Dynamics of a Competitive Lotka–Volterra Systems in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{3}$\end{document}

We describe the dynamics of the 3-dimensional competitive Lotka–Volterra systems x˙=x(a−x−y−z),y˙=y(b−x−y−z),z˙=z(c−x−y−z),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \dot{x}=x(a-x-y-z),\quad \dot{y}=y(b-x-y-z),\quad \dot{z}=z(c-x-y-z), $$\end{document} providing the phase portraits for all the values of the parameters a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a$\end{document}, b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b$\end{document} and c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c$\end{document} with 0<a<b<c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0< a< b< c$\end{document} in the positive octant of the Poincaré ball.


Introduction and Statement of the Main Results
We say that a polynomial vector field X = (P (x, y, z), Q(x, y, z), R(x, y, z)) in R 3 is quadratic if the maximum of the degrees of the polynomials P , Q and R is 2. A Lotka-Volterra system in R 3 is a quadratic polynomial vector field X with x a factor of P , y a factor of Q, and z a factor of R.
The Lotka-Volterra systems, were initially proposed as a model for studying the interactions between species in two dimension, developed independently by Alfred J. Lotka in 1925 [18] and by Vito Volterra in 1926 [25]. Later on in 1936 Kolmogorov [13] extended these systems to arbitrary dimension and arbitrary degree, these kinds of extended differential systems are called Kolmogorov systems.
The Lotka-Volterra systems can model many natural phenomena such as the time evolution of conflicting species in biology [20], chemical reactions [9], hydrodynamics [5], economics [24], the coupling of waves in laser physics [14], the evolution of electrons, ions and neutral species in plasma physics [15], etc. The interest in the 3-dimensional Lotka-Volterra systems becomes more important after the work of Brenig and Goriely [3,4], because they proved that many other differential systems, coming from physics, biology, chemistry and economics, can be transformed, using a quasimonomial formalism, into 3dimensional Lotka-Volterra systems.
In general the dynamics of the Lotka-Volterra systems are far from being understood, although some dynamics for special families of these systems have been revealed (see [1,2,16,17,26,27]). Thus for instance, the theory on cooperative or competitive systems was developed by Hirsch in the papers [10,11], where he proved that these systems generically exhibits a global attractor which lies on a 2-dimensional manifold.
In this work we consider the following class of competitive Lotka-Volterra systemṡ which depends on the three parameters a, b and c such that 0 < a < b < c. Here the dot denotes derivative with respect to the time t . We recall that a general 3-dimensional Lotka-Volterra systemẋ is called competitive if all its parameters a, a i , b, b i , c, c i for i = 1, 2, 3 are positive.
The competitive systems are of interest for their applications in nature, these system can describe the dynamics of two or more species competing for the same limited food source; for the territory (which is related to food resources too), or in some way inhibit each other their growth. In competitive systems it is assumed always that the growth of each species is directly proportional to the number of individuals of the species (see [8]). For more details on the competitive systems see the mentioned papers of Hirsch [10,11].
For a survey of competitive and related types of systems see the book by Freedman (1980) [8]. Only very special competitive systems have the desirable property that every trajectory converges to an equilibrium as t → +∞.
The region of ecological interest in competitive systems is the first octant of R 3 and the infinity of this region, that will be studied using the Poincaré compactification (see Sect. 2).
An invariant is a first integral depending on the time. The invariants of the form f (x, y, z)e st with s ∈ R \ {0} are called Darboux invariants. The existence of a Darboux invariant allows to obtain information on the α-limits and ω-limits of the orbits of the differential system. So when they exist, they play an important role in the study of the dynamics of a differential system.
Note that system (1) has the Darboux invariant because on the solutions (x(t), y(t), z(t)) of system (1) we have For more information about the Darboux invariants see Theorem 8.7 of [7]. Note that if 2c = a + b, then I is a first integral of system (1), but this situation will not be considered here because under the assumption that 0 < a < b < c we always have that 2c > a + b.
The objective of this work is to describe the dynamics of the Lotka-Volterra systems (1) in the positive octant of R 3 adding the infinity, i.e. we are interest in describing the dynamics of system (1) on the region Roughly speaking the Poincaré ball D 3 is the closed unit ball centered at the origin of R 3 , where its interior is identified with R 3 and its boundary S 2 is defined as the infinity R 3 , in the sense that in the space R 3 we can go to or come from the infinity in as many directions as points has the 2-dimensional sphere S 2 . A polynomial differential system in R 3 , i.e. in the interior of D 3 can be extended to the its boundary S 2 in a unique analytic way, this extension is called the Poincaré compactification of a polynomial differential system, because it was done by first time by Poincaré in [23] for the polynomial differential systems in R 2 , for its extension to polynomial differential systems in R 3 see [6]. For a brief introduction to Poincaré compactification see Sect. 2.
We say that two compactified polynomial differential systems on the Poincaré ball D 3 are topologically equivalent if there is a homeomorphism of D 3 which send orbits of one system into orbits of the other system preserving or reversing the orientation of all+ the orbits.
In the next theorem we describe the phase portrait on the positive octant of the Poincaré ball, i.e. on D, of the 3-dimensional competitive Lotka-Volterra systems (1).

