷^𝑞𝑥₃(𝑡)=𝑥₃(𝑡)−𝑏₃+𝑎₃₁𝑥₁(𝑡)−𝑎₃₂𝑥₂(𝑡)−𝑎₃₃𝑥₃(𝑡), (1) with the initial values 𝑥_𝑖(𝑡)|_𝑡=0=𝑥_𝑖(0), 𝑖=1,2,3, where 0<𝑞≤1; especially when 𝑞=1, the system (1) is a classical integer-order system. All constant coefficients 𝑎_𝑖𝑗 and 𝑏_𝑖 (𝑖,𝑗=1,2,3) can be arbitrary positive real numbers. 𝑥₁(𝑡)≥0 represents the density of prey species at time 𝑡, and 𝑥₂(𝑡)≥0, 𝑥₃(𝑡)≥0 represent the densities of predator species at time 𝑡. In this case, system (1) can be regarded as a fractional Lotka-Volterra predator-prey system with interspecific competition.

In the following, existence and uniqueness of solutions for system (1) are given. In addition, the important results related to the stability of the fractional systems are presented to provide the theoretical bases for the further study.

Here, the fractional Lotka-Volterra system (1) can be rewritten in the form 𝐷^𝑞𝑋(𝑡)=𝐴𝑋(𝑡)+𝑥₁(𝑡)𝐵₁𝑋(𝑡)+𝑥₂(𝑡)𝐵₂𝑋(𝑡)+𝑥₃(𝑡)𝐵₃𝑋(𝑡),𝑋(0)=𝑋₀, (2) where 0<𝑞≤1, 𝑡∈(0,𝑇], and 𝑋(𝑡)=𝑥₁(𝑡)𝑥₂(𝑡)𝑥₃(𝑡),𝑋₀=𝑥₁(0)𝑥₂(0)𝑥₃(0),𝑏𝐴=¹000−𝑏₂000−𝑏₃,𝐵₁=−𝑎₁₁−𝑎₁₂−𝑎₁₃000000,𝐵₂=𝑎000₂₁−𝑎₂₂−𝑎₂₃,𝐵000₃=𝑎000000₃₁−𝑎₃₂−𝑎₃₃. (3)

Definition 1. For 𝑋(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡))^𝑇, let 𝐶^∗[0,𝑇] be the set of continuous column vectors 𝑋(𝑡) on the interval [0,𝑇]. The norm of 𝑋(𝑡)∈𝐶^∗[0,𝑇] is given by ‖𝑋(𝑡)‖=∑³_𝑖=1sup_𝑡|𝑥_𝑖(𝑡)|.

Theorem 2. System (2) has a unique solution if 𝑋(𝑡)∈𝐶^∗[0,𝑇].

