Volume 9, 2007 Conference in Honor of Claude Lobry, 2008

1. Avant Propos

Mary Teuw Niane ; Gauthier Sallet ; Tewik Sari ; Hamidou Touré.
Les amis de Claude Lobry ont organisé du 10 au 14 septembre 2007 à l’université Gaston Berger de Saint Louis une conférence en son honneur. Les apports scientifiques de Claude Lobry ont été non seulement multiformes et pluridisciplinaires, mais il a souvent été un précurseur dans nombre d’activités. Cette conférence s’est tenue en Afrique, à la demande des mathématiciens africains, en raison de l’activité particulière de Claude Lobry pour le développement des mathématiques en Afrique depuis sa prise de fonction comme Directeur du CIMPA en 1995 jusqu’à nos jours. Son livre « Les mathématiques : une nécessité pour le développement » est un vibrant plaidoyer pour le développement des mathématiques en Afrique

2. Claude Lobry, un mathématicien militant

Gauthier Sallet.
We show in this communication, that Claude Lobry has always been contributing to mathematics and simultaneously promoting actions to develop a certain idea of acting in mathematics

3. Ockham’s razor: Deriving cyclic evolutions from viability and inertia constraints

Jean-Pierre Aubin.
This article deals with a theme to which Claude Lobry has been interested for a long time: what is the nature of mathematics motivated by biological sciences? It starts by presenting the subjective opinions of its author, illustrated by the simplest application one can think of: demonstrating that it is possible to produce cyclic evolutions on the simple basis of viability and inertia constraints, without using periodic differential equations. It is not impossible that this approach is foreign to an explanation of biological clocks (or economic cycles) in another field.

4. Quantitative analysis of metabolic networks and design of minimal bioreaction models

Georges Bastin.
This tutorial paper is concerned with the design of macroscopic bioreaction models on the basis a quantitative analysis of the underlying cell metabolism. The paper starts with a review of two fundamental algebraic techniques for the quantitative analysis of metabolic networks : (i) the decomposition of complex metabolic networks into elementary pathways (or elementary modes), (ii) the metabolic flux analysis which aims at computing the entire intracellular flux distribution from a limited number of flux meaurements. Then it is discussed how these two fundamental techniques can be exploited to design minimal bioreaction models by using a systematic model reduction approach that automatically produces a family of equivalent minimal models which are fully compatible with the underlying metabolism and consistent with the available experimental data. The theory is illustrated with an experimental case-study on CHO cells.

5. Generic analysis of the response of calcifying microalgae to an elevation of pCO2 : qualitative vs quantitative analysis

Olivier Bernard ; Antoine Sciandre.
Calcifying microalgae can play a key role in atmospheric CO2 trapping through large scale precipitation of calcium carbonate in the oceans. However, recent experiments revealed that the associated fluxes may be slow down by an increase in atmospheric CO2 concentration. In this paper we design models to account for the decrease in calcification and photosynthesis rates observed after an increase of pCO2 in Emiliania huxleyi chemostat cultures. Since the involved mechanisms are still not completely understood, we consider various models, each of them being based on a different hypothesis. These models are kept at a very general level, by maintaining the growth and calcification functions in a generic form, i.e. independent on the exact shape of these functions and on parameter values. The analysis is thus performed using these generic functions where the only hypothesis is an increase of these rates with respect to the regulating carbon species. As a result, each model responds differently to a pCO2 elevation. Surprisingly, the only models whose behaviour are in agreement with the experimental results correspond to carbonate as the regulating species for photosynthesis. Finally we show that the models whose qualitative behaviour are wrong could be considered as acceptable on the basis of a quantitative prediction error criterion.

6. Survey of recent results of multi-compartments intra-host models of malaria and HIV

Samuel Bowong ; Jean-Luc Dimi ; Jean-Claude Kamgang ; Joseph Mbang ; Jean Jules Tewa.
We present the recent results obtained for the within-host models of malaria and HIV. We briefly recall the Anderson-May-Gupter model. We also recall the Van Den Driessche method of computation for the basic reproduction ratio R0 ; here, a very simple formula is given for a new class of models. The global analysis of these models can be founded in [1, 2, 3, 5]. The results we recall here are for a model of one strain of parasites and many classes of age, a general model of n strains of parasites and k classes of age, a S E1 E2 · · ·En I S model with one linear chain of compartments and finally a general S Ei1 Ei2 · · ·Ein I S model with k linear chains of compartments. When R0 <=1, the authors prove that there is a trivial equilibria calling disease free equilibrium (DFE) which is globally asymptotically stable (GAS) on the non-negative orthant , and when R0 > 1, they prove the existence of a unique endemic equilibrium in the non-negative orthant and give an explicit formula. They provided a weak condition for the global stability of endemic equilibrium

7. Complexity in a prey-predator model

Baba I. Camara ; Moulay A. Aziz Alaoui.
In this paper we consider a predator-prey model given by a reaction-diffusion system. It incorporates the Holling-type-II and a modified Leslie-Gower functional response. We focus on qualitaive analysis, bifurcation mecanisms and patterns formation.

