Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 35, pp. 1–38.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

PHASE PORTRAITS OF A FAMILY OF KOLMOGOROV

SYSTEMS DEPENDING ON SIX PARAMETERS

ÉRIKA DIZ-PITA, JAUME LLIBRE, M. VICTORIA OTERO-ESPINAR

Abstract. We consider a general 3-dimensional Lotka-Volterra system with
a rational first integral of degree two of the form H = xiyjzk. The restriction

of this Lotka-Volterra system to each surface H(x, y, z) = h varying h ∈ R
provide Kolmogorov systems. With the additional assumption that they have a
Darboux invariant of the form x`ymest they reduce to the Kolmogorov systems

ẋ = x
(
a0 − µ(c1x+ c2z

2 + c3z)
)
,

ż = z
(
c0 + c1x+ c2z

2 + c3z
)
.

We classify the phase portraits in the Poincaré disc of all these Kolmogorov

systems which depend on six parameters.

1. Introduction

The Lotka-Volterra systems have been used for modeling many natural phe-
nomena, such as the time evolution of conflicting species in biology [20], chemical
reactions, plasma physics [15] or hydrodynamics [6], just as other problems from
social science and economics.

These systems, which are polynomial differential equations of degree two, were
initially proposed, independently, by Lotka in 1925 and Volterra in 1926, both in
the context of competing species. Later on Lotka-Volterra systems were generalized
and considered in arbitrary dimension, i.e.

ẋi = xi

(
ai0 +

n∑
j=1

aijxj

)
, i = 1, . . . , n.

Consequently the applications of these systems started to multiply. Moreover Kol-
mogorov in [14] extended the Lotka-Volterra systems as follows

ẋi = xiPi(x1, . . . , xn), i = 1, . . . , n,

where Pi are polynomials of degre at most m. These kind of systems are now
known as Kolmogorov systems. They have in particular all the applications of the
Lotka-Volterra systems as for instance in the study of the black holes in cosmology,
see [1].

2010 Mathematics Subject Classification. 34C05.
Key words and phrases. Kolmogorov system; Lotka-Volterra system; phase portrait;

Poincaré disc.
c©2021 Texas State University.

Submitted June 1, 2020. Published May 3, 2021.

1



2 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

The global qualitative dynamics of the Lotka-Volterra systems in dimension
two has been completely studied in [24], where all possible phase portraits on the
Poincaré disc have been classified.

There are few results about the global dynamics of the Lotka-Volterra systems in
dimension three. Our objective is to study the phase portraits of the 3-dimensional
Lotka-Volterra systems

ẋ = x(a0 + a1x+ a2y + a3z),

ẏ = y(b0 + b1x+ b2y + b3z),

ż = z(c0 + c1x+ c2y + c3z),

(1.1)

which have a rational first integral of degree two of the form xiyjzk. We have used
the Darboux theory of integrability to obtain a characterization of these systems.
As a result, we have reduced the initial problem to a problem in dimension two,
the study of the global dynamics of two families of Kolmogorov systems. In this
paper we focus on the first family, which is

ẋ = x(a0 + a1x+ a2z
2 + a3z),

ż = z(c0 + c1x+ c2z
2 + c3z),

(1.2)

Kolmogorov systems (1.2) depend on eight parameters, this is a big number to
classify all their distinct topological phase portraits. Then we require that Kol-
mogorov systems (1.2) have a Darboux invariant of the form x`ymest, then these
systems are reduced to study the Kolmogorov systems

ẋ = x
(
a0 − µ(c1x+ c2z

2 + c3z)
)
,

ż = z
(
c0 + c1x+ c2z

2 + c3z
)
,

(1.3)

which now depend on six parameters. For these Kolmogorov systems we give the
topological classification of all their phase portraits in the Poincaré disc. Roughly
speaking the Poincaré disc is the closed unit disc centered at the origin of R2. Its
interior is identified with R2 and the circle of its boundary is identified with the
infinity of R2. In the plane R2 we can go or come from the infinity in as many
directions as points have the circle. The polynomial differential systems can be
extended to the closed Poincaré disc, i.e. they can be extended to infinity and in
this way we can study their dynamics in a neighborhood of infinity. This extension
is called the Poincaré compactification, for more details see subsection 2.1. Thus
our main result is the following.

Theorem 1.1. Kolmogorov systems (1.3) have 102 topologically distinct phase por-
traits in the Poincaré disc under condition (H1) given in Figure 16.

(H1) c2 6= 0, a0 ≥ 0, c1 ≥ 0, c3 ≥ 0, a0 + c0µ 6= 0, a0c1µ 6= 0, µ 6= −1

We will see that we can assume condition (H1) because Kolmogorov systems (1.3)
can be reduced to satisfy such condition either using symmetries, or eliminating
known phase portraits, or eliminating phase portraits with infinitely many finite
or infinite singular points. Some topological phase portraits have been classified in
the Poincaré disc are; see for instance [4, 13, 17].

In Section 3 using the Darboux theory of integrability we explain the reduction
from the Lotka-Volterra system (1.1) to the Kolmogorov systems (1.3). In Section
4 we give some properties of the system obtained. In Section 5 we study the local
phase portrait of the finite singular points, and in Section 6 we do the same with



EJDE-2021/35 A CLASS OF KOLMOGOROV SYSTEMS 3

the infinite singular points, applying the blow-up technique. Finally in Section 7
we prove Theorem 1.1.

2. Preliminaries

2.1. Poincaré compactification. To study the behavior of the trajectories of our
polynomial differential systems near infinity we will use the Poincaré compactifica-
tion. We provide a short summary about this method, more details can be found
in [9, Chapter 5].

Let X = (P (x, y), Q(x, y)) be a polynomial vector field of degree d defined in
R2. Consider the Poincaré sphere S2 = {y ∈ R3 : y2

1 +y2
2 +y2

3 = 1}, and its tangent
plane at the point (0, 0, 1) which is identified with R2.

We consider the central projections f+ : R2 → S2 and f− : R2 → S2. By defini-
tion, f+(x) is the intersection of the straight line passing through the point x and
the origin with the northern hemisphere of S2, and respectively for f−(x) with the
southern hemisphere. The differential Df+ and respectively Df− send the vector
field X into a vector field X on S2\S1. Note that the points at infinity of R2 are in
bijective correspondence with the points of the equator S1 of S2.

The vector fieldX can be extended analytically to a vector field on S2 multiplying
X by yd3 . We denothe this vector field by ρ(X), and it is called the Poincaré
compactification of the vector field X on R2.

For studying the dynamics of X in the neighborhood of the infinity, we must
study the dynamics of ρ(X) near S1. The sphere S2 is a 2-dimensional manifold so
we need to know the expressions of the vector field ρ(X) in the local charts (Ui, φi)
and (Vi, ψi), where Ui = {y ∈ S2 : yi > 0}, Vi = {y ∈ S2 : yi < 0}, φi : Ui −→ R2

and ψi : Vi −→ R2 for i = 1, 2, 3 with φi(y) = −ψi(y) = (ym/yi, yn/yi) for m < n
and m,n 6= i.

In the local chart (U1, φ1) the expression of ρ(X) is

u̇ = vd
[
− uP

(1

v
,
u

v

)
+Q

(1

v
,
u

v

)]
, v̇ = −vd+1P

(1

v
,
u

v

)
. (2.1)

In the local chart (U2, φ2) the expression of ρ(X) is

u̇ = vd
[
P
(1

v
,
u

v

)
− uQ

(1

v
,
u

v

)]
, v̇ = −vd+1P

(1

v
,
u

v

)
, (2.2)

and in the local chart (U3, φ3) the expression of ρ(X) is

u̇ = P (u, v), v̇ = Q(u, v). (2.3)

In the charts (Vi, ψi), with i = 1, 2, 3, the expression for ρ(X) is the same as in the
charts (Ui, φi) multiplied by (−1)d−1.

The equator S1 is invariant by the vector field ρ(X) and all the singular points
of ρ(X) which lie in this equator are called the infinite singular points of X. If
y ∈ S1 is an infinite singular point, then −y is also an infinite singular point and
they have the same (respectively opposite) stability if the degree of vector field is
odd (respectively even).

The image of the closed northern hemisphere of S2 onto the plane y3 = 0 under
the orthogonal projection π is called the Poincaré disc D2. Since the orbits of ρ(X)
on S2 are symmetric with respect to the origin of R3, we only need to consider the
flow of ρ(X) in the closed northern hemisphere, and we can project the phase por-
trait of ρ(X) on the northern hemisphere onto the Poincaré disc. We shall present
the phase portraits of the polynomial differential systems (1.3) in the Poincaré disc.



4 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

2.2. Topological equivalence between two polynomial vector fields. Two
polynomial vector fields X1 and X2 on R2 are topologically equivalent if there exists
a homeomorphism on the Poincaré disc which preserves the infinity S1 and sends
the trajectories of the flow of π(ρ(X1)) to the trajectories of the flow of π(ρ(X2)),
preserving or reversing the orientation of all the orbits.

A separatrix of the Poincaré compactification π(ρ(X)) is an orbit at the infinity
S1, or a finite singular point, or a limit cycle, or an orbit on the boundary of a
hyperbolic sector at a finite or an infinite singular point. The set of all separatrices
of π(ρ(X)) is closed and we denote it by ΣX .

An open connected component of D2\ΣX is a canonical region of π(ρ(X)). The
separatrix configuration of π(ρ(X)) is the union of an orbit of each canonical region

with the set ΣX , and it is denoted by Σ
′

X . We denote by S (respectively R) the
number of separatrices (respectively canonical regions) of a vector field π(ρ(X)).

We say that two separatrix configurations Σ
′

X1
and Σ

′

X2
are topologically equi-

valent if there is a homeomorphism h : D2 → D2 such that h(Σ
′

X1
) = Σ

′

X2
.

The following theorem of Markus [21], Neumann [22] and Peixoto [23] allows us
to investigate only the separatrix configuration of a polynomial differential system
in order to determine its phase portrait in the Poincaré disc.

Theorem 2.1. The phase portraits in the Poincaré disc of two compactified polyno-
mial vector fields π(ρ(X1)) and π(ρ(X2)) with finitely many separatrices are topo-

logically equivalent if and only if their separatrix configurations Σ
′

X1
and Σ

′

X2
are

topologically equivalent.

2.3. Blow-up technique. There exist classification theorems for hyperbolic and
semi-hyperbolic singular points, and also for nilpotent singular points which can be
found in [9, Chapter 2]. The centers are more difficult to study, see for instance
[9, Chapter 4]. Whereas to study a singular point for which the Jacobian matrix
is identically zero, the only possibility is studying each singular point case by case.
The main technique to perform the desingularization of a linearly zero singular
point is the blow-up technique. We give a short summary about this method, more
details can be found in [2].

Roughly speaking the idea behind the blow up technique is to explode, through
a change of variables that is not a diffeomorphism, the singularity to a line. Then,
for studying the original singular point, one studies the new singular points that
appear on this line, and this is simpler. If some of these new singular points are
linearly zero, the process is repeated. Dumortier proved that this iterative process
of desingularization is finite, see [8].

Consider a real planar polynomial differential system

ẋ = P (x, y) = Pm(x, y) + . . . ,

ẏ = Q(x, y) = Qm(x, y) + . . . ,
(2.4)

where P and Q are coprime polynomials, Pm and Qm are homogeneous polynomials
of degree m ∈ N and the dots mean higher order terms in x and y. Note that we
are assuming that the origin is a singular point because m > 0. We define the
characteristic polynomial of (2.4) as

F(x, y) := xQm(x, y)− yPm(x, y), (2.5)

and we say that the origin is a nondicritical singular point if F 6≡ 0 and a dicritical
singular point if F ≡ 0. In this last case Pm = xWm−1 and Qm = yWm−1, where



EJDE-2021/35 A CLASS OF KOLMOGOROV SYSTEMS 5

Wm−1 6≡ 0 is a homogeneous polynomial of degree m − 1. If y − vx is a factor of
Wm−1 and v = tan θ∗, θ∗ ∈ [0, 2π), then θ∗ is a singular direction.

The homogeneous directional blow up in the vertical direction is the mapping
(x, y) → (x, z) = (x, y/x), where z is a new variable. This map transforms the
origin of (2.4) into the line x = 0, which is called the exceptional divisor. The
expression of system (2.4) after the blow up in the vertical direction is

ẋ = P (x, xz), ż =
Q(x, xz)− zP (x, xz)

x
, (2.6)

that is always well-defined since we are assuming that the origin is a singularity.
After the blow up, we cancel an appearing common factor xm−1 (xm if F ≡ 0).
Moreover, the mapping swaps the second and the third quadrants in the vertical
directional blow up. [2, Propositions 2.1 and 2.2] provide the relationship between
the original singular point of system (2.4) and the new singularities of system (2.6).
For additional details see [3].

