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.

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.

Chaotic attractors in Atkinson-Allen model of four competing species.

We study the occurrence of chaos in the Atkinson-Allen model of four competing species, which plays the role as a discrete-time Lotka-Volterra-type model. We show that in this model chaos can be generated by a cascade of quasiperiod-doubling bifurcations starting from a supercritical Neimark-Sacker bifurcation of the unique positive fixed point. The chaotic attractor is contained in a globally attracting invariant manifold of codimension one, known as the carrying simplex.

Interactions between different predator-prey states: a method for the derivation of the functional and numerical response.

In this paper we introduce a formal method for the derivation of a predator's functional response from a system of fast state transitions of the prey or predator on a time scale during which the total prey and predator densities remain constant. Such derivation permits an explicit interpretation of the structure and parameters of the functional response in terms of individual behaviour. The same method is also used here to derive the corresponding numerical response of the predator as well as of the prey.

The Evolution of Immigration Strategies Facilitates Niche Expansion by Divergent Adaptation in a Structured Metapopulation Model.

Local adaptation and habitat choice are two key factors that control the distribution and diversification of species. Here we model habitat choice mechanistically as the outcome of dispersal with nonrandom immigration. We consider a structured metapopulation with a continuous distribution of patch types and determine the evolutionarily stable immigration strategy as the function linking patch type to the probability of settling in the patch on encounter.

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).

Stability analysis of a state-dependent delay differential equation for cell maturation: analytical and numerical methods.

We consider a mathematical model describing the maturation process of stem cells up to fully mature cells. The model is formulated as a differential equation with state-dependent delay, where maturity is described as a continuous variable. The maturation rate of cells may be regulated by the amount of mature cells and, moreover, it may depend on cell maturity: we investigate how the stability of equilibria is affected by the choice of the maturation rate.

On the evolution of patch-type dependent immigration.

Empirical studies of dispersal indicate that decisions to immigrate are patch-type dependent; yet theoretical models usually ignore this fact. Here, we investigate the evolution of patch-type dependent immigration of a population inhabiting and dispersing in a heterogeneous landscape, which is structured by patches of low and high reward. We model the decision to immigrate in detail from a mechanistic underpinning.

Modeling the Population-Level Processes of Biodiversity Gain and Loss at Geological Timescales.

The path of species diversification is commonly observed by inspecting the fossil record. Yet, how species diversity changes at geological timescales relate to lower-level processes remains poorly understood. Here we use mathematical models of spatially structured populations to show that natural selection and gradual environmental change give rise to discontinuous phenotype changes that can be connected to speciation and extinction at the macroevolutionary level.

Adaptive dynamics on an environmental gradient that changes over a geological time-scale.

The standard adaptive dynamics framework assumes two timescales, i.e. fast population dynamics and slow evolutionary dynamics.

Evolutionarily stable mating decisions for sequentially searching females and the stability of reproductive isolation by assortative mating.

We consider mating strategies for females who search for males sequentially during a season of limited length. We show that the best strategy rejects a given male type if encountered before a time-threshold but accepts him after. For frequency-independent benefits, we obtain the optimal time-thresholds explicitly for both discrete and continuous distributions of males, and allow for mistakes being made in assessing the correct male type.

The DeAngelis-Beddington functional response and the evolution of timidity of the prey.

We study the evolution of "timidity" of the prey (i.e., its readiness to seek refuge) in a predator-prey model with the DeAngelis-Beddington functional response.

Estimating the transmission dynamics of Streptococcus pneumoniae from strain prevalence data.

Streptococcus pneumoniae is a typical commensal bacterium causing severe diseases. Its prevalence is high among young children attending day care units, due to lower levels of acquired immunity and a high rate of infectious contacts between the attendees. Understanding the population dynamics of different strains of S.

A mechanistic derivation of the DeAngelis-Beddington functional response.

We give a derivation of the DeAngelis-Beddington functional response in terms of mechanisms at the individual level, and for the first time involving prey refuges instead of the usual interference between predators.

Body condition dependent dispersal in a heterogeneous environment.

We find the evolutionarily stable dispersal behaviour of a population that inhabits a heterogeneous environment where patches differ in safety (the probability that a juvenile individual survives until reproduction) and productivity (the total competitive weight of offspring produced by the local individual), assuming that these characteristics do not change over time. The body condition of clonally produced offspring varies within and between families. Offspring compete for patches in a weighted lottery, and dispersal is driven by kin competition.

Variability within families and the evolution of body-condition-dependent dispersal.

In a population where body condition varies between and within families, we investigate the evolution of dispersal as a function of body condition ('strength', e.g. body size).

On the evolution of the timing of reproduction with non-equilibrium resident dynamics.

We study the evolution of an individual's reproductive strategy in a mechanistic modeling framework. We assume that the total number of juveniles one adult individual can produce is a finite constant, and we study how this number should be distributed during the season, given the types of inter-individual interactions and mortality processes included in the model. The evolution of the timing of reproduction in this modeling framework has already been studied earlier in the case of equilibrium resident dynamics, but we generalize the situation to also fluctuating population dynamics.

Necessary and sufficient conditions for the existence of an optimisation principle in evolution.

This paper presents a necessary condition for the existence of a numerical quantity optimised by evolution by natural selection, which also turns out to be a sufficient condition under rather general conditions. As a corollary, a related criterion with a particularly intuitive graphical interpretation in terms of pairwise invadability plots is obtained.

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.

Evolution of density-dependent dispersal in a structured metapopulation.

We study the evolution of density-dependent dispersal in a structured metapopulation subject to local catastrophes that eradicate local populations. To this end we use the theory of structured metapopulation dynamics and the theory of adaptive dynamics. The set of evolutionarily possible dispersal functions (i.

Ecology and evolution of symbiosis in metapopulations.

We present a model for symbionts in plant host metapopulation. Symbionts are assumed not only to form a systemic infection throughout the host and pass into the host seeds, but also to reproduce and infect new plants by spores. Thus, we study a metapopulation of qualitatively identical patches coupled through seeds and spores dispersal.