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

Claudio A. Buzzi; Robson A. T. Santos; Jaume Llibre

doi:10.1080/14029251.2020.1700635

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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: 10.1080/14029251.2020.1700635 How 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.
Copyright: © 2020 The Authors. Published by Atlantis and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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(1−x−ay−bz),y˙=y(1−bx−y−az),z˙=z(1−ax−by−z),(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 ẋ = 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)∈ℝ3:x2+y2+z2≤1,x≥0,y≥0,z≥0}

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.

(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.
(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 of ℝ³) 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^∞.
(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.

(a)
If a ≤ −2 or a ≥ 1 then we have:
1. (i)
  The α-limit set of p(γ) is either the origin of ℝ³, 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)).
(b)
If −2 < a < 1 then we have:
1. (i)
  The α-limit set of p(γ) is one of the singular points p_j 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(1−x+z+a(−y+z))=0,P2(x,y,z)=y(1+x−y+a(−z+x))=0,P3(x,y,z)=z(1+y−z+a(−x+y))=0,

namely

p0=(0,0,0) p1=(1,0,0), p2=(0,1,0), p3=(0,0,1),p4=(0,1−aA,2+aA), p5=(2+aA,0,1−aA), p6=(1−aA,2+aA,0),

where A = 1 + a + a².

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

(i)
If a ≤ −2 or a ≥ 1 system (1.1) has only four finite equilibrium points: p₀, p₁, p₂ and p₃.
(ii)
If −2 < a < 1 system (1.1) has the seven finite equilibrium points p_j 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 p₄ → p₃, p₅ → p₁ and p₆ → p₂; while if a → −2 we have that p₄ → p₂, p₅ → p₃ and p₆ → p₁. 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+az1−z12+az12−z1z2−2az1z2,z˙2=z2−az2+z1z2+2az1z2−2z22−az22,z˙3=z3+az1z3−z2z3−az2z3−z32.(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=z1−az1−2z12−az12+z1z2+2az1z2,z˙2=2z2+az2−z1z2−2az1z2−z22+az22,z˙3=z3−z1z3−az1z3+az2z3−z32.(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+az1−z12+az12−z1z2−2az1z2,z˙2=z2−az2+z1z2+2az1z2−2z22−az22,z˙3=z3+az1z3−z2z3−az2z3−z32.

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)/(1−a),0,0), pxz∞=(0,(1−a)/(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)/(1−a),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)/(1−a),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 pxz∞→px∞, pyz∞→pz∞, and pxy∞→py∞; while if a → −2 we have that pxz∞→pz∞, pyz∞→py∞, and pxy∞→px∞. 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)+(a−1)z1−(1+2a)z2),z˙2=z2((1−a)+(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)e^st, 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⁻³^t.

Proof.

It is immediate to check that

dIdt=∂I∂xx˙+∂I∂yy˙+∂I∂zz˙+∂I∂t=0,

where ẋ, ẏ 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)e^st be a Darboux invariant of system (1.1). Let p ∈ ℝ³ and φ_p(t) the solution of system (1.1) such that φ_p(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 of ℝ³.

For a proof of Proposition 4.2 see [7].

Lemma 4.1.

Let p(γ) = {φ_p(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 φ_p(0) = p and limt→+∞x(t)y(t)z(t)=+∞. Then ω(p) ⊂ R^∞.

Proof.

Let q ∈ ω(p). Then there exists a sequence (t_n) with t_n → +∞, such that φ_p(t_n) = (x(t_n), y(t_n), z(t_n)) → q when n → +∞. By Proposition 4.2 we know that q ∈ {(x, y, z) ∈ R : xyz = 0} ∪ R^∞. Suppose that q ∉ R^∞ and take ε > 0. Then there exist M > 0 and n₀ ∈ ℕ such that xyz ≤ M for all x,y,z∈B(q,ε)¯ and ϕp(tn)∈B(q,ε)¯ for all n ≥ n₀. Therefore x(t_n)y(t_n)z(t_n) ≤ M for all n ≥ n₀. 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 t ≥ t₀, which is a contradiction. Therefore q ∈ R^∞.