Finally, to study the behavior of the solutions around the origin of system (2.4), it
is necessary to study the singular points of system (2.6) on the exceptional divisor.
They correspond to either characteristic directions in the nondicritical case, or
singular directions in the dicritical case. It may happen that some of these singular
points are linearly zero, in which case we have to repeat the process. As we said
before, it is proved in [8] that this chain of blow ups is finite.

2.4. Indices of planar singular points. Given an isolated singularity q of a
vector field X, defined on an open subset of R2 or S2, we define the index of q by
means of the Poincaré Index Formula. We assume that q has the finite sectorial
decomposition property. Let e, h and p denote the number of elliptic, hyperbolic
and parabolic sectors of q, respectively, and suppose that e+ h+ p > 0. Then the
index of q is iq = 1 + (e− h)/2, and it is always an integer.

We recall that the Poincaré compactification of a vector field in R2 introduced
in Subsection 2.1 is a tangent vector field on the sphere S2, so the next result will
be very useful in our study.

Theorem 2.2 (Poincaré-Hopf Theorem). For every tangent vector field on S2 with
a finite number of singular points, the sum of their indices is 2.

2.5. Invariants and Application of the Darboux Theory. The Darboux Theo-
ry of Integrability provides a link between the integrability of polynomial vector
fields and the number of invariant algebraic curves that they have. The basic re-
sults on dimension two can be found in Chapter 8 of [9], and these results have
been extended to Rn and Cn in [16, 18, 19].

We consider a real polynomial differential system in dimension three, that is a
system of the form

dx/dt = ẋ = P (x, y, z),

dy/dt = ẏ = Q(x, y, z),

dz/dt = ż = R(x, y, z),

(2.7)

where P,Q and R are polynomials in the variables x, y and z. We denote by
m = max{degP,degQ,degR} the degree of the polynomial system, and we always
assume that the polynomials P,Q and R are relatively prime in the ring of the real
polynomials in the variables x, y and z.



6 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

Theorem 2.3 (Darboux Integrability Theorem). Suppose that a polynomial system
(2.7) of degree m admits p irreducible invariant algebraic surfaces fi = 0 with
cofactors Ki for i = 1, . . . , p. Then the next statements hold.

(a) There exist λi ∈ C not all zero such that
∑p
i=1 λiKi = 0 if and only if the

function fλ1
1 . . . f

λp
p is a first integral of system (2.7).

(b) There exist λi ∈ C not all zero such that
∑p
i=1 λiKi = −s for some s ∈

R\{0} if and only if the function fλ1
1 . . . f

λp
p exp(st) is a Darboux invariant

of system (2.7).

3. Reduction of the Lotka-Volterra systems in R3 to the
Kolmogorov systems in R2

As we said our objective is to study the global dynamics of the Lotka-Volterra
systems (1.1) in dimension three, which have a rational first integral of degree
two of the form xiyjzk. The Darboux theory of integrability allow us to obtain a
characterization of these systems.

We consider the irreducible invariant algebraic surfaces f1(x, y, z) = x = 0,
f2(x, y, z) = y = 0 and f3(x, y, z) = z = 0 of system (1.1), with cofactors K1, K2

and K3, respectively. As Ki is the cofactor of fi we have that

Xfi = P
∂fi
∂x

+Q
∂fi
∂y

+R
∂fi
∂z

= Kifi.

Then for the invariant algebraic surfaces considered we obtain the cofactors K1 =
a0 + a1x + a2y + a3z, K2 = b0 + b1x + b2y + b3z, and K3 = c0 + c1x + c2y + c3z,
respectively.

Applying Theorem 2.3, since we assume that xλ1yλ2zλ3 is a first integral of
system (1.1), we obtain that there exist λi ∈ C, with i ∈ {1, 2, 3}, not all zero,

such that
∑3
i=1 λiKi = 0. Apart from the trivial solution {λ1 = 0, λ2 = 0, λ3 = 0},

there are the following three solutions of this equation:

S1 = {c0 = 0, c1 = 0, c2 = 0, c3 = 0, λ2 = 0, λ1 = 0},

S2 = {b0 = −c0λ3

λ2
, b1 = −c1λ3

λ2
, b2 = −c2λ3

λ2
, b3 = −c3λ3

λ2
, λ1 = 0},

S3 = {a0 =
−b0λ2 − c0λ3

λ1
, a1 =

−b1λ2 − c1λ3

λ1
, a2 =

−b2λ2 − c2λ3

λ1
,

a3 =
−b3λ2 − c3λ3

λ1
},

which give rise to three families of Lotka-Volterra polynomial differential systems
of degree two in R3, with a first integral of the form xλ1yλ2zλ3 .

If we consider the family given by solution S1, as the parameters ci, i = 0, . . . , 3,
are zero, we have that ż = 0 and the Lotka-Volterra system is reduced to:

ẋ = x(a0 + a1x+ a2y + a3z),

ẏ = y(b0 + b1x+ b2y + b3z),

ż = 0.

As ż = 0, z is constant and this system has H = z as a first integral. Note that if
we consider the first integral H = xλ1yλ2zλ3 , and apply the conditions given by S1,
it is λ1 = λ2 = 0, we obtain H = zλ3 , with λ3 = 2 for getting the degree two, but in
this case we will consider the simplest first integral. In each invariant plane with z



EJDE-2021/35 A CLASS OF KOLMOGOROV SYSTEMS 7

constant, we have a Lotka-Volterra polinomial differential system in R2. The phase
portrait of these systems has been studied in [24], so we are not going to deal with
this case.

In this article we study the family given by the solution S2. This solution provides
the values of parameters bi as a function of the parameters λ2, λ3 and ci, with
i = 0, . . . , 3, so we can replace them in the expression of ẏ obtaining

ẏ = y
(
− c0λ3

λ2
− c1λ3

λ2
x− c2λ3

λ2
y − c3λ3

λ2
z
)
.

If we denote λ = −λ3/λ2, then the original Lotka-Volterra system becomes

ẋ = x(a0 + a1x+ a2y + a3z),

ẏ = λy(c0 + c1x+ c2y + c3z),

ż = z(c0 + c1x+ c2y + c3z).

Given that λ1 = 0, the first integral H = xλ1yλ2zλ3 is reduced to H = yλ2zλ3 , but
if this is a first integral, also H = (yλ2zλ3)−1/λ2 = y−1z−λ3/λ2 = y−1zλ = zλ/y is
a first integral. If we want H to be rational of degree two, we must take λ = 2. In
each level H = 1/h, with h 6= 0, we will have 1/h = z2/y, so y = hz2 and then, for
each h, the initial Lotka-Volterra system on dimension three reduces to the system
on dimension two

ẋ = x(a0 + a1x+ a2hz
2 + a3z),

ż = z(c0 + c1x+ c2hz
2 + c3z).

We must study the phase portrait of the systems of this family, but it is equivalent
to study the phase portraits of the family of Kolmogorov systems in dimension two

ẋ = x(a0 + a1x+ a2z
2 + a3z),

ż = z(c0 + c1x+ c2z
2 + c3z).

(3.1)

In the particular cases in which H is zero or infinity, the differential system on
dimension three is reduced to a Lotka-Volterra system on dimension two, having in
each case z = 0 and y = 0, respectively. We recall that these systems had already
been studied in [24].

Systems (3.1) depend on eight parameters and the classification of all their dis-
tinct topological phase protraits is huge. For this reason we study the subclass
of them having a Darboux invariant of the form xλ1zλ2est. By statement (b) of
Theorem 2.3 the expression λ1Kx + λ2Kz + s must be zero, where Kx and Ky are
the cofactors of the invariant planes x = 0 and z = 0, respectively. Note that s and
λ2

1 + λ2
2 cannot be zero. We obtain the cofactors Kx = a0 + a1x+ a2z

2 + a3z and
Kz = c0 + c1x + c2z

2 + c3z and then, solving the equation λ1Kx + λ2Kz + s = 0,
we obtain the following two non-trivial solutions

S̃1 = {s = −a0λ1 − c0λ2, a1 = −c1λ2

λ1
, a2 = −c2λ2

λ1
, a3 = −c3λ2

λ1
},

S̃2 = {s = −c0λ2, c1 = 0, c2 = 0, c3 = 0, λ1 = 0}.

So we have two subsystems from the initial system (3.1). According to the condi-

tions given by solution S̃1 the first subsystem is

ẋ = x
(
a0 −

c1λ2

λ1
x− c2λ2

λ1
z2 − c3λ2

λ1
z
)
,



8 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

ż = z
(
c0 + c1x+ c2z

2 + c3z
)
.

If we denote λ2/λ1 = µ and λ1 = λ, then this subsystem becomes

ẋ = x
(
a0 − µ(c1x+ c2z

2 + c3z)
)
,

ż = z
(
c0 + c1x+ c2z

2 + c3z
)
,

(3.2)

and its Darboux invariant is xλzλµe−tλ(a0+c0µ). But if this is a Daboux invariant,
also it is xzµe−t(a0+c0µ). Note that in order that we have a Darboux invariant
a0 + c0µ cannot be zero.

If we consider now the solution S̃2 we obtain the subsystem

ẋ = x
(
a0 + a1x+ a2z

2 + a3z
)
,

ż = c0z,

which is equivalent to the previous one, taking µ = 0 and interchanging the variables
x and z, so it is sufficient to study the Kolmogorov systems (3.2) depending on six
parameters.

4. Properties of system (1.3)

In this section we state some results that will be used on the classification to
reduce the number of phase portraits appearing. Note that if c2 = 0, then the
system (1.3) is a Lotka-Volterra system in dimension 2. A global topological clas-
sification of these systems has been completed in [24], so we limit our study to the
case c2 6= 0.

We recall that for obtaining system (1.3) we have supposed that system (3.1)
has the Darboux invariant I = xzµe−t(a0+c0µ), so it is required that a0 + c0µ 6= 0.

Proposition 4.1. Consider system (1.3) and suppose that (x̃(t), z̃(t)) is a solution
of this system. If we change c1 by −c1 (respectively c3 by −c3 ), then (−x̃(t), z̃(t))
(respectively (x̃(t),−z̃(t))) is other solution of the obtained system.

Remark 4.2. By Proposition 4.1 we can limit our study to Kolmogorov systems
(1.3) with c1 and c3 non-negatives. In the cases with these parameters negatives,
we will obtain phase portraits symmetric to the ones obtained in the positive cases,
with respect to the z-axis when we change the sign of c1, and with respect to the
x-axis when we change the sign of c3.

Corollary 4.3. Consider system (1.3) and suppose (x̃(t), z̃(t)) is a solution. If
c1 = 0 (respectively c3 = 0), then (−x̃(t), z̃(t)) (respectively (x̃(t),−z̃(t))) is also a
solution.

Remark 4.4. Corollary 4.3 simplifies the study of the cases with c1 = 0 or c3 = 0,
because it proves that the phase portraits have to be symmetric with respect to the
z-axis and x-axis respectively, and this fact will be useful in obtaining the global
phase portraits from the local results.

Proposition 4.5. Let (x̃(t), z̃(t)) be a solution of system (1.3). In the next cases
we obtain another system with solution (−x̃(−t),−z̃(−t)).

(1) If a0, c0 and c2 are not zero, and we change the sign of all of them.
(2) If a0 = 0 and we change the sign of c0 and c2, which are not zero.
(3) If c0 = 0 and we change the sign of a0 and c2, which are not zero.



EJDE-2021/35 A CLASS OF KOLMOGOROV SYSTEMS 9

Remark 4.6. To classify all the phase portraits of the Kolmogorov systems (1.3),
according to the previous results, it is sufficient to consider a0 ≥ 0. And when
a0 = 0 we will consider also c0 > 0.

Remark 4.7. In short, according to the previous results and considerations, from
now on it will be sufficient to study the Kolmogorov systems (1.3) with the their
parameters satisfying

(H2) c2 6= 0, a0 ≥ 0, c1 ≥ 0, c3 ≥ 0, a0 + c0µ 6= 0 .

Theorem 4.8. For system (1.3) the next statements hold.

(1) If c1 6= 0, then on any straight line z = cte 6= 0, there exists only one
contact point.

(2) If c1 = 0, then there exist two invariant straight lines z = (
√
c23 − 4c0c2 −

c3)/(2c2) and z = −(
√
c23 − 4c0c2 + c3)/(2c2) if c23 > 4c0c2, and one invari-

ant straight line z = −c3/(2c2) if c23 = 4c0c2. There are not contact points
on any other straight line z = cte 6= 0.

Proof. First we suppose c1 6= 0 and consider a straight line z = z0 6= 0. Then the
contact points on this straight line are those on which ż = 0 and, as z0 6= 0, the
only possible contact point is the one that satisfies c0 + c1x+ c2z

2
0 + c3z0 = 0, i.e.

the point such that its first coordinate is x = −(c2z
2
0 + c3z0 + c0)/c1.

