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

Gwiazda, Piotr; Jabłoński, Jędrzej; Marciniak–Czochra, Anna; Ulikowska, Agnieszka

doi:10.1002/num.21879

Cited by 25 publications

(29 citation statements)

References 35 publications

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“…Importantly, this study also found that insufficiently frequent internalizations may cause a failure in the ODEs describing the boundary cohort (Gwiazda et al . ).…”

Section: Discussionmentioning

confidence: 97%

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

2017

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Abstract• 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 considered when choosing numerical method.

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“…Importantly, this study also found that insufficiently frequent internalizations may cause a failure in the ODEs describing the boundary cohort (Gwiazda et al . ).…”

Section: Discussionmentioning

confidence: 97%

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

2017

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“…In case of male and female populations it only differs by the evolution of localisations in the boundary cohorts. This simplification was already proposed in [3] for the single species case, whose convergence was proven in [9]. Here compared to [10], we simplified the EBT scheme further by choosing the localisation of the couples in the boundary cohorts consistently in terms of the localisations of the male and female populations.…”

Section: Relation Between the Methodsmentioning

confidence: 99%

“…The method has been popular for many years, even its convergence was proven only recently in [3], and in [9] where the rate of convergence was also shown. Results on the convergence were obtained using a theoretical approach to stability, where the underlying model is embedded in a space of nonnegative Radon measures (M + (R + )) equipped with a bounded Lipschitz distance (flat metric).…”

Section: Introductionmentioning

confidence: 99%

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

Carrillo¹,

Gwiazda²,

Kropielnicka³

et al. 2019

SIAM J. Numer. Anal.

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The Escalator Boxcar Train (EBT) method is a well known and widely used numerical method for onedimensional structured population models of McKendrick-von Foerster type. Recently the method, in its full generality, has been applied to aged-structured two-sex population model (Fredrickson-Hoppensteadt model), which consists of three coupled hyperbolic partial differential equations with nonlocal boundary conditions. We derive the simplified EBT method and prove its convergence to the solution of Fredrickson-Hoppensteadt model. The convergence can be proven, however only if we analyse the whole problem in the space of nonnegative Radon measures equipped with bounded Lipschitz distance (flat metric). Numerical simulations are presented to illustrate the results.Functions c m , c f and c c describe the rates of disappearance of individuals, where disappearance of males or females is related to death, while couples disappearance reflects divorce or death of one of spouses. Functions b m and b f are birth rates of males and females. Observe that the mentioned coefficients depend on ecological pressure in a nonlinear manner, that is they are nonlocal operators depending on the distribution of males, females and couples.The marriage function T models the number of new marriages of males and females of age x and y, respectively, at time t. It also depends nonlinearly on the distribution of individuals. The choice of this function is a subject of ongoing discussions, see [14,15,21,23], due to the properties like heterosexuality, homogeneity, consistency or competition. In this paper we follow the formulation proposed in [18], namelyx, y)dy is the amount of unmarried males and u f (t, y) − ∞ 0 u c (t, x, y)dx is the number of unmarried females. The functions h, g ∈ L 1 (R + ) ∩ L ∞ (R + ) describes the distribution of eligible males/females on the marriage market. We further assume that youngsters do not marry below a certain age a, i.e. h(x) = g(y) = 0 for x, y ∈ [0, a 0 ) .( 1.3)The regularity of the remaining coefficients and their nonlinear dependences are presented in detail in Section 3.2. The aim of the paper is analysis and convergence of a simplified EBT scheme, see Subsection 2.1, corresponding to (1.1). More precisely, we show convergence of the numerical integrator embedding the underlying problem (1.1) and its numerical scheme in a space of measures. The nonlinear age-structured, two-sex population model was presented and analysed in a space of nonnegative Radon measures in [25], where the well possedness of the problem was proved.Recently, the EBT method has been derived in [10] for a partially linearised system of aged-structured equationsLemma 3.3. Let µ = J i=1 m i δ xi andμ = J i=1m i δx i , where J ∈ N, x i ,x i ∈ R N + and m i ,m i ∈ R + . Then, d N (µ,μ) ≤ J i=1 ( x i −x i m i + |m i −m i |) . γ + J v=Bn h(x c vj (t)) m m v (t) − J w=Bnm c vw (t) + J w=Bn g(y c iw (t)) m f w (t) − J v=Bnm c vw (t).

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“…Remark: For the boundary condition (5) the system is not explicitly solvable since the boundary condition is dependent on the solution itself. In a version of EBT, or in fact very similar in this context splitting method, we use the following approximation (for a full description see [22,37,23,38]):…”

Section: Cohort Approachmentioning

confidence: 99%

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

Gwiazda

Miasojedow

Rosińska

2016

Journal of Theoretical Biology

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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.

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

Cited by 25 publications

References 35 publications

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

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

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

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

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