Global dynamics of a Lotka–Volterra system in ℝ³

Jaume Llibre; Y. Paulina Martínez; Claudia Valls

doi:10.1080/14029251.2020.1757240

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Volume 27, Issue 3, May 2020, Pages 509 - 519

Global dynamics of a Lotka–Volterra system in ℝ³

Authors

Jaume Llibre

Departament de Matemàtiques, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain,jllibre@mat.uab.cat

Y. Paulina Martínez

Centre de Recerca Matemática, 08193 Bellaterra, Barcelona, Catalonia, Spain,yohanna.martinez@uab.cat

Claudia Valls

Center for Mathematical Analysis, Geometry and Dynamical Systems, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal,cvalls@math.tecnico.ulisboa.pt

Received 22 July 2019, Accepted 18 December 2019, Available Online 4 May 2020.

DOI: 10.1080/14029251.2020.1757240 How to use a DOI?
Keywords: Lotka–Volterra system; phase portraits; Poincaré ball
Abstract: In this work we consider the Lotka–Volterra system in ℝ³
x˙=−x(x−y−z), y˙=−y(−x+y−z), z˙=−z(−x−y+z),
introduced recently in [7], and studied also in [8] and [14]. In the first two papers the authors mainly studied the integrability of this differential system, while in the third paper they studied the system as a Hamilton-Poisson system, and also started the analysis of its dynamics. Here we provide the global phase portraits of this 3-dimensional Lotka–Volterra system in the Poincaré ball, that is in ℝ³ adding its extension to the infinity.
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 and statement of the main results

The Lotka–Volterra systems, developed independently by Alfred J. Lotka in 1925 [9] and Vito Volterra in 1926 [15], were initially proposed as models for studying the interactions in two dimensions between species. Kolmogorov [5] in 1936 extended these systems to arbitrary dimension and degree, which are now called Kolmogorov systems.

The Lotka–Volterra systems have been applied to model different natural phenomena such as the time evolution of conflicting species in biology (which began with the work of May [10]), the evolution of competition between three species (studied by May and Leonard [11]), the evolution of electrons, ions and neutral species in plasma physics [6], chemical reactions [4], hydrodynamics [1], economics [13], etc.

In this work we consider the following Lotka–Volterra system

x˙=−x(x−y−z),y˙=−y(−x+y−z),z˙=−z(−x−y+z),(1.1)

where the dot denotes derivative with respect to the time t.

The main goal of this work is to describe the global dynamics of the Lotka–Volterra system (1.1) in ℝ³ adding the infinity, i.e. we are interested in describing the phase portraits of system (1.1) on the Poincaré ball 𝔻³. A first attempt to describe these phase portraits was done in [14] where the authors gave a Hamilton-Poisson formulation of system (1.1). Leach and Miritzis in [7] proved that system (1.1) has a first integral, and in [8] the authors proved that this system has the two independent first integrals x(y − z) and y(z − x). The existence of these first integrals has some consequences on the phase portrait of the system. Thus on the invariant plane x = 0 one gets that all trajectories are contained in the hyperbolas yz = constant. On the other hand, a generic trajectory is contained in the elliptic curves intersection of the surfaces x(y − z) = constant and y(z − x) = constant. In any case these informations are not sufficient for obtaining the global phase portrait of system (1.1) in the Poincaré ball.

Our objective is to establish the α- and ω-limit of all the orbits of system (1.1) and to characterize the phase portrait of this system in the Poincaré ball 𝔻³. We recall that the Poincaré ball can be seen as the closed unit ball centered at the origin of ℝ³, where its interior is identified with ℝ³ and its boundary 𝕊² is identified with the infinity of ℝ³ (in the sense that in ℝ³ we can go or come from infinity in as many directions as points has the 2-dimensional sphere 𝕊²).

