Dating and localizing an invasion from post-introduction data and a coupled reaction-diffusion-absorption model

Invasion of new territories by alien organisms is of primary concern for environmental and health agencies and has been a core topic in mathematical modeling, in particular in the intents of reconstructing the past dynamics of the alien organisms and predicting their future spatial extents. Partial differential equations offer a rich and flexible modeling framework that has been applied to a large number of invasions. In this article, we are specifically interested in dating and localizing the introduction that led to an invasion using mathematical modeling, post-introduction data and an adequate statistical inference procedure. We adopt a mechanistic-statistical approach grounded on a coupled reaction-diffusion-absorption model representing the dynamics of an organism in an heterogeneous domain with respect to growth. Initial conditions (including the date and site of the introduction) and model parameters related to diffusion, reproduction and mortality are jointly estimated in the Bayesian framework by using an adaptive importance sampling algorithm. This framework is applied to the invasion of \textit{Xylella fastidiosa}, a phytopathogenic bacterium detected in South Corsica in 2015, France

Quartic formulation of Coulomb 3D frictional contact

In this paper, we focus on the problem of a single contact with Coulomb friction, which we reduce to a root-nding problem on a degree 4 polynomial. This formulation give us the exact number of solutions, as well as their analytical form when they exist.Dans ce document, nous reformulons le problème unitaire (1 con- tact frottant) en la recherche de zéro d'un polynôme du quatrième degré. Si elles existent, cette formulation donne la forme analytique des solutions

The spatio-temporal dynamics of neutral genetic diversity

International audienceThe notions of pulled and pushed solutions of reaction-dispersal equations introduced by Garnier et al. (2012) and Roques et al. (2012) are based on a decomposition of the solutions into several components. In the framework of population dynamics, this decomposition is related to the spatio-temporal evolution of the genetic structure of a population. The pulled solutions describe a rapid erosion of neutral genetic diversity, while the pushed solutions are associated with a maintenance of diversity. This paper is a survey of the most recent applications of these notions to several standard models of population dynamics, including reaction-diffusion equations and systems and integro-differential equations. We describe several counterintuitive results, where unfavorable factors for the persistence and spreading of a population tend to promote diversity in this population. In particular, we show that the Allee effect, the existence of a competitor species, as well as the presence of climate constraints are factors which can promote diversity during a colonization. We also show that long distance dispersal events lead to a higher diversity, whereas the existence of a nonreproductive juvenile stage does not affect the neutral diversity in a range-expanding population

Time-stepping numerical simulation of switched circuits with the nonsmooth dynamical systems approach

International audienceThe numerical integration of switching circuits is known to be a tough issue when the number of switches is large, or when sliding modes exist. Then, classical analog simulators may behave poorly, or even fail. In this paper, it is shown on two examples that the nonsmooth dynamical systems (NSDS) approach, which is made of: 1) a specific modeling of the piecewise-linear electronic devices (ideal diodes, Zener diodes, transistors); 2) the Moreau's time-stepping scheme; and 3) specific iterative one-step solvers, supersedes simulators of the simulation program with integrated circuit emphasis (SPICE) family and hybrid simulators. An academic example constructed in [Maffezzoni, , IEEE Trans. CADICS, vol 25, no. 11, Nov. 2006], so that the Newton-Raphson scheme does not converge, and the buck converter are used to make extensive comparisons between the NSDS method and other methods of the SPICE family and a hybrid-like method. The NSDS method, implemented in the siconos platform developed at INRIA, proves to be on these two examples much faster and more robust with respect to the model parameter variations

The nonsmooth dynamical systems approach for the analog simulation of switched circuits within the Siconos framework

