PHASE PORTRAITS OF QUADRATIC LOTKA – VOLTERRA SYSTEMS WITH A DARBOUX INVARIANT IN THE POINCARÉ DISC

We characterize the global phase portraits in the Poincaré disc of all the planar Lotka–Volterra quadratic polynomial differential systems having a Darboux invariant.


Introduction and statement of main results
Let R[x, y] (resp.C[x, y]) be the ring of the polynomials in the variables x and y with coefficients in R (resp.C).We consider a system of polynomial differential equations or simply a polynomial differential system in R 2 defined by (1) ẋ = P (x, y), ẏ = Q(x, y), where P, Q ∈ R[x, y] and the dot denotes derivative with respect to the independent variable t usually called the time.We say that the maximum of the degrees of the polynomials P and Q is the degree of system (1).Usually a quadratic polynomial differential system of degree 2 is called simply a quadratic system.Sometimes we shall talk about the quadratic vector field (2) X = P (x, y) ∂ ∂x + Q(x, y) ∂ ∂y .
associated to the quadratic system (1).A non-locally constant real function H of class C 1 defined on an open set U is a first integral of the polynomial vector field X if H(x(t), y(t)) is constant for all value of t for which the solution (x(t), y(t)) of X is defined on U. We observe that H is a first integral of X if and only if X H = 0 on U. Let f ∈ C[x, y] \ {0}.The algebraic curve f (x, y) = 0 is an invariant algebraic curve of the polynomial system (1) if for some polynomial k ∈ C[x, y] we have The polynomial k is called the cofactor of the invariant algebraic curve f = 0. Let g, h ∈ C[x, y] and assume that g and h are relatively prime in the ring C[x, y] or that h = 1.Then the function exp(g/h) is called an exponential factor of system (1) if for some polynomial k ∈ C[x, y] of degree at most 1 we have that A non-constant function I = I(x, y, t) is an invariant of X if That is, I is an invariant of X if it is a first integral depending on the time t.Moreover we say that I is a Darboux invariant if it is an invariant which can be written as , where f 1 = 0, . . ., f p = 0 are invariant algebraic curves of X , F 1 , . . ., F q are exponential factors of X , λ 1 , . . ., λ p , µ 1 , . . ., µ q ∈ C and s ∈ R\{0}.
The quadratic Lotka-Volterra systems are of the form (3) ẋ = x(ax + by + c), ẏ = y(Ax + By + C), where a, b, c, A, B, C are constants.In this paper we study the quadratic systems (3) that possess a Darboux invariant.
System (3) and its generalization to dimension n, since were originally studied by Lotka and Volterra in [12,18], are called Lotka-Volterra systems.Later on these differential systems were studied by Kolmogorov [9], and after his work some authors called them Kolmogorov systems.There are many natural phenomena that can be modeled by these differential systems such as the time evolution of conflicting species in biology [15], chemical reactions, plasma physics [10], hydrodynamics [2], economics, ... The knowledge of a first integral is of special interest for a differential system in the plane because it allows to compute the explicit expressions of the trajectories of the system.However, when we cannot compute a first integral of the system it is interesting to know if the system has an invariant.Roughly speaking, with a first integral we can describe completely the phase portrait of a planar differential system, while with an invariant we only can describe its asymptotic behavior, i.e. the ω-and α-limit sets of its trajectories.
Our first result is the characterization of the quadratic Lotka-Volterra systems (3) having a Darboux invariant.
Their Darboux invariants are e −C t y and e (−c l 1 −C l 2 ) t x l 1 y l 2 respectively, where l 1 and l 2 are constants.
Theorem 1 will be proved in section 2. Theorem 1 improves some preliminary results on the Darboux invariants of the quadratic Lotka-Volterra systems given in Theorem 1 of [3].
Note that if l 2 = 0 then system (5) becomes system (4) after the change of variables and parameters (x, y, A, B, C, c) → (y, x, b, a, c, C).Hence, in system (5) we have excluded the case l 2 = 0.
We remark that in Theorem 1 we do not consider the differential systems (4) and ( 5) which become linear differential systems, i.e. we only consider quadratic differential systems (4) and (5).This is obtained with the conditions a 2 + b 2 = 0 and A 2 + B 2 = 0 in the systems (4) and (5), respectively.
We observe that the conditions C = 0 and c 2 + C 2 = 0, in the systems (4) and ( 5) respectively, are necessary in order that such systems really have a Darboux invariant.
Our next results provide the global phase portraits of systems ( 4) and ( 5) in the Poincaré disc, see Theorems 2 and 3, respectively.Theorem 2. The phase portrait of any quadratic Lotka-Volterra system (4) is topologically equivalent to one of the 7 configurations of Figures 1-4.In sections 5 and 6 we shall prove Theorems 2 and 3, respectively.But before that we recall the notions of the Poincaré compactification (section 3), the definitions of separatrices and of canonical regions (section 4) that we will need later on for drawing the global phase portraits of systems ( 4) and (5).