A polynomial differential system in ℝ³ (in the interior of 𝔻³) can be extended analytically to its boundary 𝕊². This extension was done by the first time by Poincaré in [12] for polynomial differential systems in ℝ² and it is called the Poincaré compactification. In [2] the authors give an extension to polynomial differential systems in ℝ³. For a brief introduction to Poincaré compactification see the appendix.

Two compactified polynomial differential systems in the Poincaré ball 𝔻³ are topologically equivalent if there is a homeomorphism of 𝔻³ which send orbits of one system into orbits of the other system, preserving or reversing the orientation of all orbits.

We note that system (1.1) has the symmetry (x,y,z,t) → (−x,−y − z,−t). Then it is sufficient to study its dynamics for x ≥ 0.

In the next theorem we describe the phase portrait on the half-ball x ≥ 0 of the Poincaré ball for the 3-dimensional Lotka–Volterra system (1.1).

Theorem 1.1.

The dynamics of the 3-dimensional Lotka–Volterra system (1.1) in the Poincaré ball is the following.

(a)

The phase portraits on the invariant planes x = 0, y = 0 and z = 0 are shown in Figures 1 and 2.
(b)

The phase portraits on the invariant planes x = z, y = z and x = y are shown in Figure 3.
(c)

The phase portrait at infinity is topologically equivalent to the one of Figure 4.
(d)

The phase portraits on the boundary of the twelve invariant regions obtained dividing the Poincaré ball by the six previous invariant planes in the region x ≥ 0 are topologically equivalent to the ones shown in Figure 6.
(e)

The phase portraits in the interior of the twelve invariant regions provided in statement are topologically equivalent to the ones shown in Figure 7. Moreover the α- and ω-limits of each orbit in the interior of these regions are given in Table 1.

The paper is organized as follows. In section 2 we study system (1.1) on the planes x = 0, y = 0, z = 0, y = x, z = y and z = x, and we prove statements (a) and (b) of Theorem 1.1. In section 3 we study the infinite equilibria of system (1.1) and give the phase portraits on the Poincaré ball, proving statements (c), (d) and (e) of Theorem 1.1. In the appendix there is a brief description of the Poincaré compactification for polynomial differential systems in ℝ³ and in ℝ².

2. Phase Portraits of system (1.1) on the planes

In this section we study all the phase portraits of the 3-dimensional Lotka–Volterra system (1.1) on the planes x = 0, y = 0, z = 0, y = x, z = y and z = x.

First we point out that in [8] the authors proved that system (1.1) has the two independent first integrals

H1=(xyz)(x−y)(x−z)(y−z), andH2=x2y2−x2yz−xy2z+x2z2−xyz2+y2z2

Moreover the unique irreducible Darboux polynomials with non-zero cofactor of system (1.1) are:

(1)

x, y, z with cofactors −(x − y − z), −(−x + y + −z), and −(−x + −y + z), respectively.
(2)

x − z, y − z and z − x with cofactors −(x − y + z), −(−x + y + z), and −(x + y − z), respectively.

We separate the study of the dynamics in the invariant planes in two cases.

2.1. Phase portraits on the invariant planes x = 0, y = 0 and z = 0

First note that system (1.1) on the planes x = 0, y = 0 and z = 0 are equivalent. In fact, system (1.1) on x = 0 is given by

y˙=−y(y−z), z˙=−z(−y+z),(2.1)

on y = 0 by

x˙=−x(x−z), z˙=−z(−x+z),(2.2)

and on z = 0 by

x˙=−x(x−y), y˙=−y(−x+y).(2.3)

Clearly system (2.1) and system (2.2) are equivalent through the change of variables (y,z) → (x,z), and system (2.3) is equivalent to the previous ones considering the changes of variables (x,y) → (x,z), or (x,y) → (y,z) to get system (2.2), or (2.1), respectively. Hence we are only going to analyze the global phase portraits of system (2.3).

