A spatially explicit Markovian individual-based model for terrestrial plant dynamics

An individual-based model (IBM) of a spatiotemporal terrestrial ecological population is proposed. This model is spatially explicit and features the position of each individual together with another characteristic, such as the size of the individual, which evolves according to a given stochastic model. The population is locally regulated through an explicit competition kernel. The IBM is represented as a measure-valued branching/diffusing stochastic process. The approach allows (i) to describe the associated Monte Carlo simulation and (ii) to analyze the limit process under large initial population size asymptotic. The limit macroscopic model is a deterministic integro-differential equation.Comment: 31 pages, 1 figur

A mass-structured individual-based model of the chemostat: convergence and simulation

We propose a model of chemostat where the bacterial population is individually-based, each bacterium is explicitly represented and has a mass evolving continuously over time. The substrate concentration is represented as a conventional ordinary differential equation. These two components are coupled with the bacterial consumption. Mechanisms acting on the bacteria are explicitly described (growth, division and up-take). Bacteria interact via consumption. We set the exact Monte Carlo simulation algorithm of this model and its mathematical representation as a stochastic process. We prove the convergence of this process to the solution of an integro-differential equation when the population size tends to infinity. Finally, we propose several numerical simulations

Parallel and interacting Markov chains Monte Carlo method

In many situations it is important to be able to propose $N$ independent realizations of a given distribution law. We propose a strategy for making $N$ parallel Monte Carlo Markov Chains (MCMC) interact in order to get an approximation of an independent $N$-sample of a given target law. In this method each individual chain proposes candidates for all other chains. We prove that the set of interacting chains is itself a MCMC method for the product of $N$ target measures. Compared to independent parallel chains this method is more time consuming, but we show through concrete examples that it possesses many advantages: it can speed up convergence toward the target law as well as handle the multi-modal case

A modeling approach of the chemostat

Population dynamics and in particular microbial population dynamics, though they are complex but also intrinsically discrete and random, are conventionally represented as deterministic differential equations systems. We propose to revisit this approach by complementing these classic formalisms by stochastic formalisms and to explain the links between these representations in terms of mathematical analysis but also in terms of modeling and numerical simulations. We illustrate this approach on the model of chemostat.Comment: arXiv admin note: substantial text overlap with arXiv:1308.241

Stochastic models of the chemostat

We consider the modeling of the dynamics of the chemostat at its very source. The chemostat is classically represented as a system of ordinary differential equations. Our goal is to establish a stochastic model that is valid at the scale immediately preceding the one corresponding to the deterministic model. At a microscopic scale we present a pure jump stochastic model that gives rise, at the macroscopic scale, to the ordinary differential equation model. At an intermediate scale, an approximation diffusion allows us to propose a model in the form of a system of stochastic differential equations. We expound the mechanism to switch from one model to another, together with the associated simulation procedures. We also describe the domain of validity of the different models

On the variations of the principal eigenvalue with respect to a parameter in growth-fragmentation models

We study the variations of the principal eigenvalue associated to a growth-fragmentation-death equation with respect to a parameter acting on growth and fragmentation. To this aim, we use the probabilistic individual-based interpretation of the model. We study the variations of the survival probability of the stochastic model, using a generation by generation approach. Then, making use of the link between the survival probability and the principal eigenvalue established in a previous work, we deduce the variations of the eigenvalue with respect to the parameter of the model

Estimation of the parameters of a stochastic logistic growth model

We consider a stochastic logistic growth model involving both birth and death rates in the drift and diffusion coefficients for which extinction eventually occurs almost surely. The associated complete Fokker-Planck equation describing the law of the process is established and studied. We then use its solution to build a likelihood function for the unknown model parameters, when discretely sampled data is available. The existing estimation methods need adaptation in order to deal with the extinction problem. We propose such adaptations, based on the particular form of the Fokker-Planck equation, and we evaluate their performances with numerical simulations. In the same time, we explore the identifiability of the parameters which is a crucial problem for the corresponding deterministic (noise free) model

The Gauss-Galerkin approximation method in nonlinear filtering

We study an approximation method for the one-dimensional nonlinear filtering problem, with discrete time and continuous time observation. We first present the method applied to the Fokker-Planck equation. The convergence of the approximation is established. We finally present a numerical example

A Monte Carlo method without grid for a fractured porous domain model

International audienceThe double porosity model allows one to compute the pressure at a macroscopic scale in a fractured porous media, but requires the computation of some exchange coefficient characterizing the passage of the fluid from and to the porous media (the matrix) and the fractures. This coefficient may be numerically computed by some Monte Carlo method, by evaluating the time a Brownian particle spends in the matrix and the fissures. Although we simulate some stochastic processes, the approach presented here does not use approximation by random walks, and then does not require any discretization. We are interested only in the particles in the matrix. A first approximation of the exchange coefficient may then be computed. In a forthcoming paper, we will present the simulation of the particles in the fissures

Modèles à espace d'états non linéaires/non gaussiens et inférence bayésienne par méthode {MCMC} -- Une application en évaluation des stocks halieutiques

National audienceDifference equations with delay are widely used to model the evolution of the biomass of a fish stock (delay difference models). Represented as a state- space model they allow, starting from the data of the annual catches, a relevant Bayesian analysis. For this purpose we can use an hybrid MCMC method combi- ning a Metropolis-Hastings algorithm within a Gibbs sampler, namely the single- component Metropolis-Hastings algorithm