Escalator Boxcar Train Research Articles

Bayesian inference of a non-local proliferation model.

From a systems biology perspective, the majority of cancer models, although interesting and providing a qualitative explanation of some problems, have a major disadvantage in that they usually miss a genuine connection with experimental data. Having this in mind, in this paper, we aim at contributing to the improvement of many cancer models which contain a proliferation term. To this end, we propose a new non-local model of cell proliferation. We select data that are suitable to perform Bayesian inference for unknown parameters and we provide a discussion on the range of applicability of the model. Furthermore, we provide proof of the stability of posterior distributions in total variation norm which exploits the theory of spaces of measures equipped with the weighted flat norm. In a companion paper, we provide detailed proof of the well-posedness of the problem and we investigate the convergence of the escalator boxcar train (EBT) algorithm applied to solve the equation.

Measure-valued solutions to size-structured population model of prey controlled by optimally foraging predator harvester

Radon-measure-valued solutions to a size structured population model of the McKendrick–von Foerster-type are analytically studied under general assumptions on individuals’ growth, birth and mortality rates. The model is used to describe changes in size structure of zooplankton when prey size-dependent mortality rate is a consequence of a planktivorous fish foraging in low prey-density environment (commonly found in predator-controlled populations). The model of foraging is based on the optimization of the rate of net energy intake as a function of predator speed. Mortality is defined as an operator on a metric space of nonnegative Radon measures equipped with the bounded Lipschitz distance. The solutions to the size structured model of zooplankton population are studied analytically and numerically. Numerical solutions (derived using the Escalator Boxcar Train (EBT)-like schema), in particular those starting from Dirac deltas corresponding to distinct cohorts, exhibit regularization in time and convergence to the same stationary state.

A model for the dynamics of the parasitic stages of equine cyathostomins

A model was developed to reproduce the dynamics of the parasitic stages of equine cyathostomins. Based on a detailed review of published literature, a deterministic simulation model was constructed using the escalator boxcar-train approach, which allows for fully-overlapping cohorts of worms and approximately normally distributed variations in age/size classes. Key biological features include a declining establishment of ingested infective stage larvae as horses age. Development rates are constant for all the parasitic stages except the encysted early third stage larvae, for which development rates are variable to reflect the sometimes extended arrestment of this stage. For these, development is slowed in the presence of adult worms in the intestinal lumen, and when ingestion of infective larvae on herbage is high or extended. In the absence of anthelmintic treatments, the life span of adult worms is approximately 12 months, and the presence of an established adult worm burden largely blocks the transition of luminal fourth stage larvae to the adult stage, resulting in mortality of the larvae. This inhibition is removed by effective anthelmintic treatment allowing the rapid replacement of adult worms from the pool of mucosal stages.Within the model, the rate and seasonality at which infective stage larvae are ingested strongly influences the dynamics of the pre-adult stages. While the adult worm burden remains relatively stable within a year, due to the negative feedback they have on developing stages, the numbers and proportions of larval stages relative to the total worm burden increase with the numbers of infective larvae ingested. Further, within the model, the seasonal rise and fall of encysted stages is largely driven by the seasonal pattern of infective larvae on pasture. Because of this, the model reproduces the contrasting seasonal patterns of mucosal larvae, typical of temperate and tropical environments, using only the appropriate seasonality of larvae on pasture. Thus, the model reproduces output typical of different climatic regions and suggests that observed patterns of arrested development may simply reflect the numbers and seasonality of free-living stages on pasture as determined by different management practices and weather patterns.

The Escalator Boxcar Train Method for a System of Age-Structured Equations in the Space of Measures

The Escalator Boxcar Train (EBT) numerical method has been designed and widely used by theoretical biologists to compute solutions of one-dimensional structured population models of McKendrick--von...

Optimization in structure population models through the Escalator Boxcar Train

The Escalator Boxcar Train (EBT) is a tool widely used in the study of balance laws motivated by structure population dynamics. This paper proves that the approximate solutions defined through the EBT converge to exact solutions. Moreover, this method is rigorously shown to be effective also in computing optimal controls. As preliminary results, the well posedness of classes of PDEs and of ODEs comprising various biological models is also obtained. A specific application to welfare policies illustrates the whole procedure.