We start with the study of the infinite singular points. For this purpose we use the Poincaré compactification of a polynomial differential systems in ℝ², see details in chapter 5 of [3]. System (2.3) in the local chart U₁ is given by

z˙1=−2z1(−1+z1), z˙2=−z2(−1+z1),(2.4)

and in the local chart U₂ is

z˙1=−2z1(−1+z1), z˙2=−z2(−1+z1).(2.5)

From systems (2.4) and (2.5) we get that at infinity (i.e. at all points having the coordinate z₂ = 0) the origin of each chart is an equilibrium point. Moreover, in U₁ we have a second equilibrium point given by (1,0). The eigenvalues of the linear part at the origins of U₁ and U₂ are 1 and 2, so they are unstable nodes. Note that since system (2.3) has degree 2 the corresponding equilibria on V₁ and V₂ will be stable nodes. The equilibrium (1,0) in U₁ is not a hyperbolic equilibrium. It is at the infinity of the straight line x = y which is filled of equilibria as will be seen later on in the next.

Now we study the finite singular points. System (2.3) has a straight line of equilibria given by x = y. Since −x + y is a common factor of system (2.3), doing a reparametrization of the time, system (2.3) becomes

x′=x, y′=−y.(2.6)

It is known that the global dynamics of system (2.6) is the following: it has a saddle at the origin with stable separatrices on the y-axis and unstable separatrices on x-axis and at infinity it has infinite unstable nodes at the origins of U₁ and V₁ and infinite stable nodes at the origins of U₂ and V₂.

Going back through the reparametrization of time, taking into account the change of stability in the region −x + y < 0 and the straight line of finite equilibria on −x + y = 0, we can complete the global phase portrait of system (2.3) by using the Poincaré Bendixson Theorem (see [3]) connecting the separatrices of the saddle at the origin to the nodes at the infinity. Doing so we obtain the phase portrait described in Figure 1(a).

The phase portraits in the planes x = 0, y = 0 and z = 0 in ℝ³ are shown in Figure 2. This completes the proof of statement (a) of Theorem 1.1.

2.2. Phase portraits on the invariant planes x = z, y = z and y = x

Note that the dynamics of system (1.1) restricted to the plane x = z, to the plane y = z and to the plane y = x are topologically equivalent. Actually it is sufficient to study the phase portrait in the plane x = z (taking z → x) and inthis case system (1.1) takes the form

x˙=xy, y˙=−y(y−2x).(2.7)

System (2.7) has the common factor y. Eliminating this common factor by reparametrizing the time, it becomes

x′=x, y′=2x−y,(2.8)

whose phase portrait consists in a saddle at the origin with stable separatrices on the y axis and unstable separatrices on the straight line x = y.

At infinity we have that system (2.8) in the local chart U₁ is

z1′=2−2z1, z1′=−z2,

which has a unique equilibrium on (1,0) and it is a stable node. In the local chart V₁ it has another stable node. On the other hand, on the local chart U₂ system (2.8) writes as

z1′=2z1(1−z1), z2′=(1−2z1)z2.

The origin is an equilibrium point which is an unstable node. The same stability has the origin of V₂. Observe that system (2.7) has the opposite stability in the region y < 0 than system (2.8).

From the previous analysis of system (2.8), considering the reparametrization done with the factor y, and due to the fact that there are not more finite equilibria, by the Poincaré Bendixson Theorem, we have that the separatrices of the saddle at the origin connect with the nodes at infinity. We can thus conclude that the phase portrait in the Poincaré disc of system (2.7) is the one described on Figure 3. Statement (b) of Theorem 1.1 has been proved.

3. Dynamics on ℝ³

In order to obtain the phase portrait in ℝ³ of system (1.1) we first study the phase portrait at infinity using the Poincaré compactification (for details of the Poincaré compactification in ℝ³, see appendix A.1).