Proof. Let 𝐹(𝑋(𝑡))=𝐴𝑋(𝑡)+𝑥₁(𝑡)𝐵₁𝑋(𝑡)+𝑥₂(𝑡)𝐵₂𝑋(𝑡)+𝑥₃(𝑡)𝐵₃𝑋(𝑡), then 𝑋(𝑡)∈𝐶^∗[0,𝑇] implies 𝐹(𝑋(𝑡))∈𝐶^∗[0,𝑇]. In addition, take 𝑋(𝑡),𝑌(𝑡)∈𝐶^∗[0,𝑇] and 𝑋(𝑡)≠𝑌(𝑡); the following inequality holds: ‖𝐹(𝑋(𝑡(𝑌(𝑡=𝐴(𝑋(𝑡)(𝑡))−𝐹))‖−𝑌))+𝑥₁(𝑡)𝐵₁𝑋(𝑡)−𝑦₁(𝑡)𝐵₁𝑌(𝑡)+𝑥₂(𝑡)𝐵₂𝑋(𝑡)−𝑦₂(𝑡)𝐵₂𝑌(𝑡)+𝑥₃(𝑡)𝐵₃𝑋(𝑡)−𝑦₃(𝑡)𝐵₃𝑌(𝑡=𝐴(𝑋(𝑡)(𝑡)−𝑌))+𝑥₁(𝑡)𝐵₁(𝑋(𝑡)(𝑡+𝑥−𝑌))₁(𝑡)−𝑦₁(𝑡)𝐵₁𝑌(𝑡)+𝑥₂(𝑡)𝐵₂(𝑋(𝑡)(𝑡+𝑥−𝑌))₂(𝑡)−𝑦₂(𝑡)𝐵₂𝑌(𝑡)+𝑥₃(𝑡)𝐵₃(𝑋(𝑡)(𝑡+𝑥−𝑌))₃(𝑡)−𝑦₃(𝑡)𝐵₃𝑌(𝑡≤‖𝐴(𝑋(𝑡)(𝑡+𝑥)−𝑌))‖₁(𝑡)𝐵₁(𝑋(𝑡)(𝑡+𝑥−𝑌))₁(𝑡)−𝑦₁(𝑡)𝐵₁𝑌(𝑡)+𝑥₂(𝑡)𝐵₂(𝑋(𝑡)(𝑡+𝑥−𝑌))₂(𝑡)−𝑦₂(𝑡)𝐵₂𝑌(𝑡)+𝑥₃(𝑡)𝐵₃(𝑋(𝑡)(𝑡+𝑥−𝑌))₃(𝑡)−𝑦₃(𝑡)𝐵₃𝑌(𝑡≤‖𝐴‖+𝐵₁)𝑥₁(𝑡+‖𝑌(𝑡+𝐵))‖₂𝑥₂(𝑡+‖𝑌(𝑡+𝐵))‖₃𝑥₃(𝑡+‖𝑌(𝑡‖𝑋(𝑡)(𝑡‖𝑋(𝑡)(𝑡,))‖−𝑌)‖≤𝐿−𝑌)‖ (4) where 𝐿=‖𝐴‖+(‖𝐵₁‖+‖𝐵₂‖+‖𝐵₃‖)(𝑀₁+𝑀₂)>0, and 𝑀₁ and 𝑀₂ are positive and satisfy ‖𝑋(𝑡)‖≤𝑀₁, ‖𝑌(𝑡)‖≤𝑀₂ as a result of 𝑋(𝑡),𝑌(𝑡)∈𝐶^∗[0,𝑇]. Based on Theorems  2.1 and 2.2  in [14], system (2) has a unique solution.

Theorem 3 (see [15]). The linear autonomous system 𝐷^𝑞𝑥=𝐴𝑥 is asymptotically stable if and only if arg(𝜆)>𝑞𝜋2, (5) where 𝐴∈𝑅^𝑛×𝑛, 𝑞∈(0,1), and 𝜆∈𝜎(𝐴); 𝜎(𝐴) denotes the set of all eigenvalues of the matrix 𝐴.

Theorem 4. Let 𝑥^∗ be an equilibrium of the nonlinear system (1), then the equilibrium 𝑥^∗ is locally asymptotically stable if arg(𝜆)>𝑞𝜋2, (6) where 𝜆∈𝜎(𝐽(𝑥^∗)); 𝜎(𝐽(𝑥^∗)) denotes the set of all eigenvalues of the Jacobian matrix 𝐽(𝑥^∗).

Proof. The proof follows from Theorem 3 and [11].

