# Journal of Nonlinear Mathematical Physics

Volume 27, Issue 2, January 2020, Pages 267 - 278

# Final evolutions of a class of May-Leonard Lotka-Volterra systems

Authors
Claudio A. Buzzi, Robson A. T. Santos
Departamento de Matemática, Universidade Estadual Paulista, Rua Cristóvão Colombo 2265, São José do Rio Preto, 15115-000, Brazil,claudio.buzzi@unesp.brandrobson.trevizan@outlook.com
Jaume Llibre
Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edificio C Facultad de Ciencias, Bellaterra (Barcelona), Catalonia, 08193, Spain,jllibre@mat.uab.cat
Received 8 June 2019, Accepted 29 August 2019, Available Online 27 January 2020.
DOI
10.1080/14029251.2020.1700635How to use a DOI?
Keywords
May-Leonard system; Lotka-Volterra system; invariant, global dynamics
Abstract

We study a particular class of Lotka-Volterra 3-dimensional systems called May-Leonard systems, which depend on two real parameters a and b, when a + b = −1. For these values of the parameters we shall describe its global dynamics in the compactification of the non-negative octant of ℝ3 including its infinity. This can be done because this differential system possesses a Darboux invariant.

Open Access

## 1. Introduction

Polynomial ordinary differential systems are often used in various branches of applied mathematics, physics, chemist, engineering, etc. Models studying the interaction between species of predator-prey type have been extensively analyzed as the classical Lotka-Volterra systems. For more information on the Lotka-Volterra systems see for instance [8] and the references quoted there. In particular, one of these competition models between three species inside the class of 3-dimensional Lotka-Volterra systems is the May-Leonard model given by the polynomial differential system in ℝ3

x˙=x(1xaybz),y˙=y(1bxyaz),z˙=z(1axbyz),(1.1)
where a and b are real parameters and the dot denotes derivative with respect to the time t. See for more details on the May-Leonard system the papers [10] and [2] and on Lotka-Volterra systems [9], and the references quoted there.

The Lotka-Volterra systems in ℝ3 have the property that the three coordinate planes are invariant by the flow of these systems. Moreover, at points of straight line x = y = z, system (1.1) is reduced to = x−(1 + a + b)x2, because the other equations do not provide any further information. Therefore, the bisectrix of the non-negative octant is an invariant straight line for this differential system.

In this paper we describe the global dynamics of system (1.1) in function of the parameters a and b when a + b = −1. The system (1.1) is defined in ℝ3. In order to study the dynamics of its orbits at infinity we extend analytically its flow by using the Poincaré compactification of ℝ3. In the appendix we give precise definitions for this compactification. The region of interest in our study is the non-negative octant of ℝ3, i.e. where x ≥ 0, y ≥ 0, z ≥ 0. So we shall study the flow of the Poincaré compactification in the region

R={(x,y,z)3:x2+y2+z21,x0,y0,z0}
of the Poincaré ball.

We remark that the global dynamics of the May-Leonard system (1.1) with a + b = −1 can be studied because this differential system has a Darboux invariant.

The differential system (1.1) has been extensively studied in order to understand the interaction between species and try to predict possible extinction or overpopulation for example. However our interest is purely mathematical, we want to illustrate how the Darboux invariant can be used to describe the global dynamics of a differential system. Note that we are interested in the study of system (1.1) for all real values of the parameters a and b satisfying a + b = −1, and not only for their positive values. Consequently our analysis has no biological meaning. This study could be made in a similar way in the others octants of ℝ3.

## 2. Statement of the main results

We denote by X the polynomial vector field associated to the differential system (1.1), and by p(X) the Poincaré compactification of X, see the appendix in section 5. The flow of system (1.1) in the region R is described in the next two theorems. For a formal definition of topologically equivalent phase portraits see [5].

### Theorem 2.1.

For the May-Leonard differential system (1.1) in the octant R the following statements hold when a + b = −1.

1. (a)

The phase portrait of the Poincaré compactification p(X) of system (1.1) on the boundaries x = 0, y = 0 and z = 0 of R is topologically equivalent to the one described in Fig. 1(a) if a ≤ −2 or a ≥ 1, and in Fig. 2(a) if −2 < a < 1.