Proof of Theorem 1
In order to prove Theorem 1 first we recall a result from the Darboux theory of integrability that we will use, for a proof see Theorem 8.7 of [6].
Proposition 4. Supposse that a polynomial system (1) of degree n admits p irreducible invariant algebraic curves f i = 0 with cofactors k i for i = 1, . . ., p, q exponential factors F j = exp(g j /h j ) with cofactors l j for j = 1, . . ., q.If there exist λ i , µ j ∈ C not all zero such that (6) µ j l j = −s for some s ∈ R\{0}, then the (multi-valued) function Proof of Theorem 1.It is easy to verify that the straight lines x = 0 and y = 0 are invariant algebraic curves of the quadratic Lotka-Volterra systems (3) with cofactors k 1 = ax + by + c and k 2 = Ax + By + C respectively.So solving equation ( 6) with k 1 and k 2 we obtain two solutions which are either (8) From (8) we arrive to the family of quadratic Lotka-Volterra systems (4) having the Darboux invariant e −C t y when C = 0. Similarly from (9) we obtain the family (5) with the Darboux invariant e −(c l 1 +C l 2 ) t x l 1 y l 2 when −c l 1 − C l 2 = 0, as it is stated in the theorem.

Poincaré compactification
Let X be a planar vector field of degree n.The Poincaré compactified vector field p(X ) corresponding to X is an analytic vector field induced on S 2 as follows (see, for instance [8], or Chapter 5 of [6]).Let S 2 = {y = (y 1 , y 2 , y 3 ) ∈ R 3 : y 2 1 + y 2 2 + y 2 3 = 1} (the Poincaré sphere) and T y S 2 be the tangent space to S 2 at point y.Consider the central projection f : T (0,0,1) S 2 → S 2 .This map defines two copies of X , one in the northern hemisphere and the other in the southern hemisphere.Denote by X ′ the vector field Df • X defined on S 2 except on its equator S 1 = {y ∈ S 2 : y 3 = 0}.Clearly S 1 is identified to the infinity of R 2 .In order to extend X ′ to a vector field on S 2 (including S 1 ) it is necessary that X satisfies suitable conditions.In the case that X is a planar vector field of degree n then p(X ) is the only analytic extension of y n−1 3 X ′ to S 2 .On S 2 \S 1 there are two symmetric copies of X , and knowing the behavior of p(X ) around S 1 , we know the behavior of X at infinity.The projection of the closed northern hemisphere of S 2 on y 3 = 0 under (y 1 , y 2 , y 3 ) −→ (y 1 , y 2 ) is called the Poincaré disc, and it is denoted by D. The Poincaré compactification has the property that S 1 is invariant under the flow of p(X ).
In this paper we say that two polynomial vector fields X and Y on R 2 are topologically equivalent if there exists a homeomorphism on S 2 preserving the infinity S 1 carrying orbits of the flow induced by p(X ) into orbits of the flow induced by p(Y), preserving or reversing simultaneously the sense of all orbits.
The expression for p(X ) in the local chart (U 1 , F 1 ) is given by The expression for p(X and for Thus we obtain a polynomial vector field in each local chart.