We consider now the case with c1 = 0. Then looking for the points on the
straight line z = z0 6= 0 satisfying ż = 0, we obtain that they must verify the
condition c0 +c2z

2
0 +c3z0 = 0, and solving this equation we obtain that either there

are no contact points, or a full straight line of contact points , or two straight line
of contact points, depending on the solutions z0 of that equation. �

5. Study of local finite singular points

System (1.3) has the following finite singularities:

• P0 = (0, 0),
• P1 =

(
0, Rc−c3

2c2

)
and P2 =

(
0,−Rc+c3

2c2

)
if c23 > 4c0c2,

• P3 =
(
0,− c3

2c2

)
if c23 = 4c0c2,

• P4 =
(
a0
c1µ

, 0
)

if c1µ 6= 0.

We use the notation Rc =
√
c23 − 4c0c2 in order to simplify the expressions which

will appear. Moreover if a0 = 0 and c1µ = 0, all the points on the z-axis are singular
points, and the system can be reduced to a Lotka-Volterra system in dimension 2.
Therefore from now on we will consider the hypothesis

(H3) c2 6= 0, a0 ≥ 0, c1 ≥ 0, c3 ≥ 0, a0 + c0µ 6= 0, a2
0 + (c1µ)2 6= 0.

Under this assumption there are 6 different cases according to the finite singular
points existing for system (1.3), which are given in Table 1. Then we study the
possible local phase portraits in each one of the finite singular points under the
hypothesis (H3).

The origin is always an isolated singular point for system (1.3), and we have the
next classification for its phase portraits: if a0c0 6= 0 the singularity is hyperbolic
and two cases are possible, the origin is a saddle point if c0 < 0, and it is an unstable
node if c0 > 0. If a0 6= 0 and c0 = 0 the singularity is semi-hyperbolic and it has
two possibilities: if c3 6= 0 then the origin is a saddle-node, if c3 = 0 and c2 < 0 it



10 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

Table 1. The different cases for the finite singular points.

Case Conditions Finite singular points

1 c23 > 4c0c2, c1µ 6= 0. P0, P1, P2, P4.

2 c23 > 4c0c2, c1µ = 0, a0 6= 0. P0, P1, P2.

3 c23 = 4c0c2, c1µ 6= 0. P0, P3, P4.

4 c23 = 4c0c2, c1µ = 0, a0 6= 0. P0, P3.

5 c23 < 4c0c2, c1µ 6= 0. P0, P4.

6 c23 < 4c0c2, c1µ = 0, a0 6= 0. P0.

is a topological saddle, and if c3 = 0 and c2 > 0 it is a topological unstable node.
Finally if a0 = 0 the origin is a semi-hyperbolic saddle-node.

When P1 is a singular point of system (1.3), it can present different phase por-
traits. If c0 6= 0 then P1 is hyperbolic and it can present the following phase
portraits: if c2(a0 + c0µ)(Rc − c3) < 0 then P1 is a saddle, if a0 + c0µ < 0 and
c2(Rc − c3) < 0 it is a stable node, and finally if a0 + c0µ > 0 and c2(Rc − c3) > 0
it is an unstable node. The singular point P1 collides with the origin if c1 = 0.

When P2 is a singular point of system (1.3), it can present three different phase
portraits: if c2(a0 + c0µ) < 0 then P2 is a saddle, if a0 + c0µ < 0 and c2 < 0 then
it is a stable node, and if a0 + c0µ > 0 and c2 > 0 it is an unstable node.

When P3 is a singularity of system (1.3) it is a semi-hyperbolic saddle-node if
c3 6= 0, and it collides with the origin if c3 = 0.

When P4 is a singularity of system (1.3) it is hyperbolic if a0 6= 0 and can
present two different phase portraits: if (a0 + c0µ)µ > 0 then P4 is a saddle, and if
µ(a0 + c0µ) < 0 it is an stable node. If a0 = 0 the singularity P4 collides with the
origin.

Lemma 5.1. Under hypothesis (H3) there are 50 different cases according to the
local phase portrait of the finite singular points of system (1.3), which are given in
Tables 2–7.

Proof. We have to analyze cases 1 to 6 in Table 1 and determine the local phase
portraits of the singular points existing in each one of them, according to their
individual classification. We start with the first one, in which the conditions, c23 >
4c0c2 and c1µ 6= 0 hold. The singular points are P0, P1, P2 and P4. We shall
consider three subcases: a0 = 0, c0 = 0 and a0c0 6= 0.

Consider case c0 = 0 in which the origin is a saddle-node and P1 collides with
the origin. Since c0 = 0 and a0 > 0, the singular point P2 is a saddle if c2 < 0, and
an unstable node if c2 > 0. In these two cases P4 can be either a saddle if µ > 0,
or a stable node if µ < 0. This leads to cases 1.1 to 1.4 in Table 2.

We continue with the case a0 = 0 in which P0 is again a saddle-node, but in
this case it coincides with P4. Suppose that P1 is an unstable node, then we have
c0µ > 0 and c2(Rc− c3) > 0. By Remark 4.6 we will only consider the case c0 > 0.
Then if c2 > 0 and Rc − c3 > 0 taking into account the expression of Rc and
squaring both terms, we obtain that c23 − 4c0c2 > c23, so c0c2 < 0, which leads to
a contradiction. The same occurs if we suppose c2 < 0. Therefore P1 cannot be
an unstable node. If P1 is a saddle, then P2 can be a saddle or an unstable node,
but not a stable node, which is only possible if c0 < 0, by an analogous reasoning



EJDE-2021/35 A CLASS OF KOLMOGOROV SYSTEMS 11

to the previous one. If P1 is a stable node then c0µ < 0, so P2 can be a saddle or
stable node, but not an unstable node because it requires that c0µ > 0. This leads
to cases 1.5 to 1.8.

The last case is a0c0 6= 0 in which the origin is a hyperbolic singular point. We
start when P0 is a saddle, then c0 < 0. First we consider that P1 is also a saddle,
and so c2(a0 + c0µ)(Rc − c3) < 0. If P2 is a saddle then c2(a0 + c0µ) < 0, and we
obtain Rc − c3 > 0. From this we deduce like in previous cases that c0c2 < 0, but
we are supposing c0 < 0 and so c2 > 0 and a0 + c0µ < 0. From the last inequality
a0 < −c0µ, and so µ has to be positive. In short µ(a0 + c0µ) < 0, and consequently
P4 can only be a stable node. If P0 and P1 are saddles, but P2 is a stable node,
reasoning in an analogous way we obtain that P4 is again a stable node. This leads
to cases 1.9 and 1.10. Note that if P0 and P1 are saddles it is impossible for P2 to
be an unstable node. In that case we would have that c0 < 0, c2 > 0, a0 + c0µ > 0
and Rc − c3 < 0. From this last inequality we obtain that c0c2 > 0, which is a
contradiction. We consider now the cases where P0 is a saddle and P1 an unstable
node, in which the conditions c0 < 0, a0 + c0µ > 0 and c2(Rc − c3) > 0 hold. It is
obvious that P2 cannot be a stable node because it requires that a0 + c0µ < 0, so
P2 is a saddle if c2 < 0 and an unstable node if c2 > 0. In both cases P4 can be
either a saddle if µ > 0, or a stable node if µ < 0. This leads to cases 1.11 to 1.14.
Note that the case with P0 a saddle and P1 a stable node is not possible, because
we would have c0 < 0, a0 +c0µ < 0 and c2(Rc−c3) < 0. If c2 > 0 then Rc−c3 < 0,
whence we deduce c0c2 > 0 and get a contradiction. The same argument is valid if
c2 < 0. With analogous reasoning as in the case where P0 is an unstable node we
obtain the subcases 1.15 to 1.20.

Now we study case 2 of Table 1, in which c23 > 4c0c2, c1µ = 0, and a0 6= 0. We
shall consider three cases: c0 < 0, c0 > 0, and c0 = 0.

We start with case c0 < 0 in which P0 is a saddle. If P1 is a saddle, then P2 can
be a saddle or a stable node. If P2 is an unstable node, then we have the conditions
c2(a0 + c0µ)(Rc− c3) < 0, c2 > 0, and a0 + c0µ > 0, so Rc− c3 < 0, and we deduce
c0c2 > 0 which is a contradiction. P1 cannot be a stable node, because in that case
we would have the conditions c0 < 0, a0 + c0µ < 0, and c2(Rc − c3) < 0 which lead
to a contradiction in the following way: if c2 > 0 then Rc − c3 < 0 and squaring
we deduce c0c2 > 0 which is not possible because c0 < 0 and we are supposing
c2 > 0. An analogous reasoning works in the case c2 < 0. If P1 is an unstable node
then P2 can be either a saddle or an unstable node, but not a stable node because
it requires a0 + c0µ to be negative, but we already know that this expression is
positive because it is a condition in order that P1 be an unstable node. This leads
to cases 2.1 to 2.4 of Table 3.

We continue with the case c0 > 0, in which P0 is an unstable node. If P1 is a
saddle, then P2 can be a saddle or an unstable node. If P2 is a stable node then we
have the conditions c0 > 0 , c2(a0 + c0µ)(Rc − c3) < 0, a0 + c0µ < 0 and c2 < 0.
Thus we have Rc−c3 < 0 and squaring we obtain c0c2 > 0 which is a contradiction.
This leads to cases 2.5 and 2.6. If P1 is a stable node it can be proved similarly
to previous cases, that P2 cannot be an unstable node. This leads to cases 2.7
and 2.8. P1 cannot be an unstable node, because in that case we would have the
conditions c0 > 0, a0 + c0µ > 0 and c2(Rc − c3) > 0 which lead to a contradiction
in the following way: if c2 > 0 then Rc − c3 > 0, and squaring we deduce c0c2 < 0



12 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

which is not possible because c0 > 0 and we are supposing c2 > 0. An analogous
reasoning works in the case c2 < 0.

Al last we have the case c0 = 0. Necessarily c3 6= 0 so the origin is a saddle-node.
Also we have that P1 coincides with the origin. For the singular point P2 we have
that it is a saddle if c2 < 0, and an unstable node if c2 > 0. This leads to cases 2.9
and 2.10.

We study case 3 of Table 1 in which c23 = 4c0c2 and c1µ 6= 0. Then c0 = 0 if
and only if c3 = 0. We consider a0 > 0 and c0 < 0, then the origin is a saddle and
P3 a saddle-node (as c3 6= 0). The singular point P4 is either a saddle or a stable
node, depending on the sign of µ(a0 + c0µ). The same is valid in the case a0 > 0
and c0 < 0, except for the origin which is now an unstable node. We get the cases
3.1 to 3.4 of Table 4. We continue with the case in which a0 = 0 and so P0 is a
saddle-node, P4 coincides with P0 and P3 is a saddle-node. This correspond with
case 3.5. At last we have the condition c0 = 0, under which P3 coincides with P0.
If c2 < 0 then it is a topological saddle, and if c2 > 0 it is a topological unstable
node. In any case P4 can be either a saddle or a stable node. This leads to cases
3.6 to 3.9.

Now we address the case 4 of Table 1 in which c23 = 4c0c2, c1µ = 0, and a0 6= 0.
The origin is a saddle if c0 < 0, and an unstable node if c0 > 0. If c0 = 0 then
c3 = 0, so we distinguish two semi-hyperbolic possibilities for the origin: if c2 < 0
it is a topological saddle, and if c2 > 0 it is a topological unstable node. The
classification of P3 is totally determined by the one of P0, because it only depends
on whether c3 is zero or not. We get cases 4.1 to 4.4.

In case 5 of Table 1 the conditions c23 < 4c0c2 and c1µ 6= 0 hold. The singular
points are P0 and P4. From condition c23 < 4c0c2 we obtain that c0 6= 0. If a0 = 0
then the origin is a saddle-node and P4 coincides with the origin. If a0 6= 0 then
both singular points are hyperbolic, and it leads to cases 5.2 to 5.5.

Finally in case 6 of Table 1 we have the conditions c23 < 4c0c2, c1µ = 0, and
a0 6= 0. The unique singular point is the origin and as c0 cannot be zero, it is either
a saddle or an unstable node. �

6. Study of local infinite singular points

To study the behavior of the trajectories of system (1.3) near infinity we consider
its Poincaré compactification. For the moment we assume the same hypothesis (H3)
than in previous sections. According to equations (2.1) and (2.2), we obtain the
compactification in the local charts U1 and U2 respectively. From Section 2 it is
enough to study the singular points over v = 0 in the chart U1 and the origin of
the chart U2.

