Lotka-Volterra systems

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Qualitative stability patterns for Lotka-Volterra systems on rectangles

Qualitative stability patterns for Lotka-Volterra systems on rectangles

Thème BIO — Systèmes biologiques Projet Comore Rapport de recherche n° 6346 — November 2007 — 16 pages Abstract: We present a qualitative analysis of the Lotka-Volterra differential equation within rect- angles that are transverse with respect to the flow. In similar way to existing works on affine sys- tems (and positively invariant rectangles), we consider here nonlinear Lotka-Volterra n-dimensional equation, in rectangles with any kind of tranverse patterns. We give necessary and sufficient condi- tions for the existence of symmetrically transverse rectangles (containing the positive equilibrium), giving notably the method to build such rectangles. We also analyse the stability of the equilib- rium thanks to this transverse pattern. We finally propose an analysis of the dynamical behavior inside a rectangle containing the positive equilibrium, based on Lyapunov stability theory. More particularly, we make use of Lyapunov-like functions, built upon vector norms. This work is a first step towards a qualitative abstraction and simulation of Lotka-Volterra systems.
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Global behaviour of n-dimensional Lotka-Volterra systems

Global behaviour of n-dimensional Lotka-Volterra systems

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A criterion of global convergence to equilibrium for differential systems. Application to Lotka-Volterra systems

A criterion of global convergence to equilibrium for differential systems. Application to Lotka-Volterra systems

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Global stability with selection in integro-differential Lotka-Volterra systems modelling trait-structured populations

Global stability with selection in integro-differential Lotka-Volterra systems modelling trait-structured populations

(all species asymptotically survive) such that ρ converges to ρ ∞ , because we will see how it determines on which phenotypes the densities concentrate. When it is possible, we will investigate the speed at which convergence and concentration occur. An interesting question within the scope of this paper is also to see if a result of that type is sharp, i.e., to compare the assumptions needed to obtain global asymptotic stability in our generalised setting to those known for classical Lotka-Volterra equations.

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Lotka-Volterra with randomly fluctuating environments: a full description

Lotka-Volterra with randomly fluctuating environments: a full description

5 Switching between two persistent Lotka-Volterra systems Let us assume that ε 0 and ε 1 are of Type 3. In this case, one can easily get that extinction of species y is not possible if u is to close of 0 or 1; in other words, [0, 1] \ ˜ I is either empty or is an open interval which closure is contained in [0, 1]. Recall

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Convergence to equilibrium in competitive Lotka-Volterra and chemostat systems

Convergence to equilibrium in competitive Lotka-Volterra and chemostat systems

then there exists a unique ¯ n = (¯ n 1 , . . . , ¯ n N ) ∈ R N + with ¯ n 6= 0, s.t. for any solution n(t) = (n 1 , . . . , n N ) to (1.1) with initial data n i (0) > 0 for any i, n(t) −→ ¯n, as t → +∞. This result shows that, in Lotka-Volterra systems which are symmetric in the sense of (1.9), the competition between a mutant trait and a resident popu- lation leads to a unique stationary state, regardless of the initial population state. This is precisely the assumption needed in [1] to apply a limit of large population and rare mutations to an individual-based model. In particular, Thm. 2.7 of [1] applies to symmetric competitive Lotka-Volterra systems.
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Simulation de la dynamique d'une poutre de Reissner par les séries de Volterra

Simulation de la dynamique d'une poutre de Reissner par les séries de Volterra

Cette seconde relation doit être vérifiée en perma- nence, elle ne peut être approximée. Il est donc impos- sible d’appliquer les séries de Volterra (tronquées) a ce système. C’est la raison pour laquelle la prochaine sec- tion consiste en un travail de réécriture afin de ne tra- vailler que sur une seule variable u dans l’algèbre à la place de T et Ξ.