In the following, the stability of system (1) is investigated by giving some appropriate conditions. The asymptotic stability of the equilibria is demonstrated based on Theorem 4. Through simple calculation, the equilibria of system (1) are obtained and denoted as 𝑃₁=(0,0,0),𝑃₂=𝑏0,0,−³𝑎₃₃,𝑃₃=𝑏0,−²𝑎₂₂,0,𝑃₄=𝑏₁𝑎₁₁,𝑃,0,0₅=𝑐0,⁵²𝑐₅,𝑐₅₃𝑐₅,𝑃₆=𝑐₆₁𝑐₆𝑐,0,⁶³𝑐₆,𝑃₇=𝑐₇₁𝑐₇,𝑐₇₂𝑐₇,0,𝑃₈=𝑐₈₁𝑐₈,𝑐₈₂𝑐₈,𝑐₈₃𝑐₈, (7) where 𝑐₅₂=𝑏₃𝑎₂₃−𝑏₂𝑎₃₃,𝑐₅₃=𝑏₂𝑎₃₂−𝑏₃𝑎₂₂,𝑐₅=𝑎₂₂𝑎₃₃−𝑎₂₃𝑎₃₂,𝑐₆₁=𝑏₁𝑎₃₃+𝑏₃𝑎₁₃,𝑐₆₃=𝑏₁𝑎₃₁−𝑏₃𝑎₁₁,𝑐₆=𝑎₁₁𝑎₃₃+𝑎₁₃𝑎₃₁,𝑐₇₁=𝑏₁𝑎₂₂+𝑏₂𝑎₁₂,𝑐₇₂=𝑏₁𝑎₂₁−𝑏₂𝑎₁₁,𝑐₇=𝑎₁₁𝑎₂₂+𝑎₁₂𝑎₂₁,𝑐₈₁=𝑏₁𝑎₂₂𝑎₃₃−𝑏₁𝑎₂₃𝑎₃₂+𝑏₂𝑎₁₂𝑎₃₃−𝑏₂𝑎₁₃𝑎₃₂+𝑏₃𝑎₁₃𝑎₂₂−𝑏₃𝑎₁₂𝑎₂₃,𝑐₈₂=𝑏₁𝑎₂₁𝑎₃₃−𝑏₁𝑎₂₃𝑎₃₁−𝑏₂𝑎₁₁𝑎₃₃−𝑏₂𝑎₁₃𝑎₃₁+𝑏₃𝑎₁₁𝑎₂₃+𝑏₃𝑎₁₃𝑎₂₁,𝑐₈₃=𝑏₁𝑎₂₂𝑎₃₁−𝑏₁𝑎₂₁𝑎₃₂+𝑏₂𝑎₁₁𝑎₃₂+𝑏₂𝑎₁₂𝑎₃₁−𝑏₃𝑎₁₁𝑎₂₂−𝑏₃𝑎₁₂𝑎₂₁, and 𝑐₈=𝑎₁₁𝑎₂₂𝑎₃₃−𝑎₁₁𝑎₂₃𝑎₃₂+𝑎₁₂𝑎₂₁𝑎₃₃−𝑎₁₂𝑎₂₃𝑎₃₁+𝑎₁₃𝑎₂₂𝑎₃₁−𝑎₁₃𝑎₂₁𝑎₃₂.

Because of the fact that all constant coefficients of system (1) are positive, 𝑃₂, 𝑃₃, and 𝑃₅ are in contradiction with the actual situation; hence the asymptotical stability of other five equilibria will be studied in detail.

Theorem 5. For the three-dimensional fractional Lotka-Volterra system (1), the following results can be obtained.(a)𝑃₁ is unstable;(b)𝑃₄ is locally asymptotically stable if 𝑏₁/𝑎₁₁<𝑏₂/𝑎₂₁,𝑏₁/𝑎₁₁<𝑏₃/𝑎₃₁;(c)𝑃₆ is locally asymptotically stable if 𝑏₃/𝑎₃₁<𝑏₁/𝑎₁₁<𝑏₂/𝑎₂₁;(d)𝑃₇ is locally asymptotically stable if 𝑏₂/𝑎₂₁<𝑏₁/𝑎₁₁<𝑏₃/𝑎₃₁;(e)𝑃₈ is locally asymptotically stable if 𝑎₃₂/𝑎₂₂<𝑎₃₁/𝑎₂₁<𝑎₃₃/𝑎₂₃.

