Limit Cycles in a Cubic Kolmogorov System with Harvest and Two Positive Equilibrium Points

Abstract

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.

1. Introduction

In mathematical ecology, a class of systems of the form 𝑑𝑥_𝑖𝑑𝑡=𝑥_𝑖𝐹𝑥₁,…,𝑥_𝑛,𝑖=1,…,𝑛, (1) are frequently used to model the interaction of species occupying the same ecological niche. The differential equations modeling the interaction of two species 𝑑𝑥,𝑑𝑡=𝑥𝐹𝑥,𝑦𝑑𝑦,𝑑𝑡=𝑦𝐺𝑥,𝑦 (2) 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 𝐹 and 𝐺 are linear. There can of course only be one critical point in the interior of the realistic quadrant (𝑥>0,𝑦>0) in this case, but this can be a centre; however, there are no isolated periodic solutions.

If 𝐹 and 𝐺 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 𝐹 and 𝐺 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 𝐹 and 𝐺 are cubic, there are also many works to consider its limit cycles and dynamics behaviors, see [3, 4]. In [5], 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 (1,2) and (2,1) can bifurcate 1 small limit cycle under a certain condition, and 3 limit cycles can occur near point (1,1) at the same step [6]. 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 𝑑𝑥+7𝑑𝑡=−22𝛽+𝜃8𝛽+5𝜃𝑥−22𝛽+𝜃𝑥²+124𝛽−𝜃𝑥𝑦+𝑎₂₀𝑥³+126𝛽+3𝜃−4𝑎₂₀𝑥²𝑦+12−4𝛽−𝜃+2𝑎₂₀𝑥²𝑦,𝑑𝑦−7𝑑𝑡=22𝛽+𝜃8𝛽+5𝜃𝑦+22𝛽+𝜃𝑦²+121−4𝛽+𝜃𝑥𝑦+22𝑏₂₀−4𝛽−𝜃𝑥³+𝑏₂₀𝑥²𝑦+122𝛽−𝜃−4𝑏₂₀𝑥𝑦², (3) which have two positive equilibrium points (1,1) and (2,2). We use our Computer Algebra procedure Mathematic (described in [9]) to compute the focal values at the critical points (1,1) and (2,2). We will show that five limit cycles can bifurcate from points (1,1) and (2,2) 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 (1,1) and (2,2). 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 (1,1) and (2,2) 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 (1,1).