2. (b)

The phase portrait of the Poincaré compactification p(X) of system (1.1) on R = ∂R ∩ {x2 + y2 + z2 = 1} (i.e. the phase portrait at the infinity of the non-negative octant of3) is topologically equivalent to the one described in Fig. 1(b) if a ≤ −2 or a ≥ 1, Fig. 2(b) if a = −1/2, Fig. 2(c) if −2 < a < −1/2, and Fig. 2(d) if −1/2 < a < 1. In particular, there are no periodic orbits in R.

3. (c)

When a = −1/2 the planes x = y, x = z and y = z are invariant by the flow of system (1.1), and the phase portrait of the Poincaré compactification p(X) of system (1.1) on R ∩ {x = y}, R∩{x = z} and R∩{y = z} are topologically equivalent to the ones described in (a), (b) and (c) of Fig. 3 respectively.

Let p(γ) denote the orbit γ of the vector field X associated to system (1.1) in the Poincaré compactification p(X).

### Theorem 2.2.

Let γ be an orbit of system (1.1) with a + b = −1 such that p(γ) is contained in the interior of R. Then the following statements hold.

1. (a)

If a ≤ −2 or a ≥ 1 then we have:

1. (i)

The α-limit set of p(γ) is either the origin of3, or the heteroclinic loop connecting the singular points p1, p2 and p3, or the heteroclinic loop connecting the singular points px, py and pz (see Fig. 1(a)).

2. (ii)

The ω-limit set of p(γ) is either pb (the positive endpoint of the bisectrix x = y = z), or the heteroclinic loop connecting the singular points px, py and pz (see Fig. 1(b)).

2. (b)

If −2 < a < 1 then we have:

1. (i)

The α-limit set of p(γ) is one of the singular points pj for j = 0, 1... , 6 or px, py, pz, pxy, pxz, pyz (see Fig. 2(a)).

2. (ii)

The ω-limit set of p(γ) is pb, the positive endpoint of the bisectrix x = y = z (see Figure 2(b)(c)(d)).

An immediate consequence of Theorem 2.2 is the following result.

### Corollary 2.1.

All orbit γ of system (1.1) with a + b = −1 such that p(γ) is contained in the interior of R has their α-limit in xyz = 0 and their ω-limit in R.

## 3. Proof of Theorem 1

The following two lemmas will be useful to the proof of Theorem 1.

### Lemma 3.1.

The May-Leonard differential system (1.1) with a+ b = −1 has four finite equilibrium points in the case a ≤ −2 or a ≥ 1, and has seven finite equilibrium points in the case −2 < a < 1. Moreover, the local dynamics around these equilibrium points are presented in Figures 1(a) and 2(a).

### Proof.

The finite singular points of system (1.1) with a + b = −1 are the solutions of the system

P1(x,y,z)=x(1x+z+a(y+z))=0,P2(x,y,z)=y(1+xy+a(z+x))=0,P3(x,y,z)=z(1+yz+a(x+y))=0,
namely
p0=(0,0,0)p1=(1,0,0),p2=(0,1,0),p3=(0,0,1),p4=(0,1aA,2+aA),p5=(2+aA,0,1aA),p6=(1aA,2+aA,0),
where A = 1 + a + a2.

Since A > 0 for a ∈ ℝ and the region of interest is R, we have:

1. (i)

If a ≤ −2 or a ≥ 1 system (1.1) has only four finite equilibrium points: p0, p1, p2 and p3.

2. (ii)

If −2 < a < 1 system (1.1) has the seven finite equilibrium points pj for j = 0, 1... , 6.

All these finite equilibrium points are hyperbolic if a ≠ −2,1, and consequently its local phase portrait is topologically equivalent to the phase portrait of its linear part by the Hartman–Grobman Theorem, see for instance [4].

We note that when a ∈ (−2,1) and a → 1 we have that p4p3, p5p1 and p6p2; while if a → −2 we have that p4p2, p5p3 and p6p1. This behavior of these equilibria allows to determine by continuity the local phase portraits on the boundary of R of the non-hyperbolic equilibrium points p1, p2 and p3 when a = −2 and a = 1 from the global phase portraits of the boundary of R when a ∈ (−2,1).

The linear part of system (1.1) at the equilibrium p0 is the identity matrix. Therefore it is a repelling equilibrium.