On the performance of four methods for the numerical solution of ecologically realistic size‐structured population models

Summary Size‐structured population models (SSPMs) are widely used in ecology to account for intraspecific variation in body size. Three characteristic features of size‐structured populations are the dependence of life histories on the entire size distribution, intrinsic population renewal through the birth of new individuals, and the potential accumulation of individuals with similar body sizes due to determinate or stunted growth. Because of these three features, numerical methods that work well for structurally similar transport equations may fail for SSPMs and other transport‐dominated models with high ecological realism, and thus their computational performance needs to be critically evaluated. Here, we compare the performance of four numerical solution schemes, the fixed‐mesh upwind (FMU) method, the moving‐mesh upwind (MMU) method, the characteristic method (CM), and the Escalator Boxcar Train (EBT) method, in numerically solving three reference problems that are representative of ecological systems in the animal and plant kingdoms. The MMU method is here applied for the first time to SSPMs, whereas the three other methods have been employed by other authors. Our results show that the EBT method performs best, except for one of the three reference problems, in which size‐asymmetric competition affects individual growth rates. For that reference problem, the FMU method performs best, closely followed by the MMU method. Surprisingly, the CM method does not perform well for any of the three reference problems. We conclude that life‐history features should be carefully considered when choosing the numerical method for analyzing ecologically realistic size‐structured population models.

A deterministic eco-genetic model for the short-term evolution of exploited fish stocks

Eco-genetic models describe the contemporary evolution of quantitative genetic traits by integrating the key ecological aspects with the modes of inheritance. Because of the flexibility in incorporating physiological, ecological, and genetic detail, eco-genetic models are typically individual-based and stochastic. Here we propose a deterministic eco-genetic population model to study the evolution of maturation schedules in exploited fish stocks. It is the first deterministic model addressing the evolution of the probabilistic maturation reaction norm (PMRN)—the genetic trait recognized to control maturation in fish species. The PMRN is the probability of interseasonal maturation as a function of age and size of juvenile individuals. Being independent of the growth trajectory followed by the individual, it can be considered a non-plastic phenotype, under limited change in temperature and climate. We consider the continuous age and size structures of the fish stock and explicitly represent foraging on a single food resource and mating under 1:1 sex ratio. Inheritance follows a single-locus-two-alleles Mendelian rule to limit the number of genotypes. We adapt the standard escalator boxcar train method for the simulation of physiologically structured population models to the case of two-dimensional age distributions—mature individuals being also distributed according to the age at maturation. The model simulations qualitatively reproduce empirical trends, specifically the genotypic redistribution toward early-maturing types under increasing exploitation and the eventual collapse of the stock. Interestingly, increased effort targeted at large sizes seems to erode the genetic diversity, an effect missed by stochastic models that might serve as warning to fishery managers and policy makers.

Bayesian inference for age-structured population model of infectious disease with application to varicella in Poland

The dynamics of the infectious disease transmission are often best understood by taking into account the structure of population with respect to specific features, for example age or immunity level. The practical utility of such models depends on the appropriate calibration with the observed data. Here, we discuss the Bayesian approach to data assimilation in the case of a two-state age-structured model. Such models are frequently used to explore the disease dynamics (i.e. force of infection) based on prevalence data collected at several time points. We demonstrate that, in the case when the explicit solution to the model equation is known, accounting for the data collection process in the Bayesian framework allows us to obtain an unbiased posterior distribution for the parameters determining the force of infection. We further show analytically and through numerical tests that the posterior distribution of these parameters is stable with respect to a cohort approximation (Escalator Boxcar Train) of the solution. Finally, we apply the technique to calibrate the model based on observed sero-prevalence of varicella in Poland.

