Hopf bifurcation problems near double positive equilibrium points for a class of quartic Kolmogorov model

: The Kolmogorov model is a class of significant ecological models and is initially introduced to describe the interaction between two species occupying the same ecological habitat. Limit cycle bifurcation problem is close to Hilbertis 16th problem. In this paper, we focus on investigating bifurcation of limit cycle for a class of quartic Kolmogorov model with two positive equilibrium points. Using the singular values method, we obtain the Lyapunov constants for each positive equilibrium point and investigate their limit cycle bifurcations behavior. Furthermore, based on the analysis of their Lyapunov constants’ structure and Hopf bifurcation, we give the condition that each one positive equilibrium point of studied model can bifurcate 5 limit cycles, which include 3 stable limit cycles.


Introduction
The following differential autonomous systems in a planar vector field ẋ = F(x, y), ẏ = G(x, y) have been widely studied and a great deal of attentions have been paid to this problem in many literatures.This activity reflects the breadth of interest in Hilbert's 16th problem and the fact that the above systems are often used as mathematical models to describe real-life problems.Hilbert's 16th problem is to find the maximum number of limit cycles of system (1).The predator-prey system, the competition system and the cooperation model are the three most basic types of systems in mathematical ecology.Theoretically, many natural predator-prey systems can be discussed and investigated by some kinds of ecological models.The qualitative properties of differential systems are often used to describe the characteristics of ecosystems, as they have been investigated in some literatures, for example, Liénard systems ( [2,3,19,21]), Kolmogorov systems ( [2-12, 16-20, 22-26]) and some other differential systems ( [13][14][15]) and so on.The Kolmogorov systems (introduced by A. Kolmogorov in 1936 [12]), as a class of significant ecological models, were initially introduced to describe the interaction between two species occupying the same ecological habitat.The form Kolmogorov models are as follows: dx dt = x f (x, y), dy dt = y g(x, y), (2) in which f (x, y) and g(x, y) are polynomials in x and y.The variables x and y are often described as the number of species in two ecological populations, dx dt and dy dt represent the growth rates of x and y.Hence, attention is often restricted to the behavior of orbits in the 'realistic quadrant' {(x, y) : x > 0, y > 0}.Particular significance in applications is the existence of limit cycles and the number of limit cycles that can occur near positive equilibrium points, because a limit cycle corresponds to an equilibrium state of the system.The existence and stability of limit cycles are closely related to the positive equilibrium points.Hence, many references studying Kolmogorov models pay more attention to the limit cycles problem.
On the qualitative analysis and the bifurcation of limit cycles for cubic planar Kolmogorov systems, some papers are as follows: Ref. [1] characterized the center conditions for a cubic Kolmogorov differential system and obtained the condition that the positive equilibrium point can become a fine focus of order five.Ref. [23,24] studied a class of cubic Kolmogorov systems that can bifurcate three limit cycles from the positive equilibrium point (1,1).Ref. [4] investigated the limit cycles bifurcation problem for a class of cubic Kolmogorov system and showed that this class of the Kolmogorov system could bifurcate five limit cycles including 3 stable cycles.Ref. [22] showed that a class of the cubic Kolmogorov system could bifurcate 6 limit cycles.Ref. [25] obtained the condition of integrability and non-linearizability of weak saddles for a cubic Kolmogorov model.Ref. [6] investigated limit cycles in a class of the quartic Kolmogorov model with three positive equilibrium points.Ref. [7] studied the three-Dimensional Hopf Bifurcation for a class of the cubic Kolmogorov model.Ref. [19] investigated the integrability of a class of 3-dimensional Kolmogorov system and provided the phase portraits.
In addition to cubic Kolmogorov systems, there is also a lot of literature on the bifurcation of limit cycles for some generalized Kolmogorov systems and higher order Kolmogorov systems.For example: Ref. [10,26] studied a general Kolmogorov model and obtained the condition for the existence and uniqueness of limit cycles and it classified a series of models.Ref. [8] investigated the bifurcation of limit cycles for a class of the quartic Kolmogorov model with two symmetrical positive singular points.Ref. [9] investigated the Hopf bifurcation problem about small amplitude limit cycles and the local bifurcation of critical periods for a quartic Kolmogrov system at the single positive equilibrium point (1,1) and proved that the maximum number of small amplitude limit cycles bifurcating from the equilibrium point (1,1) was 7. Ref. [20] considered the Kolmogorov system of degree 3 in R 2 and R 3 , and showed it had an equilibrium point in the positive quadrant and octant, and provided sufficient conditions in order that the equilibrium point will be a Hopf point for the planar case and a zero-Hopf point for the spatial one, and studied the limit cycles bifurcating from these equilibria using averaging theory of second and first order.
As far as limit cycles of Kolmogorov models are concerned, many good results have been obtained, especially in lower degree system by analyzing a sole positive equilibrium point's state.However, results for Simultaneous limit cycles bifurcating from several different equilibrium points' is less seen, and perhaps it is difficult to investigate this kind of problem.From an ecological perspective, investigation about multiple positive equilibrium points is meaningful, the equilibrium point (a, b) represents that the ratio of the density about predator and prey is a : b, and so it is possible for several equilibrium points to occur in ecosystem.
In this paper, we study a class of the following quartic Kolmogorov models in which a, b, A 10 and B 10 ∈ R.
Clearly, model (3) has two positive equilibrium points namely, (1,1) and (2,1).We will focus on the limit cycles bifurcations of the two positive equilibrium points.By analyzing and proving carefully, we obtain that each one of the two positive equilibrium points can be a 5th-order fine focus.Furthermore, we find the condition that each positive equilibrium point can bifurcate five limit cycles, of which three limit cycles are stable.Our results are concise (especial that in the expressions of focal values) and the proof about existence of limit cycles is algebraic and symbolic.
This paper includes 4 sections.In Section 2, we introduce the method to compute focal value offered by [18].In Section 3, we respectively compute the focal values of the two positive equilibrium points of model ( 3) and obtain the condition that they can be two 5th-order fine focuses.In Section 4, we discuss the bifurcation of limit cycles of model (3) and obtain that each one of the two positive equilibrium points of model (3) can have five small limit cycles; we give an example that three stable limit cycles can occur near each positive equilibrium point under a certain condition.