In chart U1 system (1.3) writes

u̇ = c2(µ+ 1)u3 + c3(µ+ 1)u2v + (c0 − a0)uv2 + c1(µ+ 1)uv,

v̇ = c2µu
2v + c3µuv

2 − a0v
3 + c1µv

2.
(6.1)

Taking v = 0 we obtain u̇
∣∣
v=0

= c2(µ+1)u3 and v̇
∣∣
v=0

= 0. Therefore if µ = −1 all
points at infinity are singular points, and we will not deal with this situation in this
paper. In other case, if µ 6= −1 the only singular point is the origin of U1, which
we denote by O1. The linear part of system (6.1) at the origin is identically zero,
so we must use the blow-up technique to study it, leading to the following result



EJDE-2021/35 A CLASS OF KOLMOGOROV SYSTEMS 13

Table 2. Classification in case 1 of Table 1 according to the local
phase portraits of finite singular points.

Case 1: c23 > 4c0c2, c1µ 6= 0.

Sub. Conditions Classification

1.1 a0 > 0, c0 = 0, µ > 0, c2 < 0. P0 ≡ P1 saddle-node, P2 saddle,
P4 saddle.

1.2 a0 > 0, c0 = 0, µ > 0, c2 > 0. P0 ≡ P1 saddle-node,
P2 unstable node, P4 saddle.

1.3 a0 > 0, c0 = 0, µ < 0, c2 < 0. P0 ≡ P1 saddle-node, P2 saddle,
P4 stable node.

1.4 a0 > 0, c0 = 0, µ < 0, c2 > 0. P0 ≡ P1 saddle-node,
P2 unstable node, P4 stable node.

1.5 a0 = 0, c0 > 0, c2µ < 0, Rc − c3 > 0. P0 ≡ P4 saddle-node, P1 saddle,
P2 saddle.

1.6 a0 = 0, c0 > 0, Rc − c3 < 0, µ > 0,
c2 > 0.

P0 ≡ P4 saddle-node, P1 saddle,
P2 unstable node.

1.7 a0 = 0, c0 > 0, µ < 0, Rc − c3 < 0,
c2 > 0.

P0 ≡ P4 saddle-node,
P1 stable node, P2 saddle.

1.8 a0 = 0, c0 > 0, µ < 0, c2 < 0,
Rc − c3 > 0.

P0 ≡ P4 saddle-node,
P1 stable node, P2 stable node.

1.9 a0 > 0, c0 < 0, µ > 0, (a0 + c0µ) < 0,
c2 > 0, Rc − c3 > 0.

P0 saddle, P1 saddle, P2 saddle,
P4 stable node.

1.10 a0 > 0, c0 < 0, c2 < 0, µ > 0,
a0 + c0µ < 0, Rc − c3 < 0.

P0 saddle, P1 saddle,
P2 stable node, P4 stable node.

1.11 a0 > 0, c0 < 0, c2 < 0, µ > 0,
a0 + c0µ > 0, (Rc − c3) < 0.

P0 saddle, P1 unstable node,
P2 saddle, P4 saddle.

1.12 a0 > 0, c0 < 0, c2 < 0, µ < 0,
a0 + c0µ > 0, (Rc − c3) < 0.

P0 saddle, P1 unstable node,
P2 saddle, P4 stable node.

1.13 a0 > 0, c0 < 0, µ > 0, a0 + c0µ > 0,
c2 > 0, Rc − c3 > 0.

P0 saddle, P1 unstable node,
P2 unstable node, P4 saddle.

1.14 a0 > 0, c0 < 0, µ < 0, a0 + c0µ > 0,
c2 > 0, Rc − c3 > 0.

P0 saddle, P1 unstable node,
P2 unstable node, P4 stable node.

1.15 a0 > 0, c0 > 0, µ(a0 + c0µ) > 0,
c2(a0 + c0µ) < 0, Rc − c3 > 0.

P0 unstable node, P1 saddle,
P2 saddle, P4 saddle.

1.16 a0 > 0, c0 > 0, µ(a0 + c0µ) < 0,
c2(a0 + c0µ) < 0, Rc − c3 > 0.

P0 unstable node, P1 saddle,
P2 saddle, P4 stable node.

1.17 a0 > 0, c0 > 0, µ > 0, Rc − c3 < 0,
a0 + c0µ > 0, c2 > 0.

P0 unstable node, P1 saddle,
P2 unstable node, P4 saddle.

1.18 a0 > 0, c0 > 0, µ < 0, Rc − c3 < 0,
a0 + c0µ > 0, c2 > 0.

P0 unstable node, P1 saddle,
P2 unstable node, P4 stable node.

1.19 a0 > 0, c0 > 0, µ < 0, a0 + c0µ < 0,
Rc − c3 < 0, c2 > 0.

P0 unstable node, P1 stable node,
P2 saddle, P4 saddle.

1.20 a0 > 0, c0 > 0, µ < 0, a0 + c0µ < 0,
c2 < 0, Rc − c3 > 0.

P0 unstable node, P1 stable node,
P2 stable node, P4 saddle.

which is proved in Subsections 6.1 and 6.2. From now on we include the condition



14 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

Table 3. Classification in case 2 of Table 1 according to the local
phase portraits of finite singular points.

Case 2: c23 > 4c0c2, c1µ = 0, a0 > 0.

Sub. Conditions Classification

2.1 c0 < 0, c2(a0 + c0µ) < 0, Rc − c3 > 0. P0 saddle, P1 saddle, P2 saddle.

2.2 c0 < 0, Rc − c3 < 0, a0 + c0µ < 0,
c2 < 0.

P0 saddle, P1 saddle,
P2 stable node.

2.3 c0 < 0, a0 + c0µ > 0, Rc − c3 < 0,
c2 < 0.

P0 saddle, P1 unstable node,
P2 saddle.

2.4 c0 < 0, a0 + c0µ > 0, c2 > 0,
Rc − c3 > 0.

P0 saddle, P1 unstable node,
P2 unstable node.

2.5 c0 > 0, c2(a0 + c0µ) < 0, Rc − c3 > 0. P0 unstable node, P1 saddle,
P2 saddle.

2.6 c0 > 0, Rc − c3 < 0, a0 + c0µ > 0,
c2 > 0.

P0 unstable node, P1 saddle,
P2 unstable node.

2.7 c0 > 0, a0 + c0µ < 0, Rc − c3 < 0,
c2 > 0.

P0 unstable node, P1 stable node,
P2 saddle.

2.8 c0 > 0, a0 + c0µ < 0, c2 < 0,
Rc − c3 > 0.

P0 unstable node, P1 stable node,
P2 stable node.

2.9 c0 = 0, a0 > 0, c2 < 0. P0 ≡ P1 saddle-node, P2 saddle.

2.10 c0 = 0, a0 > 0, c2 > 0. P0 ≡ P1 saddle-node,
P2 unstable node.

µ 6= −1 in our hypothesis, so we will work under the condition (H1):

c2 6= 0, a0 ≥ 0, c1 ≥ 0, c3 ≥ 0, a0 + c0µ 6= 0, a0c1µ 6= 0, µ 6= −1.

Lemma 6.1. Under assumption (H1) the origin of the chart U1 is an infinite
singular point of system (1.3), and it has 47 distinct local phase portraits described
in Figure 1.

For system (6.1), if c1 6= 0 the characteristic polynomial is F = −c1uv2 6≡ 0, so
the origin is a nondicrital singular point. If c1 = 0 the characteristic polynomial is
F = −c3u2v− c2u3v− c0uv3, which cannot be identically zero because c2 6= 0. We
will study this two cases separately.

6.1. Case c1 non-zero. Consider c1 6= 0. We introduce the new variable w1 by
means of the variable change uw1 = v, and get the system

u̇ = (c0 − a0)u3w2
1 + c3(µ+ 1)u3w1 + c2(µ+ 1)u3 + c1(µ+ 1)u2w1,

ẇ1 = −c0u2w3
1 − c3u2w2

1 − c2u2w1 − c1uw2
1.

(6.2)

Now we cancel the common factor u, getting the system

u̇ = (c0 − a0)u2w2
1 + c3(µ+ 1)u2w1 + c2(µ+ 1)u2 + c1(µ+ 1)uw1,

ẇ1 = −c0uw3
1 − c3uw2

1 − c2uw1 − c1w2
1,

(6.3)

for which the only singular point on the exceptional divisor is the origin, and it is
linearly zero, so we have to repeat the process. Now the characteristic polynomial
is F = −c2(µ + 2)u2w1 − c1(µ + 2)uw2

1, so the origin is a nondicritical singular



EJDE-2021/35 A CLASS OF KOLMOGOROV SYSTEMS 15

Table 4. Classification in case 3 of Table 1 according to the local
phase portraits of finite singular points.

Case 3: c23 = 4c0c2, c1µ 6= 0.

Sub. Conditions Classification

3.1 a0 > 0, c0 < 0, µ(a0 + c0µ) > 0. P0 saddle, P3 saddle-node,
P4 saddle.

3.2 a0 > 0, c0 < 0, µ(a0 + c0µ) < 0. P0 saddle, P3 saddle-node,
P4 stable node.

3.3 a0 > 0, c0 > 0, µ(a0 + c0µ) > 0. P0 unstable node, P3 saddle-node,
P4 saddle.

3.4 a0 > 0, c0 > 0, µ(a0 + c0µ) < 0. P0 unstable node, P3 saddle-node,
P4 stable node.

3.5 a0 = 0, c0 > 0. P0 ≡ P4 saddle-node,
P3 saddle-node.

3.6 c0 = 0, a0 > 0, c2 < 0, µ > 0. P0 ≡ P3 topological saddle,
P4 saddle.

3.7 c0 = 0, a0 > 0, c2 < 0, µ < 0. P0 ≡ P3 topological saddle,
P4 stable node.

3.8 c0 = 0, a0 > 0, c2 > 0, µ > 0. P0 ≡ P3 topological unstable node,
P4 saddle.

3.9 c0 = 0, a0 > 0, c2 > 0, µ < 0. P0 ≡ P3 topological unstable node,
P4 stable node.

Table 5. Classification in case 4 of Table 1 according to the local
phase portraits of finite singular points.

Case 4: c23 = 4c0c2, c1µ = 0, a0 > 0.

Sub. Conditions Classification

4.1 c0 < 0. P0 saddle, P3 saddle-node.

4.2 c0 > 0. P0 unstable node, P3 saddle-node.

4.3 c0 = 0, c2 < 0. P0 ≡ P3 topological saddle.

4.4 c0 = 0, c2 > 0. P0 ≡ P3 topological unstable node.

Table 6. Classification in case 5 of Table 1 according to the local
phase portraits of finite singular points.

Case 5: c23 < 4c0c2, c1µ 6= 0.

Sub. Conditions Classification

5.1 a0 = 0. P0 ≡ P4 saddle-node.

5.2 a0 > 0, c0 < 0, µ(a0 + c0µ) > 0. P0 saddle, P4 saddle.

5.3 a0 > 0, c0 < 0, µ(a0 + c0µ) < 0. P0 saddle, P4 stable node.

5.4 a0 > 0, c0 > 0, µ(a0 + c0µ) > 0. P0 unstable node, P4 saddle.

5.5 a0 > 0, c0 > 0, µ(a0 + c0µ) < 0. P0 unstable node, P4 stable node.

point if µ 6= −2, and it is dicritical if µ = −2. In both cases we introduce the new



16 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

Table 7. Classification in case 6 of Table 1 according to the local
phase portraits of finite singular points.

Case 6: c23 < 4c0c2, c1µ = 0, a0 > 0.

Sub. Conditions Classification

6.1 c0 < 0. P0 saddle.

6.2 c0 > 0. P0 unstable node.

variable w2 doing the change uw2 = w1, obtaining the system

u̇ = (c0 − a0)u4w2
2 + c3(µ+ 1)u3w2 + c2(µ+ 1)u2 + c1(µ+ 1)u2w2,

ẇ2 = (a0 − 2c0)u3w3
2 − c3(µ+ 2)u2w2

2 − c1(µ+ 2)uw2
2 − c2(µ+ 2)uw2.

(6.4)

In the nondicritical case we have to cancel the common factor u obtaining

u̇ = (c0 − a0)u3w2
2 + c3(µ+ 1)u2w2 + c2(µ+ 1)u+ c1(µ+ 1)uw2,

ẇ2 = (a0 − 2c0)u2w3
2 − c3(µ+ 2)uw2

2 − c1(µ+ 2)w2
2 − c2(µ+ 2)w2.

(6.5)

But in the dicritical case, when µ = −2, we must cancel the common factor u2 from
system (6.4), and we obtain the system

u̇ = (c0 − a0)u2w2
2 − c3uw2 − c2 − c1w2,

ẇ2 = (a0 − 2c0)uw3
2.

(6.6)

6.1.1. Nondicritical case. In this case it is necessary to study the singular points of
system (6.5) on the exceptional divisor. The origin is always a singular point, and
we denote it by Q0. As µ+ 2 6= 0 there is another singular point, Q1 = (0,−c2/c1)
and we determine their local phase portraits.