The Escalator Boxcar Train method for a system of age-structured equations

The Escalator Boxcar Train method (EBT) is a numerical method for structured population models of McKendrick -- von Foerster type. Those models consist of a certain class of hyperbolic partial differential equations and describe time evolution of the distribution density of the structure variable describing a feature of individuals in the population. The method was introduced in late eighties and widely used in theoretical biology, but its convergence was proven only in recent years using the framework of measure-valued solutions. Till now the EBT method was developed only for scalar equation models. In this paper we derive a full numerical EBT scheme for age-structured, two-sex population model (Fredrickson-Hoppensteadt model), which consists of three coupled hyperbolic partial differential equations with nonlocal boundary conditions. It is the first step towards extending the EBT method to systems of structured population equations.

Splitting-particle methods for structured population models: Convergence and applications

We propose a new numerical scheme designed for a wide class of structured population models based on the idea of operator splitting and particle approximations. This scheme is related to the Escalator Boxcar Train (EBT) method commonly used in biology, which is in essence an analogue of particle methods used in physics. Our method exploits the split-up technique, thanks to which the transport step and the nonlocal integral terms in the equation can be separately considered. The order of convergence of the proposed method is obtained in the natural space of finite non-negative Radon measures equipped with the flat metric. This convergence is studied even adding reconstruction and approximation steps in the particle simulation to keep the number of approximation particles under control. We validate our scheme in several test cases showing the theoretical convergence error. Finally, we use the scheme in situations in which the EBT method does not apply showing the flexibility of this new method to cope with the different terms in general structured population models.

Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded Lipschitz distance

Recently developed theoretical framework for analysis of structured population dynamics in the spaces of nonnegative Radon measures with a suitable metric provides a rigorous tool to study numerical schemes based on particle methods. The approach is based on the idea of tracing growth and transport of measures which approximate the solution of original partial differential equation. In this article, we present analytical and numerical study of two versions of Escalator Boxcar Train algorithm which has been widely applied in theoretical biology, and compare it to the recently developed split‐up algorithm. The novelty of this article is in showing well‐posedness and convergence rates of the schemes using the concept of semiflows on metric spaces. Theoretical results are validated by numerical simulations of test cases, in which distances between simulated and exact solutions are computed using flat metric. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1797–1820, 2014

On the Convergence of the Escalator Boxcar Train

The Escalator Boxcar Train (EBT) is a numerical method that is widely used in theoretical biology to investigate the dynamics of physiologically structured population models, i.e., models in which individuals differ by size or other physiological characteristics. The method was developed more than two decades ago, but has so far resisted attempts to give a formal proof of convergence. Using a modern framework of measure-valued solutions, we investigate the EBT method and show that the sequence of approximating solution measures generated by the EBT method converges weakly to the true solution measure under weak conditions on the growth rate, birth rate, and mortality rate. In rigorously establishing the convergence of the EBT method, our results pave the way for wider acceptance of the EBT method beyond theoretical biology and constitutes an important step towards integration with established numerical schemes.

Optimal Forest Management in the Presence of Intraspecific Competition

The Escalator Boxcar Train method is used to solve the distributed optimal control problems of forest management numerically. It takes into account intraspecific competition for scarce resources such as light, space, and nutrients during reproduction, growth, and mortality. It provides an alternative to gradient projection methods and Markov processes. It is implemented with standard software. The application is on the optimal forest management regime in the presence of intraspecific competition.

A model approach to evaluate the effect of temperature and food concentration on individual life-history and population dynamics of Daphnia

