Von Foerster Equation Research Articles

Dynamic population modeling of bacterioplankton community response to gelatinous marine zooplankton bloom collapse and its impact on marine nutrient balance

The diverse microbial community in the ocean, encompassing various metabolic types, interacts with the wide array of compounds in the dissolved organic matter (DOM) pool, thereby influencing the ocean’s biogeochemical state and, consequently, the global climate. Our understanding of the interactions between specific DOM constituents and microbial consortia remains limited, necessitating further refinement to achieve a mechanistic comprehension of the relationship between the DOM field and the microbial metabolic network. Attaining this level of understanding is crucial for accurately predicting the marine ecosystem’s response to natural and anthropogenic perturbations. To address this gap, we developed a bacterial population model based on the von Foerster equation. This model aims to describe the complex microbial-mediated degradation of gelatinous zooplankton (hereinafter ‘jellyfish’) detritus, as an important, but largely overlooked source of DOM in the ocean. By considering microbial growth and decay, as well as DOM uptake, and nutrient release, the model is able to describe the microbial community’s life cycle, and the biochemical transformations of the jellyfish-derived organic matter. We fitted the model to results of laboratory microcosm experiments conducted to simulate scenarios experienced by ambient microbiomes during decay of two different jellyfish species in the northern Adriatic Sea. By interpreting the fitted parameters, we highlight the differences in the microbial response to different jellyfish species, namely how these affect the microbial community composition and the release of nutrients. This model has been specifically designed for integration with ocean circulation models to create a comprehensive physical-biogeochemical ocean model. Such an extended model can be utilized for multi-scale simulations to assess the system’s response to jellyfish and jellyfish-derived organic matter. Given that jellyfish blooms may become more prevalent under future ocean scenarios, this modeling approach is essential for understanding their potential impact on marine ecosystems.

On the dynamics of predator–prey models with role reversal

We present a novel mathematical model that incorporates role reversal in predator–prey interactions. In particular, we consider the case where adult preys attack and kill juvenile predators. The model is derived from the McKendrick–Von Foerster equation and includes a maturation delay for juvenile predators. We prove that the initial value problem of the modeling framework is globally wellposed and establish conditions for the existence of positive equilibrium points. The model is compared with a version without role reversal, and we use the implicit function theorem to analyze the impact of weak role reversal on the equilibrium points of the model. This approach is necessitated as finding explicit expressions for the equilibrium points in the model with role reversal is particularly challenging. Phase portraits and bifurcation diagrams are used to illustrate our theoretical findings. Our results suggest that the maturation delay of predators and the handling time are the key factors in the dynamics of the model with role reversal, thus highlighting the importance of considering these phenomena in predator–prey interactions. This study provides a novel theoretical observation that the role reversal mechanism in predator–prey interactions has the potential to prevent cyclic population dynamics. The model outcome has been validated with the help of available data for the forage-piscivore fish trade-offs. Experimental studies are needed to validate model predictions and determine its relevance in real-world ecological systems.

Краевая задача для нагруженного уравнения Маккендрика – фон Ферстера дробного порядка

The paper considers the loaded McKendrick-von Foerster equation of fractional order, which characterizes the population dynamics with age structure taking migration into account. The boundary value problem in the rectangular domain is studied. The solution is found by reduction to the Volterra integral equation of the 2nd kind. The existence and uniqueness theorem of the problem under study is provedn.

From individual-based epidemic models to McKendrick-von Foerster PDEs: a guide to modeling and inferring COVID-19 dynamics.

We present a unifying, tractable approach for studying the spread of viruses causing complex diseases requiring to be modeled using a large number of types (e.g., infective stage, clinical state, risk factor class). We show that recording each infected individual’s infection age, i.e., the time elapsed since infection, has three benefits. First, regardless of the number of types, the age distribution of the population can be described by means of a first-order, one-dimensional partial differential equation (PDE) known as the McKendrick-von Foerster equation. The frequency of type i is simply obtained by integrating the probability of being in state i at a given age against the age distribution. This representation induces a simple methodology based on the additional assumption of Poisson sampling to infer and forecast the epidemic. We illustrate this technique using French data from the COVID-19 epidemic. Second, our approach generalizes and simplifies standard compartmental models using high-dimensional systems of ordinary differential equations (ODEs) to account for disease complexity. We show that such models can always be rewritten in our framework, thus, providing a low-dimensional yet equivalent representation of these complex models. Third, beyond the simplicity of the approach, we show that our population model naturally appears as a universal scaling limit of a large class of fully stochastic individual-based epidemic models, where the initial condition of the PDE emerges as the limiting age structure of an exponentially growing population starting from a single individual.