Through the transformations 𝑥=𝑥−1,  𝑦=𝑦−1,  𝜏=𝛽𝑡, we still denote 𝑥,𝑦 by 𝑥,𝑦 for convenience. The point (1,1) will be moved into (0,0) of the new system, and the system will be changed into the following system: 𝑑𝑥𝑑𝜏=4𝑥²−𝑦−4𝑥𝑦−3𝑥²𝑦+2𝑦²+2𝑥𝑦²−𝑎₂₀𝑥²𝛽−𝑎₂₀𝑥³𝛽+2𝑎₂₀𝑥𝑦𝛽+2𝑎₂₀𝑥²𝑦𝛽−𝑎₂₀𝑦²𝛽−𝑎₂₀𝑥𝑦²𝛽+2𝑥²𝜃𝛽−3𝑥𝑦𝜃−2𝛽3𝑥²𝑦𝜃+𝑦2𝛽²𝜃+2𝛽𝑥𝑦²𝜃,2𝛽𝑑𝑦𝑑𝜏=𝑥−2𝑦²−𝑥𝑦²+2𝑦³−𝑏₂₀𝑥²𝛽+2𝑏₂₀𝑥𝑦𝛽−𝑏₂₀𝑥²𝑦𝛽−𝑏₂₀𝑦²𝛽+2𝑏₂₀𝑥𝑦²𝛽−𝑏₂₀𝑦³𝛽+𝑥𝑦𝜃−2𝛽3𝑦²𝜃+2𝛽𝑥𝑦²𝜃+𝑦2𝛽³𝜃.2𝛽 (4) Furthermore, by the transformations √𝑧=𝑥+𝑖𝑦,𝑤=𝑥−𝑖𝑦,𝑇=𝑖𝜏,𝑖=−1, (5) system (4) can be transformed into the following system: 𝑑𝑧1𝑑𝑇=𝑧+(8𝛽−22𝑎₂₀+2𝑖𝑏₂₀−(6−2𝑖)𝛽−(3−2𝑖)𝜃𝑧²−2−4𝑖𝑎₂₀+4𝑏₂₀+(4+12𝑖)𝛽+(3+5𝑖)𝜃𝑧𝑤+22𝑎₂₀+2𝑖𝑏₂₀−(2+2𝑖)𝛽−𝑖𝜃𝑤²+−2𝑎₂₀−2𝑏₂₀+(4+4𝑖)𝛽+(1+𝑖)𝜃𝑧³+2(−1+2𝑖)𝑎₂₀+(1+2𝑖)𝑏₂₀+(1−4𝑖)𝛽+(1−𝑖)𝜃𝑧²𝑤+(2+4𝑖)𝑎₂₀+(2−4𝑖)𝑏₂₀−(1−𝑖)4𝛽+𝜃𝑧𝑤²−2−𝑎₂₀+𝑏₂₀+𝛽+𝜃𝑤³;𝑑𝑤1𝑑𝑇=−𝑤−(8𝛽−22𝑎₂₀−2𝑖𝑏₂₀−(6+2𝑖)𝛽−(3+2𝑖)𝜃𝑤²−24𝑖𝑎₂₀+4𝑏₂₀+(4−12𝑖)𝛽+(3−5𝑖)𝜃𝑧𝑤+22𝑎₂₀−2𝑖𝑏₂₀−(2−2𝑖)𝛽+𝑖𝜃𝑧²+−2𝑎₂₀−2𝑏₂₀+(4−4𝑖)𝛽+(1−𝑖)𝜃𝑤³+2(−1−2𝑖)𝑎₂₀+(1−2𝑖)𝑏₂₀+(1+4𝑖)𝛽+(1+𝑖)𝜃𝑤²𝑧+(2−4𝑖)𝑎₂₀+(2+4𝑖)𝑏₂₀−(1+𝑖)4𝛽+𝜃𝑤𝑧²−2−𝑎₂₀+𝑏₂₀+𝛽+𝜃𝑧³; (6) applying the recursive formulae in Theorem 2.5 in [9], 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: 𝑢₁=𝑖8𝛽²−2𝑎₂₀−2𝑏₂₀×+4𝛽+𝜃−4𝑎₂₀+4𝑏₂₀,𝑢+12𝛽+9𝜃₂=𝑖192𝛽³−2𝑎₂₀−2𝑏₂₀×+4𝛽+𝜃344𝛽²+750𝛽𝜃+325𝜃²,𝑢₃𝑖=−12288𝛽⁵−2𝑎₂₀−2𝑏₂₀+4𝛽+𝜃𝑓₁,𝑢₄𝑖=−5898240𝛽⁷−2𝑎₂₀−2𝑏₂₀+4𝛽+𝜃𝑓₂, (7) where 𝑓₁=−213760𝑏²₂₀𝛽²−213760𝑏₂₀𝛽³+147392𝛽⁴−323520𝑏²₂₀𝛽𝜃−697600𝑏₂₀𝛽²𝜃+84104𝛽³𝜃−116000𝑏²₂₀𝜃²−682160𝑏₂₀𝛽𝜃²−554530𝛽²𝜃²−203000𝑏₂₀𝜃³−544505𝛽𝜃³−128125𝜃⁴;𝑓₂=14520162123939840𝛽¹⁰+77088521496444928𝛽⁹𝜃+146627049158313472𝛽⁸𝜃²+98147703728382592𝛽⁷𝜃³−45397952769129120𝛽⁶𝜃⁴−116390817537863400𝛽⁵𝜃⁵−79417641637292700𝛽⁴𝜃⁶−27193429543260550𝛽³𝜃⁷−5139731868326875𝛽²𝜃⁸−582232780375000𝛽𝜃⁹−38940972656250𝜃¹⁰. (8)

While 𝑢₁=𝑢₂=𝑢₃=0,𝑢₄≠0, the following theorem holds.

Theorem 2. The origin of system (6) is a 4-order weak focus if and only if 𝑎₂₀=144𝑏₂₀+12𝛽+9𝜃,344𝛽²+750𝛽𝜃+325𝜃²𝑓=0,₁=0. (9)