Proof. For 𝑃₁=(0,0,0), its Jacobian matrix is 𝐽𝑃₁=𝑏₁000−𝑏₂000−𝑏₃, (8) and the eigenvalues of 𝐽(𝑃₁) satisfy 𝜆₁=𝑏₁>0,𝜆₂=−𝑏₂<0, and 𝜆₃=−𝑏₃<0; hence the equilibrium 𝑃₁ is unstable.
For 𝑃₄, its Jacobian matrix is 𝐽𝑃₄=−𝑏₁−𝑏₁𝑎₁₂𝑎₁₁−𝑏₁𝑎₁₃𝑎₁₁0𝑏₁𝑎₂₁−𝑏₂𝑎₁₁𝑎₁₁0𝑏00¹𝑎₃₁−𝑏₃𝑎₁₁𝑎₁₁, (9) and the eigenvalues of 𝐽(𝑃₄) satisfy 𝜆₁=−𝑏₁<0, 𝜆₂=(𝑏₁𝑎₂₁−𝑏₂𝑎₁₁)/𝑎₁₁<0, and 𝜆₃=(𝑏₁𝑎₃₁−𝑏₃𝑎₁₁)/𝑎₁₁<0; hence the equilibrium 𝑃₄ is locally asymptotically stable.
For 𝑃₆, use the notations below: 𝐽𝑃₆=𝐴₁₁𝐴₁₂𝐴₁₃0𝐴₂₂0𝐴₃₁𝐴₃₂𝐴₃₃, (10) and its characteristic equation is 𝜆−𝐴₂₂𝜆²−𝐴₁₁+𝐴₃₃𝜆+𝐴₁₁𝐴₃₃−𝐴₁₃𝐴₃₁=0. (11) Based on the condition from (c), the following formulas can be easily got 𝐴₁₁=𝑐₁₁𝑐₀<0,𝐴₁₃=𝑐₁₃𝑐₀𝐴<0,₃₁=𝑐₃₁𝑐₀>0,𝐴₂₂=𝑐₂₂𝑐₀𝐴<0,₃₃=𝑐₃₃𝑐₀<0, (12) where 𝑐₁₁=−𝑏₁𝑎₁₁𝑎₃₃−𝑏₃𝑎₁₁𝑎₁₃,𝑐₁₃=−𝑏₁𝑎₁₃𝑎₃₃−𝑏₃𝑎₁₃𝑎₁₃,𝑐₃₁=𝑏₁𝑎₃₁𝑎₃₁−𝑏₃𝑎₁₁𝑎₃₁,𝑐₂₂=𝑎₃₃(𝑏₁𝑎₂₁−𝑏₂𝑎₁₁𝑎)+₂₃(𝑏₃𝑎₁₁−𝑏₁𝑎₃₁)+𝑎₁₃(𝑏₃𝑎₂₁−𝑏₂𝑎₃₁),𝑐₃₃=−𝑏₁𝑎₃₁𝑎₃₃+𝑏₃𝑎₁₁𝑎₃₃, and 𝑐₀=a₁₁𝑎₃₃+𝑎₁₃𝑎₃₁. Then, the following results can be obtained: 𝜆₁=𝐴₂₂<0,𝜆₂+𝜆₃=𝐴₁₁+𝐴₃₃𝜆<0,₂𝜆₃=𝐴₁₁𝐴₃₃−𝐴₁₃𝐴₃₁>0. (13) Hence the equilibrium 𝑃₆ is locally asymptotically stable.
Similarly, it can be readily derived that the equilibrium 𝑃₇ is locally asymptotically stable.
For 𝑃₈, let 𝑃₈=(𝑥^∗₁,𝑥^∗₂,𝑥^∗₃); the Jacobian matrix of 𝑃₈ can be written as 𝐽𝑃₈=−𝑎₁₁𝑥^∗₁−𝑎₁₂𝑥^∗₁−𝑎₁₃𝑥^∗₁𝑎₂₁𝑥^∗₂−𝑎₂₂𝑥^∗₂−𝑎₂₃𝑥^∗₂𝑎₃₁𝑥^∗₃−𝑎₃₂𝑥^∗₃−𝑎₃₃𝑥^∗₃=𝐵₁₁𝐵₁₂𝐵₁₃𝐵₂₁𝐵₂₂𝐵₂₃B₃₁𝐵₃₂𝐵₃₃, (14) and its characteristic equation is 𝜆³+𝐶₁𝜆²+𝐶₂𝜆+𝐶₃=0, (15) where 𝐶₁=−(𝐵₁₁+𝐵₂₂+𝐵₃₃),𝐶₂=𝐵₁₁𝐵₂₂+𝐵₁₁𝐵₃₃+𝐵₂₂𝐵₃₃−𝐵₂₃𝐵₃₂−𝐵₁₂𝐵₂₁−𝐵₁₃𝐵₃₁, and 𝐶₃=−𝐵₁₁𝐵₂₂𝐵₃₃+𝐵₁₁𝐵₂₃𝐵₃₂+𝐵₁₂𝐵₂₁𝐵₃₃+𝐵₁₃𝐵₂₂𝐵₃₁−𝐵₁₂𝐵₂₃𝐵₃₁−𝐵₁₃𝐵₂₁𝐵₃₂. For simplicity, the equivalent characteristic equation is introduced as follows: 𝜆³+𝑎₁𝜆²+𝑎₂𝜆+𝑎₃=(𝜆−𝑎)𝜆²−𝑏𝜆+𝑐=0. (16) On the basis of the above equivalent substitutions, the following inequalities can be gained: 𝑎₁=𝑎₁₁𝑥^∗₁+𝑎₂₂𝑥^∗₂+𝑎₃₃𝑥^∗₃𝑎>0,₂=𝑎₁₁𝑎₂₂+𝑎₁₂𝑎₂₁𝑥^∗₁𝑥^∗₂+𝑎₁₁𝑎₃₃+𝑎₁₃𝑎₃₁𝑥^∗₁𝑥^∗₃+𝑎₂₂𝑎₃₃−𝑎₂₃𝑎₃₂𝑥^∗₂𝑥^∗₃𝑎>0,₃=𝑎₁₁𝑎₂₂𝑎₃₃−𝑎₂₃𝑎₃₂+𝑎₁₂𝑎₂₁𝑎₃₃−𝑎₂₃𝑎₃₁+𝑎₁₃𝑎₂₂𝑎₃₁−𝑎₂₁𝑎₃₂𝑥^∗₁𝑥^∗₂𝑥^∗₃𝑎>0,₁𝑎₂−𝑎₃=𝑑₁𝑥^∗₁𝑥^∗₂𝑥^∗₃+𝑑₂𝑥^∗₁𝑥^∗₂²+𝑑₃𝑥^∗₁𝑥^∗₃²+𝑑₄𝑥^∗₁²𝑥^∗₂+𝑑₅𝑥^∗₁²𝑥^∗₃+𝑑₆𝑎₂₂𝑥^∗₂²𝑥^∗₃+𝑎₃₃𝑥^∗₂𝑥^∗₃²>0,𝑎+𝑏=−𝑎₁<0,𝑎𝑏+𝑐=𝑎₂>0,𝑎𝑐=−𝑎₃<0,−𝑎²𝑏−𝑎𝑏²−𝑏𝑐=𝑎₁𝑎₂−𝑎₃>0, (17) where 𝑑₁=2𝑎₁₁𝑎₂₂𝑎₃₃+𝑎₁₂𝑎₂₃𝑎₃₁+𝑎₁₃𝑎₂₁𝑎₃₂,𝑑₂=𝑎₁₁𝑎²₂₂+𝑎₁₂𝑎₂₁𝑎₂₂,𝑑₃=𝑎₁₁𝑎²₃₃+𝑎₁₃𝑎₃₁𝑎₃₃,𝑑₄=𝑎²₁₁𝑎₂₂+𝑎₁₁𝑎₁₂𝑎₂₁,𝑑₅=𝑎²₁₁𝑎₃₃+𝑎₁₁𝑎₁₃𝑎₃₁, and 𝑑₆=𝑎₂₂𝑎₃₃−𝑎₂₃𝑎₃₂. Using the proof by contradiction, it can be concluded that the eigenvalues of 𝐽(𝑃₈) satisfy 𝜆₁𝜆=𝑎<0,₂+𝜆₃𝜆=𝑏<0,₂𝜆₃𝜆=𝑐>0,arg_𝑖>𝑞𝜋2,𝑖=2,3. (18) Hence the equilibrium 𝑃₈ is locally asymptotically stable.