The origin Q0 is always hyperbolic. It is a saddle if µ ∈ (−∞,−2) ∪ (−1,+∞),
a stable node if c2 > 0 and µ ∈ (−2,−1), and an unstable node if c2 < 0 and
µ ∈ (−2,−1).

The singular point Q1 is semi-hyperbolic. If c2(a0 + c0µ) > 0 then Q1 is a
topological saddle, if c2(µ+2) > 0 and (µ+2)(a0 +c0µ) < 0 then it is a topological
unstable node, and if c2(µ + 2) < 0 and (µ + 2)(a0 + c0µ) > 0 it is a topological
stable node. These conditions come together in the next 7 subcases.

(1) If µ ∈ (−∞,−2) ∪ (−1,+∞) and c2(a0 + c0µ) > 0, then Q0 is a saddle
and Q1 a topological saddle. To determine the phase portrait around the w2-axis
for system (6.5), we must fix the sign of c2, which determines the position of the
singular point Q1, and also the sign of µ+1, which determines the sense of the flow
along the x-axis. Thus we deal with the following subcases.

Subcase (1.1). Let µ < −2 (so µ + 1 < 0) and c2 > 0. Then the singular point
Q1 is on the negative part of the w2-axis and the expression u̇

∣∣
w2=0

= c2(µ + 1)u

determines the sense of the flow, so the phase portrait is the one in Figure 2(a).
To return to system (6.4) we multiply by u, thus the orbits in the second and

third quadrants change their orientation. Moreover all the points on the w2-axis
become singular points. The resultant phase portrait is given in Figure 2(b).

When going back to the (u,w1)-plane the second and the third quadrants swap
from the (u,w2)-plane, and the exceptional divisor shrinks to a point, and hence the
orbits are slightly modified. Attending to the expresions of u̇

∣∣
w1=0

= c2(µ+1)u2 and

ẇ1

∣∣
u=0

= −c1w2
1, we know the sense of the flow along the axes. Following the results



EJDE-2021/35 A CLASS OF KOLMOGOROV SYSTEMS 17

(L1) (L2) (L3) (L4) (L5)

(L6) (L7) (L8) (L9) (L10)

(L11) (L12) (L13) (L14) (L15)

(L16) (L17) (L18) (L19) (L20)

(L21) (L22) (L23) (L24) (L25)

(L26) (L27) (L28) (L29) (L30)

Figure 1. Local phase portraits of the infinite singular point O1

(to be continued)

mentioned in Subsection 2.3, the separatrix of the singular point Q1 = (0,−c2/c1)
in the (u,w2)-plane, becomes the separatrix with slope −c2/c1 in the (u,w1)-plane.



18 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

(L31) (L32) (L33) (L34) (L35)

(L36) (L37) (L38) (L39) (L40)

(L41) (L42) (L43) (L44) (L45)

(L46) (L47)

Figure 1. (cont.) Local phase portraits of the infinite singular
point O1.

We get the phase portrait given in Figure 2(c), and multiplying again by u, the one
given in Figure 2(d).

w2

u

w2

u

(a) (b) (c) (d)

Figure 2. Desingularization of the origin of system (6.1) with
c1 6= 0. Nondicritical case (1.1).

Finally we must go back to the (u, v)-plane, swapping the second and the third
quadrants and contracting the exceptional divisor to the origin. The orbits tending



EJDE-2021/35 A CLASS OF KOLMOGOROV SYSTEMS 19

to the origin in forward or backward time, became orbits tending to the origin in
forward or backward time with slope zero, i.e. tangent to the u-axis. According to
the expressions u̇

∣∣
v=0

= c1µv
2 − a0v

3 and u̇
∣∣
v=0

= c2(µ + 1)u3, which determine
the sense of the flow along the axes, we obtain the local phase portrait at the origin
for system (6.1) given in Figure 1(L1).

Subcase (1.2) If we maintain µ < −2 but take c2 < 0, the reasoning is essentially
similar to the one we have given in the previous case, and we obtain the phase
portrait (L2) of Figure 1.

Subcase (1.3) Let µ > −1 and c2 > 0. This determines the position of the
singular point Q1 and the sense of the flow along the axes, so around the w2-axis
we obtain the phase portrait given in Figure 3(a).

As in the previous subcase we multiply by u obtaining the phase portrait given in
Figure 3(b), as the orbits in the second and third quadrants change their orientation
and all the point in the w2-axis become singular points.

To undo the variable change we analyze the sense of the flow along the axes
according to the expression u̇

∣∣
w1=0

= c2(µ + 1)u2, which determines that the flow

goes in the positive sense of the u-axis, and ẇ1

∣∣
u=0

= −c1w2
1 which determines that

the flow goes in the negative sense of the w1-axis. Moreover we swap the second
and third quadrants, and press the exceptional divisor into the origin, modifying
the orbits. We obtain the phase portrait given in Figure 3(c). Multiplying again
by u we obtain the phase portrait 3(d).

Now we have to undo de second variable change. We note that u̇
∣∣
v=0

= c2(µ +

1)u3, so the flow gets away from the origin along the u-axis, nevertheless the sense
of the flow along the v-axis is not determined by v̇

∣∣
u=0

= c1µv
2 − a0v

3, it depends
on the constant µ. If µ > 0 the flow goes in the positive sense of the v-axis, if
µ < 0 in the opposite sense and, if µ = 0 the flow goes to the origin. Thus we must
distinguish three subcases and in each of them, modifying the orbits properly, we
obtain, respectively, the phase portraits given in Figure 1(L3), (L4) and (L5).

Subcase (1.4). Let µ > −1 and c2 < 0. By a similar reasoning to the previous
one, we obtain the phase portraits (L6) and (L7) of Figure 1.

w2

u

w2

u

w1

u

w1

u

(a) (b) (c) (d)

Figure 3. Desingularization of the origin of system (6.1) with
c1 6= 0. Nondicritical case (1.3).

(2) If µ ∈ (−∞,−2) ∪ (−1,+∞), c2(µ+ 2) > 0 and (µ+ 2)(a0 + c0µ) < 0, then
Q0 is a saddle and Q1 a topological unstable node. We must distinguish two cases
according to the sign of c2.

Subcase (2.1). We consider c2 < 0 so µ < −2 and a0 + c0µ > 0. The singular
point Q1 is on the positive w2-axis and it is an unstable node, so the sense of the



20 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

flow along the axes is determined, and we obtain the phase portrait given in Figure
4(a). Multiplying by u we obtain Figure 4(b).

We see that for system (6.3) the flow goes in the negative sense along the w1-axis
and in the positive sense along the u-axis, according to the expressions u̇

∣∣
w1=0

=

c2(µ + 1)u2 and ẇ1

∣∣
u=0

= −c1w2
1. We undo the variable change modifying the

orbits properly, and we note that it must exist an hyperbolic or elliptic sector in
both first and third quadrants, thus it can appear the configuration given in Figure
4(c) or the one given in Figure 5(a). From the first of them multiplying by u we
obtain 4(d), and if we undo the variable change in a similarly way than in the
previous cases, we obtain the phase portrait in Figure 1(L8).

If we consider hyperbolic sectors we continue undoing the blow up from Figure
5(a), obtaining successively the phase portraits 5(b) and 5(c). However in our study
we have proved, by means of the index theory, that only the case with elliptic sectors
is feasible in the global phase portraits obtained. More detailed explanations will be
given in Section 7 but, roughly speaking we know that the index of the vector field
on the sphere must be 2, and this index is the sum of the indices of all singularities,
which depend on the sectors that they have, so if the index is 2 considering two
elliptic sectors in a particular singular point, it cannot be 2 if we change those
sectors for hyperbolic ones. In conclusion the only phase portrait that will appear
in this case is (L8) of Figure 1.

w2

u

w2

u

w1

u

w1

u

(a) (b) (c) (d)

Figure 4. Desingularization of the origin of system (6.1) with
c1 6= 0. Nondicritical case (2.1).

From now on we will omit the reasonings about how to undo the variable changes
for obtaining the final phase portrait, because they are similar to the ones of the
previous cases. The results obtained are the following.

w1

u

w1

u

(a) (b) (c)

Figure 5. Desingularization of the origin of system (6.1) with
c1 6= 0. Alternative to nondicritical case (2.1).



EJDE-2021/35 A CLASS OF KOLMOGOROV SYSTEMS 21

Subcase (2.2). Let c2 > 0 so µ > −1 and a0 + c0µ < 0. We obtain the phase
portraits (L9) and (L10) of Figure 1. In (L9) it is possible to consider hyperbolic
sectors instead of the elliptic ones, but applying index theory to the global phase
portraits obtained in our study, we note that only the phase portrait with elliptic
sectors is feasible.

(3) If µ ∈ (−∞,−2) ∪ (−1,+∞), c2(µ+ 2) < 0 and (µ+ 2)(a0 + c0µ) > 0, then
Q0 is a saddle and Q1 a topological stable node. If c2 > 0 we obtain the phase
portrait (L11) of Figure 1, and if c2 < 0 we obtain the phase portraits (L12), (L13)
and (L14) of Figure 1. In (L11) and (L12) it would be possible that the elliptic
sectors appearing were hyperbolic sectors, but again we have proved that only the
elliptic option is feasible according to the index theory.

(4) If c2 > 0, µ ∈ (−2,−1) and a0 + c0µ > 0, then Q0 is a stable node and Q1 a
topological saddle. We obtain the phase portrait (L15) of Figure 1.

(5) If c2 > 0, µ ∈ (−2,−1) and a0 + c0µ < 0, then Q0 is a stable node and Q1 a
topological unstable node. We obtain the phase portrait (L11) of Figure 1.

(6) If c2 < 0, µ ∈ (−2,−1) and a0 + c0µ < 0, then Q0 is an unstable node and
Q1 a topological saddle. We obtain the phase portrait (L16) of Figure 1.

(7) If c2 < 0, µ ∈ (−2,−1) and a0 + c0µ > 0, then Q0 is an unstable node and
Q1 a topological stable node. We obtain the phase portrait (L8) of Figure 1.

6.1.2. Dicritical case. Now we must study the singular points on the exceptional
divisor of system (6.6). In this case there is only one singular point, R = (0,−c2/c1)
which is non-degenerated. We shall distinguish several subcases.
• If c23 < −4c2(a0 − 2c0) and c2c3 < 0, then P is a stable focus. We shall distin-

guish two cases depending on the sign of the parameter c2, because it determines
if the singular point is on the positive or negative u-axis. We consider c2 > 0. In
Figure 6 the blowing-down process is represented. The phase portrait around the
u-axis is the one given in Figure 6(a), multiplying by u2 we obtain (b), undoing the
second variable change we obtain (c), multiplying by u we obtain (d) and finally,
undoing the first variable change we obtain the phase portrait (L11) of Figure 1.
Taking c2 < 0 and by the same method we obtain the phase portrait (L8) of Figure
1.

w2

u

w2

u

w1

u u

w1

(a) (b) (c) (d)

Figure 6. Desingularization of the origin of system (6.1) with
c1 6= 0. Dicritical case (1), c2 > 0.

From now on we omit explanations in cases in which similar arguments are
valid, and same results are obtained. In order to simplify the notation we define
β =

√
c23 + 4c2(a0 − 2c0).



22 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

• If c23 < −4c2(a0−2c0) and c2c3 > 0, then P is an unstable focus. The reasoning
is analogous to the one of the previous case and we obtain the same phase portraits:
(L11) if c2 > 0, and (L8) if c2 < 0.
• If c23 = −4c2(a0 − 2c0) and c2c3 < 0 or if c23 > −4c2(a0 − 2c0), c2(c3 − β) < 0

and c2(c3 + β) < 0, then P is a stable node. If c23 = −4c2(a0 − 2c0) and c2c3 > 0
or if c23 > −4c2(a0 − 2c0), c2(c3 − β) > 0 and c2(c3 + β) > 0, then P is an unstable
node. In both cases the phase portrait obtained is again (L11) if c2 > 0, and (L8)
if c2 < 0.
• If c3 = 0 and c2(a0 − 2c0) < 0 then P is a linear center. In this case the

singular point P could be a center or a focus, but the final phase portrait obtained
when P is a center is the same as the one we obtained previously for the case with
a focus, so the result is (L11) if c2 > 0, and (L8) if c2 < 0.
• If c23 > −4c2(a0−2c0) and (c3−β)(c3+β) < 0, or if c3 = 0 and c2(a0−2c0) > 0,

then P is a saddle. The blowing-down considering c2 > 0 is represented in Figure
7. The final result is (L17) of Figure 1. If we take c2 < 0 we obtain the phase
portrait (L18).

w2

u

w2

u

w1

u

w1

u

System (6.6) System (6.4) System (6.3) System (6.2)

Figure 7. Desingularization of the origin of system (6.1) with
c1 6= 0. Dicritical case (5), c2 > 0.