Proof. 𝑢₁=𝑢₂=𝑢₃=0 implies that the relations above among parameters hold. Further, when 𝑎₂₀=(1/4)(4𝑏₂₀+12𝛽+9𝜃),344𝛽²+750𝛽𝜃+325𝜃²=0, 𝑓₁=−3164304640𝑏²₂₀[phone]√1153𝑏²₂₀−505511440𝑏₂₀𝜃+16321424√1153𝑏₂₀𝜃−1443860355𝜃²[phone]√1153𝜃²;𝑓₂=−748902307758080𝑏⁴₂₀[phone]√1153𝑏⁴₂₀−239280807055360𝑏³₂₀𝜃+7725648121856√1153𝑏³₂₀𝜃+5835163629391040𝑏²₂₀𝜃²−171639395144768√1153𝑏²₂₀𝜃²[phone]𝑏₂₀𝜃³−28776579351136√1153𝑏₂₀𝜃³+3030081181205925𝜃⁴−89269930142715√1153𝜃⁴, (10) we have Resultant𝑓₁,𝑓₂,𝑏₂₀=2.87343×10⁵³𝜃⁸≠0, (11) where Resultant[𝑔(𝑥),𝑓(𝑥),𝑥] denotes the resultant of 𝑓(𝑥),𝑔(𝑥) with respect to 𝑥. So 𝑓₂≠0 when 𝑓₁=0. Namely, point (1,1) is a fourth-order weak focus.

We next study bifurcation of limit cycles of the perturbed system of (4). When condition (9) holds, we can obtain 𝐷𝑢𝐽=¹,𝑎₂₀𝐷𝑢₁,𝛽𝐷𝑢₁,𝑏₂₀𝐷𝑢₂,𝑎₂₀𝐷𝑢₂,𝛽𝐷𝑢₂,𝑏₂₀𝐷𝑢₃,𝑎₂₀𝐷𝑢₃,𝛽𝐷𝑢₃,𝑏₂₀≠0. (12)

In fact, 𝐽=7001704448𝑖234375−75+√1153¹¹𝜃⁷×688𝑏₂₀+227𝜃+5√1153𝜃²×−17721692278080𝑏²₂₀[phone]√1153𝑏²₂₀−2991374746080𝑏₂₀𝜃+85097939808√1153𝑏₂₀𝜃−2771947690805𝜃²[phone]√1153𝜃². (13) So Resultant[𝐽,𝑓₃,𝑏₂₀]≠0, namely, 𝐽≠0, when 𝑎₂₀=(1/4)(4𝑏₂₀+12𝛽+9𝜃),344𝛽²+750𝛽𝜃+325𝜃²=0.

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 (1,1).

With a similar method, we could discuss the bifurcation of limit cycles from point (2,2). Through the transformations 𝑥=𝑥−2,𝑦=𝑦−2,𝜏=𝜃𝑡, we still denote 𝑥,𝑦 by 𝑥,𝑦 for convenience. Then point (1,1) will be moved into point (0,0) of the new system, and the system will be changed into the following system: 𝑑𝑥=𝑥𝑑𝜏2²−𝑦−7𝑥𝑦2−3𝑥²𝑦2+𝑦²+𝑥𝑦²2−2𝑎₂₀𝑥²𝜃−𝑎₂₀𝑥³𝜃+4𝑎₂₀𝑥𝑦𝜃+2𝑎₂₀𝑥²𝑦𝜃−2𝑎₂₀𝑦²𝜃−𝑎₂₀𝑥𝑦²𝜃+𝑥²𝛽𝜃−6𝑥𝑦𝛽𝜃−3𝑥²𝑦𝛽𝜃+4𝑦²𝛽𝜃+2𝑥𝑦²𝛽𝜃,𝑑𝑦𝑑𝜏=𝑥+3𝑥𝑦2+𝑦²2+𝑥𝑦²2+𝑦³2−2𝑏₂₀𝑥²𝜃+4𝑏₂₀𝑥𝑦𝜃−𝑏₂₀𝑥²𝑦𝜃−2𝑏₂₀𝑦²𝜃+2𝑏₂₀𝑥𝑦²𝜃−𝑏₂₀𝑦³𝜃−2𝑥𝑦𝛽𝜃+3𝑦²𝛽𝜃−𝑥𝑦²𝛽𝜃+2𝑦³𝛽𝜃. (14) Furthermore, by the transformations 𝑧=𝑥+𝑖𝑦,𝑤=𝑥−𝑖𝑦,𝑇=𝑖𝜏,𝑖=√−1, (15) system (14) can be transformed into the following system: 𝑑𝑧1𝑑𝑇=𝑧+(28𝜃−4𝑎₂₀−4𝑖𝑏₂₀+(3+5𝑖)𝛽+(3−𝑖)𝜃𝑧²+28𝑖𝑎₂₀−8𝑏₂₀+(6−10𝑖)𝛽+(1−3𝑖)𝜃𝑧𝑤−24𝑎₂₀+4𝑖𝑏₂₀−(9−𝑖)𝛽−(4−2𝑖)𝜃𝑤²+−2𝑎₂₀−2𝑏₂₀+4(1+4𝑖)𝛽+(1+𝑖)𝜃𝑧³+2(−1+2𝑖)𝑎₂₀+(1+2𝑖)𝑏₂₀+(1−4𝑖)𝛽+(1−𝑖)𝜃𝑧²𝑤+2(1+2𝑖)𝑎₂₀+2(1−2𝑖)𝑏₂₀−(1−𝑖)4𝛽+𝜃𝑧𝑤²−2−𝑎₂₀+𝑏₂₀+𝛽+𝜃𝑤³;𝑑𝑤1𝑑𝑇=−𝑤−28𝜃−4𝑎₂₀+4𝑖𝑏₂₀+(3−5𝑖)𝛽+(3+𝑖)𝜃𝑤²+2−8𝑖𝑎₂₀−8𝑏₂₀+(6+10𝑖)𝛽+(1+3𝑖)𝜃𝑧𝑤−24𝑎₂₀−4𝑖𝑏₂₀−(9+𝑖)𝛽−(4+2𝑖)𝜃𝑧²+−2𝑎₂₀−2𝑏₂₀+4(1−4𝑖)𝛽+(1−𝑖)𝜃𝑤³+2(−1−2𝑖)𝑎₂₀+(1−2𝑖)𝑏₂₀+(1+4𝑖)𝛽+(1+𝑖)𝜃𝑤²𝑧+2(1−2𝑖)𝑎₂₀+2(1+2𝑖)𝑏₂₀−(1+𝑖)4𝛽+𝜃𝑤𝑧²−2−𝑎₂₀+𝑏₂₀+𝛽+𝜃𝑧³. (16) Direct computation can give the following theorem.

