This is a preprint of: “Global dynamics of a Lotka-Volterra system in R3 ”, Jaume Llibre, Yohanna Paulina Martinez Mancilla, Clàudia Valls, J. Nonlinear Math. Phys., vol. 27(3), 509–519, 2020. DOI: [10.1080/14029251.2020.1757240] GLOBAL DYNAMICS OF A LOTKA–VOLTERRA SYSTEM IN R3 JAUME LLIBRE1 , Y. PAULINA MARTÍNEZ2 AND CLAUDIA VALLS3 Abstract. In this work we consider the Lotka-Volterra system in R3 ẋ = −x(x − y − z), ẏ = −y(−x + y − z), ż = −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. 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 phe- nomena such as the time evolution of conflicting species in biology (which began with the work of May [11]), the evolution of competition between three species (studied by May and Leonard [10]), 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 ẋ = −x(x − y − z), (1) ẏ = −y(−x + y − z), ż = −z(−x − y + z), 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) in R3 adding the infinity, i.e. we are interested in describing the phase portraits of system (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). Leach and Miritzis in [7] proved that system (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 2010 Mathematics Subject Classification. Primary 37C10, Secondary 34C05. Key words and phrases. Lotka–Volterra system, phase portraits, Poincaré ball. 1 2 J. LLIBRE, Y.P. MARTÍNEZ AND C. VALLS 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) in the Poincaré ball. Our objective is to establish the α- and ω-limit of all the orbits of system (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 an- alytically 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 com- pactification. 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) 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). Theorem 1. The dynamics of the 3–dimensional Lotka–Volterra system (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 (d) 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) 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. In section 3 we study the infinite equilibria of system (1) and give the phase portraits on the Poincaré ball, proving statements (c), (d) and (e) of Theorem GLOBAL DYNAMICS OF A LOTKA–VOLTERRA SYSTEM IN R3 3 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) on the planes In this section we study all the phase portraits of the 3–dimensional Lotka– Volterra system (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) has the two independent first integrals H1 = (xyz)(x − y)(x − z)(y − z), and H2 = x2 y 2 − x2 yz − xy 2 z + x2 z 2 − xyz 2 + y 2 z 2 . Moreover the unique irreducible Darboux polynomials with non-zero cofactor of system (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) on the planes x = 0, y = 0 and z = 0 are equivalent. In fact, system (1) on x = 0 is given by (2) ẏ = −y(y − z), ż = −z(−y + z), on y = 0 by (3) ẋ = −x(x − z), ż = −z(−x + z), and on z = 0 by (4) ẋ = −x(x − y), ẏ = −y(−x + y). Clearly system (2) and system (3) are equivalent through the change of variables (y, z) → (x, z), and system (4) is equivalent to the previous ones considering the changes of variables (x, y) → (x, z), or (x, y) → (y, z) to get system (3), or (2), respectively. Hence we are only going to analyze the global phase portraits of system (4). 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 (4) in the local chart U1 is given by (5) ż1 = −2z1 (−1 + z1 ), ż2 = −z2 (−1 + z1 ), and in the local chart U2 is (6) ż1 = −2z1 (−1 + z1 ), ż2 = −z2 (−1 + z1 ). From systems (5) and (6) 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 4 J. LLIBRE, Y.P. MARTÍNEZ AND C. VALLS part at the origins of U1 and U2 are 1 and 2, so they are unstable nodes. Note that since system (4) 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 (4) has a straight line of equi- libria given by x = y. Since −x + y is a common factor of system (4), doing a reparametrization of the time, system (4) becomes (7) x0 = x, y 0 = −y. It is known that the global dynamics of system (7) 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 (4) 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). (a) (b) (c) Figure 1. Phase portraits of system (1) in the planes x = 0, y = 0 and z = 0, respectively. 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. 2.2. Phase portraits on the invariant planes x = z, y = z and y = x. Note that the dynamics of system (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) takes the form (8) ẋ = xy, ẏ = −y(y − 2x). GLOBAL DYNAMICS OF A LOTKA–VOLTERRA SYSTEM IN R3 5 Figure 2. Phase portrait of system (1) on the planes x = 0, y = 0, z = 0 inside the Poincaré ball. System (8) has the common factor y. Eliminating this common factor by reparametriz- ing the time, it becomes (9) x0 = x, y 0 = 2x − y, 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 (9) in the local chart U1 is z10 = 2 − 2z1 , z20 = −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 (9) writes as z10 = 2z1 (1 − z1 ), z20 = (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 (8) has the opposite stability in the region y < 0 than system (9). From the previous analysis of system (9), 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 (8) is the one described on Figure 3. Statement (b) of Theorem 1 has been proved. 6 J. LLIBRE, Y.P. MARTÍNEZ AND C. VALLS (a) (b) (c) Figure 3. Phase portraits of system (1) in the planes z = x, y = z and x = y, respectively. 