6,859 research outputs found
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
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
The Operating Diagram for a Two-Step Anaerobic Digestion Model
The Anaerobic Digestion Model No. 1 (ADM1) is a complex model which is widely
accepted as a common platform for anaerobic process modeling and simulation.
However, it has a large number of parameters and states that hinder its
analytical study. Here, we consider the two-step reduced model of anaerobic
digestion (AM2) which is a four-dimensional system of ordinary differential
equations. The AM2 model is able to adequately capture the main dynamical
behavior of the full anaerobic digestion model ADM1 and has the advantage that
a complete analysis for the existence and local stability of its steady states
is available. We describe its operating diagram, which is the bifurcation
diagram which gives the behavior of the system with respect to the operating
parameters represented by the dilution rate and the input concentrations of the
substrates. This diagram, is very useful to understand the model from both the
mathematical and biological points of view
Competitive exclusion for chemostat equations with variable yields
In this paper, we study the global dynamics of a chemostat model with a
single nutrient and several competing species. Growth rates are not required to
be proportional to food uptakes. The model was studied by Fiedler and Hsu [J.
Math. Biol. (2009) 59:233-253]. These authors prove the nonexistence of
periodic orbits, by means of a multi-dimensional Bendixon-Dulac criterion. Our
approach is based on the construction of Lyapunov functions. The Lyapunov
functions extend those used by Hsu [SIAM J. Appl. Math. (1978) 34:760-763] and
by Wolkowicz and Lu [SIAM J. Appl. Math. (1997) 57:1019-1043] in the case when
growth rates are proportional to food uptakes
A method for the reconstruction of unknown non-monotonic growth functions in the chemostat
We propose an adaptive control law that allows one to identify unstable
steady states of the open-loop system in the single-species chemostat model
without the knowledge of the growth function. We then show how one can use this
control law to trace out (reconstruct) the whole graph of the growth function.
The process of tracing out the graph can be performed either continuously or
step-wise. We present and compare both approaches. Even in the case of two
species in competition, which is not directly accessible with our approach due
to lack of controllability, feedback control improves identifiability of the
non-dominant growth rate.Comment: expansion of ideas from proceedings paper (17 pages, 8 figures),
proceedings paper is version v
Modeling and analysis of random and stochastic input flows in the chemostat model
In this paper we study a new way to model noisy input flows in the chemostat model, based on the Ornstein-Uhlenbeck process. We introduce a parameter β as drift in the Langevin equation, that allows to bridge a gap between a pure Wiener process, which is a common way to model random disturbances, and no noise at all. The value of the parameter β is related to the amplitude of the deviations observed on the realizations. We show that this modeling approach is well suited to represent noise on an input variable that has to take non-negative values for almost any time.European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Ministerio de EconomĂa y Competitividad (MINECO). EspañaJunta de AndalucĂ
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