Theorem 1
For every 3-dimensional competitive Lotka-Volterra system (1) the following statements hold. (b) All the points at the infinity in the positive octant of the Poincaré ball are equilibrium points. Every one of these infinite equilibria is the α-limit of a unique orbit not contained at infinity.
(c) The plane (x, y, z) = bcx + acy + abz − abc = 0 is invariant by the flow of system. The phase portrait of the system on this plane is topologically equivalent to the one described in Fig. 1 (d) Let φ t (q) be the solution of the system such that φ 0 (q) = q with q in the interior of the positive octant and satisfying that (q) = 0. This work is organized as follows. In Sect. 2 we present some basic definitions and some preliminary results necessary to prove Theorem 1. In Sect. 3 we prove Theorem 1.

Poincaré Compactification of R 3
In order to give a detailed proof of Theorem 1 we need the Poincaré compactification.
We consider a polynomial vector field X = (P , Q, R) associated to the polynomial differential systemẋ 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 R 3 .
We consider the local charts (U k , φ k ) and (V k , ψ k ) for k = 1, 2 on the disc D 3 defined by where the diffeomorphisms φ k : U k → R 3 for k = 1, 2, 3 are Note that the coordinates (z 1 , z 2 , z 3 ) have different meaning depending on local chart, but the points of the infinity, i.e. the points of the boundary S 2 of D 3 all have the coordinate z 3 = 0. Now we give the expression of the compactified vector filed 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 U 1 of D 3 is In a similar way the expression of p(X ) in U 2 is where z = (z 1 /z 3 , 1/z 3 , z 2 /z 3 ).
The singular points of p(X ) which are on the boundary S 2 of D 3 (at z 3 = 0) are called infinite singular points, and we call finite singular points to the ones which are in the interior of D 3 .
From equations (3), (4) and (5) it follows that the infinity S 2 of the Poincaré disc 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 U 1 , in U 2 with z 1 = 0, and the origin of the local chart U 3 in case that this be 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 S 2 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 R 3 see [6].
As we said in the introduction two compactified polynomial differential systems on the Poincaré ball D 3 are topologically equivalent if there is a homeomorphism of D 3 sending orbits of one to the other system, either preserving or reversing the orientation of all the orbits.

Poincaré Compactification of R 2
See chapter 5 of [7] for the expressions of the compactified vector field of a polynomial differential system in R 3 . Now we shall see how to characterize the phase portrait of a compactified vector field p(X) in the Poincaré disc D 2 defined by the invariant planes x = 0, y = 0 and z = 0.
Let Y be the restriction of the Lotka-Volterra system (1) on some of the invariant planes x = 0, y = 0 and z = 0. A separatrix of p(Y ) is an orbit which is either an equilibrium point, or a trajectory which lies in the boundary of a hyperbolic sector of a finite or an infinite equilibrium point, or any orbit contained in S 1 (the boundary of D 2 , i.e. the infinity of the plane), or a limit cycle. Neumann [21] proved that the set formed by all separatrices of p(Y ), denoted by S(p(Y )) is closed.
The open connected components of D 2 \ S(p(Y )) are called canonical regions of Y or of p(Y ). A separatrix configuration is the union of S(p(Y )) plus one orbit chosen in each canonical region. Two separatrix configurations S(p(Y )) and S(p(Y)) are topologically equivalent if there is an orientation preserving or reversing homeomorphism which maps the trajectories of S(p(Y )) into the trajectories of S(p(Y)). The following result is due to Markus [19], Neumann [21] and Peixoto [22], who found it independently.

Phase Portraits of System (1)
In this section we study all the phase portraits of the 3-dimensional Lotka-Volterra systems (1) in the positive octant of the Poincaré ball. Initially we study the finite and infinite singular points as function of the parameters a, b and c. First note that system (1) has always the three invariant planes x = 0, y = 0 and z = 0. We start with the study of the infinite singular points, for this purpose we use the Poincaré compactification. System (1) in local chart U 1 iṡ in the local chart U 2 iṡ and in the local chart U 3 iṡ From systems (6), (7) and (8) we get that all the points at infinity (i.e. all the points having the coordinate z 3 = 0) are equilibrium points. Moreover, the eigenvalues of the linear part of those systems at these equilibrium points have always one positive eigenvalue, and the eigenvalue 0 with multiplicity two. So by the theory of normal hyperbolicity (see for instance [12]) every equilibrium point is the α-limit of a unique orbit. Roughly speaking, the whole infinity of the positive octant in the Poincaré ball is a repeller. In short we have proved statement (b) of Theorem 1. Now we study the finite singular points. System (1) has four finite equilibria: namely e 1 = (0, 0, 0), e 2 = (a, 0, 0), e 3 = (0, b, 0) and e 4 = (0, 0, c).