System (1.1) in the local charts U₁, U₂ and U₃ are exactly the same, except for the meaning of the coordinates (z₁,z₂,z₃) (see (A.1), (A.2) and (A.3)). At infinity we have the following system in the local charts

z1=−2z1(−1+z1), z2=−2z2(−1+z2), z3=−z3(−1+z1+z2).(3.1)

Considering (3.1) on the local chart U₁, we have the four infinite equilibria (i.e, on z₃ = 0): the origin, p^1=(0,0,0), p^2=(1,0,0), p^3=(0,1,0) and p^4=(1,1,0). In the local chart U₂ the infinite equilibria on z₁ = z₃ = 0 are the origin of U₂, p^5=(0,0,0) and p^6=(0,1,0). Finally, in the local chart U₃ we have that the origin p^7=(0,0,0) is an infinite equilibria.

All the infinite equilibria are at the infinity of the planes studied in section 2 and so their phase portraits are known. Due to the fact that the linear part of system (3.1) is

(−2(z1−1)−2z1000−2(z2−1)−2z20−z3−z3−z1−z2+1)

which is a diagonal matrix, it is easy to conclude that p^1, p^5 and p^7 are unstable nodes, p^4 is a stable node, that the eigenvalues of p^2 are −2,2,0, and the eigenvalues of p^3 and p^6 are 2,−2,0. Thus, p^2, p^3 and p^6 are saddles in the plane z₃ = 0.

In summary in the Poincaré ball we have 14 infinite equilibria. They are p₁ = (1,0,0), p₂ = (1,1,0), p₃ = (1,0,1), p₄ = (1,1,1), p₅ = (0,1,0), p₆ = (0,1,1) and p₇ = (0,0,1) on the charts U₁,U₂ and U₃ and their corresponding antipodal equilibria on the local charts V_i (that we will denote by q_i the antipodal point of p_i) for i = 1,2,3.

Using the Poincaré-Bendixson Theorem we can characterize the connections between the separatrices of the saddles at p₂, p₃ and p₆ with the infinite nodes. Furthermore, these separatrices are on the invariant planes studied in section 2. The phase portrait at infinity is the one shown in Figure 4. This completes the proof of statement (c) of Theorem 1.1.

On the other hand, the finite equilibria of system (1.1) are the origin and the straight lines of equilibria already mentioned in section 2. Then we can describe the phase portrait in the Poincaré ball.

In order to give an efficient description of the global dynamics of system (1.1) we separate the Poincaré ball in regions defined by the intersections of the planes x = 0, y = 0, z = 0, x = y, x = z and y = z. There are 24 regions, all invariant by the flow of system (1.1). Due to the symmetry (x,y,z,t) → (−x,−y,−z, −t) it is sufficient to analyze the invariant regions contained in x ≥ 0. These regions are

R1={(x,y,z):0≤y<x<z}, R7={(x,y,z):y≤0≤x<z},R2={(x,y,z):0≤x<y<z}, R8={(x,y,z):y≤0≤z<x},R3={(x,y,z):0≤x<z<y}, R9={(x,y,z):z≤0≤y<x},R4={(x,y,z):0≤z<x<y}, R10={(x,y,z):z≤0≤x<y},R5={(x,y,z):0≤z<y<x}, R11={(x,y,z):z<y≤0≤x},R6={(x,y,z):0≤y<z<x}, R12={(x,y,z):y<z≤0≤x}.

The mentioned regions R₁-R₁₂ are shown in Figure 5. We note that the differential system is invariant under the cyclic permutation of the variables x,y,z, and consequently the dynamics on several of the regions R_k for k = 1,...,12 in which is divided the half-space x ≥ 0 can be obtained one from the others. But in what follows we prefer to provide the dynamics in all these twelve regions because this facilitates understanding the motion between these regions.

To provide the phase portrait in each of these regions we describe first the phase portrait in the faces of each region, and after that the phase portrait in the interior of them. Note that in each region the isolated equilibria are on the vertices of the region and a segment filled of equilibria is on an edge of the region. We recall that one orbit in each face and one orbit in the interior are sufficient for determining the phase portraits in each invariant region. The edges of the regions are separatrices or segments filled of equilibria.