Straightened characteristics of McKendrick-von Foerster equation

We study the McKendrick-von Foerster equation with renewal (that is the age-structured model, with total population dependent coefficient and nonlinearity). By using a change of variables, the model is then transformed to a standard age-structured model in which the total population dependent coefficient of the transport term reduces to a constant 1. We use this transformation to get existence, uniqueness of solutions of the problem in a semigroup setting. Since straight lines are more convenient in the exact and approximate solution of PDEs, we provide sufficient conditions of reducing more general equations. We give a difference scheme to find approximate solutions of the age-structured model. Finally, some numerical simulations are presented to demonstrate the convergence and stability of the difference scheme.

Numerical solution to a nonlinear McKendrick-Von Foerster equation with diffusion

Numerical solution to a nonlinear McKendrick-Von Foerster equation with diffusion

Априорная оценка решения нелокальной краевой задачи для уравнения Маккендрика - фон Фёрстера дробного порядка

In this paper, by the method of energy inequalities, an a priori estimate for the solution of the nonlocal boundary value problem is obtained for the generalized Mackendrick - von Foerster equation with the Caputo operator with respect to the time variable.

Risk factors in the illness-death model: Simulation study and the partial differential equation about incidence and prevalence.

Recently, we developed a partial differential equation (PDE) that relates the age-specific prevalence of a chronic disease with the age-specific incidence and mortality rates in the illness-death model (IDM). With a view to planning population-wide interventions, the question arises how prevalence can be calculated if the distribution of a risk-factor in the population shifts. To study the impact of such possible interventions, it is important to deal with the resulting changes of risk-factors that affect the rates in the IDM. The aim of this work is to show how the PDE can be used to study such effects on the age-specific prevalence of a chronic disease, to demonstrate its applicability and to compare the results to a discrete event simulation (DES), a frequently used simulation technique. This is done for the first time based on the PDE which only needs data on population-wide epidemiological indices and is related to the von Foerster equation. In a simulation study, we analyse the effect of a hypothetical intervention against type 2 diabetes. We compare the age-specific prevalence obtained from a DES with the results predicted from modifying the rates in the PDE. The DES is based on 10000 subjects and estimates the effect of changes in the distributions of risk-factors. With respect to the PDE, the change of the distribution of risk factors is synthesized to an effective rate that can be used directly in the PDE. Both methods, DES and effective rate method (ERM) are capable of predicting the impact of the hypothetical intervention. The age-specific prevalences resulting from the DES and the ERM are consistent. Although DES is common in simulating effects of hypothetical interventions, the ERM is a suitable alternative. ERM fits well into the analytical theory of the IDM and the related PDE and comes with less computational effort.

The Chapman-Richards Distribution and its Relationship to the Generalized Beta

BackgroundThe Chapman-Richards distribution is developed as a special case of the equilibrium solution to the McKendrick-Von Foerster equation. The Chapman-Richards distribution incorporates the vital rate assumptions of the Chapman-Richards growth function, constant mortality and recruitment into the mathematical form of the distribution. Therefore, unlike ‘assumed’ distribution models, it is intrinsically linked with the underlying vital rates for the forest area under consideration.MethodsIt is shown that the Chapman-Richards distribution can be recast as a subset of the generalized beta distribution of the first kind, a rich family of assumed probability distribution models with known properties. These known properties for the generalized beta are then immediately available for the Chapman-Richards distribution, such as the form of the compatible basal area-size distribution. A simple two-stage procedure is proposed for the estimation of the model parameters and simulation experiments are conducted to validate the procedure for four different possible distribution shapes.ResultsThe simulations explore the efficacy of the two-stage estimation procedure; these cover the estimation of the growth equation and mortality—recruitment derives from the equilibrium assumption. The parameter estimates are shown to depend on both the sample size and the amount of noise imparted to the synthetic measurements. The results vary somewhat by distribution shape, with the smaller, noisier samples providing less reliable estimates of the vital rates and final distribution forms.ConclusionsThe Chapman-Richards distribution in its original form, or recast as a generalized beta form, presents a potentially useful model integrating vital rates and stand diameters into a flexible family of resultant distributions shapes. The data requirements are modest, and parameter estimation is straightforward provided the minimal recommended sample sizes are obtained.