3. Dynamics on R3 In order to obtain the phase portrait in R3 of system (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) in the local charts U1 , U2 and U3 are exactly the same, except for the meaning of the coordinates (z1 , z2 , z3 ) (see (11), (12) and (13)). At infinity we have the following system in the local charts (10) z1 = −2z1 (−1 + z1 ), z2 = −2z2 (−1 + z2 ), z3 = −z3 (−1 + z1 + z2 ). Considering (10) on the local chart U1 , we have the four infinite equilibria (i.e, on z3 = 0): the origin, p̂1 = (0, 0, 0), p̂2 = (1, 0, 0), p̂3 = (0, 1, 0) and p̂4 = (1, 1, 0). In the local chart U2 the infinite equilibria on z1 = z3 = 0 are the origin of U2 , p̂5 = (0, 0, 0) and p̂6 = (0, 1, 0). Finally, in the local chart U3 we have that the origin p̂7 = (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 (10) is   −2(z1 − 1) − 2z1 0 0  0 −2(z2 − 1) − 2z2 0  −z3 −z3 −z1 − z2 + 1 which is a diagonal matrix, it is easy to conclude that p̂1 , p̂5 and p̂7 are unstable nodes, p̂4 is a stable node, that the eigenvalues of p̂2 are −2, 2, 0, and the eigenvalues of p̂3 and p̂6 are 2, −2, 0. Thus, p̂2 , p̂3 and p̂6 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 Vi (that we will denote by qi the antipodal point of pi ) for i = 1, 2, 3. Using the Poincaré-Bendixson Theorem we can characterize the connections be- tween the separatrices of the saddles at p2 , p3 and p6 with the infinite nodes. GLOBAL DYNAMICS OF A LOTKA–VOLTERRA SYSTEM IN R3 7 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. Figure 4. Phase portrait of system (1) at infinity on the boundary of the Poincaré ball for x ≥ 0. On the other hand, the finite equilibria of system (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) 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). 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 differen- tial system is invariant under the cyclic permutation of the variables x, y, z, and consequently the dynamics on several of the regions Rk for k = 1, . . . , 12 in which is divided the half-space x ≥ 0 can be obtained one from the others. But in what 8 J. LLIBRE, Y.P. MARTÍNEZ AND C. VALLS follows we prefer to provide the dynamics in all these twelve regions because this facilitates understanding the motion between these regions. Figure 5. Regions R1 − R12 on the Poincaré ball. 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. GLOBAL DYNAMICS OF A LOTKA–VOLTERRA SYSTEM IN R3 9 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 and complete the proof of Theorem 1. Appendix 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. We consider a polynomial vector field X = (P, Q, R) associated to the polynomial differential system ẋ1 = P (x1 , x2 , x3 ), ẋ2 = Q(x1 , x2 , x3 ), ẋ3 = 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 poly- nomial differential system in R3 . We consider the local charts (Uk , φk ) and (Vk , ψk ) 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 φk : Uk → R3 for k = 1, 2, 3 are     x2 x3 1 x1 x3 1 φ1 (x) = , , = (z1 , z2 , z3 ), φ2 (x) = , , = (z1 , z2 , z3 ), x1 x1 x1   x2 x2 x2 x1 x2 1 φ3 (x) = , , = (z1 , z2 , z3 ), x3 x3 x3 and ψk (x) = −φk (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 poly- nomial vector field X = (P, Q, R) in each local chart. The expression of the com- pactified analytical vector field p(X ) of X of degree n on the local chart U1 of D3 10 J. LLIBRE, Y.P. MARTÍNEZ AND C. VALLS Figure 6. Phase portraits of system (1) at the boundaries of the 12 regions contained in x ≥ 0 obtained dividing the Poincaré ball by the six invariant planes, x = 0, y = 0, z = 0, x = y, x = z and y = z. is (11) z3n (−z1 P (z) + Q(z), −z2 P (z) + R(z), −z3 P (z)) , where z = (1/z3 , z1 /z3 , z2 /z3 ). In a similar way the expression of p(X ) in U2 is (12) z3n (−z1 Q(z) + P (z), −z2 Q(z) + R(z), −z2 Q(z)) , where z = (z1 /z3 , 1/z3 , z2 /z3 ). Finally the vector field p(X ) in U3 is (13) z3n (−z1 R(z) + P (z), −z2 R(z) + Q(z), −z2 R(z)) , 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 . GLOBAL DYNAMICS OF A LOTKA–VOLTERRA SYSTEM IN R3 11 Figure 7. Phase portraits of system (1) in the interior of the invariant regions. From equations (11), (12) and (13) 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 Vk is the same as in Uk 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. Remark. For the expressions of the compactified vector field of a polynomial diffe- rential system in R3 see chapter 5 of [3] . 12 J. LLIBRE, Y.P. MARTÍNEZ AND C. VALLS 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 Economı́a, 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. References [1] F. H. Busse. Transition to turbulence via the statistical limit cycle route. In H. Haken, editor, Chaos and Order in Nature, pages 36–44, Berlin, Heidelberg, 1981. 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Accademia Nazionale dei Lincei,, CCCCXXIII(II):558–560, 1926. 1 Departament de Matemàtiques, Facultat de Ciències Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat GLOBAL DYNAMICS OF A LOTKA–VOLTERRA SYSTEM IN R3 13 2 Centre de Recerca Matemática, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: yohanna.martinez@uab.cat 3 Center for Mathematical Analysis, Geometry and Dynamical Systems, Departa- mento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal Email address: cvalls@math.tecnico.ulisboa.pt