Journal ArticleDOI

# Passivity of Lotka–Volterra and quasi-polynomial systems

01 Apr 2021-Nonlinearity (IOP Publishing)-Vol. 34, Iss: 4, pp 1880-1899

### 1. Introduction

• Lotka–Volterra systems are widely-used models to describe the dynamic behavior of interactive species or agents [1].
• In the paper [4] the equivalences among the different stability types of Lotka–Volterra systems were discussed.
• This advantageous property is a basis of several results in the field, see e.g. [5–7].
• Passivity theory offers a possibility to overcome this difficulty, that is one of the motivations of their present work.
• It allows a system categorisation in terms of energy transfer between the system and its environment.

### 2.1. Passive systems

• The input-affine system (1) is passive w.r.t.
• If the system (1) is zero state detectable and passive w.r.t.
• Assume that the system (1) is passive w.r.t.

### 2.2. Lotka–Volterra systems

• The notion of Volterra–Lyapunov stability (see [15]) differs from the common Lyapunov stability definition invoked in theorem 2.
• The Volterra–Lyapunov matrix stability is used here to conclude on the asymptotic stability of Lotka–Volterra systems. .
• The strict inequality (8) is a sufficient asymptotic stability condition for the autonomous Lotka–Volterra system.
• If the coefficient matrix M is rank-deficient (i.e. not invertible), this condition cannot be applied.

### 3.1. Passivity of Lotka–Volterra systems

• For passivity analysis define the error state: e = x − x∗, (16) where x∗ is a strictly positive equilibrium point satisfying (5).
• −Ky asymptotically stabilises the equilibrium state x∗.
• It was exploited that the sum of a negative definite matrix and a negative semidefinite matrix having the same dimensions is a negative definite matrix.
• Note, that the full rank of matrix M is not necessary for passivity.

### 3.2. Special Lotka–Volterra systems

• As follows, the passivity properties of two important classes of Lotka–Volterra systems are analysed.
• The coefficients m12, m21 ∈ R>0 determine the inter-influence of the predator–prey population.

### 3.3. Passivity of quasi-polynomial systems

• The generalisation of the passivity-related results for Lotka–Volterra systems to the QP case is based on the joint storage function family of the two system classes.
• The passivity of the QP system (30) can be investigated in the monimial space, i.e. through the Lotka–Volterra variables, by analysing the Lotka–Volterra model that corresponds to (30).
• The error state of the QP model is then defined using the Lotka–Volterra states [i.e. the QP quasimonomials of the QP system (30)].
• This way, the passivity of the QP model can be handled using the above results on the passivity of Lotka–Volterra models and on the fact, that QP and Lotka–Voltera models share their storage functions (6) and (14).

### 4. Feedback equivalence of Lotka–Volterra systems to passive systems

• In the case of a Lotka–Volterra system that is not passive or it is passive only for some set of its parameters, one can use the notion of feedback equivalence to ensure and investigate the passivity.
• The notion of feedback equivalence introduces a class of input-affine systems, that are not necessarily passive, but they can be made passive by applying suitable state feedback to them.
• In the general case, the skew-symmetry property of the interconnection matrix of the controlled system can be assured by choosing u = Kpe + up, Kp ∈ Rm×m. (33) With this input the interconnection matrix (35) Accordingly, the Lotka–Volterra system is equivalent to a passive system where the input is defined in equations (33) and (35).
• This method includes extra parameters into the Lotka–Volterra coefficient matrix M in a bilinear way and turns the LMI (32) into a bilinear matrix inequality, which belongs to an NP-complete problem class.
• This method is not necessary in the present case.

### 5. Disturbance attenuation

• Passivity theory opens the possibility to design powerful controllers not only for stabilising but also for disturbance rejection purposes [20].
• This section proposes a physically meaningful yet realisable controller for disturbance attenuation of Lotka–Volterra systems.