Separatrices and canonical regions
Let p(X ) be the Poincaré compactification in the Poincaré disc D of the polynomial differential system (1) defined in R 2 , and let Φ be its analytic flow.Following Markus [14] and Neumann [16] we denote by (U, Φ) the flow of a differential system on an invariant set U ⊂ D under the flow Φ.The flow (U, Φ) is said to be parallel if it is topologically equivalent to one of the following flows: (i) The flow defined in R 2 by the differential system ẋ = 1, ẏ = 0, called strip flow.(ii) The flow defined in R 2 \ {0} by the differential system in polar coordinates ṙ = 0, θ = 1, called annular flow.(iii) The flow defined in R 2 \ {0} by the differential system in polar coordinates ṙ = r, θ = 0, called spiral or radial flow.It is known that the separatrices of a polynomial vector field p(X ) in the Poincaré disc D are: (I) all the orbits of p(X ) which are in the boundary S 1 of the Poincaré disc (i.e. at the infinity of R 2 ), (II) all the finite singular points of p(X ), (III) all the limit cycles of p(X ), and (IV) all the separatrices of the hyperbolic sectors of the finite and infinite singular points of p(X ).
Let S be the union of the separatrices of the flow (D, Φ) defined by p(X ) in the Poincaré disc D. It is easy to check that S is an invariant closed set.If N is a connected component of D \ S, then N is also an invariant set under the flow Φ of p(X ), and the flow (N, Φ| N ) is called a canonical region of the flow (D, Φ) .For a proof of this proposition see [16] or [11].The separatrix configuration S c of a flow (D, Φ) is the union of all the separatrices S of the flow together with an orbit belonging to each canonical region.The separatrix configuration S c of the flow (D, Φ) is said to be topologically equivalent to the separatrix configuration S * c of the flow (D, Φ * ) if there exists a homeomorphism from D to D which transforms orbits of S c into orbits of S * c preserving or reversing the sense of all orbits.
Theorem 6 (Markus-Neumann-Peixoto).Let (D, Φ) and (D, Φ * ) be two compactified Poincaré flows with finitely many separatrices coming from two polynomial vector fields (2).Then they are topologically equivalent if and only if their separatrix configurations are topologically equivalent.
For a proof of this result we refer the reader to [14,16,17].It follows from the previous theorem that in order to classify the phase portraits in the Poincaré disc of a planar polynomial differential system having finitely many separatrices finite and infinite, it is enough to describe their separatrix configuration.
Finally, Bautin [1] showed that the Lotka-Volterra systems cannot have a limit cycle, therefore we only look for the other types of separatrices.

Proof of Theorem 2
We consider the family of quadratic Lotka-Volterra systems (4).First, we calculate its finite singularities and we find that if ac = 0 then there are two singular points (0, 0) and (−c/a, 0).In the cases a = 0 and c = 0, or a = 0 and c = 0, we have only the singular point (0, 0); and if a = 0 and c = 0 there exist infinitely many singular points of the form (x, 0) with x ∈ R. Recall that C = 0 in order to guarantee the existence of a Darboux invariant.
As we are interested in classifying the phase portraits in the Poincaré disc of system (4), we introduce a rescaling of the variables and of the time to reduce the number of its parameters as much as possible, and hence the phase portraits obtained for the new system will be topologically equivalent to the corresponding phase portraits of system (4).Let (10) x = αX, y = βY and t = γT where α, β, and γ are parameters.
Let b = 0, then system ( 12) is ( 13) Therefore system (13) possesses infinitely many finite singularities, the points (X, 0).Resolving this quadratic system we obtain solutions So the origin is a singular point on U 2 .By Theorem 2.19 of [6], it is a semi-hyperbolic saddle-node with stable separatrix tangent at (0, 0) to the positive v-axis.The two unstable separatrices are tangent to the u-axis in (0, 0).Therefore, by joining all the previous information we obtain the phase portrait of quadratic system (11) on the Poincaré disc when b = 0, which is topologically equivalent to the one shown in Figure 1.
In the figures we shall use the following notation.The singular points are denoted by dots, the separatrices which are non singular points are denoted by thicker lines, and an orbit inside a canonical region is represented by a thinner line.
System (14) written in the variables X and Y , according to (10), becomes  15) is ( 16) This system has only a finite singularity, the origin, which is a semi-hyperbolic saddlenode with stable separatrix tangent at (0, 0) to the negative X-axis and two unstable separatrices tangent at (0, 0) to the Y-axis.The expression for the Poincaré compactification p(X ) in the local chart the origin is the unique singular point, which is a hyperbolic stable node at infinity.Since on U 1 the degree of p(X ) is 2, the diametrally opposite point is a hyperbolic unstable node in V 1 .
The expression for p(X So the origin of U 2 is a singular point linearly zero.Since system ( 17) is homogeneous we apply the classification due to Date [5], see the appendix.Using the definitions given at the appendix, we have that D = 0, [h κλ ] = 0 0 0 −1/9 = 0 and H = −1/4 = 0.So system (17) belongs to the type II (1).Moreover the value of the covariant K 2 is 27/8 > 0. Therefore we obtain that system (17) is of Type II(1-1).In consequence the phase portrait in the Poincaré disc of system ( 14) if b = 0 is topologically equivalent to the one showed in Figure 2 with b = 0.
If b = 0 we have that system (15) is X ′ = X(X +Y ), Y ′ = Y .This system possesses only a finite singularity, the origin which is a semi-hyperbolic saddle-node with stable separatrix tangent at (0, 0) to the negative X-axis and unstable separatrices tangent to the Y-axis at (0, 0).The expression for p(X ) in the local chart Thus there are two infinite singularities on U 1 , (0, 0) and (−1, 0).The origin is a hyperbolic stable node.The singular point (−1, 0) is a hyperbolic saddlenode whose stable separatrix is tangent at (−1, 0) to the positive v-axis.The unstable separatrices are tangent at (−1, 0) to the u-axis.