Theorem 4. The first four singular point quantities at the origin of system (16) are as follows: 𝑢₁=𝑖4𝜃²−2𝑎₂₀−2𝑏₂₀+4𝛽+𝜃8𝑎₂₀−8𝑏₂₀,𝑢−3𝜃₂𝑖=−48𝜃³−2𝑎₂₀−2𝑏₂₀+4𝛽+𝜃56𝛽²+57𝛽𝜃+10𝜃²,𝑢₃=𝑖6144𝜃⁵−2𝑎₂₀−2𝑏₂₀+4𝛽+𝜃𝑓₃,𝑢₄𝑖=−5898240𝛽⁷−2𝑎₂₀−2𝑏₂₀+4𝛽+𝜃𝑓₄, (17) where 𝑓₃=−143360𝑏²₂₀𝛽²+286720𝑏₂₀𝛽³−206080𝛽⁴−168960𝑏²₂₀𝛽𝜃+355840𝑏₂₀𝛽²𝜃−264320𝛽³𝜃−40960𝑏²₂₀𝜃²+103040𝑏₂₀𝛽𝜃²−82336𝛽²𝜃²+5120𝑏₂₀𝜃³−1939𝛽𝜃³+980𝜃⁴,𝑓₄=2967207936000𝛽¹⁰−9714651627520𝛽⁹𝜃−65013770321920𝛽⁸𝜃²−124569363533824𝛽⁷𝜃³−124914242976768𝛽⁶𝜃⁴−75962577303936𝛽⁵𝜃⁵−29760585599448𝛽⁴𝜃⁶−7669544070125𝛽³𝜃⁷−1275738647714𝛽²𝜃⁸−125233021760𝛽𝜃⁹−5502919200𝜃¹⁰. (18)

While solving 𝑢₁=𝑢₂=𝑢₃=0 and 𝑢₄≠0, we can immediately obtain the following theorem.

Theorem 5. The origin of system (16) is a 4-order weak focus if and only if 𝑎₂₀=188𝑏₂₀+3𝜃,56𝛽²+57𝛽𝜃+10𝜃²𝑓=0,₃=0. (19)

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 (2,2).

The results also yield that when point (1,1) is a fourth-order weak focus, point (2,2) of system is a first-order weak focus at the same time. Point (1,1) is a first-order weak focus when point (2,2) is a fourth-order weak focus. Namely, by a simultaneously perturbation, five limit cycles could be bifurcated with two different distribution; see Figure 1.

(a)

(b)

Figure 1 

Distribution of limit cycles.

3. Center Conditions of Two Positive Equilibrium Points

If all Lyapunov constants are equal to zero, it will be a center condition of a critical point. So Theorems 1 and 4 imply the following theorem.