### 5.1. The control problem

• In realistic Lotka–Volterra systems the natural rate cannot be considered constant, its nominal value may change in time.
• The deviation of the rate vector from its nominal value can be viewed as an additive, non-constant, bounded disturbance.

### 5.3. Disturbance attenuation

• Let the prescribed attenuation gain be γ.
• It yields that, if the controller gain matrix is chosen such that k > 1 2 ( 1 + 1 γ ) , (47) then the disturbance attenuation control objective is achieved.

### 6.1. Lotka–Volterra model

• A three-dimensional Lotka–Volterra predator–prey system is considered which describes the behavior of three coexisting species: two predators and one prey.
• The equilibrium states of the predators’ populations always satisfy m12x∗2 + m13x∗3 = l1.
• Without control, the extinction of one of the predator populations can be observed.
• The simulation results presented in figures 3 and 4 show that the proposed disturbance attenuation approach ensures the convergence of the controlled states to the setpoint.

### 6.2. Quasi-polynomial model with m = n

• The examined QP system belongs to the same equivalence class as the Lotka–Volterra model (51) of the previous example.
• For the sake of simplicity, the numerical parameter values of section 6.1 has been used here.
• This also means, that they are dynamically similar.
• These quasimonomials correspond to the Lotka–Volterra system (51)’s state variables x1, x2 and x3, respectively.
• The simulation results presented in figures 7 and 8 show that the proposed disturbance attenuation approach ensures the convergence of the controlled states to the setpoint.

### 6.3. Quasi-polynomial model with m = n

• Usually, the Lotka–Volterra and the QP systems has state spaces of different dimensions [7].
• The typical situation is when the QP system is embedded into the Lotka–Volterra form.
• The Lotka–Volterra embedding inflates the originally n dimensional state space to an m dimensional Lotka–Volterra dynamics.
• The simulation results presented in figures 11 and 12 show that the proposed disturbance attenuation approach ensures the convergence of the controlled states to the setpoint which was determined from the Lotka–Volterra case using the quasi-monomial transformation corresponding to the QP system.
• Similarly to the Lotka–Volterra case, smaller prescribed disturbance attenuation level ensures smaller steady state error.

### 7. Discussion and conclusions

• Using the classical logarithmic Lyapunov function of autonomous Lotka–Volterra and QP systems the authors derive passivity conditions of the open version of these systems.
• This means that one considers the possibility of manipulating the death/birth rate vectors of the species as inputs.
• The condition of passivity in (21) is less strict than the classical one by requiring only semidefiniteness.
• By using the input-extended model of the system, it was shown that the asymptotic stability i.e. the convergence into the equilibrium point, can be achieved by using diagonal state feedback having arbitrarily small gains.
• Moreover, passivity theory provides a novel and powerful approach to design linear static feedback control laws for Lotka–Volterra systems.

Did you find this useful? Give us your feedback

Content maybe subject to copyright    Report

Nonlinearity
PAPER • OPEN ACCESS
Passivity of Lotka–Volterra and quasi-polynomial systems
To cite this article: Lrinc Márton et al 2021 Nonlinearity 34 1880
View the article online for updates and enhancements.