8. Computational probability modeling and Bayesian inference

Fabien Campillo ; Rivo Rakotozafy ; Vivien Rossi.
Computational probabilistic modeling and Bayesian inference has met a great success over the past fifteen years through the development of Monte Carlo methods and the ever increasing performance of computers. Through methods such as Monte Carlo Markov chain and sequential Monte Carlo Bayesian inference effectively combines with Markovian modelling. This approach has been very successful in ecology and agronomy. We analyze the development of this approach applied to a few examples of natural resources management.

9. Two simple growth models in the chemostat

Jean-Luc Gouzé ; Valérie Lemesle.
In a chemostat, transient oscillations are often experimentally observed during cell growth. The aim of this paper is to propose simple autonomous models which are able (or not) to generate these oscillations, and to investigate them analytically. Our point of view is based on a simplification of the cell cycle in which there are two states (mature and immature) with the transfer between the two dependent on the available resources. We built two similar models, one with cell biomass and the other with cell number density. We prove that the first one oscillates, but not the second. This paper is dedicated to Claude Lobry, who helped us to build a first version of these models.

10. Uncoupling Isaacs’equations in nonzero-sum two-player differential games : The example of conflict over parental care

Frédéric Hamelin ; Pierre Bernhard.
We use a recently uncovered decoupling of Isaacs PDE’s of some mixed closed loop Nash equilibria to give a rather complete analysis of the classical problem of conflict over parental care in behavioural ecology, for a more general set up than had been considered heretofore.

11. La méthode des élucidations successives

Claude Lobry.
In the process of elaboration of a model one emphasize on the necessity of confronting the model with the reality which it is supposed to represent. There is another aspect of the modelling process, to my opinion also essential, about which one usually do not speak. It consists in a logico-linguistic work where formal models are used to produce prediction which are not confronted with the reality but serve for falsifying assertions which nevertheless seemed to be derived from the not formalized model. More exactly a first informal model is described in the natural language and, considered in the natural language, seems to say some thing but in a more or less clear way. Then we translate the informal model into a formal model (mathematical model or computer model) where what was argumentation becomes demonstration.The formal model so serves for raising ambiguities of the natural language. But conversely a too much formalized text quickly loses any sense for a human brain what makes necessary the return for a less formal language. It is these successive "translations" between more or less formal languages that I try to analyze on two examples, the first one in population dynamics, the second in mathematics.

12. Applications de méthodes d’agrégation de variables à l’analyse de modèles spatiaux de dynamique des populations

Tri Nguyen-Huu ; Pierre Auger.
Models in population dynamics can deal with an important number of parameters and variables, which can make them difficult to analyse. Aggregation of variables allow reducing complexity of such models by building simplified models governing fewer variables by use of the existence of different time scales associated to the processes governing the whole system. Those reduced models allows analysing and describing the global dynamics of the system. We present those methods for time discrete models and illustrate their use for the study of spatial host-parasitoids models.

13. Continuous branching processes : the discrete hidden in the continuous : Dedicated to Claude Lobry

Etienne Pardoux.
Feller diffusion is a continuous branching process. The branching property tells us that for t > 0 fixed, when indexed by the initial condition, it is a subordinator (i. e. a positive–valued Lévy process), which is fact is a compound Poisson process. The number of points of this Poisson process can be interpreted as the number of individuals whose progeny survives during a number of generations of the order of t × N, where N denotes the size of the population, in the limit N ―>µ. This fact follows from recent results of Bertoin, Fontbona, Martinez [1]. We compare them with older results of de O’Connell [7] and [8]. We believe that this comparison is useful for better understanding these results. There is no new result in this presentation.

14. Practical coexistence in the chemostat with arbitrarily close growth functions

Alain Rapaport ; Denis Dochain ; Jérôme Harmand.
We show that the coexistence of different species in competition for a common resource may be substantially long when their growth functions are arbitrarily closed. The transient behavior is analyzed in terms of slow-fast dynamics. We prove that non-dominant species can first increase before decreasing, depending on their initial proportions.

15. On a 1D-Shallow Water Model: Existence of solution and numerical simulations

Olivier Besson ; Soulèye Kane ; Mamadou Sy.
The study of a 1D-shallow water model, obtained in a height-flow formulation, is presented. It takes viscosity into account and can be used for the flood prediction in rivers. For a linearized system, the existence and uniqueness of a global solution is proved. Finally, various numerical results are presented regarding the linear and non linear case.