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.

3. Bifurcation Analysis of a Three-Dimensional Fractional Lotka-Volterra Predator-Prey System

Consider a three-dimensional fractional Lotka-Volterra system: 𝐷^𝑞𝑥₁(𝑡)=𝑥₁(𝑡)𝑏₁−𝑎₁₁𝑥₁(𝑡)−𝑎₁₂𝑥₂(𝑡)−𝑎₁₃𝑥₃(𝑡,𝐷_𝑞)𝑥₂(𝑡)=𝑥₂(𝑡)−𝑏₂+𝑎₂₁𝑥₁(𝑡)−𝑎₂₂𝑥₂(𝑡),𝐷^𝑞𝑥₃(𝑡)=𝑥₃(𝑡)−𝑏₃+𝑎₃₁𝑥₁(𝑡)−𝑎₃₃𝑥₃(𝑡), (19) with the initial values 𝑥_𝑖(𝑡)|_𝑡=0=𝑥_𝑖(0), 𝑖=1,2,3, where 0<𝑞<1, 𝑎₁₁<0, and the other constant coefficients are positive. 𝑥₁(𝑡)≥0 represents the density of prey species at time 𝑡, and 𝑥₂(𝑡)≥0, 𝑥₃(𝑡)≥0 represent the densities of predator species at time 𝑡. In this case, system (19) can be regarded as a fractional Lotka-Volterra predator-prey system.

