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Journal ArticleDOI

A survey on stably dissipative Lotka-Volterra systems with an application to infinite dimensional Volterra equations

01 Apr 2014-Publicacions Matematiques (Universitat Autònoma de Barcelona, Departament de Matemàtiques)-Vol. 58, pp 421-452

AbstractFor stably dissipative Lotka{Volterra equations the dynamics on the attractor 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.

Summary (2 min read)

1. Introduction

  • He investigated in considerably detail the association of two species, one of which (the predator) feeds on the other (the prey) and for this case the authors have the nowadays called Lotka–Volterra equations (see [29, p. 14, eq(4)]).
  • The dynamics of systems of type (1) are far from understood, although special classes of these Lotka–Volterra systems have been studied.
  • While seeking a variational principle for the system, he was successful in finding a Hamiltonian formulation in the case where the interaction matrix is skew-symmetric, at the expense of doubling the number of dimensions (see Section 2 for details).
  • It was already observed in [23] that there may exist periodic orbits on (non-trivial) attractors.

2. Basic notions

  • Here, the authors will recall some basic notions and facts concerning general Lotka–Volterra systems which will be useful in the next sections.
  • For a more detailed account of general properties of Lotka–Volterra systems the authors refer to the book by Hofbauer and Sigmund [9].
  • Similarly, to exclude α-limit points one uses the Liapunov function −V .
  • On the other hand, the following result shows that the average behavior of the orbits is related to the values of the fixed points (see [2]).

3. Conservative systems

  • In the case were system (1) is conservative Volterra was able to introduce a Hamiltonian structure for the system by doubling the number of variables.
  • The authors recall now Volterra’s construction, so they assume that system (1) is conservative and a choice of gauge has been made so that the matrix (ajk) is skew-symmetric.
  • The full justification of this procedure will be given later in the section.
  • Now, if one introduces another set of variables Recall that the modern approach to Hamiltonian systems is based on the following generalization of the notion of a Poisson bracket (see for example [1], [17]).

4. Dissipative systems

  • Since the authors want their results to persist under small perturbation they introduce the following definition.
  • Note that the authors only allow perturbations that have the same graph as the original system.
  • Also they use instead the name stably admissible.
  • For stably dissipative systems this choice can be improved [24]: Lemma 4.2.
  • It will be convenient to modify slightly the notion of graph associated with the system the authors introduced above.

5. Examples

  • Example 5.1. In [3, p. 159, eq. (33)], the authors give a detailed analysis of a 4-dimensional Lotka–Volterra system in order to illustrate the complexity of the dynamics that can occur on the attractor.
  • These equations were extensively analysed by Duarte, Fernandes and Oliva in [3] where the flow generated by (51) was discussed using a reduction procedure to establish the existence of invariant sets with a Hamiltonian structure.
  • Theorem 5. Consider the system of Lotka–Volterra equations (61) coupled with the linear equations (62) and assume that: (i) the Lotka– Volterra system (61) has a singular point q ∈ Rm+ ; and, (ii) the interaction matrix A is stably dissipative.
  • The paper [31] by Zhao and Luo proves necessary and sufficient conditions for a matrix to be stably dissipative.

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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)
#
,
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
s1
· · · 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.

Citations
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Abstract: Autonomous sustained oscillations are ubiquitous in living and nonliving systems. As open systems, far from thermodynamic equilibrium, they defy entropic laws which mandate convergence to stationarity. We present structural conditions on network cycles which support global Hopf bifurcation, i.e. global bifurcation of non-stationary time-periodic solutions from stationary solutions. Specifically, we show how monotone feedback cycles of the linearization at stationary solutions give rise to global Hopf bifurcation, for sufficiently dominant coefficients along the cycle. We include four example networks which feature such strong feedback cycles of length three and larger: Oregonator chemical reaction networks, Lotka-Volterra ecological population dynamics, citric acid cycles, and a circadian gene regulatory network in mammals. Reaction kinetics in our approach are not limited to mass action or Michaelis-Menten type.