The eigenvalues of linear part at equilibrium points p1, p2 and p3 are −1, 1− a, 2+ a. Therefore when a < −2 or a > 1 these equilibria have a 2-dimensional stable manifold and an 1-dimensional unstable one; and for −2 < a < 1 these equilibria have a 2-dimensional unstable manifold and an 1-dimensional stable one.

When −2 < a < 1 the eigenvalues of linear part at equilibrium points p4, p5 and p6 are 3, −1 and (−2 + a + a2)/A. Since −2 + a + a2 < 0 for −2 < a < 1 then these equilibria have a 2-dimensional stable manifold and an 1-dimensional unstable one. Moreover, p4 (respectively p5 and p6) is an attractor restricted to the invariant boundary x = 0 (respectively y = 0 and z = 0). Now we explain in few words, what we mean by saying that the local phase portraits, for a = −2 and a = 1, can be determined by continuity. For example, if a ∈ (−2,1) then, on the plane x = 0, we have that p4 is a node and p2 is a saddle. If a tends to −2 then p4 tends to p2 and we have a saddle-node bifurcation. For a = −2 we have that p4 = p2 is a saddle-node singularity with two hyperbolic sectors in the half-plane {x = 0,y < 0} and a parabolic sector in the half-plane {x = 0,y > 0}. So, in a neighborhood of p2 contained in the half-plane {x = 0, y > 0}, the local phase portraits are the same for a = −2 or a < −2 in the non-negative octant. The same analysis can be done for the other points p1 and p3, and for the case when a tends to 1.

### Lemma 3.2.

The May-Leonard differential system (1.1) with a + b = −1 has four infinite equilibrium points in the case a ≤ −2 or a ≥ 1, and has seven infinite equilibrium points in the case −2 < a < 1. Moreover, the local dynamics around these equilibrium points are presented in Figures 1(b), 2(b), 2(c) and 2(d).

### Proof.

Now we shall study the infinite equilibrium points. For study the dynamics on the infinity R of R we shall use the Poincaré compactification of the differential system (1.1). See appendix for details. Thus the differential system (1.1) in the local chart U1 becomes

z˙1=2z1+az1z12+az12z1z22az1z2,z˙2=z2az2+z1z2+2az1z22z22az22,z˙3=z3+az1z3z2z3az2z3z32.(3.1)

So system (1.1), with a + b = −1 and satisfying a ≤ −2 or a ≥ 1, has two equilibrium points at infinity: (0, 0, 0) and (1, 1, 0). We call the first one px the positive endpoint of the x-axis, and the second one pb the positive endpoint of the bisectrix x = y = z. The linear part at the equilibrium px has the eigenvalues 1 − a and 2 + a at infinity and eigenvalue 1 in its finite direction. Therefore, on the infinity px is a saddle such that its stable separatrix is contained in the z1-axis when a ≤ −2 (respectively z2-axis when a ≥ 1).

The eigenvalues of linear part at equilibrium pb are (3±i3(1+2a))/2 and 0. Therefore, on the infinity pb is a stable focus turning clockwise if a ≤ −2 (respectively counterclockwise if a ≥ 1).

Now system (1.1) in the local chart U2 writes

z˙1=z1az12z12az12+z1z2+2az1z2,z˙2=2z2+az2z1z22az1z2z22+az22,z˙3=z3z1z3az1z3+az2z3z32.(3.2)

Since the local chart U2 covers the end part of the plane x = 0 at infinity of the non-negative octant of ℝ3, we are interested only in the equilibrium points which are on z1 = 0 and z3 = 0. In this case, with a + b = −1 and satisfying a ≤ −2 or a ≥ 1, there is one equilibrium point at infinity: (0, 0, 0). We call this equilibrium py the positive endpoint of the y-axis. The eigenvalues of linear part at equilibrium py are 1 − a and 2 + a at infinity and eigenvalue 1 in its finite direction. Therefore, on the infinity py is a saddle such that its stable separatrix is contained in the z2-axis when a ≤ −2 (respectively z1-axis when a ≥ 1).

Now for a ≤ −2 or a ≥ 1 we only need to study the equilibrium point at the endpoint of positive z-half-axis, i.e. the equilibrium point at the origin of the local chart U3. We call this equilibrium point pz. In the local chart U3 system (1.1) becomes

z˙1=2z1+az1z12+az12z1z22az1z2,z˙2=z2az2+z1z2+2az1z22z22az22,z˙3=z3+az1z3z2z3az2z3z32.

