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Parameter estimation in non-linear mixed effects models with SAEM algorithm: extension from ODE to PDE

E. Grenier, V. Louvet, P. Vigneaux (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Parameter estimation in non linear mixed effects models requires a large number of evaluations of the model to study. For ordinary differential equations, the overall computation time remains reasonable. However when the model itself is complex (for instance when it is a set of partial differential equations) it may be time consuming to evaluate it for a single set of parameters. The procedures of population parametrization (for instance using SAEM algorithms) are then very long and in some cases...

Parameter Identification of a Fed-Batch Cultivation of S. Cerevisiae using Genetic Algorithms

Angelova, Maria, Tzonkov, Stoyan, Pencheva, Tania (2010)

Serdica Journal of Computing

Fermentation processes as objects of modelling and high-quality control are characterized with interdependence and time-varying of process variables that lead to non-linear models with a very complex structure. This is why the conventional optimization methods cannot lead to a satisfied solution. As an alternative, genetic algorithms, like the stochastic global optimization method, can be applied to overcome these limitations. The application of genetic algorithms is a precondition for robustness...

Parametric inference for mixed models defined by stochastic differential equations

Sophie Donnet, Adeline Samson (2008)

ESAIM: Probability and Statistics

Non-linear mixed models defined by stochastic differential equations (SDEs) are considered: the parameters of the diffusion process are random variables and vary among the individuals. A maximum likelihood estimation method based on the Stochastic Approximation EM algorithm, is proposed. This estimation method uses the Euler-Maruyama approximation of the diffusion, achieved using latent auxiliary data introduced to complete the diffusion process between each pair of measurement instants. A tuned...

Particle Dynamics Methods of Blood Flow Simulations

A. Tosenberger, V. Salnikov, N. Bessonov, E. Babushkina, V. Volpert (2011)

Mathematical Modelling of Natural Phenomena

Various particle methods are widely used to model dynamics of complex media. In this work molecular dynamics and dissipative particles dynamics are applied to model blood flows composed of plasma and erythrocytes. The properties of the homogeneous particle fluid are studied. Capillary flows with erythrocytes are investigated.

Past, Present and Future of Brain Stimulation

J. Modolo, R. Edwards, J. Campagnaud, B. Bhattacharya, A. Beuter (2010)

Mathematical Modelling of Natural Phenomena

Recent technological advances including brain imaging (higher resolution in space and time), miniaturization of integrated circuits (nanotechnologies), and acceleration of computation speed (Moore’s Law), combined with interpenetration between neuroscience, mathematics, and physics have led to the development of more biologically plausible computational models and novel therapeutic strategies. Today, mathematical models of irreversible medical conditions...

Pattern and Waves for a Model in Population Dynamics with Nonlocal Consumption of Resources

S. Genieys, V. Volpert, P. Auger (2010)

Mathematical Modelling of Natural Phenomena

We study a reaction-diffusion equation with an integral term describing nonlocal consumption of resources in population dynamics. We show that a homogeneous equilibrium can lose its stability resulting in appearance of stationary spatial structures. They can be related to the emergence of biological species due to the intra-specific competition and random mutations. Various types of travelling waves are observed.

Pattern Formation Induced by Time-Dependent Advection

A. V. Straube, A. Pikovsky (2010)

Mathematical Modelling of Natural Phenomena

We study pattern-forming instabilities in reaction-advection-diffusion systems. We develop an approach based on Lyapunov-Bloch exponents to figure out the impact of a spatially periodic mixing flow on the stability of a spatially homogeneous state. We deal with the flows periodic in space that may have arbitrary time dependence. We propose a discrete in time model, where reaction, advection, and diffusion act as successive operators, and show that...

Pattern Formation of Competing Microorganisms in Sediments

Y. Schmitz, M. Baurmann, B. Engelen, U. Feudel (2010)

Mathematical Modelling of Natural Phenomena

We present a three species model describing the degradation of substrate by two competing populations of microorganisms in a marine sediment. Considering diffusion to be the main transport process, we obtain a reaction diffusion system (RDS) which we study in terms of spontaneous pattern formation. We find that the conditions for patterns to evolve are likely to be fulfilled in the sediment. Additionally, we present simulations that are consistent with experimental data from the literature. We...

Patterns, Memory and Periodicity in Two-Neuron Delayed Recurrent Inhibitory Loops

J. Ma, J. Wu (2010)

Mathematical Modelling of Natural Phenomena

We study the coexistence of multiple periodic solutions for an analogue of the integrate-and-fire neuron model of two-neuron recurrent inhibitory loops with delayed feedback, which incorporates the firing process and absolute refractory period. Upon receiving an excitatory signal from the excitatory neuron, the inhibitory neuron emits a spike with a pattern-related delay, in addition to the synaptic delay. We present a theoretical framework to view...

Patterns of Zooplankton Functional Response in Communities with Vertical Heterogeneity: a Model Study

A. Morozov, E. Arashkevich (2008)

Mathematical Modelling of Natural Phenomena

Parameterization of zooplankton functional response is crucial for constructing plankton models. Theoretical studies predict enhancing of system stability in case the response is of sigmoid type. Experiments on feeding in laboratories tell us in favor of non-sigmoid types for most herbivorous zooplankton species. However, recent field observations show that the overall functional response of zooplankton in the whole euphotic zone can exhibit a sigmoid behavior even when the response for the same...

PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic

Benoît Perthame (2004)

Applications of Mathematics

Modeling the movement of cells (bacteria, amoeba) is a long standing subject and partial differential equations have been used several times. The most classical and successful system was proposed by Patlak and Keller & Segel and is formed of parabolic or elliptic equations coupled through a drift term. This model exhibits a very deep mathematical structure because smooth solutions exist for small initial norm (in the appropriate space) and blow-up for large norms. This reflects experiments on...

Periodic dynamics in a model of immune system

Marek Bodnar, Urszula Foryś (2000)

Applicationes Mathematicae

The aim of this paper is to study periodic solutions of Marchuk's model, i.e. the system of ordinary differential equations with time delay describing the immune reactions. The Hopf bifurcation theorem is used to show the existence of a periodic solution for some values of the delay. Periodic dynamics caused by periodic immune reactivity or periodic initial data functions are compared. Autocorrelation functions are used to check the periodicity or quasiperiodicity of behaviour.

Periodic solution to a multispecies predator-prey competition dynamic system with Beddington-DeAngelis functional response and time delay

Xiaojie Lin, Zengji Du, Yansen Lv (2013)

Applications of Mathematics

In this paper, we are concerned with a delayed multispecies competition predator-prey dynamic system with Beddington-DeAngelis functional response. Some sufficient conditions which guarantee the existence of a positive periodic solution for the system are obtained by applying the Mawhin coincidence theory. The interesting thing is that the result is related to the delays, which is different from the corresponding ones known from literature (the results are delay-independent).

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