On the basis of Theorem 2, it is not difficult to prove that system (19) has a unique solution in a similar way.

It is clear that there are eight equilibria for system (19). Here, we focus on the bifurcation investigation of the equilibrium 𝑥^∗ which can be called a positive equilibrium when some conditions are satisfied. The equilibrium 𝑥^∗ is obtained as 𝑥^∗=𝑥^∗₁,𝑥^∗₂,𝑥^∗₃=𝑑₁₁𝑑,𝑑₂₂𝑑,𝑑₃₃𝑑, (20) where 𝑑₁₁=𝑏₁𝑎₂₂𝑎₃₃+𝑏₂𝑎₁₂𝑎₃₃+𝑏₃𝑎₁₃𝑎₂₂,𝑑₂₂=𝑏₁𝑎₂₁𝑎₃₃−𝑏₂(𝑎₁₁𝑎₃₃+𝑎₁₃𝑎₃₁)+𝑏₃𝑎₁₃𝑎₂₁,𝑑₃₃=𝑏₁𝑎₂₂𝑎₃₁+𝑏₂𝑎₁₂𝑎₃₁−𝑏₃(𝑎₁₁𝑎₂₂+𝑎₁₂𝑎₂₁), and 𝑑=𝑎₁₁𝑎₂₂𝑎₃₃+𝑎₁₂𝑎₂₁𝑎₃₃+𝑎₁₃𝑎₂₂𝑎₃₁. And its Jacobian matrix can be expressed as 𝐽𝑥^∗=−𝑎₁₁𝑥^∗₁−𝑎₁₂𝑥^∗₁−𝑎₁₃𝑥^∗₁𝑎₂₁𝑥^∗₂−𝑎₂₂𝑥^∗₂0𝑎₃₁𝑥^∗₃0−𝑎₃₃𝑥^∗₃. (21) Furthermore, the eigenvalues of 𝐽(𝑥^∗) satisfy the characteristic equation 𝜆−𝛽𝜆²−𝛼+𝛽𝜆+𝛼𝛽−𝛾=0, (22) where 𝛼=−𝑎₁₁𝑥^∗₁, 𝛽=−𝑎₂₂𝑥^∗₂=−𝑎₃₃𝑥^∗₃, and 𝛾=−𝑎₁₂𝑎₂₁𝑥^∗₁𝑥^∗₂−𝑎₁₃𝑎₃₁𝑥^∗₁𝑥^∗₃.

In the following, by choosing the fractional order 𝑞 as the bifurcation parameter and analyzing the associated characteristic equation (22) of system (19) at the positive equilibrium, we investigate the bifurcation phenomena of the positive equilibrium of system (19) and obtain the conditions under which system (19) undergoes a Hopf bifurcation.

Proposition 6. The positive equilibrium 𝑥^∗ of system (19) is locally asymptotically stable if and only if all the following conditions are satisfied:(i)𝛽<0,(ii)𝛼𝛽−𝛾>0, and(iii)𝛼+𝛽<2𝑐𝑜𝑠(𝑞𝜋/2)𝛼𝛽−𝛾.

Proof. For the characteristic equation (22), the root 𝜆₁=𝛽<0, and 𝜆₂,𝜆₃ satisfy the equation 𝜆²−(𝛼+𝛽)𝜆+𝛼𝛽−𝛾=0. It is clear that |arg(𝜆_2,3)|>𝑞𝜋/2 if and only if the conditions (ii) and (iii) hold. Based on Theorem 4, Proposition 6 proves to be true.

In addition, by analyzing the condition (iii) of Proposition 6 in detail, the following results can be gained.

Proposition 7. With respect to system (19), if 𝛽<0 and 𝛼𝛽−𝛾>0, the following statements can be obtained.(a)If 𝛼+𝛽≤0, the equilibrium 𝑥^∗ is locally asymptotically stable, for any 𝑞∈(0,1).(b)If 0<𝛼+𝛽<2𝛼𝛽−𝛾, the equilibrium 𝑥^∗ is locally asymptotically stable if and only if 𝑞∈(0,𝑞^∗), where 𝑞^∗=(2/𝜋)arccos((𝛼+𝛽)/2𝛼𝛽−𝛾).(c)If 𝛼+𝛽≥2𝛼𝛽−𝛾, the equilibrium 𝑥^∗ is unstable for any 𝑞∈(0,1).