6.2. Case c1 zero. We consider system (6.1) and do the same variable change
that we did in the case with c1 6= 0, the result is obviously system (6.2) but taking
c1 = 0, i.e.

u̇ = (c0 − a0)u3w2
1 + c3(µ+ 1)u3w1 + c2(µ+ 1)u3,

ẇ1 = −c0u2w3
1 − c3u2w2

1 − c2u2w1.
(6.7)

In this case we can cancel a common factor u2 getting the system

u̇ = (c0 − a0)uw2
1 + c3(µ+ 1)uw1 + c2(µ+ 1)u,

ẇ1 = −c0w3
1 − c3w2

1 − c2w1,
(6.8)

for which we must study the singular points on the exceptional divisor, i.e. on the
straight line u = 0.

The origin S0 = (0, 0) is always a singular point. The other singular points on
this line are those for which w1 is a solution of c0w

2
1 + c3w1 + c2 = 0. If c0 6= 0

and c23 > 4c0c2 then S1 = (0,−(Rc + c3)/(2c0)) and S2 = (0, (Rc − c3)/(2c0)) are
singular points. If c0 6= 0 and c23 = 4c0c2, then S3 = (0,−c3/2c0) is a singular
point, and finally, if c0 and c3 are non-zero, S4 = (0,−c2/c3) is a singular point.



EJDE-2021/35 A CLASS OF KOLMOGOROV SYSTEMS 23

Table 8. Cases with the singular points on the exceptional divisor
of system (6.8).

Case Conditions Singular points

A c0 = 0, c3 = 0. S0.

B c0 = 0, c3 6= 0. S0, S4.

C c0 6= 0, c23 < 4c0c2. S0.

D c0 6= 0, c23 = 4c0c2. S0, S3.

E c0 6= 0, c23 > 4c0c2. S0, S1, S2.

In summary we shall study the five cases given in Table 8. For doing this we
study separately the local phase portrait of each singular point assuming in each
case the necessary hypothesis for its existence.

The singular points S0, S1 and S2 are hyperbolic. S0 is a saddle if µ > −1, a
stable node if c2 > 0 and µ < −1, and an unstable node if c2 < 0 and µ < −1. S1 is a
saddle if c0(a0+c0µ) < 0, a stable node if c0 > 0 and (a0+c0µ) > 0, and an unstable
node if c0 < 0 and (a0+c0µ) < 0. S2 is a saddle if c0(a0+c0µ)(Rc−c3) < 0, a stable
node if c0(Rc− c3) > 0 and (a0 + c0µ) > 0, and an unstable node if c0(Rc− c3) < 0
and (a0 + c0µ) < 0.

The singular point S3 is a semi-hyperbolic saddle-node and finally, S4 is a hy-
perbolic saddle if c2 > 0, and a hyperbolic stable node if c2 < 0.

Using these informations we study the next cases from the five given in Table 8.
First of all we study case (A) in which the only singular point is the origin, so

we have the next three possibilities.
If c0 = c3 = 0 and µ > −1, then S0 is a saddle. To determine the phase portrait

around the w1-axis for system 6.8, we must fix the sign of c2, which determines the
sense of the flow along the axes. Considering c2 > 0 we obtain the phase portrait
given in Figure (1)(L19), and with c2 < 0 we obtain the phase portrait (L20).

If c0 = c3 = 0, µ < −1 and c2 > 0, then S0 is a stable node, and we obtain the
phase portrait (L21) of Figure (1).

If c0 = c3 = 0, µ < −1 and c2 < 0, then S0 is an unstable node, and we obtain
the phase portrait (L22) of Figure (1).

In case (B), fixed the phase portrait of S4, only two phase portraits will be
possible for the origin, as the sign of c2 is determined, and so we obtain the four
following cases.

If c0 = 0, c3 6= 0, µ > −1 and c2 > 0, then S0 and S4 are both saddle points and
from the blowing-down we obtain the phase portrait (L23) of Figure (1).

If c0 = 0, c3 6= 0, µ > −1 and c2 < 0, then S0 is a saddle and S4 a stable node.
We obtain the phase portrait (L24) of Figure (1).

If c0 = 0, c3 6= 0, µ < −1 and c2 > 0, then S0 is a stable node and S4 a saddle.
We obtain the phase portrait (L25) of Figure (1).

If c0 = 0, c3 6= 0, µ < −1 and c2 < 0, then S0 is an unstable node and S4 a
stable node. We obtain the phase portrait (L26) of Figure (1).

Again in case (C) the only singular point is the origin so we distinguish three
cases, and obtain the same local phase portrait that in case (A), but under different
conditions. If c0 6= 0, c23 < 4c0c2 and µ > −1, then S0 is a saddle. Attending to



24 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

the sign of c2, which determines the sense of the flow on the axes, we consider the
following cases: if c2 > 0 we obtain the phase portrait (L19) of Figure (1), and if
c2 < 0 we obtain the phase portrait (L20) of Figure (1).

If c0 6= 0, c23 < 4c0c2, µ < −1 and c2 > 0, then S0 is a stable node. We obtain
the phase portrait (L21) of Figure (1).

If c0 6= 0, c23 < 4c0c2, µ < −1 and c2 < 0, then S0 is an unstable node. We
obtain the phase portrait (L22) of Figure (1).

In case (D) apart from the origin, there exists the singular point S3, which is
always a saddle node, so again we obtain only three cases.

If c0 6= 0, c23 = 4c0c2 and µ > −1, then S0 is a saddle and S3 a saddle-node.
We must distinguish four subcases according to the signs of c0 and a0 + c0µ, which
determine the position of the saddle-node S3 and its sectors.

If c0 > 0 and a0 + c0µ > 0 we obtain the phase portrait (L27) of Figure (1), if
c0 > 0 and a0 + c0µ < 0 we obtain the phase portrait (L28) of Figure (1), if c0 < 0
and a0 + c0µ > 0 we have the phase portrait (L29), and if c0 < 0 and a0 + c0µ < 0
the phase portrait (L30).

If c0 6= 0, c23 = 4c0c2, µ < −1 and c2 > 0, then S0 is a stable node and S3

a saddle-node. We distinguish two subcases setting the sign of a0 + c0µ which
determines the position of the sectors of the saddle-node S3. If a0 + c0µ > 0 we
obtain the phase portrait (L31) of Figure (1), and if a0 + c0µ < 0 we obtain the
phase portrait (L32) of Figure (1).

If c0 6= 0, c23 = 4c0c2, µ < −1 and c2 < 0, then S0 is an unstable node and S3

a saddle-node. The only possibility is that a0 + c0µ > 0, and we obtain the phase
portrait (L33) of Figure (1).

In case (E) there exist three singular points, with three possible phase portraits
for each of them, however, many of the combinations are not possible, and only 13
cases will be feasible.

First, due to the conditions which define the local phase portrait in each singular
point, it is obvious that if S1 is a stable node, then S2 cannot be an unstable node,
and if S1 is an unstable node, S2 cannot be a stable node, due to the sign of a0+c0µ.

If S0 and S2 were stable nodes and S1 a saddle, the conditions c2 > 0, Rc−c3 < 0,
and c0 < 0 will hold. Squaring both terms in the condition Rc < c3 we obtain
c23 − 4c0c2 < c23, and then c0c2 > 0, which is a contradiction. The same reasoning
is valid in the next two cases.

If S0 and S2 are unstable nodes and S1 a saddle, then the conditions c2 < 0,
Rc − c3 < 0 and c0 > 0 hold, and if S0 is an unstable node, S1 a stable node and
S2 a saddle, then the same three conditions hold.

If S0, S1 and S2 are stable nodes, the conditions c2 > 0, Rc − c3 > 0 and
c0 > 0 hold. Now we take condition Rc < c3 and squaring both terms we obtain
c23 − 4c0c2 < c23, and then c0c2 > 0, which is a contradiction.

If S0 is a stable node and S1 an unstable node, the conditions µ < −1, c0 < 0
and a0 + c0µ < 0 hold. Then according to the signs of c0 and µ which are fixed,
a0 < −c0µ < 0 which contradicts the hypothesis (H1). The same reasoning is valid
if S0 and S1 are unstable nodes, because the same conditions hold. Now we will
study the feasible cases.

(E1) If c0 6= 0, c23 > 4c0c2, µ > −1, c0(a0 + c0µ)(2c0c2 − c23 − c3Rc) > 0 and
c0(a0 + c0µ)(Rc − c3)(2c0c2 − c23 + c3Rc) > 0, then S0, S1 and S2 are saddles. We
must distinguish two subcases depending on the position of the singular points S1



EJDE-2021/35 A CLASS OF KOLMOGOROV SYSTEMS 25

and S2 on the w1-axis. First if S1 is on the negative w1-axis and S2 on the positive
w1-axis, that corresponds with conditions c0 > 0, Rc−c3 > 0 and c2 < 0, we obtain
the phase portrait (L34) of Figure 1. Note that if Rc − c3 > 0, the singular points
S1 and S2 are one on the positive part of the axis and the other on the negative
part, but in any case the absolute value of the second coordinate of S1 is greater or
equal than the absolute value of the second coordinate of S2, and this determines
the relation between the slopes of orbits in the phase portraits. Conversely if we
have S1 on the positive w1-axis and S2 on the negative one, i.e. under the conditions
c0 < 0, Rc − c3 > 0 and c2 > 0, we obtain the phase portrait (L35) of Figure (1).

(E2) If c0 6= 0, c23 > 4c0c2, µ > −1, c0(a0 + c0µ)(2c0c2 − c23 − c3Rc) > 0,
c0(Rc−c3) > 0 and (a0+c0µ)(2c0c2−c23+c3Rc) < 0, then S0 and S1 are saddles and
S2 is a stable node. If c0 > 0 then Rc−c3 > 0 and so c23−4c0c2 > c23 and c0c2 < 0. If
a0 +c0µ > 0 then 2c0c2−c23−c3Rc > 0 which is not possible because 2c0c2 < 0 and
we subtract two positive terms. Conversely if a0 + c0µ < 0 then 2c0c2 − c23 < c3Rc
and 2c0c2−c23 > −c3Rc, so |2c0c2−c23| < c3Rc. Squaring we obtain 4c20c

2
2 < 0 which

is not possible. If c0 < 0 then Rc − c3 < 0 and we deduce c2 < 0. If a0 + c0µ < 0
then 2c0c2 − c23 − c3Rc > 0, but c23 − 2c0c2 > c23 − 4c0c2 > 0 so 2c0c2 − c23 < 0 and
subtracting c3Rc > 0 the result cannot be positive. In conclusion we deduce that
c0, c2 < 0 and a0 +c0µ > 0. Hence we have −(Rc+c3)/(2c0) > (Rc−c3)/(2c0) > 0.
This determines the only possible position of the singular points which are both in
the positive w1-axis. Undoing the blow up we obtain the phase portrait (L36) of
Figure 1.

(E3) If c0 6= 0, c23 > 4c0c2, µ > −1, c0(a0 + c0µ)(2c0c2 −2 −c3Rc) > 0, c0(Rc −
c3) < 0 and (a0+c0µ)(2c0c2−c23+c3Rc) > 0, then S0 and S1 are saddles and S2 is an
unstable node. Therefore we deduce that 0 > (Rc − c3)/(2c0) > −(Rc + c3)/(2c0),
so both singular points are on the negative w1 axis, S1 under S2. We obtain the
phase portrait (L37) of Figure (1).

(E4) If c0 6= 0, c23 > 4c0c2, µ > −1, c0 > 0, (a0 + c0µ)(2c0c2 − c23 − c3Rc) < 0
and c0(Rc − c3)(a0 + c0µ)(2c0c2 − c23 + c3Rc) > 0, then S0 and S2 are saddles and
S1 is a stable node. Then we deduce that −(Rc + c3)/(2c0) < (Rc − c3)/(c0) < 0,
so both singular points are on the negative w1 axis, S1 under S2. We obtain the
phase portrait (L38) of Figure (1).

(E5) If c0 6= 0, c23 > 4c0c2, µ > −1, c0 > 0, (a0 + c0µ)(2c0c2 − c23 − c3Rc) < 0,
c0(Rc − c3) > 0 and (a0 + c0µ)(2c0c2 − c23+) < 0, then S0 is a saddle and S1 and
S2 are stable nodes. We obtain the phase portrait (L45) of Figure (1).

(E6) If c0 6= 0, c23 > 4c0c2, µ > −1, c0 < 0, (a0 + c0µ)(2c0c2− c23− c3Rc) > 0 and
(a0 + c0µ)(Rc − c3)(2c0c2 − c23 + c3Rc) < 0, then S0 and S2 are saddles and S1 is
an unstable node. Hence we deduce that 0 < (Rc − c3)/(2c0) < −(Rc + c3)/(2c0),
so both singular points are on the positive w1 axis, S2 under S1. We obtain the
phase portrait (L47) of Figure (1).