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Controllability of SISO Volterra Models via Diffusive Representation

Controllability of SISO Volterra Models via Diffusive Representation

Abstract: The problem under consideration is the controllability of a wide class of convolution Volterra systems, namely the class of “diffusive” systems, for which there exists an input-output state realization whose state evolves in the so-called diffusive representation space. We first show that this universal state variable is approximately controllable, and then deduce that such Volterra systems always possess suitable controllability properties, stated and proved. Then, we show how to solve the optimal null control problem in an LQ sense. A numerical example finally highlights these results.
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The Laplace transform of the integrated Volterra Wishart process

The Laplace transform of the integrated Volterra Wishart process

The Laplace transform is exponentially quadratic in the forward process (g t ) t≤T , and cannot in general be recovered from that of finite dimensional affine Volterra processes introduced in Abi Jaber et al. ( 2019a ), see Remark 2.1 . We also mention that the mod- els studied here are quadratic constructions of Gaussian processes and do not pose any difficulty regarding existence and uniqueness, in contrast for instance with conventional Wishart processes that go beyond squares of Gaussians, see Bru ( 1991 ).

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Time reversal of Volterra processes driven stochastic differential equation

Time reversal of Volterra processes driven stochastic differential equation

STOCHASTIC DIFFERENTIAL EQUATIONS L. DECREUSEFOND Abstract. We consider stochastic differential equations driven by some Volterra processes. Under time reversal, these equations are transformed into past de- pendent stochastic differential equations driven by a standard Brownian mo- tion. We are then in position to derive existence and uniqueness of solutions of the Volterra driven SDE considered at the beginning.

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On linear MMSE based turbo-equalization of nonlinear Volterra channels

On linear MMSE based turbo-equalization of nonlinear Volterra channels

+ Fig. 1. Block diagram of a satellite communication channel 2. NONLINEAR VOLTERRA CHANNEL MODEL The satellite transponder’s HPA is usually represented by an IMUX (Input MUltipleXing filter) followed by a memoryless nonlinear amplifier and an OMUX (Output MUltipleXing filter). The con- stituent IMUX filter aims at removing adjacent channel interfer- ences, whereas the OMUX filters the out-of band frequencies ensu- ing from the spectral enlargement due to the nonlinear processing within the amplifier. According to [4], we can assume that the satellite transponder corresponds to a bandpass HPA whose trans- fer function c only accounts for the module of the signal x to be amplified:
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Study of a Fractional Problem of Volterra Type

Study of a Fractional Problem of Volterra Type

Since only positive solutions are meaningful in many applications, in Chapter 4 we discuss the existence and multiplicity of positive solutions of a nonlinear fractional differential equ[r]

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Third order widely non-linear Volterra MVDR beamforming

Third order widely non-linear Volterra MVDR beamforming

For a QPSK interference, the MVDR 1 beamformer [6] does not improve the Capon beamformer because term (1) of eq.(3) does not convey any information for SO circular inter- ference. In contrast, for the partially [resp., totally] structured MVDR Volterra beamformer, term (4) [resp., terms (2) and (4)] of eq.(3), convey information for fourth-order noncircular interference. Consequently, these two MVDR Volterra beam- formers outperform the Capon beamformer. Furthermore, we note that the gain can be large (about 2dB and 6dB) in this scenario.

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Stochastic Integration with respect to Volterra processes

Stochastic Integration with respect to Volterra processes

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Nonlinear system modeling and identification using Volterra-PARAFAC models

Nonlinear system modeling and identification using Volterra-PARAFAC models

SUMMARY Discrete-time Volterra models are widely used in various application areas. Their usefulness is mainly because of their ability to approximate to an arbitrary precision any fading memory nonlinear system and to their property of linearity with respect to parameters, the kernels coefficients. The main drawback of these models is their parametric complexity implying the need to estimate a huge number of parame- ters. Considering Volterra kernels of order higher than two as symmetric tensors, we use a parallel factor (PARAFAC) decomposition of the kernels to derive Volterra-PARAFAC models that induce a substantial parametric complexity reduction. We show that these models are equivalent to a set of Wiener models in parallel. We also show that Volterra kernel expansions onto orthonormal basis functions (OBF) can be viewed as Tucker models that we shall call Volterra-OBF-Tucker models. Finally, we propose three adap- tive algorithms for identifying Volterra-PARAFAC models when input–output signals are complex-valued: the extended complex Kalman filter, the complex least mean square (CLMS) algorithm and the normalized CLMS algorithm. Some simulation results illustrate the effectiveness of the proposed identification methods. Copyright © 2011 John Wiley & Sons, Ltd.
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On linear MMSE based turbo-equalization of nonlinear Volterra channels