We present a model framework for the simulation of growth and reproduction of Daphnia at varying conditions of food concentration and temperature. The core of our framework consists of an individual level model that simulates allocation of assimilated carbon into somatic growth, maintenance costs, and reproduction on the basis of a closed carbon budget. A fixed percentage of assimilated carbon is allocated into somatic growth and maintenance costs. Special physiological adaptations in energy acquisition and usage allow realistic model performance even at very low food concentrations close to minimal food requirements. All model parameters are based on physiological measures taken from the literature. Model outputs were thoroughly validated on data from a life-table experiment with Daphnia galeata. For the first time, a successful model validation was performed at such low food concentrations. The escalator boxcar train (EBT) was used to integrate this individual level model into a stage-structured population model. In advance to previous applications of the EBT to Daphnia we included an additional clutch compartment into the model structure that accounts for the characteristic time delay between egg deposition and hatching in cladocerans. By linking two levels of biological organisation, this model approach represents a comprehensive framework for studying Daphnia both at laboratory conditions and in the field. We compared outputs of our stage-structured model with predictions by two other models having analogous parameterisation: (i) another individual level Daphnia model (Kooijman–Metz model) and (ii) a classical unstructured population model. In contrast to our Daphnia model, the Kooijman–Metz model lacks the structure to account for the optimisation of energy acquisition and maintenance requirements by individual daphnids. The unstructured population model showed different patterns of population dynamics that were not in concordance with typical patterns observed in the field. Thus, we conclude our model provides a comprehensive tool for the simulation of growth and reproduction of Daphnia and corresponding population dynamics.

Numerical Methods and Parameter Estimation of a Structured Population Model with Discrete Events in the Life History

We consider two numerical methods for the solution of a physiologically structured population (PSP) model with multiple life stages and discrete event reproduction. The model describes the dynamic behaviour of a predator–prey system consisting of rotifers predating on algae. The nitrate limited algal prey population is modelled unstructured and described by an ordinary differential equation (ODE). The formulation of the rotifer dynamics is based on a simple physiological model for their two life stages, the egg and the adult stage. An egg is produced when an energy buffer reaches a threshold value. The governing equations are coupled partial differential equations (PDE) with initial and boundary conditions. The population models together with the equation for the dynamics of the nutrient result in a chemostat model. Experimental data are used to estimate the model parameters. The results obtained with the explicit finite difference (FD) technique compare well with those of the Escalator Boxcar Train (EBT) method. This justifies the use of the fast FD method for the parameter estimation, a procedure which involves repeated solution of the model equations.

Studying the Dynamics of Structured Population Models: A Versatile Technique and Its Application to Daphnia

We introduce a versatile model formulation, called the escalator boxcar train, that can be used to study the dynamics of populations structured by age, size, or any other physiological trait. Spanning the entire spectrum between the classical, age-structured, Leslie matrix models and continuous-time, physiologically structured population models, it combines the computational ease of the matrix models with the flexibility of the continuous-time models: it allows the population dynamics to depend on an arbitrary number of individual characteristics and can account for nonlinear interactions of the population with its (fluctuating) environment, for density dependence in individual behavior, and for a continuous reproduction process. Application of the technique to a model for the dynamics of a Daphnia population reveals that a food-dependent maturation delay induces a very specific type of population cycle, in which the demography of the population plays a major role. The influence of "growth-curve plasticity," that is, the change in size at each age with a change in food density, on individual feeding behavior (at higher food densities individuals are larger and hence eat more) is a potential mechanism to stabilize the dynamics. The influence of growth-curve plasticity on reproduction (primarily the increase in maturation delay with a decrease in food density) greatly destabilizes the system.

Numerical methods for structured population models: The Escalator Boxcar Train

AbstractA numerical integration method is introduced for the class of hyperbolic partial differential equations that occur in physiologically structured population models. These equations describe the time evolution of the population density‐function over the individual state‐space Ω. Exploiting the biological interpretation of the equation, this density‐function is represented by a set of moments over a collection of subdomains in Ω, moving along the characteristics of the partial differential equation (PDE). These moments are readily interpreted as numbers of individuals in a cohort, mean individual state in a cohort, etc. The numerical method consists of approximating the differential equations.The method appears to be very efficient for this special type of PDE. It combines such desirable properties as as lack of dispersion and dissipation and a preservation of monotonicity of the density function along the characteristics with a strong relation to the biological background of the equations for these moments to arrive at a closed system of ordinary differential equations. The method also allows one to incorporate the usual type of nonlinearities that occur in physiologically structured population models. A numerical example is presented as an illustration of the application of the method.