The linear part at the equilibrium point pz has eigenvalues 1− a and 2+ a at infinity and eigenvalue 1 in its finite direction. Therefore the origin of the local chart U3 is a saddle such that its stable separatrix is contained in the z1-axis when a ≤ −2 (respectively z2-axis when a ≥ 1).

We have proved that the phase portrait of system (1.1) on R, for a + b = −1 and a ≤ −2, is the one presented in Figure 1(b). In the same figure, reversing the sense of all the orbits, we have the global dynamics on R, for a + b = −1 and a ≥ 1.

It remains to study the infinite equilibrium points of system (1.1) in case −2 < a < 1. In the local chart U1 the system (3.1) has four equilibrium points at infinity: px=(0,0,0), pxy=((2+a)/(1a),0,0), pxz=(0,(1a)/(2+a),0) and pb=(1,1,0). The eigenvalues of linear part at equilibrium px are 1 − a and 2 + a at infinity and eigenvalue 1 in its finite direction. Then this equilibrium is an unstable node. The linear part at the equilibrium pxy (respectively pxz) has the eigenvalues −(2 + a), A/(1 − a) and 3A/(1 − a) (respectively (a − 1), A/(2 + a) and 3A/(2 + a)). Therefore the equilibria pxy and pxz are saddles such that their stable separatrix is contained in the z1-axis and z2-axis respectively. The eigenvalues of the linear part at the equilibrium pb are (3±i3(1+2a))/2 and 0. Therefore, on the infinity pb is a stable focus turning clockwise if −2 < a < −1/2 (respectively counterclockwise if −1/2 < a < 1). When a = −1/2 on the infinity pb is stable node.

Now since we are interested only in the equilibrium points which are on z1 = 0 and z3 = 0, in the local chart U2 the system (3.2) has two equilibrium points: py=(0,0,0) and pyz=(0,(2+a)/(1a),0). The eigenvalues of linear part at equilibrium py are 1 − a and 2 + a at infinity and eigenvalue 1 in its finite direction. Then, on the infinity this equilibrium is an unstable node. The linear part at the equilibrium (0, (2 + a)/(1 − a), 0) has the eigenvalues −(2 + a), A/(1 − a) and 3A/(1 − a) at infinity. Therefore, on the infinity pyz=(0,(2+a)/(1a),0) is a saddle such that its stable separatrix is contained in the z2-axis.

In the local chart U3 we only need to study the equilibrium point pz=(0,0,0). The eigenvalues of linear part at this equilibrium are 1− a and 2+ a at infinity and eigenvalue 1 in its finite direction. Then, on the infinity this equilibrium is an unstable node.

We conclude that the infinite equilibrium points of system (1.1) with a + b = −1 in case −2 < a < 1 are

• px = the positive endpoint of the x-axis,

• py = the positive endpoint of the y-axis,

• pz = the positive endpoint of the z-axis,

• pxz = the positive endpoint of the straight line (a − 1)x + (2 + a)z = 0, y = 0,

• pxy = the positive endpoint of the straight line (2 + a)x + (a − 1)y = 0, z = 0,

• pyz = the positive endpoint of the straight line (2 + a)y + (a − 1)z = 0, x = 0,

• pb = the positive endpoint of the bisectrix x = y = z,

and the phase portraits of system (1.1) on R, for a + b = −1 and −2 < a < 1, are presented in Figures 2(b), 2(c) and 2(d).

We also observe that when a ∈ (−2,1) and a → 1 we have that pxzpx, pyzpz, and pxypy; while if a → −2 we have that pxzpz, pyzpy, and pxypx. So the behavior of these equilibria allows to determine by continuity the local phase portraits on the boundary of R of the non-hyperbolic equilibrium which are at the positive endpoints of the axes of coordinates when a = −2 and a = 1 from the global phase portraits of the boundary of R when a ∈ (−2,1).

We will show now that does not exist a periodic orbit on R. For this we will need Bautin’s Theorem, which is proved in [1].

### Theorem 3.1 (Bautin’s Theorem).

A quadratic polynomial differential system of the form

x˙=x(a0+a1x+a2y),y˙=x(b0+b1x+b2y),
has no limit cycles.

### Lemma 3.3

There are no periodic orbits of the Poincaré compactification of the vector field associated to system (1.1) in the Poincaré compactification on R.