16. Approche structurelle des systèmes, de la géométrie à la théorie des graphes

Jean-Michel Dion ; Christian Commault.
In this work, which was presented at the conference in honor of Claude Lobry, we focus on a structural approach of systems which was the mainstream of our research. The modeling ability of this approach and the power of the associated graph tools are enlightened. As an illustration we consider the disturbance decoupling problem by measurement feedback and solve this problem using geometric and graph techniques

17. On a Radially Symmetrical Green’s Function

Ould Ahmed Izid Bih Isselkou.
It is quite usual to transform elliptic PDE problems of second order into fixed point integral problems, via the Green’s function. But it is not easy, in general, to handle integrals involved in such a formulation. When it comes to the Laplacian operator on balls of Rn, we give here a radially symmetrical Green’s function which, under some nonlinearity assumptions, makes the Green’s Integral representation formula easier to use; we give three examples of application.

18. On some robust algorithms for the Robin inverse problem

Mohamed Jaoua ; Slim Chaabane ; Chokri Elhechmi ; Juliette Leblond ; Moncef Mahjoub ; Jonathan R. Partington.
The problem we are dealing with is to recover a Robin coefficient (or impedance) from measurements performed on some part of the boundary of a domain, in the framework of nondestructive testing by the means of Electric Impedance Tomography. The impedance can provide information on the location of a corroded area, as well as on the extent of the damage, which has possibly occurred on an unaccessible part of the boundary. Two different identification algorithms are presented and studied: the first one is based on a Kohn and Vogelius cost function, actually an energetic least squares one, which turns the inverse problem into an optimization one ; as for the second, it makes use of the best approximation in Hardy classes, in order to extend the Cauchy data to the unreachable part of the boundary, and then compute the Robin coefficient from these extended data. Special focus is put on the robustness with respect to noise, both from a mathematical and and numerical point of view. Some numerical experiments are eventually presented and compared.

19. Une approche pour la modèlisation et le contrôle des instabilités de combustion

Ioan Doré Landau ; Fethi Bouziani ; Robert R. Bitmead.
A set of two coupled genralized Vand der Pol equations is proposed as a control model for combustion instabilities. The system is analyzed using the Krylov-Bogoliubov method. The control aspects related to quenching of the oscillations are examined. The analysis results are compared with simulation results.

20. Quantum systems and control 1

Pierre Rouchon.
This paper describes several methods used by physicists for manipulations of quantum states. For each method, we explain the model, the various time-scales, the performed approximations and we propose an interpretation in terms of control theory. These various interpretations underlie open questions on controllability, feedback and estimations. For 2-level systems we consider: the Rabi oscillations in connection with averaging; the Bloch-Siegert corrections associated to the second order terms; controllability versus parametric robustness of open-loop control and an interesting controllability problem in infinite dimension with continuous spectra. For 3-level systems we consider: Raman pulses and the second order terms. For spin/spring systems we consider: composite systems made of 2-level sub-systems coupled to quantized harmonic oscillators; multi-frequency averaging in infinite dimension; controllability of 1D partial differential equation of Shrödinger type and affine versus the control; motion planning for quantum gates. For open quantum systems subject to decoherence with continuous measures we consider: quantum trajectories and jump processes for a 2-level system; Lindblad-Kossakovsky equation and their controllability.

21. Stabilité Lp exponentielle d’un système d’échangeurs thermiques avec diffusion et sans diffusion

Chen-Zhong Li ; Abdoua Tchousso ; Xiao-Dong Li ; Gauthier Sallet.
In this paper we study exponential stability of a heat exchanger system with diffusion and without diffusion in the context of Banach spaces. The heat exchanger system is governed by hyperbolic partial differential equations (PDE) and parabolic PDEs, respectively, according to the diffusion impact ignored or not in the heat exchange. The exponential stability of the model with diffusion in the Banach space (C[0, 1])4 is deduced by establishing the exponential Lp stability of the considered system, and using the sectorial operator theory. The exponential decay rate of stability is also computed for the model with diffusion. Using the perturbation theory, we establish the exponential stability of the model without diffusion in the Banach space (C[0, 1])4 with the uniform topology. However the exponential decay rate of stability without diffusion is not exactly computed, since its associated semigroup is non analytic. Indeed the purpose of our paper is to investigate the exponential stability of a heat exchanger system with diffusion and without diffusion in the real Banach space X1 = (C[0, 1])4 with the uniform norm. The exponential stability of these two models in the Hilbert space X2 = (L2(0, 1))4 has been proved in [31] by using Lyapunov’s direct method. The first step consists to study the stability problem in the real Banach space Xp = (Lp(0, 1))4 equipped with the usual Lp norm, p > 1. By passing to the limit (p ! 1) we can extend some results of exponential […]

22. Théorie générale d’équation de type hyperbolique-parabolique non linéaire

Hamidou Touré.
We develop general theory for degenerate hyperbolic-parabolic type problems using semi-group theory in Banach spaces. We establish existence, uniqness results and continuous dependance with respects to data for mild solution. Similar results are developped for weak solution of entropy type, and existence of solutions are studied.