The expression for
Therefore the origin is the unique singularity on U 2 which is a semi-hyperbolic saddlenode, its stable separatrix is tangent at origin to the positive v-axis and the unstable separatrices to the u-axis.So we get the phase portrait of system ( 14) if b = 0, which is topologically equivalent to the one shown in Figure 2    Again from (10) we obtain that system ( 18) is transformed into X ′ = X(bβγY + cγ), Y ′ = CγY , and hence we obtain the two systems We only consider the case b = 0 because for b = 0 the system is linear, then the previous system is topologically equivalent to the system X ′ = X(Y + m), Y ′ = Y , respectively, with m = c/C a nonzero constant.The only finite singular point is (0, 0) which is a hyperbolic unstable node if m > 0, or a hyperbolic saddle if m < 0.

The expression for
This system is homogeneous and it possesses a unique singular point, the origin, which is linearly zero.We apply the appendix in order to study this singularity.Thus we obtain D = 0, [h κλ ] = −1/9 0 0 0 = 0 and H = −1/4 = 0.So the system belongs to the type II(1).Moreover K 2 = 27m/8, in consequence if m > 0, the system is of the type II(1-1), and it is of type II(1-2) if m < 0.
In U 2 the expression for p(X ) is u ′ = u+(m−1)uv, v ′ = −v 2 .So U 2 contains only a singularity, the origin, which is a semi-hyperbolic saddle-node with stable separatrix tangent at the origin to the positive v-axis.The two unstable separatrices are tangent at (0, 0) to the u-axis.In Figure 3 are shown the phase portraits topologically equivalent to the ones of system (18) for b = 0.  From (10) we have that system (19) is topologically equivalent to one of the two systems We analyze first the case b = 0. Consider the system X ′ = X 2 + mX, Y ′ = Y with m = 0. Finite singularities are (0, 0) and (−m, 0).The singular point (0, 0) is a hyperbolic unstable node if m > 0 and a hyperbolic saddle if m < 0. Also (−m, 0) is a hyperbolic saddle if m > 0 and a hyperbolic unstable node if m < 0. The expression of p(X There is a unique singularity on U 1 , the origin, which is a hyperbolic stable node. So the origin is a singular point on U 2 linearly zero.Since this system is homogeneous we can use the appendix for its classification.Hence D = −m 2 /27 < 0 and so the system belongs to the type I⊖.Moreover since K 2 = 27/8 > 0, the system is of type I⊖ (1).
Joining all the above information we get the two phase portraits of the system (19) when b = 0 corresponding to the cases m < 0 or m > 0. However, it is not difficult to verify that these two configurations are topologically equivalent.In Figure 4 with b = 0 is presented a phase portrait topologically equivalent to the one of system (19) when b = 0.
If b = 0, we consider the system X ′ = X 2 + XY + mX, Y ′ = Y with m = 0.There are two finite singularities, (0, 0) and (−m, 0).The origin is a hyperbolic unstable node if m > 0, and it is a hyperbolic saddle if m < 0. The singular point (−m, 0) is a hyperbolic saddle if m > 0 and it is a hyperbolic unstable node if m < 0. The expression of p(X ) on So the singular points on U 1 are (0, 0) and (−1, 0).The origin is a stable node and (−1, 0) is a semi-hyperbolic saddle-node, its stable separatrix is tangent to the v-axis at (−1, 0) and the unstable separatrices are tangent to the u-axis at (−1, 0).
Therefore the origin of U 2 is a singular point which is a semi-hyperbolic saddle-node with stable separatrix tangent at (0, 0) to the positive v-axis.The two unstable separatrices are tangent at (0, 0) to the u-axis.So we obtain the two phase portraits for m < 0 and m > 0. It is easy to see that these two configurations are topologically equivalent.Figure 4 with b = 0 shows a phase portrait topologically equivalent to the one of system (4) in this case.This completes the proof of Theorem 2.