The linear matrix of system (1) is
Since the eigenvalues at the origin of the matrix M are a, b and c, the origin is a repeller. The eigenvalues of M at e 2 are −a, b − a and c − a, so taking into account their eigenvectors the equilibrium e 2 has an 1-dimensional stable manifold contained in the positive x-axis, and a 2-dimensional unstable manifold. Moreover the equilibrium e 2 restricted to the planes y = 0 and z = 0 is a saddle.
The eigenvalues of M at e 3 are a − b, −b and c − b, so taking into account their eigenvectors the equilibrium e 3 has a 2-dimensional stable manifold contained in the positive quadrant of the plane z = 0 (restricted to this plane e 3 is a stable node), and an 1-dimensional unstable manifold contained in the positive quadrant of the plane x = 0 (restricted to this plane e 3 is a saddle).
Finally the eigenvalues of M at e 4 are a − c, b − c and −c, therefore e 4 is an attractor. We note that the finite equilibria are all on the boundary of the positive octant of R 3 , more precisely at the origin and on the positive axes of this octant.
We shall study the phase portrait of system (1) on the three faces x = 0, y = 0 and z = 0 of the closed positive octant in the Poincaré ball. Taking into account the local phase portraits of the finite equilibria on these three faces, and that the axes x, y and z are invariant (i.e. if an orbit has point on some of these axes it is contained in that axis), then we obtain the phase portraits described in Fig. 1(a). This completes the proof of statement (a) of Theorem 1. Now we shall study the phase portrait of system (1) in the interior of the positive octant.
First we shall prove that the plane (x, y, z) = bcx + acy + abz − abc = 0 that contains the three equilibria e 2 , e 3 and e 4 is invariant by the flow of system (1). Indeed, since on the points (x, y, z) where (x, y, z) = 0 its derivative with respect to the time t also is zero, consequently the plane (x, y, z) = 0 is invariant by the flow of system (1). An easy analysis of the phase portrait of system (1) restricted to the invariant plane (x, y, z) = 0, provides that this phase portrait is the one shown in Fig. 1(b). Note that isolating z from the invariant plane = 0 and substituting it into the differential system (1) we get the system:ẋ System (10) has three equilibria: an attractor at the origin, a repeller in (a, 0), and a saddle in (0, b); the infinity is filled of equilibria. We have already studied the phase portrait on the invariant planes x = 0, y = 0 and z = 0, then we need to study the orbit of a point q = (x, y, z) for xyz = 0 such that (q) = 0. On this plane , there exist a finite saddle in (0, b), a repeller in (a, 0), and an attractor in (a, 0). This and the continuity of the solutions implies that the point q has its α-limit in (a, 0) and its ω-limit in (0, 0). Thus, we have that the phase portrait on the invariant plane ⊂ R 3 is the one shown in Fig. 1(b). Note that we must consider the correspondence (0, 0) → (0, 0, c), (a, 0) → (a, 0, 0) and (0, b) → (0, b, 0). This completes the proof of statement (c) of Theorem 1. From Fig. 1(b) it follows that the 2-dimensional unstable manifold of the equilibrium e 2 restricted to the positive octant is contained in the plane (x, y, z) = 0; and that the 2-dimensional stable manifold of the equilibrium e 3 restricted to the positive octant is contained in the plane z = 0.
The next lemma completes the phase portrait of the competitive Lotka-Volterra system (1), proving statement (d) of Theorem 1.

Lemma 3
Let φ t (q) = (x(t), y(t), z(t)) be the solution of the Lotka-Volterra system (1) such that φ t (q) = q with q = (x, y, z) in the interior of the positive octant and satisfying that (q) = 0.
Due to the fact that this Darboux invariant it is constant on the orbits, this implies that we needs satisfies that lim t→+∞ xyz −2 → 0, then the ω-limit must be in x = 0, or y = 0, or in z = ∞. Since the infinity is a repeller, the ω-limit must be the equilibrium point e 4 = (0, 0, c) which is the unique attractor in {x = 0} ∪ {y = 0}. This proves statement (a) of Lemma 3.
On the other hand the α-limit, according with the second expression in (11), is conditioned by the relation lim t→−∞ xyz −2 → ∞, then can be in x → ∞, or y → ∞, or z → 0. We recall that the plane is invariant under the flow, then a orbit passing for q such that (q) < 0 must have their α-limit on e 1 (we know that e 1 is the unique repeller (even more is the unique equilibrium) on the region (q) < 0), and an orbit passing for a point q with (q) > 0 has it α-limit at the infinity, because there are not finite equilibria on this region and the infinity is a repeller. These affirmations prove statements (b) and (c) of Lemma 3. Figure 1(c) shows one orbit in each region (q) > 0, (q) = 0 and (q) < 0.
In short the phase portrait of system (1) in the first octant is as is shown in Fig. 1(d).