1 citations


References
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Book
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Abstract: 1 Introduction to Lie Groups- 11 Manifolds- Change of Coordinates- Maps Between Manifolds- The Maximal Rank Condition- Submanifolds- Regular Submanifolds- Implicit Submanifolds- Curves and Connectedness- 12 Lie Groups- Lie Subgroups- Local Lie Groups- Local Transformation Groups- Orbits- 13 Vector Fields- Flows- Action on Functions- Differentials- Lie Brackets- Tangent Spaces and Vectors Fields on Submanifolds- Frobenius' Theorem- 14 Lie Algebras- One-Parameter Subgroups- Subalgebras- The Exponential Map- Lie Algebras of Local Lie Groups- Structure Constants- Commutator Tables- Infinitesimal Group Actions- 15 Differential Forms- Pull-Back and Change of Coordinates- Interior Products- The Differential- The de Rham Complex- Lie Derivatives- Homotopy Operators- Integration and Stokes' Theorem- Notes- Exercises- 2 Symmetry Groups of Differential Equations- 21 Symmetries of Algebraic Equations- Invariant Subsets- Invariant Functions- Infinitesimal Invariance- Local Invariance- Invariants and Functional Dependence- Methods for Constructing Invariants- 22 Groups and Differential Equations- 23 Prolongation- Systems of Differential Equations- Prolongation of Group Actions- Invariance of Differential Equations- Prolongation of Vector Fields- Infinitesimal Invariance- The Prolongation Formula- Total Derivatives- The General Prolongation Formula- Properties of Prolonged Vector Fields- Characteristics of Symmetries- 24 Calculation of Symmetry Groups- 25 Integration of Ordinary Differential Equations- First Order Equations- Higher Order Equations- Differential Invariants- Multi-parameter Symmetry Groups- Solvable Groups- Systems of Ordinary Differential Equations- 26 Nondegeneracy Conditions for Differential Equations- Local Solvability- In variance Criteria- The Cauchy-Kovalevskaya Theorem- Characteristics- Normal Systems- Prolongation of Differential Equations- Notes- Exercises- 3 Group-Invariant Solutions- 31 Construction of Group-Invariant Solutions- 32 Examples of Group-Invariant Solutions- 33 Classification of Group-Invariant Solutions- The Adjoint Representation- Classification of Subgroups and Subalgebras- Classification of Group-Invariant Solutions- 34 Quotient Manifolds- Dimensional Analysis- 35 Group-Invariant Prolongations and Reduction- Extended Jet Bundles- Differential Equations- Group Actions- The Invariant Jet Space- Connection with the Quotient Manifold- The Reduced Equation- Local Coordinates- Notes- Exercises- 4 Symmetry Groups and Conservation Laws- 41 The Calculus of Variations- The Variational Derivative- Null Lagrangians and Divergences- Invariance of the Euler Operator- 42 Variational Symmetries- Infinitesimal Criterion of Invariance- Symmetries of the Euler-Lagrange Equations- Reduction of Order- 43 Conservation Laws- Trivial Conservation Laws- Characteristics of Conservation Laws- 44 Noether's Theorem- Divergence Symmetries- Notes- Exercises- 5 Generalized Symmetries- 51 Generalized Symmetries of Differential Equations- Differential Functions- Generalized Vector Fields- Evolutionary Vector Fields- Equivalence and Trivial Symmetries- Computation of Generalized Symmetries- Group Transformations- Symmetries and Prolongations- The Lie Bracket- Evolution Equations- 52 Recursion Operators, Master Symmetries and Formal Symmetries- Frechet Derivatives- Lie Derivatives of Differential Operators- Criteria for Recursion Operators- The Korteweg-de Vries Equation- Master Symmetries- Pseudo-differential Operators- Formal Symmetries- 53 Generalized Symmetries and Conservation Laws- Adjoints of Differential Operators- Characteristics of Conservation Laws- Variational Symmetries- Group Transformations- Noether's Theorem- Self-adjoint Linear Systems- Action of Symmetries on Conservation Laws- Abnormal Systems and Noether's Second Theorem- Formal Symmetries and Conservation Laws- 54 The Variational Complex- The D-Complex- Vertical Forms- Total Derivatives of Vertical Forms- Functionals and Functional Forms- The Variational Differential- Higher Euler Operators- The Total Homotopy Operator- Notes- Exercises- 6 Finite-Dimensional Hamiltonian Systems- 61 Poisson Brackets- Hamiltonian Vector Fields- The Structure Functions- The Lie-Poisson Structure- 62 Symplectic Structures and Foliations- The Correspondence Between One-Forms and Vector Fields- Rank of a Poisson Structure- Symplectic Manifolds- Maps Between Poisson Manifolds- Poisson Submanifolds- Darboux' Theorem- The Co-adjoint Representation- 63 Symmetries, First Integrals and Reduction of Order- First Integrals- Hamiltonian Symmetry Groups- Reduction of Order in Hamiltonian Systems- Reduction Using Multi-parameter Groups- Hamiltonian Transformation Groups- The Momentum Map- Notes- Exercises- 7 Hamiltonian Methods for Evolution Equations- 71 Poisson Brackets- The Jacobi Identity- Functional Multi-vectors- 72 Symmetries and Conservation Laws- Distinguished Functionals- Lie Brackets- Conservation Laws- 73 Bi-Hamiltonian Systems- Recursion Operators- Notes- Exercises- References- Symbol Index- Author Index