(E7) If c0 6= 0, c23 > 4c0c2, µ > −1, c0 < 0, (a0 + c0µ)(2c0c2 − c23 − c3Rc) > 0,
Rc − c3 > 0 and (a0 + c0µ)(2c0c2 − c23 + c3Rc) > 0, then S0 is a saddle and S1 and
S2 are unstable nodes. We obtain the phase portrait (L46) of Figure (1).

(E8) If c0 6= 0, c23 > 4c0c2, c2 > 0, µ < −1, c0(a0 + c0µ)(2c0c2 − c23 − c3Rc) > 0
and c0(a0 + c0µ)(Rc − c3)(2c0c2 − c23 + c3Rc) > 0, then S0 is a stable node and S1

and S2 are saddles. We obtain the phase portrait (L39) of Figure (1).
(E9) If c0 6= 0, c23 > 4c0c2, c2 > 0, µ < −1, c0(a0 + c0µ)(2c0c2 − c23 − c3Rc) > 0,

c0(Rc − c3) < 0 and (a0 + c0µ)(2c0c2 − c23 + c3Rc) > 0, then S0 is a stable node,



26 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

S1 is a saddle and S2 is an unstable node. We obtain the phase portrait (L40) of
Figure (1).

(E10) If c0 6= 0, c23 > 4c0c2, c2 > 0, µ < −1, c0 > 0, (a0+c0µ)(2c0c2−c23−c3Rc) <
0 and (a0 + c0µ)(Rc − c3)(2c0c2 − c23 + c3Rc) > 0, then S0 and S1 are stable nodes
and S2 is a saddle. We obtain the phase portrait (L41) of Figure (1).

(E11) If c0 6= 0, c23 > 4c0c2, c2 < 0, µ < −1, c0(a0 + c0µ)(2c0c2− c23− c3Rc) > 0,
and c0(a0 + c0µ)(Rc − c3)(2c0c2 − c23 + c3Rc) > 0, then S0 is an unstable node and
S1 and S2 are saddles. We obtain the phase portrait (L42) of Figure (1).

(E12) If c0 6= 0, c23 > 4c0c2, c2 < 0, µ < −1, c0(a0 + c0µ)(2c0c2− c23− c3Rc) > 0,
c0(Rc − c3) > 0, and (a0 + c0µ)(2c0c2 − c23 + c3Rc) < 0, then S0 is an unstable
node, S1 is a saddle and S2 is a stable node. We obtain the phase portrait (L43)
of Figure (1).

(E13) If c0 6= 0, c23 > 4c0c2, c2 < 0, µ < −1, c0 > 0, (a0+c0µ)(2c0c2−c23−c3Rc) <
0, Rc − c3 > 0, and (a0 + c0µ)(2c0c2 − c23 + c3Rc) < 0, then S0 is an unstable node
and S1 and S2 are stable nodes. We obtain the phase portrait (L44) of Figure (1).

Note that in the phase portraits (L22), (L30), (L33), (L43), (L46) and (L47) of
Figure 1 it is possible to consider hyperbolic sectors instead of the elliptic ones,
but we have only represented the elliptic cases by the same reason given before, i.e.
because applying the index theory to the phase portraits in the sphere S2 described
in Section 7, we proved that they are the only feasible.

Once we completed the study in the local chart U1, we study the origin of chart
U2 which turned out to be much simpler. The system has the expression

u̇ = −c1(µ+ 1)u2v + (a0 − c0)uv2 − c3(µ+ 1)uv − c2(µ+ 1)u,

v̇ = −c1uv2 − c0v3 − c3v2 − c2v.
(6.9)

Lemma 6.2. The origin of chart U2 is always a hyperbolic infinite singular point
of system (1.3). It is a saddle if µ < −1, a stable node if c2 > 0 and µ > −1, and
an unstable node if c2 < 0 and µ > −1.

7. Global phase portraits

To prove the global result stated in Theorem 1.1, we will bring together the local
information obtained in the previous sections. We start our classification from the
cases in Tables 2 to 7. In some of them the conditions determine only one local
phase portrait in each one of the infinite singular points but, in many others we
shall distinguish several possibilities. In some cases the local information gives rise
to only one phase portrait, this occurs when the sepatrices can be connected in only
one way, but in others several global possibilities appear, and we shall prove which
of them are feasible.

In Table 9 we give, for each case in the Tables 2 to 7, the local phase portrait of
the infinite singularities O1 and O2 (in most cases this depends on the parameters),
and also we give the global phase portrait on the Poincaré disc obtained. And now
we detail the reasonings in some cases, although they will not be showed in all cases
to avoid repetitions.

Case 1.1. The infinite singular point O1 has the local phase portrait (L12) given
in Figure 1, and O2 is an unstable node. As we said in Section 6, the elliptic sectors
appearing in phase portrait (L12) could be hyperbolic sectors if we attend only to
local results, but now having all the global information we can prove that they are
elliptic by using the index theory. By Theorem 2.2 the sum of the indices of all



EJDE-2021/35 A CLASS OF KOLMOGOROV SYSTEMS 27

the singular points on the Poincaré sphere has to be 2. To compute this sum we
must consider that the finite singular points on the Poincaré disc appear twice on
the sphere (on the northern hemisphere and on the southern hemisphere). Thus if
we denote by indF the sum of the indices of the finite singular points, and by indI
the sum of the indices of the infinite singular points, the equality 2 indF + indI = 2
must be satisfied.

In this particular case the finite singular points are a saddle-node whose index
is 0, and two saddles whose index is −1, so indF = −2. We deduce that indI must
be 6. The infinite singular points are O1 and O2, the origins of the local charts U1

and U2, and the origins of the symmetric local charts V1 and V2, which have the
same index. Since O2 is a node, and so it has index 1, we obtain that the sum of
the indices of O2 and its symmetric must be 4, i.e. the index of O2 has to be 2.
From the Poincaré formula for the index given in subsection 2.4 we obtain

e− h
2

+ 1 = 2 =⇒ e− h = 2.

Hence only the case with tho elliptic sectors on the local phase portait (L12) is
possible, because if we had two hyperbolic sectors instead of the elliptic ones, the
index of O2 would be zero.

We recall that by an analogous application of the index theory in the corre-
sponding cases, it can be concluded that elliptic sectors appearing in the local
phase portraits (L8), (L9), (L11), (L22), (L30), (L33), (L43), (L46) and (L47) of
O1, are indeed elliptic rather than hyperbolic.

In this case 1.1 there is only one possible phase portrait on the Poincaré disc,
the one given in Figure (16) (G1).

Case 1.6. In this case O1 has the local phase portrait (L3) and O2 is a stable
node. From the local results we can obtain three possible global phase portraits
given in Figure 8.

Subcase 1 Subcase 2 Subcase 3

Figure 8. Possible global phase portraits in case 1.6.

By Theorem 4.8 on the straight lines z = z0 6= 0 cannot be more than one contact
point, but as it is shown in Figure 9, if subcases 1 and 2 are feasible, there exist
straight lines z = z0, with −(Rc + c3)/(2c2) < z0 < (Rc− c3)/(2c2), on which there
exist two contact points, so we deduce that the only possible global phase portrait
is the subcase 3, i.e. (G10) of Figure 16.

Case 1.10. In this case O1 has the local phase portrait (L6) and O2 is an unstable
node. From the local results we can obtain three possible global phase portraits,
but in two of them shown in Figure 10 we can find straight lines z = z0 6= 0 with
(Rc − c3)/(2c2) < z0 < −(Rc + c3)/(2c2), on which there are two contact points,



28 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

Subcase 1 Subcase 2

Figure 9. Straight lines with two contact points on the two first
subcases of 1.6

so according to Theorem 4.8 they are not possible. Then the only possibility is the
phase portrait (G20) of Figure 16.

Subcase 1 Subcase 2

Figure 10. Straight lines with two contact points on two subcases
of 1.10.

Case 2.2. In this case O1 has the local phase portrait (L47) and O2 is an unstable
node. From the local results and by Theorem 4.3 the phase portrait is symmetric,
we obtain three possible global phase portraits, the ones given of Figure 11.

Subcase 1 Subcase 2 Subcase 3

Figure 11. Possible global phase portraits in case (2.2).

By Theorem 4.8 we know that, under the conditions of this case, two invariant
straight lines z = ±(Rc−c3)/(2c2) must exist, and it is only possible in the subcase
1, which provides the phase portrait (G42) of Figure 16.

Case 2.6. Here we distinguish three subcases and, in two of them, three global
phase portraits appear, but in each case we use different arguments to prove which
of the options is realizable. If µ = 0, then O1 has the local phase portrait (L5) and



EJDE-2021/35 A CLASS OF KOLMOGOROV SYSTEMS 29

O2 is a stable node. We obtain three phase portraits, but we conclude that two
of them are not feasible because we can find straight lines z = z0 6= 0 with two
contact points, as it is shown in Figure 12. Therefore there is only one global phase
portrait, the (G50) of Figure 16.

Subcase 1 Subcase 2

Figure 12. Straight lines with two contact points on two subcases.

If c1 = 0 and µ > −1, then O1 has the local phase portrait (L38) and O2 is
a stable node. We obtain three possible global phase portraits, the ones given in
Figure 13. By Theorem 4.8 we know that, under the conditions of this case, two
invariant straight lines z = ±(Rc − c3)/(2c2) must exist, and it is only possible in
the subcase 3, which provides the phase portrait (G51) of Figure 16.

Subcase 1 Subcase 2 Subcase 3

Figure 13. Possible global phase portraits in case 2.6 with c1 = 0
and µ > −1.

If c1 = 0 and µ < −1, then O1 has the local phase portrait (L41) and O2 is a
saddle. In this case we obtain only one phase portrait (G52) of Figure 16.

Case 3.2. Here we distinguish three subcases and in two of them there is only one
possible global phase portait. More precisely, if µ < −1 then O1 has the local phase
portrait (L8), O2 is a saddle and the global phase portrait is (G64). If µ ∈ (−1, 0)
then O1 has the local phase portrait (L13), O2 is an unstable node and we obtain
the phase portrait (G65).

If µ > 0 then O1 has the local phase portrait (L6) and O2 is an unstable node,
but in this case we obtain three phase portraits, the ones given in Figure 14. By
Theorem 4.8, there must exist a contact point on each straight line z = z0, but if
in subcases 1 and 2 we take a straight line z = z0 with z0 > −c3/(2c2), there are
not contact points on it, so those subcases are not feasible. The only possibility is
the subcase 3, which provides the phase portrait (G63) of Figure 16.



30 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

Subcase 1 Subcase 2 Subcase 3

Figure 14. Possible global phase portraits in case 3.2 with µ > 0.

Case 4.2. Here we distinguish five different subcases. First if c1 = 0, µ < −1 and
a0 + c0µ > 0, then O1 has the local phase portrait (L31) and O2 is a saddle. In
this case we obtain the global phase portrait (G90).

If c1 = 0, µ < −1 and a0 + c0µ < 0, then O1 has the local phase portrait (L32)
and O2 is a saddle. By Corollary 4.3 the phase portrait must be symmetric so we
obtain three possibilities, given in Figure 15. We further know that there must
exist an invariant straight line z = −c3/2c2, so we can deduce that subcase 2 is not
feasible because that invariant straight line does not exist. As we also know that
this invariant straight line is a separatrix in the local phase portrait of O1, the one
appearing in (L32), the subcase 3 is not feasible, because there would exist another
separatrix over the invariant straight line that does not appear on (L32). So finally
the only possible phase portrait is (G91).

The same happens in the next cases in which we initially obtain three possibilities
but we can discard two of them with the same arguments, so finally we obtain the
next results. If µ = 0, then O1 has the local phase portrait (L5) and O2 is a
stable node and we obtain the global phase portrait (G87). If c1 = 0, µ > −1 and
a0 + c0µ > 0, then O1 has the local phase portrait (L27), O2 is a stable node, and
we obtain the phase portrait (G88). Finally if c1 = 0, µ > −1 and a0 + c0µ < 0,
then O1 has the local phase portrait (L28), O2 is a stable node, and the global
phase portait is (G89).

Subcase 1 Subcase 2 Subcase 3

Figure 15. Possible global phase portraits in case 4.2 with c1 = 0,
µ < −1 and a0 + c0µ < 0.

The same methods that we have used in the previous cases for determine which of
the global phase portraits are realizable, must be used in some other cases, namely,
1.17, 1.18 with µ ≥ −2, 1.19, 2.7, 3.3, 3.4 with µ ≥ −2, 3.5 with µ > 0, and finally
4.1 with c1 = 0, µ > −1 and a0 + c0µ < 0.