Optimal control of nonlinear renewal equation

In this article, we study the optimal control for a nonlinear renewal equation popularly known as McKendrick–von Foerster (MV) equation. We establish necessary optimality conditions using the concept of normal cone. Finally, we prove the existence and uniqueness of optimal control using the Ekeland variational principle.

A forecasting model of disease prevalence based on the McKendrick–von Foerster equation

A new model for disease prevalence based on the analytical solutions of McKendric–von Foerster's partial differential equations is developed. Derivation of the model and methods to cross check obtained results are explicitly demonstrated. Obtained equations describe the time evolution of the healthy and unhealthy age-structured sub-populations and age patterns of disease prevalence. The projection of disease prevalence into the future requires estimates of time trends of age-specific disease incidence, relative survival functions, and prevalence at the initial age and year available in the data. The computational scheme for parameter estimations using Medicare data, analytical properties of the model, application for diabetes prevalence, and relationship with partitioning models are described and discussed. The model allows natural generalization for the case of several diseases as well as for modeling time trends in cause-specific mortality rates.

Relative entropy method for measure-valued solutions in natural sciences

We describe the applications of the relative entropy framework introduced in \cite{Daf}. In particular the uniqueness of an entropy solution is proven for a scalar conservation law, using the notion of measure-valued entropy solutions. Further we survey recent results concerning measure-valued-strong uniqueness for a number of physical systems -- incompressible and compressible Euler equations, compressible Navier-Stokes, polyconvex elastodynamics and general hyperbolic conservation laws, as well as long-time asymptotics of the McKendrick-Von Foerster equation.

A size-structured matrix model to simulate dynamics of marine community size spectrum.

Several types of size-based models have been developed to model the size spectra of marine communities, in which abundance scales strongly with body size, using continuous differential equations. In this study, we develop a size-structured matrix model, which can be seen as a discretization of the Mckendrick-von Foerster equation, to simulate the dynamics of marine communities. The developed model uses a series of simple body size power functions to describe the basic processes of predator–prey interactions, reproduction, metabolism, and non-predation mortality based on the principle of mass balance. Linear size spectra with slopes of approximately –1 are successfully reproduced by this model. Several examples of numerical simulations are provided to demonstrate the model’s behavior. Size spectra with cut-offs and/or waves are also found for certain parameter values. The matrix model is flexible and can be freely manipulated by users to answer different questions and is executed over a shorter numerical calculation running time than continuous models.

A numerical scheme to the McKendrick–von Foerster equation with diffusion in age

In this paper a numerical scheme for McKendrick–von Foerster equation with diffusion in age (MV‐D) is proposed. First, we discretize the time variable to get a second‐order ordinary differential equation (ODE). At each time level, well‐posedness of this ODE is established using classical methods. Stability estimates for this semidiscrete scheme are derived. Later we construct piecewise linear (in time) functions using the solutions of the semidiscrete problems to approximate the solution to MV‐D and establish the convergence result. Numerical results are presented in some cases and compared with the corresponding analytic solutions where the latter is known explicitly.

Ergodicity, hidden bias and the growth rate gain

Many single-cell observables are highly heterogeneous. A part of this heterogeneity stems from age-related phenomena: the fact that there is a nonuniform distribution of cells with different ages. This has led to a renewed interest in analytic methodologies including use of the ‘von Foerster equation’ for predicting population growth and cell age distributions. Here we discuss how some of the most popular implementations of this machinery assume a strong condition on the ergodicity of the cell cycle duration ensemble. We show that one common definition for the term ergodicity, ‘a single individual observed over many generations recapitulates the behavior of the entire ensemble’ is implied by the other, ‘the probability of observing any state is conserved across time and over all individuals’ in an ensemble with a fixed number of individuals but that this is not true when the ensemble is growing. We further explore the impact of generational correlations between cell cycle durations on the population growth rate. Finally, we explore the ‘growth rate gain’—the phenomenon that variations in the cell cycle duration leads to an improved population-level growth rate—in this context. We highlight that, fundamentally, this effect is due to asymmetric division.

Age distribution dynamics with stochastic jumps in mortality.