As an example we consider the region R₁. This region is a tetrahedron formed by three infinite equilibria p₃, p₄ and p₇ and the finite equilibrium at the origin θ. Note that it has one edge, given by x = z and y = 0, which connects the vertex p₃ with θ filled of equilibria. The infinite equilibrium p₄ is an attractor and p₇ is a repeller. By the Poincaré-Bendixson Theorem we can establish the α- and ω-limits of all the orbits in this region.

In the faces of R₁ without a segment filled of equilibria all the orbits have their ω-limit in p₄ and their α-limit in p₇.

On the face formed by p₃, p₇ and θ the orbits have their α limit in p₇ and the ω-limit on the edge filled of equilibria.

On the face formed by p₃, p₄ and θ all orbits have their ω-limit at p₄ and their α-limit on the edge filled of equilibria.

Finally in the region R₁, by the Poincaré-Bendixson Theorem, we can conclude that an orbit in the interior of R₁ has its α-limit at p₇ and its ω-limit at p₄.

The phase portrait in the remaining regions can be characterized analogously. In Table 1 it is shown the α- and ω-limits for all orbits in the interior of each region.

	R₁, R₂	R₃, R₄	R₅, R₆	R₇, R₁₂	R₈	R₉	R₁₀, R₁₁
α-limit	p₇	p₅	p₁	p₁	p₇	p₅	p₁
ω-limit	p₄	p₄	p₄	q₅	q₅	q₇	q₇

Table 1.

α- and ω-limits in the interior each invariant region R₁-R₁₂.

In Figure 6 it is shown the phase portraits on the boundary of the invariant regions, and one orbit in the interior of each invariant region is shown in Figure 7. These previous results prove statements (d) and (e) of Theorem 1.1 and complete the proof of Theorem 1.1.

A. Poincaré Compactification

We give a description of the Poincaré compactification in order to describe the phase portraits in the Poincaré ball. The Poincaré compactification of ℝ³ is needed in the proof of Theorem 1.1.

We consider a polynomial vector field 𝒳 = (P,Q,R) associated to the polynomial differential system

x˙1=P(x1,x2,x3), x˙2=Q(x1,x2,x3), x˙3=R(x1,x2,x3).

The degree n of 𝒳 is defined as n = max{deg(P),deg(Q),deg(R)}.

Now we shall describe the equations of the Poincaré compactification of a polynomial differential system in ℝ³.

We consider the local charts (U_k,φ_k) and (V_k,ψ_k) for k = 1,2 on the disc 𝔻³ defined by

Uk={x=(x1,x2,x3)∈𝔻3:xk>0},Vk={x=(x1,x2,x3)∈𝔻3:xk<0},

where the diffeomorphisms φ_k : U_k → ℝ³ for k = 1,2,3 are

φ1(x)=(x2x1,x3x1,1x1)=(z1,z2,z3), φ2(x)=(x1x2,x3x2,1x2)=(z1,z2,z3),φ3(x)=(x1x3,x2x3,1x3)=(z1,z2,z3),

and ψ_k(x) = −φ_k(x).

Note that the coordinates (z₁,z₂,z₃) have different meaning depending on local chart, but the points at infinity, i.e. the points of the boundary 𝕊² of 𝔻³ have all the coordinate z₃ = 0.

Now we give the expression of the compactified vector field p(𝒳) of the polynomial vector field X = (P,Q,R) in each local chart. The expression of the compactified analytical vector field p(𝒳) of 𝒳 of degree n on the local chart U₁ of 𝔻³ is

z3n(−z1P(z)+Q(z),−z2P(z)+R(z),−z3P(z)),(A.1)

where z = (1/z₃,z₁/z₃,z₂/z₃).

In a similar way the expression of p(𝒳) in U₂ is

z3n(−z1Q(z)+P(z),−z2Q(z)+R(z),−z2Q(z)),(A.2)

where z = (z₁/z₃,1/z₃,z₂/z₃).