7,697 citations


"A survey on stably dissipative Lotk..." refers background in this paper

  • ...Recall that the modern approach to Hamiltonian systems is based on the following generalization of the notion of a Poisson bracket (see for example [1], [17])....

    [...]

  • ...A standard result (see [17]) in the theory of Hamiltonian systems says that a family of r-independent, Poisson commuting integrals, allows one to reduce the dimension of the system by 2r....

    [...]

  • ...This is related with a famous conjecture in the theory of Hamiltonian systems which can be stated as follows (see [20, 17]): Typically, dynamics on the common level sets of the Hamiltonian and the Casimirs are ergodic....

    [...]


Book
05 Apr 1977
Abstract: 1 Linear differential difference equations.- 1.1 Differential and difference equations.- 1.2 Retarded differential difference equations.- 1.3 Exponential estimates of x(?, f).- 1.4 The characteristic equation.- 1.5 The fundamental solution.- 1.6 The variation-of-constants formula.- 1.7 Neutral differential difference equations.- 1.8 Supplementary remarks.- 2 Retarded functional differential equations : basic theory.- 2.1 Definition.- 2.2 Existence, uniqueness, and continuous dependence.- 2.3 Continuation of solutions.- 2.4 Differentiability of solutions.- 2.5 Backward continuation.- 2.6 Caratheodory conditions.- 2.7 Supplementary remarks.- 3 Properties of the solution map.- 3.1 Finite- or infinite-dimensional problem?.- 3.2 Equivalence classes of solutions.- 3.3 Exponential decrease for linear systems.- 3.4 Unique backward extensions.- 3.5 Range in ?n.- 3.6 Compactness and representation.- 3.7 Supplementary remarks.- 4 Autonomous and periodic processes.- 4.1 Processes.- 4.2 Invariance.- 4.3 Discrete systems-maximal compact invariant sets.- 4.4 Fixed points of discrete dissipative processes.- 4.5 Stability and maximal invariant sets in processes.- 4.6 Periodic trajectories of ?-periodic processes.- 4.7 Convergent systems.- 4.8 Supplementary remarks.- 5 Stability theory.- 5.1 Definitions.- 5.2 The method of Liapunov functional.- 5.3 Liapunov functional for autonomous systems.- 5.4 Razumikhin-type theorems.- 5.5 Supplementary remarks.- 6 General linear systems.- 6.1 Global existence and exponential estimates.- 6.2 Variation-of-constants formula.- 6.3 The formal adjoint equation.- 6.4 The true adjoint.- 6.5 Boundary-value problems.- 6.6 Stability and boundedness.- 6.7 Supplementary remarks.- 7 Linear autonomous equations.- 7.1 The semigroup and infinitesimal generator.- 7.2 Spectrum of the generator-decomposition of C.- 7.3 Decomposing C with the formal adjoint equation.- 7.4 Estimates on the complementary subspace.- 7.5 An example.- 7.6 The decomposition in the variation-of-constants formula.- 7.7 Supplementary remarks.