Publ. Mat. (2014), 421–452
Proceedings of New Trends in Dynamical Systems. Salou, 2012.
DOI: 10.5565/PUBLMAT Extra14 21
A SURVEY ON STABLY DISSIPATIVE
LOTKA–VOLTERRA SYSTEMS WITH AN
APPLICATION TO INFINITE DIMENSIONAL
VOLTERRA EQUATIONS
Waldyr M. Oliva
The present paper is dedicated to Jaume Llibre, Luis Magalh˜aes and
Carlos Rocha, on the occasion of their 60th birthdays
Abstract: For stably dissipative Lotka–Volterra equations the dynamics on the at-
tractor are Hamiltonian and we argue that complex dynamics can occur. We also
present examples and properties of some infinite dimensional Volterra systems with
applications related with stably dissipative Lotka–Volterra equations. We finish by
mentioning recent contributions on the subject.
2010 Mathematics Subject Classification: 34C25, 34K15.
Key words: Lotka–Volterra, stably dissipative systems, conservative systems, Pois-
son manifolds, reducible systems.
1. Introduction
Lotka–Volterra systems were introduced in the 1920s by A. J. Lot-
ka [12] and V. Volterra [27] independently of one another in areas of
chemistry and interaction of populations, respectively.
Volterra’s researches in this field was strongly stimulated by conver-
sations with the biologist Umberto D’Ancona from Siena. He investi-
gated in considerably detail the association of two species, one of which
(the predator) feeds on the other (the prey) and for this case we have
the nowadays called Lotka–Volterra equations (see [29, p. 14, eq(4)]).
Volterra’s concern, however, was with associations of n ≥ 2 species, and
hence with the same number n of differential equations. So, Volterra
introduced the following system as a model for the competition of n bi-
ological species
(1) ˙x
j
= ε
j
x
j
+
n
X
k=1
a
jk
x
j
x
k
, j = 1, . . . , n.
422 W. M. Oliva
In this model, x
j
represents the number of individuals of species j
(so one assumes x
j
> 0), the a
jk
’s are the interaction coefficients, and
the ε
j
’s are parameters that depend on the environment. For example,
ε
j
> 0 means that species j is able to increase with food from the envi-
ronment, while ε
j
< 0 means that it cannot survive when left alone in
the environment. One can also have ε
j
= 0 which means that the popu-
lation stays constant if the species does not interact. This system (1) of
differential equations is also called a Lotka–Volterra system.
In [28] Volterra considered a more general type of equation where
some memory functions play the role of hereditary phenomena; hered-
itary here means, for instance, time of incubation or time of gestation
of the female predator. When two individuals meet and one eats the
other, the population of preys decreases immediately; on the other hand
the population of predators takes a while to increase. This delay is
interpreted as a constant number r > 0 that appears in the system of
equations (see (2)) and it is well known that we are dealing with retarded
functional differential equations (RFDEs). In [5] we see the foundations
and main results of the RFDEs. In particular, they define a flow in an in-
finite dimensional phase space. These more general equations introduced
by Volterra are the following:
(2) ˙x
j
(t) = x
j
(t)
"
ε
j
+
m
X
k=1
a
jk
x
k
(t) +
m
X
k=1
Z
0
−r
x
k
(t + θ)F
jk
(θ) dθ
#
,
where ε
j
, a
jk
are constants and x
j
= x
j
(t) > 0, for j, k = 1, . . . , m,
and F
jk
(θ) are the memory functions; the constant r, 0 < r ≤ ∞, is
called the lag or delay of the equation. See [15] and [14] for cases where
0 < r < ∞ and r = ∞, respectively.
The dynamics of systems of type (1) are far from understood, although
special classes of these Lotka–Volterra systems have been studied. We
distinguish the following classes of systems of this type:
Definition 1.1. A Lotka–Volterra system with interaction matrix A =
(a
ij
) is called
(i) cooperative (resp. competitive) if a
jk
≥ 0 (resp. a
jk
≤ 0) for
all j 6= k;
(ii) conservative if there exists a diagonal matrix D > 0 such that
AD is skew-symmetric;
(iii) dissipative if there exists a diagonal matrix D > 0 such that
AD ≤ 0.
Survey on Lotka–Volterra 423
Competitive systems and dissipative systems are mutually exclusive
classes, except for the trivial case where a
jk
= 0. General results con-
cerning competitive or cooperative systems were obtained by Smale [26]
and Hirsch [7, 8] (for recent results see [30] and references therein).
These systems typically have a global attractor consisting of equilibria
and connections between them (see e. g. [7, Theorem 1.7]).
Dissipative systems have been less studied than competitive systems,
although this class of systems goes back to the pioneer work of Volterra,
who introduced them as a natural generalization of predator-prey sys-
tems (see [29, Chapter III]). For systems where predators and preys co-
exist there is empirical and numerical evidence that periodic oscillations
occur. In fact, as it is well known, for any two dimensional predator-
prey system, the orbits are periodic. But for higher dimensional systems
the topology of orbits in phase space is much more complex, and under-
standing this topology is a challenging problem. The following theorem
(see [3]) is perhaps the first result in this direction.
