Stability and Bifurcation Analysis of Differential Equations and its ApplicationsView this Special Issue
Limit Cycles in a Cubic Kolmogorov System with Harvest and Two Positive Equilibrium Points
A class of planar cubic Kolmogorov systems with harvest and two positive equilibrium points is investigated. With the help of computer algebra system MATHEMATICA, we prove that five limit cycles can be bifurcated simultaneously from the two critical points (1, 1) and (2, 2), respectively, in the first quadrant. Moreover, the necessary conditions of centers are obtained.
In mathematical ecology, a class of systems of the form are frequently used to model the interaction of species occupying the same ecological niche. The differential equations modeling the interaction of two species being known as Kolmogorov systems have been studied extensively. It is well known that there is no limit cycles in the classical Lotka-Volterra-Gause model, where and are linear. There can of course only be one critical point in the interior of the realistic quadrant in this case, but this can be a centre; however, there are no isolated periodic solutions.
If and are quadratic, one might think by analogy that the behavior within the first quadrant is similar to that of a quadratic system. The examples we give show that this is not the case, even when and factories. There are many contributions about this system (see [1, 2]). The latter poses the question whether a predator-prey system can have two or more ecologically stable cycles. If and are cubic, there are also many works to consider its limit cycles and dynamics behaviors, see [3, 4]. In , the authors discussed a class of cubic Kolmogorov systems with three invariant algebraic curves.
Recently, a system with three positive equilibrium points has been investigated, the authors have investigated the center-focus problems and limit cycles bifurcations. They have proved that each of the two points and can bifurcate 1 small limit cycle under a certain condition, and 3 limit cycles can occur near point at the same step . Other Kolmogorov systems were also investigated recently in [7, 8]. In this paper, we will consider limit cycles which bifurcate from a class of systems of the form which have two positive equilibrium points and . We use our Computer Algebra procedure Mathematic (described in ) to compute the focal values at the critical points and . We will show that five limit cycles can bifurcate from points and simultaneously. Furthermore, some necessary and sufficient conditions for two positive equilibrium points to be centers are also to be given.
This paper is divided into three sections. In Section 2, we use the recursive algorithm to obtain that 5 limit circles could be bifurcated from points and . In Section 3, necessary and sufficient conditions for two positive equilibrium points to be centers are proved.
2. Bifurcations of Limit Circles at Two Positive Equilibrium Points
First of all, it is easy to testify that points and are two positive equilibriums of system (3) and they are all center or focus. So we need to compute the Lyapunov constants to determine its kind of singular. Now, we consider the point .
Through the transformations , , , we still denote by for convenience. The point will be moved into of the new system, and the system will be changed into the following system: Furthermore, by the transformations system (4) can be transformed into the following system: applying the recursive formulae in Theorem 2.5 in , we compute singular point quantities and simplify them; then, we have the following theorem.
Theorem 1. The first four singular point quantities at the origin of system (6) are as follows: where
While , the following theorem holds.
Theorem 2. The origin of system (6) is a 4-order weak focus if and only if
Proof. implies that the relations above among parameters hold. Further, when , we have where denotes the resultant of with respect to . So when . Namely, point is a fourth-order weak focus.
In fact, So , namely, , when .
The statement mentioned above yields that the following theorem holds.
Theorem 3. If the origin of system (4) is a 4-order weak focus, making a small perturbation to the coefficients of system (4), then, for perturbed system (4), in a small neighborhood of the origin, there exist exactly 5 small amplitude limit cycles enclosing the point .
With a similar method, we could discuss the bifurcation of limit cycles from point . Through the transformations , we still denote by for convenience. Then point will be moved into point of the new system, and the system will be changed into the following system: Furthermore, by the transformations system (14) can be transformed into the following system: Direct computation can give the following theorem.
Theorem 4. The first four singular point quantities at the origin of system (16) are as follows: where
While solving and , we can immediately obtain the following theorem.
Theorem 5. The origin of system (16) is a 4-order weak focus if and only if
Furthermore, the following theorem also holds for system (14).
Theorem 6. If the origin of system (14) is a 4-order weak focus, making a small perturbation to the coefficients of system (14), then, for perturbed system (14), in a small neighborhood of the origin, there exist exactly 5 small amplitude limit cycles enclosing the point .
The results also yield that when point is a fourth-order weak focus, point of system is a first-order weak focus at the same time. Point is a first-order weak focus when point is a fourth-order weak focus. Namely, by a simultaneously perturbation, five limit cycles could be bifurcated with two different distribution; see Figure 1.
3. Center Conditions of Two Positive Equilibrium Points
Proof. When , system (4) could be rewritten as
system (21) becomes
which is symmetric with axis.
When , system (14) could be rewritten as Transformations bring system (24) into which is symmetric with axis. So the origin is a center of system (14).
Conflict of Interests
The authors declare that they have no conflict of interests.
This research is partially supported by the National Nature Science Foundation of China (no. 11201211, 11201138, and 11261020) and the Scientific Research Fund of Hunan Provincial Education Department (no. 12B034), and Hunan Provincial Natural Science Foundation of China (no. 13JJ3106).
X. S. Liu and Z. H. Zhuo, “Existence and uniqueness of limit cycles of a cubic Kolmogorov differential system,” Biomathematics, vol. 3, pp. 266–271, 2001.View at: Google Scholar
Y. Ye and W. Ye, “Cubic Kolmogorov differential systems with two limit cycles surrounding the same focus,” Annals of Differential Equations, vol. 2, pp. 201–207, 1985.View at: Google Scholar
E. Saez and I. Szanto, “Limit cycle s of a cubic Kolmogorov system,” Applied Mathematics Letters, vol. 9, pp. 15–18, 1996.View at: Google Scholar
Y. Liu and J. Li, “Theory of values of singular point in complex autonomous differential system,” Science China A, vol. 3, pp. 245–255, 1989.View at: Google Scholar