Estimation of Variable Cefficients in the Fokker–Planck Quations Using Moving Node Finite Elements

Three Decades of Progress in Control Sciences

Davis

2010

Self Cite

Two conceptually different approaches to incorporate growth uncertainty into sizestructured population models have recently been investigated. One entails imposing a probabilistic structure on all the possible growth rates across the entire population, which results in a growth rate distribution model. The other involves formulating growth as a Markov stochastic diffusion process, which leads to a Fokker-Planck model. Numerical computations verify that a Fokker-Planck model and a growth rate distribution model can, with properly chosen parameters, yield quite similar time dependent population densities. The relationship between the two models is based on the theoretical analysis in [7].

Section: Stochastic Formulationmentioning

confidence: 99%

“…With this assumption on the growth process, we obtain the Fokker-Planck (FP) or forward Kolmogorov model for the population density u, which was carefully derived in [22] among numerous other places and subsequently studied in many references (e.g., [1,13,16]). The equation and appropriate boundary conditions are given by…”

Section: Stochastic Formulationmentioning

confidence: 99%

A Computational Comparison of Alternatives to Including Uncertainty in Structured Population Models,

Three Decades of Progress in Control Sciences

Davis

2010

Self Cite

“…In an early biological application [41], a scalar Fokker-Planck equation was employed to study the random fluctuation of gene frequencies in natural population (where x denotes gene frequency). The Fokker-Planck equation has also been effectively used in the literature (e.g., see [26] and the reference therein) to model the dispersal behavior of a population such as studies of female cabbage root fly movement in the presence of Brassica odors, and movement of flea beetles in cultivated collard patches (in these cases x denotes the space position and g(t, x) is the mean velocity of individuals at position x at time t).…”

Section: ])mentioning

confidence: 99%

“…For example, a scalar FPPS model with nonlinear and nonlocal boundary condition was studied in [31] to understand how a constant diffusion coefficient influences the stability and the Hopf bifurcation of the positive equilibrium of the system. In another example inverse problems were considered in [26] for the estimation of temporally and spatially varying coefficients in a scalar FPPS model with nonlocal boundary conditions.…”

Section: Fokker-planck Physiologically Structured Population Modelsmentioning

confidence: 99%

Propagation of Growth Uncertainty in a Physiologically Structured Population

Math. Model. Nat. Phenom.

2012

Abstract. In this review paper we consider physiologically structured population models that have been widely studied and employed in the literature to model the dynamics of a wide variety of populations. However in a number of cases these have been found inadequate to describe some phenomena arising in certain real-world applications such as dispersion in the structure variables due to growth uncertainty/variability. Prompted by this, we described two recent approaches that have been investigated in the literature to describe this growth uncertainty/variability in a physiologically structured population. One involves formulating growth as a Markov diffusion process while the other entails imposing a probabilistic structure on the set of possible growth rates across the entire population. Both approaches lead to physiologically structured population models with nontrivial dispersion. Even though these two approaches are conceptually quite different, they were found in [17] to have a close relationship: in some cases with properly chosen parameters and coefficient functions, the resulting stochastic processes have the same probability density function at each time.

“…The fitted curves can be used to approximate the numbers of cells having divided a given number of times. variable environment vs. individual stochastic mechanisms has been treated specifically in [4,11,12,25] as well as more generically in [9,10,16,17]. While probability and stochasticity is fundamental to all of these models, the mathematical constructs used to incorporate asynchrony/variability into the modeling is strikingly different conceptually and computationally.…”

Section: Mathematical Modelsmentioning

confidence: 99%

Mathematical Models of Dividing Cell Populations: Application to CFSE Data

Math. Model. Nat. Phenom.

Thompson

2012

Abstract. Flow cytometric analysis using intracellular dyes such as CFSE is a powerful experimental tool which can be used in conjunction with mathematical modeling to quantify the dynamic behavior of a population of lymphocytes. In this survey we begin by providing an overview of the mathematically relevant aspects of the data collection procedure. We then present an overview of the large body of mathematical models, along with their assumptions and uses, which have been proposed to describe the dynamics of proliferating cell populations. While much of this body of work has been aimed at modeling the generation structure (cells per generation) of the proliferating population, several recent models have considered the more fundamental task of modeling CFSE histogram data directly. Such models are analyzed and recent results are discussed. Finally, directions for future research are suggested.