Proof of Theorem 3
We will find the phase portraits in the Poincaré disc of any quadratic Lotka-Volterra system (5).For doing this we introduce a rescaling of the variables and of the time in order to reduce the number of parameters of this system as much as possible.Obtaining the phase portraits for each of the new systems we shall get the phase portraits of the corresponding systems (5).Let x = αX, y = βY and t = γT , where α, β and γ are parameters.In the new variables X, Y and T system (5) becomes here ′ denotes the derivative with respect to the variable T .We recall that l 1 l 2 = 0 and c 2 + C 2 = 0. We consider the following cases.Note that A = B = 0 provides a linear differential system, and consequently it is not considered here.
In order to study system (20) we consider three subcases.
Then only the origin is a singular point if m = 1.However, there are infinitely many singularities if m = 1, the points (u, 0).If m = 1, (0, 0) is a hyperbolic unstable node if m < 0, it is a hyperbolic saddle if 0 < m < 1, and a hyperbolic stable node when m > 1.
In U 2 the expression of p(X ) is u ′ = (m−1)u+v, v ′ = −v.So the origin is a singular point.If m > 1, it is a hyperbolic saddle, and a hyperbolic stable node if m < 1.Since the rescaling ds = XdT changes the orientation of orbits where X is negative, we obtain that the phase portraits of system ( 22) are topologically equivalent to the ones shown in Figure 5.
The origin is the unique finite singularity of system (24), which is a semi-hyperbolic saddle-node.If m > 0, its stable separatrix is tangent at (0, 0) to the negative X-axis, and to the positive X-axis when m < 0. The expression of . Phase portraits of system (22).They have infinitely many separatrices.
there is only one singular point, the origin; if m < 0, (0, 0) is a hyperbolic unstable node, or it is a hyperbolic saddle when 0 < m < 1, or a hyperbolic stable node if m > 1.

The expression of
Therefore, if m = 1 we have that (0, 0) is a linearly zero singular point and the system is homogeneous.Applying the appendix we obtain that D = 0, [h κλ ] = 0 0 0 −m 2 /9 = 0 and H = −m 2 /4 = 0.In consequence the system belongs to the type II(1).As If 0 < m < 1 then K 2 < 0 and so the system is of type II(1-2).If m = 1, then the system belongs to the type II(1-3).
Therefore we obtain the phase portraits of system (24), which are topologically equivalent to the phase portrait for b = 0 of Figure 2 when m < 0 or m > 1, and the others are topologically equivalent to the ones represented in Figure 6.
The origin is a hyperbolic saddle if m 2 < 0, and it is a hyperbolic unstable node if , it is a stable node.Now we study system (25) at infinity.The expression of p(X ) in the local chart Therefore when m 1 = 1, there is a unique singular point at U 1 the origin, which is hyperbolic.If m 1 < 0, (0, 0) is an unstable node.If 0 < m 1 < 1, the origin is a saddle; and (0, 0) is a hyperbolic stable node when m 1 > 1.However, there are infinitely many singularities at U 1 when m 1 = 1, the points (u, 0).
. So (0, 0) is a linearly zero singularity.Since system (26) is homogeneous we use the appendix for its classification.Then, since D = −m 2 1 m 2 2 /27 < 0 we get that system (26) belongs to the type I⊖.Computing the values of the covariants , we obtain the following classification for system (26).