London Mathematical Society
Nonlinearity
Nonlinearity 34 (2021) 1880–1899 https://doi.org/1 0.1088/1361-6544/abd52b
Passivity of LotkaVolterra and
quasi-polynomial systems
orinc rton
1,
, Katalin M Hangos
2,3
and Attila Magyar
3
1
Department of Electrical Engineering, Sapientia Hungarian University of
Transylvania, Tirgu Mures, Romania
2
Systems and Control Laboratory, Institute for Computer Science and Control,
Budapest, Hungary
3
Department of Electrical Engineering and Information Systems, University of
Pannonia, Veszpr
´
em, Hungary
E-mail: martonl@ms.sapientia.ro
Received 21 May 2020, revised 7 December 2020
Accepted for publication 18 December 2020
Published 18 February 2021
Abstract
This study approaches the stability analysis and controller design of
LotkaVolterra and quasi-polynomial systems from the perspective of passiv-
ity theory. The passivity based approach requires to extend the autonomous
system model with a suitable input structure. The condition of passivity for
LotkaVolterra systems is less strict than the classic asymptotic stability cri-
terion. It is shown that each LotkaVolterra system is feedback equivalent to
a passive system and a passifying state feedback controller is proposed. The
passivity based approach enables the design of novel state feedback controllers
to Lotka–Volterra systems. The asymptotic stability can be achieved by apply-
ing an additional diagonal state feedback having arbitrarily small gains. This
result was further explored to achieve rate disturbance attenuation in controlled
LotkaVolterra systems. By exploiting the dynamical similarities between the
LotkaVolterra and quasi-polynomial systems, it was shown that the passivity
related results, developed for LotkaVolterra systems, are also valid for a large
class of quasi-polynomial systems. The methods and tools developed have been
illustrated through simulation case studies.
Keywords: passivity, LotkaVolterra system, quasi-polynomial system, distur-
bance attenuation, stability
Recommended by Professor Alain Goriely.
Author to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons
Attribution 3.0 licence. Any further distribution of this work must maintain attribution
to the author(s) and the title of the work, journal citation and DOI.
1361-6544/21/041880+20\$33.00 © 2021 IOP Publishing Ltd & London Mathematical Society Printed in the UK 1880

Nonlinearity 34 (2021) 1880 LMártonet al
Mathematics Subject Classication numbers: 93C10, 93D15.
(Some gures may appear in colour only in the online journal)
1. Introduction
LotkaVolterra systems are widely-used models to describe the dynamic behavior of interac-
tive species or agents [1]. Their properties are continually studied by many researchers, see e.g.
[2, 3]. In the paper [4] the equivalences among the different stability types of LotkaVolterra
systems were discussed.
The family of quasi-polynomial (QP) systems is regarded as a generalisation of the
LotkaVolterra form. One important property of the QP system class is that it admits a
partitioning, where each class of equivalence shares the basic dynamical properties with a
LotkaVolterra model. This advantageous property is a basis of several results in the eld, see
e.g. [57].
The classical stability result developed for LotkaVolterra systems relates the positive def-
initeness of a linear matrix inequality with the asymptotic stability to a positive equilibrium
point of the system [8]. This is not applicable when the LotkaVolterra system model is rank
decient, i.e. it originates from a QP system.
Passivity theory offers a possibility to overcome this difculty, that is one of the motivations
of our present work. The idea of applying passivity theory to LotkaVolterra systems is not
new. Early results related to the passivity-based control in a class of LotkaVolterra systems
can be found in [9]. Other related feedback control approaches for LotkaVolterra systems
were presented e.g. in [1012].
Passivity is an important inputoutput property of many physical systems. It allows a system
categorisation in terms of energy transfer between the system and its environment. Roughly
speaking a system is passive if it cannot store more energy than it is supplied to it from the envi-
ronment. The internal energy of the system is characterised by a non-negative, state-dependent
storage function assigned to the system. In the passivity theory framework, the rate of the sup-
plied energy is taken as the inner product of the ‘power-coupled’ input and output vectors of
the system.
Stability analysis and control design are two important applications of the passivity
theory [13]. The passivity property of input afne systems involves the stability of the
autonomous part under mild conditions. The passivity can also be applied to achieve desired
dynamic behavior (e.g. disturbance rejection) for the system by appropriately manipulating the
system’s input.
The models of many physical systems do not possess the passivity property. However, there
exist such system models that can be transformed into passive systems using feedback. A
system is called feedback equivalent to a passive system if it can be rendered to a passive
system by performing a proper static, state-dependent, afne transformation of the original
input [13].
Motivated by the above results, the contributions of this paper are as follows. Using the clas-
sical logarithmic Lyapunov function of autonomous LotkaVolterra and QP systems we derive
passivity conditions of the open version of these systems and show that each LotkaVolterra
system is feedback equivalent to a passive system. The results were extended to the class of
QP systems, too. In addition, a control design method is presented to attenuate the effect of
death rate or birth rate disturbances in controlled LotkaVolterra systems.
1881