- 8 Linear periodic systems.- 8.1 General theory.- 8.2 Decomposition.- 8.3 Supplementary remarks.- 9 Perturbed linear systems.- 9.1 Forced linear systems.- 9.2 Bounded, almost-periodic, and periodic solutions stable and unstable manifolds.- 9.3 Periodic solutions-critical cases.- 9.4 Averaging.- 9.5 Asymptotic behavior.- 9.6 Boundary-value problems.- 9.7 Supplementary remarks.- 10 Behavior near equilibrium and periodic orbits for autonomous equations.- 10.1 The saddle-point property near equilibrium.- 10.2 Nondegenerate periodic orbits.- 10.3 Hyperbolic periodic orbits.- 10.4 Supplementary remarks.- 11 Periodic solutions of autonomous equations.- 11.1 Hopf bifurcation.- 11.2 A periodicity theorem.- 11.3 Range of the period.- 11.4 The equation $$\dot x(t) = - \alpha x(t - 1)[1 + x(t)]$$.- 11.5 The equation $$\dot x(t) = - \alpha x(t - 1)[1 - {x^2}(t)]$$.- 11.6 The equation $$\ddot x(t) + f(x(t))\dot x(t) + g(x(t - r)) = 0$$.- 11.7 Supplementary remarks.- 12 Equations of neutral type.- 12.1 Definition of a neutral equation.- 12.2 Fundamental properties.- 12.3 Linear autonomous D operators.- 12.4 Stable D operators.- 12.5 Strongly stable D operators.- 12.6 Properties of equations with stable D operators.- 12.7 Stability theory.- 12.8 General linear equations.- 12.9 Stability of autonomous perturbed linear systems.- 12.10 Linear autonomous and periodic equations.- 12.11 Nonhomogeneous linear equations.- 12.12 Supplementary remarks.- 13 Global theory.- 13.1 Generic properties of retarded equations.- 13.2 The set of global solutions.- 13.3 Equations on manifolds : definitions.- 13.4 Retraded equations on compact manifolds.- 13.5 Further properties of the attractor.- 13.6 Supplementary remarks.- Appendix Stability of characteristic equations.

5,581 citations


"A survey on stably dissipative Lotk..." refers background in this paper

  • ...In [5] we see the foundations and main results of the RFDEs....

    [...]

  • ...Equation (41) is a functional differential equation, so in standard notation this equation has the form ẋ(t)=f(xt), where xt(θ)=x(t+θ) for θ ∈ (−∞, 0] and, in an appropriate phase space, it generates a nonlinear dynamical system ([5], [6])....

    [...]



Book
15 Oct 2018
TL;DR: The author has the problem of evolution always before him, and considers analytically the effect on population of a change in the behaviour of individuals in Elements of Physical Biology.

1,839 citations


Frequently Asked Questions (1)
Q1. What have the authors contributed in "A survey on stably dissipative lotka–volterra systems with an application to infinite dimensional volterra equations" ?

The authors also present examples and properties of some infinite dimensional Volterra systems with applications related with stably dissipative Lotka–Volterra equations.