Condition
Sign of K 2 and K 3 Type Phase portrait ) of Fig. 7, (5) of Fig. 7, In consequence we obtain the phase portraits of system (25), which are topologically equivalent to the phase portrait for b = 0 of Figure 4 when m 1 < 0 and m 1 < m 2 or when m 1 > 1 and m 1 > m 2 .Or to the phase portrait for m < 0 of Figure 5 when m 1 = m 2 < 0, to the phase portrait for 0 < m < 1 of Figure 5 when 0 < m 1 = m 2 < 1, to the phase portrait for m = 1 of Figure 5 when m 1 = m 2 = 1, and to the phase portrait for m > 1 of Figure 5 when m 1 = m 2 > 1 and the others are topologically equivalent to the ones of Figure 7. (2) 1 < m1 < m2, S: 14, CR: 3.
On the local chart U 1 the expression of p(X ) is u ′ = (m + 1)u 2 + v 2 , v ′ = muv.So if m = −1 the origin is the unique singular point and it is linearly zero.Since this system is homogeneous, we use the appendix for its classification.Thus we obtain D = 4/27 > 0, and so the system is of type I⊕.Moreover since K 2 = 27m 2 (m + 1)/8 we have that if m > −1, then K 2 > 0 and so the system is of type I⊕(1).In the case m < −1 we obtain K 2 < 0, therefore the system is of type I⊕ (2).If m = −1, then K 2 = 0, and the system is of type I⊕ (3).
The expression of p(X . So (0, 0) is a singular point which is a hyperbolic saddle if m < −1 and it is a hyperbolic stable node when m > −1.In consequence, we obtain the phase portraits of system (29), which are topologically equivalent to the ones shown in Figure 8.  where m 1 = l 2 /l 1 = 0 and m 2 = C/c = 0.The finite singularities of system (30) are the hyperbolic singular points (0, 0) and (0, −m 2 ).The point (0, 0) is a saddle if m 2 < 0, and it is an unstable node when m 2 > 0.
Then m 1 = −1/m 2 and system (30) becomes (31) We analyze the reduced system X ′ = X/m 2 , Y ′ = Y whose unique singular point is the origin.If m 2 > 0, then (0, 0) is a hyperbolic unstable node, and if m 2 < 0, it is a hyperbolic saddle.On the local chart U 1 , the expression of p(X ) is u ′ = (1 − 1/m 2 )u, v ′ = −v/m 2 then (0, 0) is the unique singularity on U 1 , which is a hyperbolic unstable node if m 2 < 0; or it is a hyperbolic stable node if 0 < m 2 < 1, or a hyperbolic saddle when m 2 > 1.
The expression of p(X ) on U 2 is u ′ = (1/m 2 − 1)u, v ′ = −v.Therefore (0, 0) is a singular point, and it is hyperbolic.Moreover (0, 0) is a stable node if m 2 < 0 or m 2 > 1, and it is a saddle if 0 < m 2 < 1.In consequence we obtain the phase portraits of X ′ = X/m 2 , Y ′ = Y .And so changing the orientation of orbits when Y < −m 2 we get the phase portraits of system (31).
which is a homogeneous system.So we can apply the classification of Date to system (32) (see appendix).Realizing the calculations we obtain D = −m 2 2 /27 < 0, thus system (32) belongs to the type I⊖.Since where m = l 1 /l 2 = 0.So system (35) by interchanging the variables X and Y coincides with system (33).
Given a quadratic differential system by means of these invariants and covariants we can characterize to which of the 10 types given in Theorem 7 it belongs.More precisely,

Theorem 3 .
The phase portrait of any quadratic Lotka-Volterra system (5) is topologically equivalent either to one of the 24 configurations of Figures 5-10, or to the configuration b = 0 of Figure 2, or to the configuration b = 0 of Figure 4.

Proposition 5 .
If the number of separatrices of the flow (D, Φ) is finite, then every canonical region of the flow (D, Φ) is parallel.

Figure 2 .
Figure 2. Phase portraits of system(14).S is the number of all separatrices and CR is the number of all canonical regions.

Figure 11 .
Figure 11.Phase portraits for quadratic homogeneous vector fields.