### Proof.

The differential system (1.1) in the local chart U1 is given by (3.1). Making z3 = 0 to determine the dynamics on R we have the system

z˙1=z1((2+a)+(a1)z1(1+2a)z2),z˙2=z2((1a)+(1+2a)z1(2+a)z2).(3.3)

By Bautin’s Theorem, the system (3.3) has no limit cycles. In addition, system (3.3) has only two equilibrium points: (z1, z2) = (0, 0) and (z1, z2) = (1,1). The linear part at the equilibrium (0,0) has the eigenvalues 1 − a and 2 + a, and the eigenvalues of linear part at equilibrium (1,1) are (3±i3(1+2a))/2. Therefore, both (0,0) and (1,1) are not centers. Thus, the only possibility of periodic orbit in R would be to have a limit cycle, which we have already seen is not possible.

So, using Lemmas 3.1, 3.2 and 3.3, the proof of statements (a) and (b) of Theorem 2.1 is complete.

Now since the bisectrix x = y = z is an invariant straight line for the system, it is easy to check for a = −1/2 that the global phase portrait on the invariant planes R ∩ {x = y}, R ∩ {x = z} and R ∩ {y = z} are topologically equivalents to the ones described in (a), (b) and (c) of Fig. 3 respectively. This completes the proof of Theorem 2.1.

## 4. Proof of Theorem 2

We say that a C1 function I(x, y, z, t) is an invariant of the polynomial differential system (1.1) if I(x(t), y(t), z(t), t) is constant, for all the values of t for which the solution (x(t), y(t), z(t)) of (1.1) is defined. When the function I is independent of the time, then it is called a first integral of differential system (1.1). Also if an invariant I(x, y, z, t) is of the form f(x, y, z)est, then it is called a Darboux invariant.

### Proposition 4.1.

System (1.1), for a + b = −1, has the Darboux invariant I = I(t, x, y, z) = xyze−3t.

### Proof.

It is immediate to check that

dIdt=Ixx˙+Iyy˙+Izz˙+It=0,
where , and ż are given in (1.1). Therefore I is a Darboux invariant of system (1.1).

For knowing how to obtain the Darboux invariant given in Proposition 4.1 see statement (vi) of Theorem 8.7 of [5], there the theory is described for polynomial vector fields in ℝ2, but the results and the proofs extend to ℝ3.

### Proposition 4.2.

Let I(x, y, z, t) = f(x, y, z)est be a Darboux invariant of system (1.1). Let p ∈ ℝ3 and φp(t) the solution of system (1.1) such that φp(0) = p. Then

α(p),ω(p){f(x,y,z)=0}𝕊2¯.

Here α(p) and ω(p) denote the α-limit and ω-limit sets of p respectively, and 𝕊2 denotes the boundary of the Poincaré ball, i.e. the infinity of3.

For a proof of Proposition 4.2 see [7].

### Lemma 4.1.

Let p(γ) = {φp(t) = (x(t), y(t), z(t)) : t ∈ ℝ} be the orbit of the Poincaré compactification of system (1.1), for a + b = −1, such that φp(0) = p and limt+x(t)y(t)z(t)=+. Then ω(p) ⊂ R.

### Proof.

Let qω(p). Then there exists a sequence (tn) with tn → +∞, such that φp(tn) = (x(tn), y(tn), z(tn)) → q when n → +∞. By Proposition 4.2 we know that q ∈ {(x, y, z) ∈ R : xyz = 0} ∪ R. Suppose that qR and take ε > 0. Then there exist M > 0 and n0 ∈ ℕ such that xyzM for all x,y,zB(q,ε)¯ and ϕp(tn)B(q,ε)¯ for all nn0. Therefore x(tn)y(tn)z(tn) ≤ M for all nn0. On the other hand, as limt+x(t)y(t)z(t)=+ then exists t0 ∈ ℝ such that x(t)y(t)z(t) ≥ M for all tt0, which is a contradiction. Therefore qR.