Theorem 7. If the origins of systems (4) and (14) are centers if and only if 𝑎₂₀=12−2𝑏₂₀+4𝛽+𝜃. (20)

Proof. When 𝑎₂₀=(1/2)(−2𝑏₂₀+4𝛽+𝜃), system (4) could be rewritten as 𝑑𝑥1𝑑𝜏=−𝑦−2𝛽−2𝑏₂₀𝑥²−2𝑏₂₀𝑥³+4𝑏₂₀𝑥𝑦+4𝑏₂₀𝑥²𝑦−2𝑏₂₀𝑦²−2𝑏₂₀𝑥𝑦²−4𝑥²𝛽+4𝑥³𝛽−2𝑥²𝑦𝛽−3𝑥²𝜃+𝑥³𝜃+𝑥𝑦𝜃+𝑥²𝑦𝜃,𝑑𝑦1𝑑𝜏=𝑥−2𝛽2𝑏₂₀𝑥²−4𝑏₂₀𝑥𝑦+2𝑏₂₀𝑥²𝑦+2𝑏₂₀𝑦²−4𝑏₂₀𝑥𝑦²+2𝑏₂₀𝑦³−2𝑥𝛽+4𝑦²𝛽+2𝑥𝑦²𝛽−4𝑦³𝛽−𝑥𝑦𝜃+3𝑦²𝜃−𝑥𝑦²𝜃−𝑦³𝜃. (21) By transformation 𝑥=−𝑢+𝑣2(𝑣−1),𝑦=−−𝑢+𝑣2()()𝑣−1,𝜏=𝑣−1²𝑡, (22) system (21) becomes 𝑑𝑢1𝑑𝜏=−𝑣−4𝛽−8𝑏₂₀𝑢²+4𝑏₂₀𝑢⁴+4𝑏₂₀𝑢²𝑣−3𝑢⁴𝛽+4𝑣𝛽+11𝑢²𝑣𝛽−12𝑣²𝛽−9𝑢²𝑣²𝛽+9𝑣³𝛽−4𝑢²𝛽+11𝑢²𝑣𝜃−2𝑣²𝜃−8𝑢²𝑣²𝜃+3𝑣³𝜃,𝑑𝑣=𝑢(𝑑𝜏−1+𝑣)−4𝑏₂₀𝑢²−4𝛽+3𝑢²𝛽+9𝑣²𝛽−6𝑣𝜃+8𝑣²𝜃×4𝛽⁻¹ (23) which is symmetric with 𝑣-axis.
When 𝑎₂₀=(1/2)(−2𝑏₂₀+4𝛽+𝜃), system (14) could be rewritten as 𝑑𝑥1𝑑𝜏=−𝑦−2𝜃−4𝑏₂₀𝑥²−2𝑏₂₀𝑥³+8𝑏₂₀𝑥𝑦+4𝑏₂₀𝑥²𝑦−4𝑏₂₀𝑦²−2𝑏₂₀𝑥𝑦²+6𝑥²𝛽+4𝑥³𝛽−4𝑥𝑦𝛽−2𝑥²𝑦𝛽+𝑥²𝜃+𝑥³𝜃+3𝑥𝑦𝜃+𝑥²𝑦𝜃,𝑑𝑦1𝑑𝜏=𝑥+2𝜃−4𝑏₂₀𝑥²+8𝑏₂₀𝑥𝑦−2𝑏₂₀𝑥²𝑦−4𝑏₂₀𝑦²+4𝑏₂₀𝑥𝑦²−2𝑏₂₀𝑦³−4𝑥𝑦𝛽+6𝑦²𝛽−2𝑥𝑦²𝛽+4𝑦³𝛽+3𝑥𝑦𝜃+𝑦²𝜃+𝑥𝑦²𝜃+𝑦³𝜃. (24) Transformations 𝑥=−𝑢+𝑣2(𝑣−1),𝑦=−−𝑢+𝑣2()()𝑣−1,𝜏=𝑣−1²𝑡, (25) bring system (24) into 𝑑𝑢1𝑑𝜏=−𝑣−4𝜃−16𝑏₂₀𝑢²+4𝑏₂₀𝑢⁴+12𝑏₂₀𝑢²𝑣+10𝑢²𝛽−3𝑢⁴𝛽−15𝑢²𝑣𝛽+2𝑣²𝛽+7𝑢²𝑣²𝛽−𝑣³𝛽+2𝑢²𝜃−7𝑢²𝑣𝜃−4𝑣²𝜃+4𝑢²𝑣²𝜃+𝑣³𝜃,𝑑𝑣=𝑢()𝑑𝜏−1+𝑣4𝑏₂₀𝑢²−3𝑢²𝛽−12𝑣𝛽+7𝑣²𝛽+4𝜃−10𝑣𝜃+4𝑣²𝜃×(4𝜃)⁻¹ (26) which is symmetric with 𝑣-axis. So the origin is a center of system (14).

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgments

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).