Nonlinearity 34 (2021) 1880 LMártonet al
2. Basic notions
2.1. Passive systems
In this section relevant denitions and theorems from passivity theory are reviewed, see e.g.
[13, 14].
We consider a dynamic system which is modeled using a non-autonomous, input afne
ordinary differential equation (ODE) in the form
˙
x = f(x) + G(x)u, x(0) = x
0
,
y = h(x),
(1)
where x = x(t) R
n
, t 0 is state vector, x
0
R
n
, y, u R
m
are the output- and input vectors,
f(·), h(·), G(·) are smooth mappings with appropriate dimensions, and f(0) = 0, h(0) = 0 where
0 = (0 ...0)
T
with appropriate dimension.
The condition f (0) = 0 implies that x = 0 is an equilibrium point of the autonomous system
˙
x = f(x).
We assign to the system (1) a continuously differentiable, nonnegative storage function
S(x):R
n
R
0
such that S(0) = 0.
Definition 1. The system (1)ispassive with respect to the storage function S if
˙
S y
T
u, u, x.
Theorem 1. The input-afne system (1) is passive w.r.t. S if and only if the following
conditions hold:
S
x
f(x) 0, (2)
S
x
G(x) = h(x)
T
. (3)
Definition 2. The system (1)iszero state detectable if y(t) = 0 and u(t) = 0, t 0, imply
that lim
t→∞
x(t) = 0.
Theorem 2. If the system (1) is zero state detectable and passive w.r.t. S, then the equilib-
rium state x = 0 of the unforced system
˙
x = f(x) is Lyapunov stable, i.e. >0, δ() > 0
such that x
0
δ() implies x(t) <t > 0.
Theorem 3. Assume that the system (1) is passive w.r.t. S such that
2
S
x
2
exists and it is
continuous. If rank{
h
x
G(x)} is constant in a neighborhood of 0, then the system has a vector
relative degree {1, 1, ...,1} at x = 0.
Theorem 4. Consider that the system (1) is zero state detectable and passive w.r.t. S. Then
the control law u = Ky, where K = diag(k
i
) R
m×m
, k
i
> 0, asymptotically stabilises the
equilibrium state x = 0, i.e. lim
t→∞
x(t) = 0.
2.2. LotkaVolterra systems
The dynamic behavior of an autonomous LotkaVolterra system is described by the following
ODE [1, 8]:
˙
x = diag (x)(Mx + l), x(0) = x
0
,(4)
1882

Nonlinearity 34 (2021) 1880 LMártonet al
where x R
m
0
is the state vector in which each entry represents a species population; the
matrix M = (m
ij
) R
m×m
describes the interactions among the species; l R
m
is the natural
rate vector, x
0
R
m
>0
is the constant vector of the initial states.
Equilibrium points: a natural equilibrium point of the system (4)isx
= 0.
If rank[M l] = rank M, the system admits other equilibrium points, which satisfy the
equation
Mx
= l. (5)
Let us assume that the system admits a strictly positive equilibrium point x
= (x
i
),
x
i
R
>0
, i = 1, ..., m. The stability of the system (4) around the positive equilibrium can
be analysed using the storage function:
S =
m
i=1
c
i
x
i
x
i
x
i
ln
x
i
x
i
,(6)
where c
i
R
>0
, i = 1, ..., m.
The time derivative of the storage function reads as [8]:
˙
S =
1
2
(x x
)
T
(MC + CM
T
)(x x
). (7)
Here C = diag (c
1
c
2
...c
m
).
Definition 3 [15]. AmatrixM is VolterraLyapunov stable if there exist a positive denite,
diagonal matrix C such that
MC + CM
T
< 0. (8)
Remarks:
The notion of VolterraLyapunov stability (see [15]) differs from the common Lyapunov
stability denition invoked in theorem 2. The VolterraLyapunov matrix stability is used
here to conclude on the asymptotic stability of LotkaVolterra systems.
The stability of LotkaVolterra systems is independent of the offset vector l.
The strict inequality (8) is a sufcient asymptotic stability condition for the autonomous
LotkaVolterra system. However, if the coefcient matrix M is rank-decient (i.e. not
invertible), this condition cannot be applied.
2.3. Quasi-polynomial systems
A generalisation of the LotkaVolterra system class is the so-called QP or generalised
Lotka–Volterra model (9):
˙
z = diag (z)(Ax(z) + λ), z(0) = z
0
. (9)
In the model (9) z R
m
>0
denotes the state vector, A = (a
ij
) R
n×m
.LetB = (b
ij
) R
m×n
.
The vector x(z) of the so-called quasi-monomials are dened as below:
x
j
=
n
k=1
z
B
jk
k
, j = 1, ..., m. (10)
It is an important property of QP systems, that the set of quasi-monomials x(z)admita
LotkaVolterra dynamics having the following parameters [16]:
1883

