165 research outputs found
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 independent
realizations of a given distribution law. We propose a strategy for making
parallel Monte Carlo Markov Chains (MCMC) interact in order to get an
approximation of an independent -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
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
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