On linear MMSE based turbo-equalization of nonlinear Volterra channels

Nonlinear satellite distortions can be treated either at the transmitter with the so-called pre-compensation/pre-distortion techniques or at the receiver using equalization. In this paper, we are mainly inter- ested in iterative equalization techniques. Several channel models have been investigated to cope with nonlinear interference. A first approach was introduced in [2] [3] where the nonlinear chan- nel was considered as an additive gaussian noise associated with a signal attenuation called warping. This approximation is no longer accurate for high order mutli-level modulations since outer rings constellation symbols experience stronger distortions that inner rings symbols. Yet, the most common approach to model the nonlinear distortions is the use of the Volterra series representations originally
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Application des séries de Volterra non entière à l'identification de systèmes thermiques en conditions extrêmes

Application des séries de Volterra non entière à l'identification de systèmes thermiques en conditions extrêmes

La diffusion thermique dans un échantillon de fer ARMCO est identifié, dans ce papier, pour de larges variations de température en utilisant un modèle basé sur les séries de Volterra non [r]

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Integral operator Riccati equations arising in stochastic Volterra control problems

Integral operator Riccati equations arising in stochastic Volterra control problems

[8] F. Flandoli. Direct solution of a Riccati equation arising in a stochastic control problem with control and observation on the boundary. Applied Mathematics and Optimization, 14(1):107– 129, 1986. [9] G. Gripenberg, S.-O. Londen, and O. Staffans. Volterra integral and functional equations, volume 34 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1990.

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Modèles de volterra à complexité réduite : estimation paramétrique et application à l'égalisation des canaux de communication

Modèles de volterra à complexité réduite : estimation paramétrique et application à l'égalisation des canaux de communication

Dans le chapitre 1, il a été établi que l’estimation paramétrique d’un modèle de Vol- terra quadratique peut être complètement découplée, lorsque l’entrée considérée est indépendante et distribuée de manière identique (i.i.d., en anglais) et suit une distri- bution de probabilité symétrique. C’est le cas d’une entrée gaussienne ou encore de la plupart des signaux utilisés en télécommunications (PAM,QAM,PSK). Cependant, cette propriété ne se vérifie plus lorsqu’on augmente l’ordre de non-linéarité du mo- dèle. En général, l’estimation des coefficients des noyaux de Volterra au sens de l’er- reur quadratique moyenne minimale requiert le calcul de l’inverse de la matrice des moments de l’entrée et l’estimation de ces moments jusqu’à l’ordre six pour le cas du modèle cubique et jusqu’à l’ordre dix pour le modèle d’ordre cinq. Une simplification se basant sur l’hypothèse d’une entrée i.i.d. a été proposée par [129] pour le modèle cubique. Dans le cas des modèles de Volterra passe-bande, des simplifications ont aussi été proposées par [131] pour le modèle cubique et par [24] pour le modèle d’ordre cinq. Sous la même hypothèse d’entrée i.i.d. nous montrons comment l’estimation peut être simplifiée en utilisant le formalisme des polynômes orthogonaux.
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SAIC pour une interférence circulaire au second-ordre par filtrage de Volterra complexe d'ordre 3

SAIC pour une interférence circulaire au second-ordre par filtrage de Volterra complexe d'ordre 3

SAIC pour une interf´erence circulaire au second-ordre par filtrage de Volterra complexe d’ordre 3 Mustapha S ADOK 1,2 , Jean Pierre D ELMAS 1 , Pascal C HEVALIER 3 1 Telecom SudParis, UMR CNRS 5157, Universit´e Paris Saclay Evry, France 2 Institut National des T´el´ecommunications et TIC, LaRATIC, Oran, Alg´erie

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