Theorem 1.2. Consider a Lotka–Volterra system (1) restricted to the
flow invariant set R
n
+
≡ {(x
1
, . . . , x
n
) ∈ R
n
: x
j
> 0, j = 1, . . . , n},
and assume that (i) the system has a singular point, and (ii) is stably
dissipative. Then there exists a global attractor and the dynamics on the
attractor are Hamiltonian.
By “stably dissipative” we mean that the system is dissipative and
every system close to it is also dissipative. As we mentioned before,
the notion of dissipative system is due to Volterra. Stably dissipative
systems where first studied by Redheffer and collaborators (see [21, 22,
23, 24, 25]) under the name “stably admissible”. They gave a beautiful
description of the attractor (see Section 4 below) which we will use to
prove Theorem 1.2. The hypothesis on the existence of a singular point
is equivalent to the assumption that some orbit has a α- or ω-limit point
in R
n
+
.
One of Volterra’s main goals in introducing these equations was the
“mechanization” of biology, and he made quite an effort in trying to
pursue this program. While seeking a variational principle for the sys-
tem, he was successful in finding a Hamiltonian formulation in the case
where the interaction matrix is skew-symmetric, at the expense of dou-
bling the number of dimensions (see Section 2 for details). Along the
way, a polemic with Levi-Civita arose, an account of which can be found
in [10]. In this paper we shall give a different solution to the problem
of putting system (1) into a Hamiltonian frame. In modern language,
our approach is related to Volterra’s approach by a reduction procedure.
424 W. M. Oliva
This Hamiltonian frame is the basis for the Hamiltonian structure refered
to in Theorem 1.2.
Once the Hamiltonian character of the dynamics is established, one
would like to understand (i) what type of attractors one can get and
(ii) what kind of Hamiltonian dynamics one can have on the attractor.
It will follow from our work that this amounts to classify the dynamics
of Lotka–Volterra systems with skew-symmetric matrix whose associated
graph is a forest. We do not know of such classification but we shall argue
that these dynamics can be rather complex.
In the simplest situation, the attractor will consist of the unique fixed
point in R
n
+
and the dynamics will be trivial. It was already observed
in [23] that there may exist periodic orbits on (non-trivial) attractors.
On the other hand, if the attractor is an integrable Hamiltonian system
then one can expect the orbits to be almost periodic. As we will see be-
low, through a detailed study of a 4-dimensional chain of predator-prey
systems, non-integrable Hamiltonian system can indeed occur. There-
fore, typically, the dynamics of dissipative Lotka–Volterra systems are
extremely complex. This is related with a famous conjecture in the the-
ory of Hamiltonian systems which can be stated as follows (see [20, 17]):
Typically, dynamics on the common level sets of the Hamil-
tonian and the Casimirs are ergodic.
This survey is organized in two parts. In the first part we deal with
basic notions of general systems of type (1) and prove Theorem 1.2. In
the second part we present some examples and also applications of the
stably dissipative systems to Volterra’s systems of type (2), with and
without delays. We finish by mentioning some recent contributions on
the subject.
Part I. General Theory
2. Basic notions
Here, we will recall some basic notions and facts concerning general
Lotka–Volterra systems which will be useful in the next sections. All
of these notions can be traced back to Volterra. For a more detailed
account of general properties of Lotka–Volterra systems we refer to the
book by Hofbauer and Sigmund [9].
For fixed d
j
6= 0, the transformation
(3) y
j
=
1
d
j
x
j
, j = 1, . . . , n,
Survey on Lotka–Volterra 425
takes the Volterra system (1) with interaction matrix A, into a new
Volterra system with interaction matrix AD
(4) ˙y
j
= ε
j
y
j
+
n
X
k=1
d
k
a
jk
y
j
y
k
, j = 1, . . . , n.
We can therefore think of (3) as a gauge symmetry of the system. A
choice of representative (a
jk
) in a class of equivalence under gauge trans-
formations will be called a choice of gauge. Since will often take as phase
space R
n
+
, we consider only gauge transformations with d
j
> 0 in order
to preserve phase space. Note also that the classes of Lotka–Volterra
systems introduced in Definition 1.1 above are all gauge invariant.
Many properties of a Lotka–Volterra system can be expressed geo-
metrically in terms of its associated graph G(A, ε). This is the labeled
graph, where with each species j we associate a vertex
f
labeled with ε
j
and we draw an edge connecting vertex j to vertex k whenever a
jk
6= 0.
ε
5
f
ε
4
f
ε
8
f
f
ε
1
f
ε
2
f
ε
3
f
ε
6
f
ε
7
f
ε
9
Figure 1. Graph G(A, ε) associated with a system of
type (1).
For example, if two systems are gauge equivalent, they have the same
unlabeled graph (but not conversely). Also, conservative systems can
be caracterized in terms of its graph as it follows from the following
proposition also due to Volterra [29, Chapter III, §12]:
Proposition 2.1. A Lotka–Volterra system with interaction matrix A =
(a
jk
) is conservative if, and only if, a
jj
= 0,
(5) a
jk
6= 0 =⇒ a
jk
a
kj
< 0, j 6= k,
and
(6) a
i
1
i
2
a
i
2
i
3
· · · a
i
s
i
1
= (−1)
s
a
i
s
i
s−1
· · · a
i
2
i
1
a
i
1
i
s
for every finite sequence of integers (i
1
, . . . , i
s
), with i
r
∈ {1, . . . , n} for
r = 1, . . . , s.