Volume 20 - 2015 - Special issue - Colloquium in Honor of Éric Benoît

1. Avant propos

Nadir Sari ; Guy Wallet.
Avant propos du colloque en l'honneur d'Eric Benoît : Des dynamiques singulièrement perturbées aux dynamiques des populations.

2. Surstable solutions with a singularity at a turning point

Eric Benoît.
Dans cet article, on va établir de façon très simple, presque sans analyse complexe, la première condition de style Matkowsky nécessaire pour qu'une solution soit un canard avec singularités. Le but est d'étudier des équations différentielles singulièrement perturbées analytiques présentant un point tournant. Des illustrations numériques sont présentées.

3. Fitting coefficients of differential systems with Monte Carlo methods

Christian Chan Shio ; Francine Diener.
We consider the problem of estimating the coefficients in a system of differential equations when a trajectory of the system is known at a set of times. To do this, we use a simple Monte Carlo sampling method, known as the rejection sampling algorithm. Unlike deterministic methods, it does not provide a point estimate of the coefficients directly, but rather a collection of values that "fits" the known data well. An examination of the properties of the method allows us not only to better understand how to choose the different parameters when implementing the method, but also to introduce a more efficient method by using a new two-step approach which we call sequential rejection sampling. Several examples are presented to illustrate the performance of both the original and the new methods.

4. Canard-induced loss of stability across a homoclinic bifurcation

Mathieu Desroches ; Jean-Pierre Françoise ; Lucile Mégret.
his article deals with slow-fast systems and is, in some sense, a first approach to a general problem, namely to investigate the possibility of bifurcations which display a dramatic change in the phase portrait in a very small (on the order of 10−7 in the example presented here) change of a parameter. We provide evidence of existence of such a very rapid loss of stability on a specific example of a singular perturbation setting. This example is strongly inspired of the explosion of canard cycles first discovered and studied by E Benoît, J.-L. Callot, F. Diener and M. Diener. After some presentation of the integrable case to be perturbed, we present the numerical evidences for this rapid loss of stability using numerical continuation. We discuss then the possibility to estimate accurately the value of the parameter for which this bifurcation occurs.

5. Composite Asymptotic Expansions and Difference Equations

Augustin Fruchard ; Reinhard Schäfke.
Difference equations in the complex domain of the form y(x+ϵ)−y(x)=ϵf(y(x))/y(x) are considered. The step size ϵ>0 is a small parameter, and the equation has a singularity at y=0. Solutions near the singularity are described using composite asymptotic expansions. More precisely, it is shown that the derivative v′ of the inverse function v of a solution (the so-called Fatou coordinate) admits a Gevrey asymptotic expansion in powers of the square root of ϵ, denoted by η, involving functions of y and of Y=y/η. This also yields Gevrey asymptotic expansions of the so-called Écalle-Voronin invariants of the equation which are functions of epsilon. An application coming from the theory of complex iteration is presented.

6. Migrations in the Rosenzweig-MacArthur model and the "atto-fox" problem

Claude Lobry ; Tewfik Sari.
The Rosenzweig-MacArthur model is a system of two ODEs used in population dynamics to modelize the predator-prey relationship. For certain values of the parameters the differential system exhibits a unique stable limit cycle. When the dynamics of the prey is faster than the dynamics of the predator, during oscillations along the limit cycle, the density of preys take so small values that it cannot modelize any actual population. This phenomenon is known as the "atto-fox" problem. In this paper we assume that the populations are living in two patches and are able to migrate from one patch to another. We give conditions for which the migration can prevent the density of prey being too small.

7. Analysis of a model describing stage-structured population dynamics using hawk-dove tactics

Ali Moussaoui ; Nguyen Ngoc Doanh ; Pierre Auger.
The purpose of this paper is to investigate the effects of conflicting tactics of resource acquisition on stage structured population dynamics. We present a population subdivided into two distinct stages (immature and mature). We assume that immature individual survival is density dependent. We also assume that mature individuals acquire resources required to survive and reproduce by using two contrasted behavioral tactics (hawk versus dove). Mature individual survival thus is assumed to depend on the average cost of fights while individual fecundity depends on the average gain in the competition to access the resource. Our model includes two parts: a fast part that describes the encounters and fights involves a game dynamic model based upon the replicator equations, and a slow part that describes the long-term effects of conflicting tactics on the population dynamics. The existence of two time scales let us investigate the complete system from a reduced one, which describes the dynamics of the total immature and mature densities at the slow time scale. Our analysis shows that an increase in resource value may decrease total population density, because it promotes individual (i.e. selfish) behavior. Our results may therefore find practical implications in animal conservation or biological control for instance.

8. Monotone Dynamical Systems and Some Models of Wolbachia in Aedes aegypti Populations

Gauthier Sallet ; A.H.B. Silva Moacyr.
We present a model of infection by Wolbachia of an Aedes aegypti population. This model is designed to take into account both the biology of this infection and any available experimental data obtained in the field. The objective is to use this model for predicting the sustainable introduction of this bacteria. We provide a complete mathematical analysis of the model proposed and give the basic reproduction ratio R0 for Wolbachia. We observe a bistability phenomenon. Two equilibria are asymptotically stable : an equilibrium where all the population is uninfected and an equilibrium where all the population is infected. A third unstable equilibrium exists. We provide an lower bound for the basin of attraction of the desired infected equilibrium. We are in a backward bifurcation situation. The bistable situations occurs with natural biological values for the parameters.

9. Canards, canard cascades and black swans

Vladimir Sobolev ; Elena Shchepakina.
The paper is devoted to the investigation of the slow integral manifolds of variable stability. The existence of non periodic canards, canard cascades and black swans is stated. The theoretical developments are illustrated by several examples.

10. A discrete free boundary problem

Imme Berg.
We study the free boundary problem in a nonstandard setting of infinitesimal discretisations of the heat equation. In particular we derive regularity results of solutions and the free boundary, in terms of S-continuity and S-differentiability

11. Choice sequence and nonstandard extension of type theory

Guy Wallet.
Partant de son travail Mathematics of Infinity (1989), Martin-Löf a développé l'idée d'un lien conceptuel profond entre les notions de suite de choix et d'objet mathématique nonstandard. Précisément, il a défini une extension nonstandard de la théorie des types en ajoutant une série d'axiomes nonstandard conçue comme une sorte de suite de choix. Enfin, dans une communication de 1999, il a présenté les grandes lignes d'une théorie des types nonstandard plus générale et munie d'un fort contenu computationnel. Le présent travail est une tentative de donner un développement complet d'une théorie de ce genre. Cependant, dans le but de garder un fort contrôle sur la théorie résultante et notablement pour éviter quelques problèmes en rapport avec l'égalité définitionnelle, le champ des axiomes nonstandard est moins général que ceux proposés dans sa communication de 1999. L'étude présente est poussée jusqu'à l' introduction d'une notion de proposition externe qui joue le même rôle que les propriétés externes si utiles dans l'analyse nonstandard usuelle. Du fait que ce texte débute par une introduction à la théorie des types de Martin-Löf, il peut intéresser les mathématiciens non familiers avec ce sujet.