EJDE-2021/35 A CLASS OF KOLMOGOROV SYSTEMS 31

(G1) [R=8, S=21] (G2) [R=6, S=19] (G3) [R=5, S=18] (G4) [R=6, S=19] (G5) [R=5, S=18]

(G6) [R=4, S=17] (G7) [R=6, S=19] (G8) [R=7, S=20] (G9) [R=7, S=20] (G10) [R=6, S=19]

(G11) [R=6, S=19] (G12) [R=5, S=18] (G13) [R=5, S=18] (G14) [R=4, S=17] (G15) [R=6, S=19]

(G16) [R=7, S=20] (G17) [R=8, S=23] (G18) [R=6, S=21] (G19) [R=7, S=22] (G20) [R=6, S=21]

(G21) [R=5, S=20] (G22) [R=6, S=21] (G23) [R=5, S=20] (G24) [R=4, S=19] (G25) [R=6, S=21]

(G26) [R=7, S=22] (G27) [R=9, S=24] (G28) [R=7, S=22] (G29) [R=6, S=21] (G30) [R=6, S=21]

Figure 16. Global phase portraits of system (1.3) in the Poincaré
disc (to be continued).

Remark 7.1. Global phase portraits (G43), (G48), (G57), (G84) and (G92) can
appear both under condition c1 = 0 or under c1 6= 0 so, as we are interested
in the topological classification we represent only the non-symmetric case respect
to z-axis, but also the symmetric is possible. The same situation occurs with



32 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

(G31) [R=5, S=20] (G32) [R=4, S=19] (G33) [R=6, S=21] hfil (G34) [R=7, S=22] (G35) [R=5, S=20]

(G36) [R=6, S=21] (G37) [R=5, S=20] (G38) [R=5, S=20] (G39) [R=6, S=21] (G40) [R=7, S=22]

(G41) [R=8, S=21] (G42) [R=7, S=20] (G43) [R=6, S=19] (G44) [R=6, S=19] (G45) [R=4, S=17]

(G46) [R=6, S=19] (G47) [R=8, S=21] hfil (G48) [R=6, S=19] (G49) [R=6, S=19] (G50) [R=5, S=18]

(G51) [R=6, S=19] (G52) [R=4, S=17] (G53) [R=6, S=19] (G54)[R=7, S=20] (G55) [R=4, S=17]

(G56) [R=6, S=19] (G57) [R=6, S=17] (G58) [R=6, S=17] (G59) [R=4, S=15] (G60) [R=4, S=15]

Figure 16. (cont.) Global phase portraits of system (1.3) in the
Poincaré disc (to be continued).

the global phase portraits (G9), (G13)–(G16), (G18), (G19), (G23)–(G27), (G31)–
(G36), (G41), (G45)–(G49), (G55), (G56), (G76)–(G83), (G92)–(G102), which can
appear under the condition c3 = 0 or c3 6= 0 so they can present the symmetric or
the non-symmetric form respect to x-axis, altought we only represent one of them
because we are only interested on the topological classification.



EJDE-2021/35 A CLASS OF KOLMOGOROV SYSTEMS 33

(G61) [R=6, S=17] (G62) [R=8, S=21] (G63) [R=6, S=19] (G64) [R=5, S=18] (G65) [R=6, S=19]

(G66) [R=7, S=20] (G67) [R=6, S=19] (G68) [R=6, S=19] (G69) [R=5, S=18] (G70) [R=5, S=18]

(G71) [R=6, S=19] (G72) [R=7, S=20] (G73) [R=6, S=17] (G74) [R=6, S=17] (G75) [R=5, S=16]

(G76) [R=6, S=17] (G77) [R=3, S=14] (G78) [R=4, S=15] (G79) [R=5, S=16] (G80) [R=3, S=14]

(G81) [R=3, S=14] (G82) [R=4, S=15] (G83) [R=5, S=16] (G84) [R=6, S=17] (G85) [R=5, S=16]

(G86) [R=6, S=17] (G87) [R=5, S=16] (G88) [R=4, S=15] (G89) [R=4, S=15] (G90) [R=4, S=15]

Figure 16. (cont.) Global phase portraits of system (1.3) in the
Poincaré disc (to be continued).

Acknowledgements. E. Diz-Pita and M. V. Otero-Espinar were partially sup-
ported by the Ministerio de Economı́a, Industria y Competitividad, Agencia Es-
tatal de Investigación (Spain), grant MTM2016-79661-P (European FEDER sup-
port included, UE) and the Conselleŕıa de Educación, Universidade e Formación
Profesional (Xunta de Galicia), grant ED431C 2019/10 with FEDER funds. E.



34 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

(G91) [R=5, S=16] (G92) [R=4, S=13] (G93) [R=4, S=13] (G94) [R=3, S=12] (G95) [R=2, S=11]

(G96) [R=2, S=11] (G97) [R=5, S=14] (G98) [R=4, S=13] (G99) [R=3, S=12] (G100) [R=5, S=16]

(G101) [R=4, S=15] (G102) [R=5, S=16]

Figure 16. (cont.) Global phase portraits of system (1.3) in the
Poincaré disc.

Diz-Pita was also supported by the Ministerio de Educacion, Cultura y Deporte de
España, contract FPU17/02125.

J. Llibre was partially supported by the Ministerio de Ciencia, Innovación y Uni-
versidades, Agencia Estatal de Investigación grants 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.

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Érika Diz-Pita

Departamento de Estat́ıstica, Análise Matemática e Optimización, Universidade de San-
tiago de Compostela, 15782 Santiago de Compostela, Spain

Email address: erikadiz.pita@usc.es

Jaume Llibre

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra,

Barcelona, Spain
Email address: jllibre@mat.uab.cat

M. Victoria Otero-Espinar
Departamento de Estat́ıstica, Análise Matemática e Optimización, Universidade de San-

tiago de Compostela, 15782 Santiago de Compostela, Spain
Email address: mvictoria.otero@usc.es



36 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

Table 9. Classification of global phase portrais of system (1.3).

Case Conditions O1 O2 Global

1.1 L12 Unstable node G1
1.2 L3 Stable node G2

1.3
µ < −1 L8 Saddle G3

µ ∈ (−1, 0) L13 Unstable node G4

1.4

µ < −2 L1 Saddle G5
µ ∈ (−1, 0) L4 Saddle G7
µ ∈ (−2,−1) L15 Saddle G7
µ = −2 L17 Saddle G8

1.5 L12 Unstable node G9
1.6 L3 Stable node G10

1.7
µ ∈ (−1, 0) L10 Stable node G11
µ < −1 L11 Saddle G12

1.8

µ < −2 L2 Saddle G13
µ ∈ (−1, 0) L7 Unstable node G14
µ ∈ (−2,−1) L16 Saddle G15
µ = −2 L18 Saddle G16

1.9 L9 Stable node G19
1.10 L6 Unstable node G20
1.11 L12 Unstable node G17

1.12
µ < −1 L8 Saddle G21

µ ∈ (−1, 0) L13 Unstable node G22
1.13 L3 Stable node G18

1.14

µ < −2 L1 Saddle G23
µ ∈ (−1, 0) L4 Stable node G24
µ ∈ (−2,−1) L15 Saddle G25
µ = −2 L17 Saddle G26

1.15 L12 Unstable node G27

1.16
µ < −1 L8 Saddle G35

µ ∈ (−1, 0) L13 Unstable node G36
1.17 L3 Stable node G28

1.18

µ < −2 L1 Saddle G37
µ ∈ (−1, 0) L4 Stable node G38
µ ∈ (−2,−1) L15 Saddle G39
µ = −2 L17 Saddle G40

1.19
µ ∈ (−1, 0) L10 Saddle G29
µ < −1 L11 Saddle G30

1.20

µ < −2 L2 Saddle G31
µ ∈ (−1, 0) L7 Unstable node G32
µ ∈ (−2,−1) L16 Saddle G33
µ = −2 L18 Saddle G34

2.1 L46 Stable node G41
2.2 L47 Unstable node G42

2.3
µ = 0 L14

Unstable node G43
c1 = 0, µ > −1 L36
c1 = 0, µ < −1 L43 Saddle G44



EJDE-2021/35 A CLASS OF KOLMOGOROV SYSTEMS 37

Table 9. (cont.) Classification of global phase portrais of system (1.3).

Case Conditions O1 O2 Global

2.4
µ = 0 L5 Stable node G45

c1 = 0, µ > −1 L35 Stable node G46
c1 = 0, µ < −1 L39 Saddle G47

2.5
µ = 0 L14

Unstable node G48
c1 = 0, µ > −1 L45
c1 = 0, µ < −1 L44 Saddle G49

2.6
µ = 0 L5 Stable node G50

c1 = 0, µ > −1 L38 Stable node G51
c1 = 0, µ < −1 L41 Saddle G52

2.7
µ > −1 L37 Stable node G53
µ < −1 L40 Saddle G54

2.8
µ > −1 L34 Unstable node G55
µ < −1 L42 Saddle G56

2.9
µ = 0 L14

Unstable node G57
c1 = 0, µ > −1 L24
c1 = 0, µ < −1 L26 Saddle G58

2.10
µ = 0 L5 Stable node G59

c1 = 0, µ > −1 L23 Stable node G60
c1 = 0, µ < −1 L25 Saddle G61

3.1 L12 Unstable node G62

3.2
µ > 0 L6 Unstable node G63
µ < −1 L8 Stable node G64

µ ∈ (−1, 0) L13 Unstable node G65

3.3
µ > 0 L3 Stable node G66
µ < −1 L11 Stable node G68

µ ∈ (−1, 0) L10 Stable node G67

3.4

µ < −2 L1 Saddle G69
µ ∈ (−1, 0) L4 Stable node G70
µ ∈ (−2,−1) L15 Saddle G71
µ = −2 L17 Saddle G72

3.5
µ > 0 L3 Stable node G7

µ ∈ (−1, 0) L10 Stable node G74
µ < −1 L11 Saddle G75

3.6 L12 Unstable node G76

3.7
µ < −1 L8 Saddle G77

µ ∈ (−1, 0) L13 Unstable node G78
3.8 L3 Stable node G79

3.9

µ < −2 L1 Saddle G80
µ ∈ (−1, 0) L4 Stable node G81
µ ∈ (−2,−1) L15 Saddle G82
µ = −2 L17 Saddle G83



38 E. DIZ-PITA, J. LLIBRE, M. V. OTERO-ESPINAR EJDE-2021/35

Table 9. (cont.) Classification of global phase portrais of system (1.3).

Case Conditions O1 O2 Global

4.1

µ = 0 L14
Unstable node G84

c1 = 0, µ > −1, a0 + c0µ > 0 L29
c1 = 0, µ > −1, a0 + c0µ < 0 L30 Unstable node G85

c1 = 0, µ < −1 L33 Saddle G86

4.2

µ = 0 L5 Stable node G87
c1 = 0, µ > −1, a0 + c0µ > 0 L27 Stable node G88
c1 = 0, µ > −1, a0 + c0µ < 0 L28 Stable node G89
c1 = 0, µ < −1, a0 + c0µ > 0 L31 Saddle G90
c1 = 0, µ < −1, a0 + c0µ < 0 L32 Saddle G91

4.3
µ = 0 L14

Unstable node G92
c1 = 0, µ > −1 L20
c1 = 0, µ < −1 L22 Saddle G93

4.4
µ = 0 L5 Stable node G94

c1 = 0, µ > −1 L19 Stable node G95
c1 = 0, µ < −1 L21 Saddle G96

5.1
µ > 0 L3 Stable node G97

µ ∈ (−1, 0) L10 Stable node G98
µ < −1 L11 Saddle G99

5.2 L12 Unstable node G76

5.3
µ > 0 L6 Unstable node G100
µ < −1 L8 Saddle G77

µ ∈ (−1, 0) L13 Unstable node G78

5.4
µ > 0 L3 Stable node G79

µ ∈ (−1, 0) L10 Stable node G101
µ < −1 L11 Saddle G102

5.5

µ < −2 L1 Saddle G80
µ ∈ (−1, 0) L4 Stable node G81
µ ∈ (−2,−1) L15 Saddle G82
µ = −2 L17 Saddle G83

6.1
µ = 0 L14

Unstable node G92
c1 = 0, µ > −1 L20
c1 = 0, µ < −1 L22 Saddle G93

6.2
µ = 0 L5 Stable node G94

c1 = 0, µ > −1 L19 Stable node G95
c1 = 0, µ < −1 L21 Saddle G96


	1. Introduction
	2. Preliminaries
	2.1. Poincaré compactification
	2.2. Topological equivalence between two polynomial vector fields
	2.3. Blow-up technique
	2.4. Indices of planar singular points
	2.5. Invariants and Application of the Darboux Theory

	3. Reduction of the Lotka-Volterra systems in R3 to the Kolmogorov systems in R2
	4. Properties of system (1.3)
	5. Study of local finite singular points
	6. Study of local infinite singular points
	6.1. Case c1 non-zero
	6.2. Case c1 zero

	7. Global phase portraits
	Acknowledgements

	References