On hierarchical competition through reduction of individual growth.

We consider a population organised hierarchically with respect to size in such a way that the growth rate of each individual depends only on the presence of larger individuals. As a concrete example one might think of a forest, in which the incidence of light on a tree (and hence how fast it grows) is affected by shading by taller trees. The classic formulation of a model for such a size-structured population employs a first order quasi-linear partial differential equation equipped with a non-local boundary condition.

A systematic procedure for incorporating separable static heterogeneity into compartmental epidemic models.

In this paper, we show how to modify a compartmental epidemic model, without changing the dimension, such that separable static heterogeneity is taken into account. The derivation is based on the Kermack-McKendrick renewal equation.

One Dimensional Reduction of a Renewal Equation for a Measure-Valued Function of Time Describing Population Dynamics.

Despite their relevance in mathematical biology, there are, as yet, few general results about the asymptotic behaviour of measure valued solutions of renewal equations on the basis of assumptions concerning the kernel. We characterise, via their kernels, a class of renewal equations whose measure-valued solution can be expressed in terms of the solution of a scalar renewal equation. The asymptotic behaviour of the solution of the scalar renewal equation, is studied via Feller's classical renewal theorem and, from it, the large time behaviour of the solution of the original renewal equation is derived.

The discrete-time Kermack-McKendrick model: A versatile and computationally attractive framework for modeling epidemics.

The COVID-19 pandemic has led to numerous mathematical models for the spread of infection, the majority of which are large compartmental models that implicitly constrain the generation-time distribution. On the other hand, the continuous-time Kermack-McKendrick epidemic model of 1927 (KM27) allows an arbitrary generation-time distribution, but it suffers from the drawback that its numerical implementation is rather cumbersome. Here, we introduce a discrete-time version of KM27 that is as general and flexible, and yet is very easy to implement computationally.

Correction to: Finite dimensional state representation of physiologically structured populations.

In the original publication of the article, the Subsection 2.1.2 was published incorrectly.

Should a viral genome stay in the host cell or leave? A quantitative dynamics study of how hepatitis C virus deals with this dilemma.

Virus proliferation involves gene replication inside infected cells and transmission to new target cells. Once positive-strand RNA virus has infected a cell, the viral genome serves as a template for copying ("stay-strategy") or is packaged into a progeny virion that will be released extracellularly ("leave-strategy"). The balance between genome replication and virion release determines virus production and transmission efficacy.

Finite dimensional state representation of physiologically structured populations.

In a physiologically structured population model (PSPM) individuals are characterised by continuous variables, like age and size, collectively called their i-state. The world in which these individuals live is characterised by another set of variables, collectively called the environmental condition. The model consists of submodels for (i) the dynamics of the i-state, e.

On models of physiologically structured populations and their reduction to ordinary differential equations.

Considering the environmental condition as a given function of time, we formulate a physiologically structured population model as a linear non-autonomous integral equation for the, in general distributed, population level birth rate. We take this renewal equation as the starting point for addressing the following question: When does a physiologically structured population model allow reduction to an ODE without loss of relevant information? We formulate a precise condition for models in which the state of individuals changes deterministically, that is, according to an ODE. Specialising to a one-dimensional individual state, like size, we present various sufficient conditions in terms of individual growth-, death-, and reproduction rates, giving special attention to cell fission into two equal parts and to the catalogue derived in an other paper of ours (submitted).

The winner takes it all: how semelparous insects can become periodical.

The aim of this short note is to give a simple explanation for the remarkable periodicity of Magicicada species, which appear as adults only every 13 or 17 years, depending on the region. We show that a combination of two types of density dependence may drive, for large classes of initial conditions, all but 1 year class to extinction. Competition for food leads to negative density dependence in the form of a uniform (i.

Karl-Peter Hadeler: His legacy in mathematical biology.

Karl-Peter Hadeler is a first-generation pioneer in mathematical biology. His work inspired the contributions to this special issue. In this preface we give a brief biographical sketch of K.

Multiple coexistence equilibria in a two parasitoid-one host model.

Briggs et al. (1993) introduced a host-parasitoid model for the dynamics of a system with two parasitoids that attack different juvenile stages of a common host. Their main result was that coexistence of the parasitoids is only possible when there is sufficient variability in the maturation delays of the host juvenile stages.

Dangerous connections: on binding site models of infectious disease dynamics.

We formulate models for the spread of infection on networks that are amenable to analysis in the large population limit. We distinguish three different levels: (1) binding sites, (2) individuals, and (3) the population. In the tradition of physiologically structured population models, the formulation starts on the individual level.

The many guises of R0 (a didactic note).

The basic reproduction number R0 is, by definition, the expected life time number of offspring of a newborn individual. An operationalization entails a specification of what events are considered as "reproduction" and what events are considered as "transitions from one individual-state to another". Thus, an element of choice can creep into the concretization of the definition.

On the characteristic equation λ = α1 + (α2 + α3λ)e(-λ) and its use in the context of a cell population model.

In this paper we characterize the stability boundary in the (α1, α2)-plane, for fixed α3 with −1 < α3 < +1, for the characteristic equation from the title. Subsequently we describe a nonlinear cell population model involving quiescence and show that this characteristic equation governs the (in)stability of the nontrivial steady state. By relating the parameters of the cell model to the αi we are able to derive some biological conclusions.

SI infection on a dynamic partnership network: characterization of R0.

We model the spread of an SI (Susceptible → Infectious) sexually transmitted infection on a dynamic homosexual network. The network consists of individuals with a dynamically varying number of partners. There is demographic turnover due to individuals entering the population at a constant rate and leaving the population after an exponentially distributed time.

An extension of the classification of evolutionarily singular strategies in Adaptive Dynamics.

The existing classification of evolutionarily singular strategies in Adaptive Dynamics (Geritz et al. in Evol Ecol 12:35-57, 1998; Metz et al. in Stochastic and spatial structures of dynamical systems, pp 183-231, 1996) assumes an invasion exponent that is differentiable twice as a function of both the resident and the invading trait.

A numerical approach for investigating the stability of equilibria for structured population models.

We are interested in the asymptotic stability of equilibria of structured populations modelled in terms of systems of Volterra functional equations coupled with delay differential equations. The standard approach based on studying the characteristic equation of the linearized system is often involved or even unattainable. Therefore, we propose and investigate a numerical method to compute the eigenvalues of the associated infinitesimal generator.

Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example.

We consider the interaction between a general size-structured consumer population and an unstructured resource. We show that stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations (a renewal equation for the consumer population birth rate coupled to a delay differential equation for the resource concentration). As many results for such systems are available (Diekmann et al.

Superinfections can induce evolutionarily stable coexistence of pathogens.

Parasites reproduce and are subject to natural selection at several different, but intertwined, levels. In the recent paper, Gilchrist and Coombs (Theor. Popul.

Comment on "Linking population-level models with growing networks: a class of epidemic models".

In this Comment, we show that the expression for as proposed by R. Breban, R. Vardavas, and S.

The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach.

Our starting point is a selection-mutation equation describing the adaptive dynamics of a quantitative trait under the influence of an ecological feedback loop. Based on the assumption of small (but frequent) mutations we employ asymptotic analysis to derive a Hamilton-Jacobi equation. Well-established and powerful numerical tools for solving the Hamilton-Jacobi equations then allow us to easily compute the evolution of the trait in a monomorphic population when this evolution is continuous but also when the trait exhibits a jump.