The numerical integration of switching circuits is known to be a tough issue when the number of switches is high, or when sliding modes exist. Then classical analog simulators may behave poorly, or even fail. In this paper it is shown on two examples that the nonsmooth dynamical systems (NSDS) approach, which is made of 1) a specific modelling of the piecewise- linear electronic devices (ideal diodes, Zener diodes, transistors), 2) the Moreau's time-stepping scheme, and 3) specific iterative one-step solvers, supersedes simulators of the SPICE family and hybrid simulators. An academic example constructed in [Maffezzoni et al, IEEE Trans. on CADICS, Vol 25, No 11, November 2006], so that the Newton-Raphson scheme does not converge, and the buck converter, are used to make extensive comparisons between the NSDS method and other methods of the SPICE family and a hybrid-like method. The NSDS method, implemented in the Siconos platform developed at INRIA, proves to be on these two examples much faster and more robust with respect to the models parameters variations

An introduction to Siconos

In this document, a brief overview of the Siconos Platform is given. One of the goal is to give a flavor on a simple example of the ability of the platform to model and simulate the so-called non smooth dynamical systems (NSDS). In particular, some examples of Lagrangian mechanical systems with contact and friction or electrical circuits with ideal and piecewise linear components (diodes, MOS transistors, \ldots) are commented. Finally, the Siconos software is presented, starting from its architecture to a non exhaustive presentation of its components and functionalities. The aim of this document is not to serve as a reference guide but more as a illustrative introduction document to promote the use of the platform

Time integration of nonsmooth mechanical systems with unilateral contact. Conservation and stability of position and velocity constraints in discrete time

International audienceThis work addresses the problem of the numerical time-integration of nonsmooth mechanical systems subjected to unilateral contacts and impacts. The considered systems may be the standard multi-body systems or the space-discretized continuous systems obtained by using FEM approach. Up to now, two main numerical schemes are available to perform this task: the Moreau-Jean scheme which solves the constraints at the velocity level together with a Newton impact law and the Schatzman-Paoli scheme which directly considers the constraints at the position level. In both schemes, the position and velocity constraints are not both satisfied in discrete time. The aim of this work is to propose a new scheme inspired by the GGL approach in DAE that solves, in discrete time, the constraints on both position and velocity levels. The stability and the local order of the scheme will be discussed. Some comparisons with recent works on the adaption of Newmark's scheme will be presented

Modelling population dynamics in realistic landscapes with linear elements: a mechanistic-statistical reaction-diffusion approach

We propose and develop a general approach based on reaction-diffusion equations for modelling a species dynamics in a realistic two-dimensional (2D) landscape crossed by linear one-dimensional (1D) corridors, such as roads, hedgerows or rivers. Our approach is based on a hybrid “2D/1D model”, i.e, a system of 2D and 1D reaction-diffusion equations with homogeneous coefficients, in which each equation describes the population dynamics in a given 2D or 1D element of the landscape. Using the example of the range expansion of the tiger mosquito Aedes albopictus in France and its main highways as 1D corridors, we show that the model can be fitted to realistic observation data. We develop a mechanistic-statistical approach, based on the coupling between a model of population dynamics and a probabilistic model of the observation process. This allows us to bridge the gap between the data (3 levels of infestation, at the scale of a French department) and the output of the model (population densities at each point of the landscape), and to estimate the model parameter values using a maximum-likelihood approach. Using classical model comparison criteria, we obtain a better fit and a better predictive power with the 2D/1D model than with a standard homogeneous reaction-diffusion model. This shows the potential importance of taking into account the effect of the corridors (highways in the present case) on species dynamics. With regard to the particular case of A. albopictus, the conclusion that highways played an important role in species range expansion in mainland France is consistent with recent findings from the literature

Quartic formulation of Coulomb 3D frictional contact

In this paper, we focus on the problem of a single contact with Coulomb friction, which we reduce to a root-nding problem on a degree 4 polynomial. This formulation give us the exact number of solutions, as well as their analytical form when they exist.Dans ce document, nous reformulons le problème unitaire (1 con- tact frottant) en la recherche de zéro d'un polynôme du quatrième degré. Si elles existent, cette formulation donne la forme analytique des solutions