A kind of method to calculate focal values
In order to use the algorithm of the singular point value to compute focal values and construct the Poincaré succession function, we need to give some properties of focal values and singular point values.
Consider the following system where X k (x, y), Y k (x, y) are homogeneous polynomials of degree k on x, y.Under the polar coordinates x = rcosθ, y = rsinθ, system (4) takes the form For sufficiently small h, let be the Poincaré succession function and the solution of Eq (6) satisfies the initial value condition Lemma 2.1.For system (4) and any positive integer m, among v 2m (2π), v k (2π) and v k (π), there exists expression of the relation where In addition to indicating that v 2m = 0 under the conditions

plays an important role in construction of Poincaré succession function.
Definition 2.1.For system (4), in the expression (6) then the origin is called the fine focus (weak focus) of order k, and the quantity of v 2k+1 (2π) is called the kth focal values at the origin and for any positive integer k, v 2k+1 (2π) = 0, then the origin is called a center.

By means of transformation
system (4)| δ=0 can be transformed into the following systems where z, w, T are complex variables and It is obvious that the coefficients of system (10) satisfy conjugate condition ∀a αβ = b αβ , we call that system (4)| δ=0 and (10) are concomitant.Lemma 2.2.(See [16]) For system (10), we can derive successively the terms of the following formal series where c kk ∈ R, k = 1, 2, • • • , and to any integer m, µ m is determined by following recursion formulas Lemma 2.3.(See [18]) For systems (5)| δ=0 , (10) and any positive integer m, the following assertion holds where be polynomial functions of coefficients of system (10).

Focal values of the two positive equilibrium points of model (3)
For the linearized system of model (3), its coefficient matrix in point (x 0 , y 0 ) is as follows: For the two positive equilibrium points (1,1) and (2,1) of model ( 3), their coefficient matrixes of the linearized system of model (3) become Clearly, A (1,1) and A (2,1) have the same two eigenvalues ±i.Hence, model (3) can be changed into the system (4) by making some appropriate transformations.
For convenience, we will respectively compute the focal values of each positive equilibrium point of model (3).