Proof.

[Proof of Theorem 2.2] Let p(γ) = {φ_p(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 φ_p(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

limt→−∞x(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 p_j 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 p_j 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 pb∞∈R∞ 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 pb∞∈R∞. 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:

(i)
H1=xyz(x+y+z)3 if a + b = 2 and a ≠ 1,
(ii)
H2=y(x−z)x(y−z) if a = b ≠ 1,
(iii)
H3=xz and H4=yz if a = b = 1.

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

Appendix

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(P_i) : i = 1, 2, 3}.

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

𝔿+={y∈𝕊3:y4>0} and 𝔿−={y∈𝕊3:y4<0}

be the northern and southern hemispheres, respectively. The tangent space to 𝕊³ at the point y is denoted by T_y𝕊³. Then we identify the tangent hyperplane

T(0,0,0,1)𝕊3={(x1,x2,x3,1)∈ℝ4:(x1,x2,x3)∈ℝ3}

with ℝ³.

We consider the central projections

f+:ℝ3=T(0,0,0,1)𝕊3→𝔿+ and f−:ℝ3=T(0,0,0,1)𝕊3→𝔿−,

defined by

f+(x)=1Δx(x1,x2,x3,1) and f−(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 Df₋ ○ X 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 Df₋ ○ X. 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)=y4n−1X¯(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 (U_i, F_i), (V_i, G_i) where U_i = {y ∈ 𝕊³ : y_i > 0} and V_i = {y ∈ 𝕊³ : y_i < 0} for i = 1,2,3,4; the diffeomorphisms F_i : U_i → ℝ³ and G_i : V_i → 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 P_i = P_i(1/z₃, z₁/z₃, z₂/z₃), and the expressions of p(X) in U₂ is

z3n(−z1P2+P1,−z2P2+P3,−z3P2),(A.2)

where P_i = P_i(z₁/z₃, 1/z₃, z₂/z₃) in U₂, and in U₃ is

z3n(Δz)n−1(−z1P3+P1,−z2P3+P2,−z3P3),(A.3)

where P_i = P_i(z₁/z₃, z₂/z₃, 1/z₃) in U₃.

The expression for p(X) in U₄ is z3n+1(P1,P2,P3) where the component P_i = P_i(z₁, z₂, z₃). The expression for p(X) in the local chart V_i is the same as in U_i 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

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[2]G. Blé, V. Castellanos, J. Llibre, and I. Quilantán, Integrability and global dynamics of the May-Leonard model, Nonlinear Anal, Vol. 14, 2013, pp. 280-293.

[3]A. Cima and J. Llibre, Bounded polynomial vector fields, Trans. Amer. Math. Soc, Vol. 318, 1990, pp. 557-579.

[4]P. Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc, Vol. 11, pp. 610-620.

[5]F. Dumortier, J. Llibre, and J.C. Artés, Qualitative Theory of Planar Differential Systems, Universitext Springer, New York, 2006.

[6]P.G.L. Leach and J. Miritzis, Analytic behaviour of competition among three species, J. Nonlinear Math. Phys, Vol. 13, 2006, pp. 535-548.

[7]J. Llibre and R.D.S. Oliveira, Quadratic systems with invariant straight lines of total multiplicity two having Darboux invariants, Communications in Contemporary Mathematics, Vol. 17, 2015, pp. 145001. pp 17

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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: 10.1080/14029251.2020.1700635 How to use a DOI?
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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  -

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Journal of Nonlinear Mathematical Physics

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

1. Introduction

2. Statement of the main results

Theorem 2.1.

Theorem 2.2.

Corollary 2.1.

3. Proof of Theorem 1

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Theorem 3.1 (Bautin’s Theorem).

Lemma 3.3

Proof.

4. Proof of Theorem 2

Proposition 4.1.

Proof.

Proposition 4.2.

Lemma 4.1.

Proof.

Proof.

Remark 4.1.

Appendix

A. The Poincaré compactification in ℝ3

Acknowledgments

References

Cite this article

A. The Poincaré compactification in ℝ³