### Proof.

[Proof of Theorem 2.2] Let p(γ) = {φp(t) = (x(t), y(t), z(t)) : t ∈ ℝ} be the orbit of the Poincaré compactification of system (1.1) with a+b = −1 such that φp(0) = p with p in the interior of R. We recall that all the orbits of a differential system defined on a compact set are defined for all t ∈ ℝ. By Propositions 4.1 and 4.2 the α- and ω-limit set of p(γ) is contained in boundary of R, i.e. in {(x, y, z) ∈ R : xyz = 0} ∪ R. Furthermore, by Proposition 4.1, I(t, x(t), y(t), z(t)) = k constant with k > 0. So

x(t)y(t)z(t)=ke3t,(4.1)
for all t. Taking limit in (4.1) when t → −∞, we obtain
limtx(t)y(t)z(t)=0.

This implies that the orbit p(γ) tends to the set xyz = 0 when t → −∞. Looking at the dynamics of the flow of the compactified vector field on xyz = 0 described in Fig. 1(a) if a ≤ −2 or a ≥ 1, and in Fig. 2(a) if −2 < a < 1, we consider the following two cases.

• Case 1. Suppose that a ≤ −2 or a ≥ 1. Therefore, by Theorem 2.1(a) and Fig. 1(a) we have that the α-limit set of p(γ) can be the equilibrium point p0 because the eigenvalue at p0 is 1 with multiplicity three. The equilibrium points p1, p2, p3, px, py and pz can not be α-limit set of p(γ) in the case that p(γ) is a characteristic orbit, because the unstable separatrix of the saddles are contained in the faces of R. But other possible sets to be α-limit of the orbit p(γ) in the interior of R is either the heteroclinic loop connecting the equilibrium points p1, p2 and p3, or the heteroclinic loop connecting the equilibrium points px, py and pz. So statement (a)(i) of Theorem 2.2 is proved.

• Case 2. Assume that −2 < a < 1. By Theorem 2.1(a) and Fig. 2(a) we have that the singular points p0, px, py and pz are repelling. Furthermore, the singular points pj for j = 1,2,... ,6 and pxy, pxz, pyz are of saddle type such that their 2-dimensional unstable manifold intersect the interior of R. Therefore, the α-limit set of p(γ) can be one of the singular points pj for j = 0,1... ,6 or px, py, pz, pxy, pxz, pyz. So statement (b)(i) of Theorem 2.2 is proved.

Now we study the ω-limit set of p(γ). In a similar way taking limit in (4.1) when t → +∞ we obtain

limt+x(t)y(t)z(t)=+.

So by Proposition 4.2 and by Lemma 4.1 we conclude that the ω-limit set of p(γ) is contained in R. Looking at the dynamics of the flow of the compactified vector field on R described in Fig. 1(b) if a ≤ −2 or a ≥ 1, we conclude that the ω-limit set of p(γ) is the infinite singular point pbR or the heteroclinic loop connecting the singular points px, py and pz. So, the proof of statement (a)(ii) of Theorem 2.2 is complete. Furthermore, if −2 < a < 1, looking in Figs. 2(b)(c)(d) we conclude that the ω-limit set of p(γ) is the infinite singular point pbR. Hence statement (b)(ii) of Theorem 2.2 is proved.

### Remark 4.1.

An interesting question is: Can we say something for other values of the parameters a and b? As we have mentioned at the beginning of Section 4, both Darboux invariant and first integral are invariants to the system (1.1). Leach and Miritzis [6] obtained the following first integrals:

1. (i)

H1=xyz(x+y+z)3 if a + b = 2 and a ≠ 1,

2. (ii)

H2=y(xz)x(yz) if a = b ≠ 1,

3. (iii)

H3=xz and H4=yz if a = b = 1.

The global dynamics in these cases was studied in [2].

## A. The Poincaré compactification in ℝ3

For more details on the Poincaré compactification in ℝ3 see [3]. In ℝ3 we consider the polynomial differential system

x˙=P1(x,y,z),y˙=P2(x,y,z),z˙=P3(x,y,z),
or equivalently its associated polynomial vector field X = (P1, P2, P3). The degree n of X is defined as n = max{deg(Pi) : i = 1, 2, 3}.

Let 𝕊3 = {y = (y1, y2, y3, y4) ∈ ℝ4 : ‖y‖ = 1} be the unit sphere in ℝ4, and

𝔿+={y𝕊3:y4>0}and𝔿={y𝕊3:y4<0}
be the northern and southern hemispheres, respectively. The tangent space to 𝕊3 at the point y is denoted by Ty𝕊3. Then we identify the tangent hyperplane
T(0,0,0,1)𝕊3={(x1,x2,x3,1)4:(x1,x2,x3)3}
with ℝ3.