By means of transformation
model ( 3) takes the following form (17) and the equilibrium point (1, 1) of model ( 3) becomes the origin of ( 17) correspondingly.
Under the transformation system (17) becomes the following complex system in which Obviously, system (19) belongs to the type of system (10).Then, we can compute the focal values of the origin of (19) (namely the focal values of the equilibrium point (1, 1) of model ( 3)) by using the method of Section 2. According to the recursion formulas ( 13) and ( 14) offered by Lemma 2.2, the following theorem holds.For convenience, here we note that B 10 = λ − 6b.
According to the relation between model (3) and system (17) and system (19), from Theorem 3.1 and Lemma 2.3, the following theorem is visible.
From the quality of resultant, if f (x, y) = 0, g(x, y) = 0 have solutions, then the resultant of f (x, y), g(x, y) with respect to x or y will vanish.When computer soft Mathematica 6.0 is used, f (x, y), g(x, y) with respect to x is shown as Resultant [ f, g, x].Hence, v 3 = v 5 = v 7 = v 9 = 0 hold if and only if Eq (20) holds and While Eq (21) hold if and only if By computing, we obtain in which g(λ) is a 242 degrees function on λ .It can be seen that Eq (22) has some real solutions such as λ =  3) can be a 5th-order fine focus at most.

Focal values of the positive equilibrium points (2,1) of model (3)
Next, we compute the focal values of the positive equilibrium point (2,1) of model (3).By means of transformation model ( 3) takes the following form: (24) and the equilibrium point (2, 1) of model ( 3) becomes the origin of system (24) correspondingly.Under the transformation system (24) becomes its concomitant complex system, i.e., in which Obviously, system (26) belongs to the class of system (10).Next, we begin to compute the focal values of the origin of system (24) (namely the focal values of the equilibrium point (2, 1) of model ( 3)).According to the recursion formulas of Lemma 2.2, we have the following result.For convenience, we note that B 10 = λ 1 − 3b − 17 6 .
Theorem 3.4.The first five singular point values at the origin of system (24) are as follows:

The bifurcations of limit cycles of model (3)
After finding focal values of two positive equilibrium points of model ( 3), we will consider the limit cycle bifurcation near (1, 1) and (2, 1) of perturbed model (3).Lemma 4.1.Let J 1 be the Jacobin of the function group (v 3 , v 5 , v 7 , v 9 ) with respect to the variables (a, λ, A 10 , b), if the equilibrium (1, 1) of model ( 3) is a 5th-order fine focus, then J 1 0.
Proof.Suppose that J 1 = 0, next we deduce a contradictory result.The Jacobin of the function group (v 3 , v 5 , v 7 , v 9 ) with respect to the variables (a, λ, A 10 , b) has the following form Obviously, J 1 is a function with respect to the variables (a, λ, A 10 , b).Because the equilibrium point (1, 1) of model ( 3) is a 5th-order fine focus, then v 3 = v 5 = v 7 = v 9 = 0. Suppose that J 1 = 0, then the resultant of J 1 , v i , (i ∈ {3, 5, 7, 9}) with respect to the variable b will become 0.
Let Similarly, we can obtain the following lemma.
Proof.From lemma 4.1, J 1 0, while (1, 1) is a 5-th order fine focus of system (3), according to the theory of reference [14], the conclusion of Theorem 4.1 holds.Proof end.Similarly, we have the following theorem.
Next we will give a case that (1, 1) of the model (3) can bifurcate 5 limit cycles of which 3 limit cycles are stable.
According to existence theorem of implicit function and the result of Lemma 4.1, Eq (30) has a group of solutions as follows: From ( 30) and (31), clearly the following theorem holds.
Similarly, we can obtain the following theorem.

Conclusions
The work of this paper focuses on investigating the limit cycle bifurcation of a class of the quartic Kolmogorov model, which is an interesting and significant ecological model both in theory and applications.We used the singular values method to compute focal value.First, we give the relation between focal value and singular point value at the origin, which is necessary for us to investigate bifurcations of limit cycles.Sceond, we give the singular point values' recursive formulas and respectively compute the focal values of the two positive equilibrium points of model ( 3) and obtain the condition that they can be two 5th-order fine focuses.Next, we discuss the bifurcation of limit cycle of model (3) and obtain that each one of the two positive equilibrium points of model (3) can have five small limit cycles.Then, we show a case that each positive equilibrium point can bifurcate 3 stable limit cycles at most by making use of algebraic and symbolic proof.