Proof. The conclusions (a) and (c) are obvious. For the statement (b), due to 0<𝛼+𝛽<2𝛼𝛽−𝛾, the equation 𝜆²−(𝛼+𝛽)𝜆+𝛼𝛽−𝛾=0 has two complex roots 𝜆₂,𝜆₃, and their real part is (𝛼+𝛽)/2>0. Then |arg(𝜆_𝑖)|=arccos((𝛼+𝛽)/2𝛼𝛽−𝛾),𝑖=2,3. Besides, according to the condition arccos((𝛼+𝛽)/2𝛼𝛽−𝛾)=𝑞^∗𝜋/2,𝑞∈(0,𝑞^∗) if and only if |arg(𝜆_𝑖)|>𝑞𝜋/2,𝑖=2,3. Based on Theorem 4, it is concluded that Proposition 7 is true.

Remark 8. It is apparent that the critical value satisfies 𝑞^∗∈(0,1). When 𝑞∈(0,𝑞^∗), 𝑥^∗ is locally asymptotically stable; when 𝑞∈(𝑞^∗,1), and specially 𝑞=1, 𝑥^∗ is unstable. That is to say, it has verified that fractional differential equations are, at least, as stable as their integer-order counterparts [4].

Remark 9. Under the situation of statement (b), a bifurcation phenomenon must happen at the critical value 𝑞^∗. However, it is difficult to confirm precise bifurcation type. As an interesting bifurcation behavior, Hopf’s bifurcation is expected to take place.

According to Proposition 7, if some appropriate conditions about the constant coefficients of system (19) can be found so that statement (b) is satisfied, system (19) will undergo a bifurcation phenomenon. And the critical value 𝑞^∗ of the bifurcation parameter 𝑞 can be expressed by the constant coefficients of system (19). From this, the following theorem is specifically proposed.

Theorem 10. With respect to system (19), if the following conditions are satisfied:(i)𝑏₂=𝑏₃, 𝑎₂₂=𝑎₃₃, 𝑎₂₁=𝑎₃₁=−𝑎₁₁;(ii)𝑎₁₂+𝑎₁₃−𝑎₂₂>0,then the positive equilibrium 𝑥^∗ is locally asymptotically stable if and only if 𝑞∈(0,𝑞^∗), where 𝑞^∗=2𝜋arccos𝑏₂2𝑏₁+𝑏₂𝑏₁𝑎₂₂+𝑏₂𝑎₁₂+𝑏₂𝑎₁₃𝑎₁₂+𝑎₁₃−𝑎₂₂⁻¹. (23)

Proof. According to the condition (i), the equilibrium 𝑥^∗ can be expressed as 𝑥^∗=𝑏₁𝑎₂₂+𝑏₂𝑎₁₂+𝑏₂𝑎₁₃𝑎₁₁𝑎₂₂−𝑎₁₂−𝑎₁₃,𝑏₁+𝑏₂𝑎₁₂+𝑎₁₃−𝑎₂₂,𝑏₁+𝑏₂𝑎₁₂+𝑎₁₃−𝑎₂₂. (24) For (22), the following results can be obtained 𝛽=−𝑎₂₂𝑥^∗₂<0,𝛼+𝛽=𝑏₂𝑒>0,𝛼𝛽−𝛾=¹𝑎₁₂+𝑎₁₃−𝑎₂₂4−>0,𝛼𝛽−𝛾𝛼+𝛽²=4𝑒₁𝑎₁₂+𝑎₁₃−𝑎₂₂−𝑏²₂=𝑒₂𝑎₁₂+𝑎₁₃−𝑎₂₂>0, (25) where 𝑒₁=(𝑏₁+𝑏₂)(𝑏₁𝑎₂₂+𝑏₂𝑎₁₂+𝑏₂𝑎₁₃) and 𝑒₂=𝑎₂₂(4𝑏²₁+𝑏²₂)+3𝑏²₂(𝑎₁₂+𝑎₁₃)+4𝑏₁𝑏₂(𝑎₁₂+𝑎₁₃+𝑎₂₂). Obviously, the above conclusions satisfy statement (b) of Proposition 7, then it can be derived that 𝑞^∗=2𝜋arccos𝑏₂2𝑏₁+𝑏₂𝑏₁𝑎₂₂+𝑏₂𝑎₁₂+𝑏₂𝑎₁₃𝑎₁₂+𝑎₁₃−𝑎₂₂⁻¹. (26) Hence, the positive equilibrium 𝑥^∗ of system (19) is locally asymptotically stable if and only if 𝑞∈(0,𝑞^∗).

