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Stability and Bifurcation of Two Kinds of Three-Dimensional Fractional Lotka-Volterra Systems
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 . 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  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 , 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 . The stability and solutions of fractional predator-prey and rabies models are discussed in . 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 . The critical values of the fractional order are identified for which Hopf’s bifurcation may occur based on the stability analysis in . 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 with the initial values , , where ; especially when , the system (1) is a classical integer-order system. All constant coefficients and () can be arbitrary positive real numbers. represents the density of prey species at time , and , represent the densities of predator species at time . 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 where , , and
Definition 1. For , let be the set of continuous column vectors on the interval . The norm of is given by .
Theorem 2. System (2) has a unique solution if .
Proof. Let , then implies . In addition, take and ; the following inequality holds: where , and and are positive and satisfy , as a result of . Based on Theorems 2.1 and 2.2 in , system (2) has a unique solution.
Theorem 3 (see ). The linear autonomous system is asymptotically stable if and only if where , , and ; denotes the set of all eigenvalues of the matrix .
Theorem 4. Let be an equilibrium of the nonlinear system (1), then the equilibrium is locally asymptotically stable if where ; denotes the set of all eigenvalues of the Jacobian matrix .
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 where , and .
Because of the fact that all constant coefficients of system (1) are positive, , , and 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) is unstable;(b) is locally asymptotically stable if ;(c) is locally asymptotically stable if ;(d) is locally asymptotically stable if ;(e) is locally asymptotically stable if .
Proof. For , its Jacobian matrix is
and the eigenvalues of satisfy , and ; hence the equilibrium is unstable.
For , its Jacobian matrix is and the eigenvalues of satisfy , , and ; hence the equilibrium is locally asymptotically stable.
For , use the notations below: and its characteristic equation is Based on the condition from (c), the following formulas can be easily got where , and . Then, the following results can be obtained: Hence the equilibrium is locally asymptotically stable.
Similarly, it can be readily derived that the equilibrium is locally asymptotically stable.
For , let ; the Jacobian matrix of can be written as and its characteristic equation is where , and . For simplicity, the equivalent characteristic equation is introduced as follows: On the basis of the above equivalent substitutions, the following inequalities can be gained: where , and . Using the proof by contradiction, it can be concluded that the eigenvalues of satisfy Hence the equilibrium 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: with the initial values , , where , , and the other constant coefficients are positive. represents the density of prey species at time , and , represent the densities of predator species at time . In this case, system (19) can be regarded as a fractional Lotka-Volterra predator-prey system.
It is clear that there are eight equilibria for system (19). Here, we focus on the bifurcation investigation of the equilibrium which can be called a positive equilibrium when some conditions are satisfied. The equilibrium is obtained as where , and . And its Jacobian matrix can be expressed as Furthermore, the eigenvalues of satisfy the characteristic equation where , , and .
In the following, by choosing the fractional order 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 of system (19) is locally asymptotically stable if and only if all the following conditions are satisfied:(i),(ii), and(iii).
Proof. For the characteristic equation (22), the root , and satisfy the equation . It is clear that 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 and , the following statements can be obtained.(a)If , the equilibrium is locally asymptotically stable, for any .(b)If , the equilibrium is locally asymptotically stable if and only if , where .(c)If , the equilibrium is unstable for any .
Proof. The conclusions (a) and (c) are obvious. For the statement (b), due to , the equation has two complex roots , and their real part is . Then . Besides, according to the condition if and only if . Based on Theorem 4, it is concluded that Proposition 7 is true.
Remark 8. It is apparent that the critical value satisfies . When , is locally asymptotically stable; when , and specially , is unstable. That is to say, it has verified that fractional differential equations are, at least, as stable as their integer-order counterparts .
Remark 9. Under the situation of statement (b), a bifurcation phenomenon must happen at the critical value . 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 of the bifurcation parameter 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), , ;(ii),then the positive equilibrium is locally asymptotically stable if and only if , where
Proof. According to the condition (i), the equilibrium can be expressed as For (22), the following results can be obtained where and . Obviously, the above conclusions satisfy statement (b) of Proposition 7, then it can be derived that Hence, the positive equilibrium of system (19) is locally asymptotically stable if and only if .
According to the statement of Theorem 10, it can be concluded that the positive equilibrium is locally asymptotically stable if and only if . At , the Hopf bifurcation is expected to take place. As increases above the critical value , the positive equilibrium is unstable and a limit cycle is expected to appear in the proximity of 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 . 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 . 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 , a fractional Van der Pol system , fractional Chua and Chen’s systems [19, 20], and a fractional financial system .
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 , , respectively. Taking , and , and choosing the initial values , and , the equilibrium is . Then, Figure 1 shows that the equilibrium is locally asymptotically stable. Namely, the fifth conclusion of Theorem 5 is verified. Similarly, the other conclusions of Theorem 5 can be confirmed.
For system (19), the approximate solutions are displayed in Figures 2, 3, and 4 for the step size 0.001 and different values of , , and . Taking , and , and choosing the initial values , the positive equilibrium is , and the critical value is . Indeed, Figures 2–4 present the fact that the positive equilibrium is locally asymptotically stable when , and when increases across , an asymptotically stable limit cycle appears in a neighborhood of the positive equilibrium .
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 is chosen as the bifurcation parameter and the expression of the critical value 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.
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.
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