While deterministic age distribution models have been extensively studied and applied in various disciplines, little work has been devoted to understanding the role of stochasticity in birth and mortality terms. In this paper, we analyse a stochastic M'Kendrick-von Foerster equation in which jumps in mortality represent intense losses of population due to external events. We present explicit solutions for the probability density functions of the age distribution and the total population and for the temporal dynamics of their moments. We also derive the dynamics of the mean age of the population and its harmonic mean. The framework is then used to calculate the age distribution of salt in the soil root zone, where the accumulation of salt by atmospheric deposition is counteracted by plant uptake and by jump losses due to percolation events.

A demographic study of the exponential distribution applied to uneven-aged forests

A demographic approach based on a size-structured version of the McKendrick-Von Foerster equation is used to demonstrate a theoretical link between the population size distribution and the underlying vital rates (recruitment, mortality and diameter growth) for the population of individuals whose diameter distribution is negative exponential. This model supports the conclusion that the negative exponential distribution as applied to balanced natural uneven-aged stands is a stable equilibrium model under appropriate assumptions. These assumptions include constant recruitment of stems into the smallest class that balances total mortality, and a simple relation between per capita mortality and diameter growth. A simple maximum likelihood-based solution to parameter estimation in the inverse problem is developed allowing the estimation of recruitment and mortality given reasonable sample of diameters, along with an estimate of population size and diameter growth rate. Two sets of stand dynamics equations are developed that are based on (i) the form of the underlying negative exponential distribution, and (ii) more generally, from the derivation of the McKendrick-Von Foerster equation. Applications of the methods and their assumptions are discussed with regard to both managed and old growth uneven-aged stands. Stands or forests that are close to negative exponential structure and are judged to be reasonably close to steady state will have vital rates that support this model. In contrast, the negative exponential is likely more important and applicable as a pragmatic target distribution when used in managed forests.

Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions

In this paper, we consider a particular type of nonlinear McKend-rick-von Foerster equation with a diffusion term and Robin boundary condition. We prove the existence of a global solution to this equation. The steady state solutions to the equations that we consider have a very important role to play in the study of long time behavior of the solution. Therefore we address the issues pertaining to the existence of solution to the corresponding state equation. Furthermore, we establish that the solution of McKendrick-von Foerster equation with diffusion converges pointwise to the solution of its steady state equations as time tends to infinity.

A Hierarchical Kinetic Theory of Birth, Death and Fission in Age-Structured Interacting Populations.

We develop mathematical models describing the evolution of stochastic age-structured populations. After reviewing existing approaches, we formulate a complete kinetic framework for age-structured interacting populations undergoing birth, death and fission processes in spatially dependent environments. We define the full probability density for the population-size age chart and find results under specific conditions. Connections with more classical models are also explicitly derived. In particular, we show that factorial moments for non-interacting processes are described by a natural generalization of the McKendrick-von Foerster equation, which describes mean-field deterministic behavior. Our approach utilizes mixed-type, multidimensional probability distributions similar to those employed in the study of gas kinetics and with terms that satisfy BBGKY-like equation hierarchies.

Kinetic theory of age-structured stochastic birth-death processes.

Classical age-structured mass-action models such as the McKendrick-von Foerster equation have been extensively studied but are unable to describe stochastic fluctuations or population-size-dependent birth and death rates. Stochastic theories that treat semi-Markov age-dependent processes using, e.g., the Bellman-Harris equation do not resolve a population's age structure and are unable to quantify population-size dependencies. Conversely, current theories that include size-dependent population dynamics (e.g., mathematical models that include carrying capacity such as the logistic equation) cannot be easily extended to take into account age-dependent birth and death rates. In this paper, we present a systematic derivation of a new, fully stochastic kinetic theory for interacting age-structured populations. By defining multiparticle probability density functions, we derive a hierarchy of kinetic equations for the stochastic evolution of an aging population undergoing birth and death. We show that the fully stochastic age-dependent birth-death process precludes factorization of the corresponding probability densities, which then must be solved by using a Bogoliubov--Born--Green--Kirkwood--Yvon-like hierarchy. Explicit solutions are derived in three limits: no birth, no death, and steady state. These are then compared with their corresponding mean-field results. Our results generalize both deterministic models and existing master equation approaches by providing an intuitive and efficient way to simultaneously model age- and population-dependent stochastic dynamics applicable to the study of demography, stem cell dynamics, and disease evolution.