23. Nearly recombining processes and the calculation of expectations

Imme Van den Berg ; Elsa Amaro.
In the context of Nonstandard Analysis, we study stochastic difference equations with infinitesimal time-steps. In particular we give a necessary and sufficient condition for a solution to be nearly-equivalent to a recombining stochastic process. The characterization is based upon a partial differential equation involving the trend and the conditional variance of the original process. An analogy with Ito’s Lemma is pointed out. As an application we obtain a method for approximation of expectations, in terms of two ordinary differential equations, also involving the trend and the conditional variance of the original process, and of Gaussian integrals.

24. Questioning the signal to noise ratioin digital communications

Michel Fliess.
The signal to noise ratio, which plays such an important rôle in information theory, is shown to become pointless for digital communications where the demodulation is achieved via new fast estimation techniques. Operational calculus, differential algebra, noncommutative algebra and nonstandard analysis are the main mathematical tools.

25. Sur le retard à la bifurcation

Augustin Fruchard ; Reinhard Schäfke.
We give a non-exhaustive overview of the problem of bifurcation delay from its appearance in France at the end of the 80ies to the most recent contributions. We present the bifurcation delay for differential equations as well as for discrete dynamical systems.

26. Canard solutions and bifurcations in smooth models of plane structure variable systems

Luis Gonzaga Albuquerque.
Systems that operate in different modes with quick transition are usually studied through discontinuous systems. We give a model of a smoothing of the transition between two vector fields along a separation line, allowing perturbations of the vector fields and of the separation line. In this model there appears a canard phenomenon in certain macroscopically indeterminate situations. This phenomenon gives a new point of view on some situations usually studied through discontinuous bifurcations. We also study the dynamics near the transition line through an associated slow-fast system and compare the slow dynamics with the classical theory, namely, sliding mode dynamics in variable structure systems and equivalent control.

27. The peaking phenomenon and singular perturbations

Claude Lobry ; Tewfik Sari.
We study the asymptotic behaviour, when the parameter " tends to 0, of a class of singularly perturbed triangular systems x˙ = f(x, y), y˙ = G(y, "). We assume that all solutions of the second equation tend to zero arbitrarily fast when " tends to 0. We assume that the origin of equation x˙ = f(x, 0) is globally asymptotically stable. Some states of the second equation may peak to very large values, before they rapidly decay to zero. Such peaking states can destabilize the first equation. The paper introduces the concept of instantaneous stability, to measure the fast decay to zero of the solutions of the second equation, and the concept of uniform infinitesimal boundedness to measure the effects of peaking on the first equation. Whe show that all the solutions of the triangular system tend to zero when " ! 0 and t ! +1. Our results are formulated in both classical mathematics and nonstandard analysis.

28. Integer calculus on the Harthong-Reeb Line

Guy Wallet.
In this work, we give a presentation of the so-called Harthong-Reeb line. Only based on integer numbers, this numerical system has the striking property to be roughly equivalent to the continuous real line. Its definition requires the use of a natural number w which is infinitely large in the meaning of nonstandard analysis. Following the idea of G. Reeb, we show how to implement in this framework the Euler scheme. Then we get an exact representation in the Harthong-Reeb line of many real functions like the exponential. Since this representation is given with the help of an explicit algorithm, it is natural to wonder about the global constructivity of this numerical system. In the conclusion, we discuss this last point and we outline some new directions for getting analogous systems which would be more constructive

29. Singular perturbations on the infinite time interval

Karim Yadi.
We consider the slow and fast systems that belong to a small neighborhood of an unperturbed problem. We study the general case where the slow equation has a compact positively invariant subset which is asymptotically stable, and meanwhile the fast equation has asymptotically stable equilibria (Tykhonov’s theory) or asymptotically stable periodic orbits (Pontryagin–Rodygin’s theory). The description of the solutions is by this way given on infinite time interval. We investigate the stability problems derived from this results by introducing the notion of practical asymptotic stability. We show that some particular subsets of the phase space of the singularly perturbed systems behave like asymptotically stable sets. Our results are formulated in classical mathematics. They are proved within Internal Set Theory which is an axiomatic approach to Nonstandard Analysis.