According to the statement of Theorem 10, it can be concluded that the positive equilibrium 𝑥^∗ is locally asymptotically stable if and only if 𝑞∈(0,𝑞^∗). At 𝑞=𝑞^∗, the Hopf bifurcation is expected to take place. As 𝑞 increases above the critical value 𝑞^∗, the positive equilibrium 𝑥^∗ is unstable and a limit cycle is expected to appear in the proximity of 𝑥^∗ due to the Hopf bifurcation phenomenon.

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 [16]. As an application, it is emphasized that the limit cycle observed in numerical simulations of a simple fractional neural network cannot be an exact periodic solution of the system [17]. In addition, there are some other papers providing the numerical evidences of limit cycles.

Remark 11. Even though exact periodic solutions do not exist in autonomous fractional systems [16, 17], limit cycles have been observed by numerical simulations in many systems such as a fractional neural system [13], a fractional Van der Pol system [18], fractional Chua and Chen’s systems [19, 20], and a fractional financial system [21].

4. Numerical Simulation

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 [22, 23] and further investigated in [24–27]. In order to verify the theoretical analysis, the following numerical results are given.

For system (1), the approximate solutions are displayed in Figure 1 for the step size 0.005 and different values of 𝑞, 𝑞=1,𝑞=0.9,𝑞=0.8, respectively. Taking 𝑏₁=12,𝑏₂=𝑎₁₁=𝑎₁₃=𝑎₂₁=𝑎₂₃=𝑎₃₁=1,𝑎₁₂=𝑎₃₂=𝑎₃₃=2, and 𝑏₃=𝑎₂₂=3, and choosing the initial values 𝑥₁(0)=20,𝑥₂(0)=10, and 𝑥₃(0)=10, the equilibrium 𝑃₈ is (53/7,15/7,1/7). Then, Figure 1 shows that the equilibrium 𝑃₈ is locally asymptotically stable. Namely, the fifth conclusion of Theorem 5 is verified. Similarly, the other conclusions of Theorem 5 can be confirmed.

Figure 1 

The trajectory of system (1) converges to the equilibrium 𝑃8=(53/7,15/7,1/7).

For system (19), the approximate solutions are displayed in Figures 2, 3, and 4 for the step size 0.001 and different values of 𝑞, 𝑞=0.82, and 𝑞=0.84. Taking 𝑏₁=𝑎₁₂=𝑎₁₃=𝑎₂₂=𝑎₃₃=1,𝑏₂=𝑏₃=𝑎₂₁=𝑎₃₁=2,   and 𝑎₁₁=−2, and choosing the initial values 𝑥₁(0)=3,𝑥₂(0)=𝑥₃(0)=4, the positive equilibrium is 𝑥^∗=(2.5,3,3), and the critical value is 𝑞^∗=0.8337. Indeed, Figures 2–4 present the fact that the positive equilibrium 𝑥^∗ is locally asymptotically stable when 𝑞=0.82∈(0,0.8337), and when 𝑞=0.84 increases across 𝑞^∗=0.8337, an asymptotically stable limit cycle appears in a neighborhood of the positive equilibrium 𝑥^∗.

(a)

(b)

(c)

Figure 2 

The solution of system (19) versus time with 𝑞=0.84.

Figure 3 

When 𝑞=0.82, the trajectory of system (19) converges to the equilibrium 𝑥∗=(2.5,3,3).

Figure 4 

When 𝑞=0.84, the trajectory of system (19) converges to an asymptotically stable limit cycle.

5. Conclusion

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 (1), the asymptotic stability of the equilibria is investigated by providing simple and reasonable sufficient conditions. And simulation results prove to be quite consistent with the theoretical findings. On the other hand, for system (19), the conditions which could lead to bifurcation phenomena are obtained. Specifically, the fractional order 𝑞∈(0,1) is chosen as the bifurcation parameter and the expression of the critical value 𝑞^∗ is precisely derived. Furthermore, the numerical result is presented to illustrate that Hopf’s bifurcation can take place.

Conflict of Interests

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

Acknowledgment

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.