We consider the central projections

f+:3=T(0,0,0,1)𝕊3𝔿+andf:3=T(0,0,0,1)𝕊3𝔿,
defined by
f+(x)=1Δx(x1,x2,x3,1)andf(x)=1Δx(x1,x2,x3,1),
where Δx=(1+i=13xi2)1/2. Through these central projection ℝ3 is identified with the northern and the southern hemispheres, respectively. The equator of the sphere 𝕊3 is 𝕊2 = {y ∈ 𝕊3 : y4 = 0}. Clearly 𝕊2 can be identified with the infinity of ℝ3.

The diffeomorphisms f+ and f define two copies of X, one Df+X in the northern hemisphere and the other DfX in the southern one. Denote by X¯ the vector field on 𝕊3 \ 𝕊2 = 𝔿+ ∪ 𝔿 such that restricted to 𝔿+ coincides with Df+X and restricted to 𝔿 coincides with DfX. We extend analytically the polynomial vector field X¯ to the equator of 𝕊3, i.e. to the infinity of ℝ3, in such a way that the flow on the boundary is invariant. This is done defining the vector field

p(X)(y)=y4n1X¯(y),
for all y ∈ 𝕊3. This extended vector field p(X) is called the Poincaré compactification of X on the Poincaré sphere 𝕊3.

In what follows we shall work with the orthogonal projection of the closed northern hemisphere to y4 = 0. Note that this projection is a closed ball B of radius one, whose interior is diffeomorphic to ℝ3 and whose boundary 𝕊2 corresponds to the infinity of ℝ3. The projected vector field on B is called the Poincaré compactification on the Poincaré ball B.

As 𝕊3 is a differentiable manifold, to compute the expression for p(X) we can consider the eight local charts (Ui, Fi), (Vi, Gi) where Ui = {y ∈ 𝕊3 : yi > 0} and Vi = {y ∈ 𝕊3 : yi < 0} for i = 1,2,3,4; the diffeomorphisms Fi : Ui → ℝ3 and Gi : Vi → ℝ3 for i = 1,2,3,4, are the inverses of the central projections from the origin to the tangent planes at the points (±1,0,0,0), (0,±1,0,0), (0,0,±1,0) and (0,0,0,±1), respectively. The expression of p(X) on the local chart U1 is

z3n(z1P1+P2,z2P1+P3,z3P1),(A.1)
where Pi = Pi(1/z3, z1/z3, z2/z3), and the expressions of p(X) in U2 is
z3n(z1P2+P1,z2P2+P3,z3P2),(A.2)
where Pi = Pi(z1/z3, 1/z3, z2/z3) in U2, and in U3 is
z3n(Δz)n1(z1P3+P1,z2P3+P2,z3P3),(A.3)
where Pi = Pi(z1/z3, z2/z3, 1/z3) in U3.

The expression for p(X) in U4 is z3n+1(P1,P2,P3) where the component Pi = Pi(z1, z2, z3). The expression for p(X) in the local chart Vi is the same as in Ui multiplied by (−1)n−1. We remark that all the points on the sphere at infinity in the coordinates of any local chart have z3 = 0.

## Acknowledgments

The first author is supported by FAPESP grant 2013/2454-1 and CAPES grant 88881.068462/2014-01. The second author is partially supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The third author is supported by CAPES grant 99999.006888/2015-01 from the program CAPES-PDSE.

## References

[1]N.N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type (R), Mat. Sb, Vol. 30, No. 72, 1952, pp. 181-196. Amer. Math. Soc. Trans. No., 100 (1954)
[5]F. Dumortier, J. Llibre, and J.C. Artés, Qualitative Theory of Planar Differential Systems, Universitext Springer, New York, 2006.
Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
27 - 2
Pages
267 - 278
Publication Date
2020/01/27
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2020.1700635How to use a DOI?
Open Access

TY  - JOUR
AU  - Claudio A. Buzzi
AU  - Robson A. T. Santos
AU  - Jaume Llibre
PY  - 2020
DA  - 2020/01/27
TI  - Final evolutions of a class of May-Leonard Lotka-Volterra systems
JO  - Journal of Nonlinear Mathematical Physics
SP  - 267
EP  - 278
VL  - 27
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2020.1700635
DO  - 10.1080/14029251.2020.1700635
ID  - Buzzi2020
ER  -