Finally the vector field p(𝒳) in U₃ is

z3n(−z1R(z)+P(z),−z2R(z)+Q(z),−z2R(z)),(A.3)

where z = (z₁/z₃,z₂/z₃,1/z₃).

The singular points of p(𝒳) which are on the boundary 𝕊² of 𝔻³ (at z₃ = 0) are called infinite singular points, and we call finite singular points the ones which are in the interior of 𝔻³.

From equations (A.1), (A.2) and (A.3) it follows that the infinity 𝕊² of the Poincaré ball is invariant under the flow of the compactified vector field p(𝒳). For studying its infinite singular points we only need to study the ones that are on the local chart U₁, in U₂ with z₁ = 0, and the origin of the local chart U₃ in case that this is a singular point.

The expression for p(𝒳) in the local chart V_k is the same as in U_k multiplied by (−1)ⁿ⁻¹. Therefore the infinite singular points appear on pairs diametrally opposite on 𝕊² with the same stability if n is odd and with the opposite stability if n is even. For more details on the Poincaré compactification in ℝ³ see [2].

As we said in the introduction two compactified polynomial differential systems in the Poincaré ball 𝔻³ are topologically equivalent if there is a homeomorphism of 𝔻³ sending orbits of one system to orbits of the other system, either preserving or reversing the orientation of the orbits.

Acknowledgements

We thank to the reviewer his/her good comments which help us to improve the presentation of our results.

The first author is partially supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación grants MTM2016 - 77278 - P (FEDER) and MDM - 2014 - 0445, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911.

The second author is supported by CONICYT PCHA / Postdoctorado en el extranjero Becas Chile / 2018 - 74190062.

The third author is partially supported by FCT/Portugal through UID/ MAT/ 04459/2013.

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[3]F. Dumortier, J. Llibre, and J.C. Artés, Qualitative theory of planar differential systems, Springer, 2006. Universitext, Berlin

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[6]G. Laval and R. Pellat, Plasma physics (Les Houches), Proc. Summer School of Theoretical Physics, 1975, pp. 261.

[7]P. Leach and J. Miritzis, Competing species: Integrability and stability, J. Nonlinear Math. Phys., Vol. 11, 2006, pp. 123-133.

[8]J. Llibre and C. Valls, Polynomial, rational and analytic first integrals for a family of 3-dimensional Lotka-Volterra systems, Zeitschrift für angewandte Mathematik und Physik, Vol. 62, No. 5, 2011, pp. 761-777.

[9]A.J. Lotka, Elements of physical biology, Science Progress in the Twentieth Century (1919–1933), Vol. 21, No. 82, 1926, pp. 341-343.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 27 - 3
Pages: 509 - 519
Publication Date: 2020/05/04
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.1080/14029251.2020.1757240 How to use a DOI?
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TY  - JOUR
AU  - Jaume Llibre
AU  - Y. Paulina Martínez
AU  - Claudia Valls
PY  - 2020
DA  - 2020/05/04
TI  - Global dynamics of a Lotka–Volterra system in ℝ³
JO  - Journal of Nonlinear Mathematical Physics
SP  - 509
EP  - 519
VL  - 27
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2020.1757240
DO  - 10.1080/14029251.2020.1757240
ID  - Llibre2020
ER  -

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Journal of Nonlinear Mathematical Physics

Global dynamics of a Lotka–Volterra system in ℝ3

1. Introduction and statement of the main results

Theorem 1.1.

2. Phase Portraits of system (1.1) on the planes

2.1. Phase portraits on the invariant planes x = 0, y = 0 and z = 0

2.2. Phase portraits on the invariant planes x = z, y = z and y = x

3. Dynamics on ℝ3

A. Poincaré Compactification

Acknowledgements

References

Cite this article

Global dynamics of a Lotka–Volterra system in ℝ³

3. Dynamics on ℝ³