##### Citations
More filters
Journal ArticleDOI
01 Sep 2022
TL;DR: In this article , a new approach to the long-standing problem of global asymptotic positive observer design for n-dimensional Lotka-Volterra (LV) systems was developed, instead of the state error dynamics, the notion of state ratio dynamics is used by exploiting the positivity property of the LV systems.
Abstract: This paper develops a new approach to the long-standing problem of global asymptotic positive observer design for n -dimensional Lotka–Volterra (LV) systems. In this approach, instead of the state error dynamics, the notion of state ratio dynamics is used by exploiting the positivity property of the LV systems. A novel class of observers is proposed and the global asymptotic convergence of such observers is studied via a well-known logarithmic Lyapunov function. The obtained convergence conditions of the proposed observer are formulated as a linear matrix inequality (LMI), which can be used to determine the unknown parameters of the observer. This design procedure is modified for state estimation of time-varying LV systems, too. The applicability of the proposed method is investigated through two numerical examples.

1 citations

##### References
More filters
Journal ArticleDOI
TL;DR: In this article, three types of stability of real matrices are compared and necessary conditions are obtained in terms of the principal submatrices of a real matrix for normal matrices and matrices whose off-diagonal elements are all positive.

147 citations

Journal ArticleDOI
TL;DR: A set of nonlinear transformations is shown to split the whole family of dynamical systems into equivalence classes and it is shown on an example how to use these canonical forms to find nontrivial integrability conditions.
Abstract: A natural embedding for most time-continuous systems is presented. A set of nonlinear transformations is shown to split the whole family of dynamical systems into equivalence classes. The degree of nonlinearity is not a relevant characteristic of these classes of ordinary differential equations (ODEs). In each class there exist two particular simple canonical forms into which any such ODE can be cast. We show on an example how to use these canonical forms to find nontrivial integrability conditions. © 1989 The American Physical Society.

90 citations

Journal ArticleDOI
TL;DR: In this article, it is shown that if we have a system of m linear equations to solve, it is a great simplification to write them in matrix form Ax = b, where A is an m x n matrix of coefficients, b is a m-dimensional vector of constants and x is an n-dimensional unknowns.
Abstract: If we have a system of m linear equations to solve, it is a great simplification to write them in matrix form Ax = b , where A is an m x n matrix of coefficients, b is an m-dimensional vector of constants and x is an n-dimensional vector of unknowns.

88 citations

Journal ArticleDOI
TL;DR: In this article, sufficient conditions for the coexistence and exclusion of a stochastic competitive Lotka-Volterra model are derived, and convergence in distribution of positive solutions of the model is also established.

87 citations

Journal ArticleDOI
TL;DR: In this paper, Figueredo et al. established sufficient conditions for the existence of a Lyapunov function for a large class of non-linear systems, the quasi-polynomial systems.

43 citations