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. Biologically, our study implies that the invasion attempts by an invader into a trimorphic population under Atkinson-Allen dynamics can lead to chaos.

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. We uncover a novel mechanism whereby coexisting strains that only slightly differ in their local adaptation trait can evolve substantially different immigration strategies. In turn, different habitat use selects for divergent adaptations in the two strains. We propose that the joint evolution of immigration and local adaptation can facilitate diversification and discuss our results in the light of niche conservatism versus niche expansion.

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.g. growth and maturation, (ii) survival, (iii) reproduction, with the relevant rates described as a function of (i-state, environmental condition), (iv) functions of (i-state, environmental condition), like biomass or feeding rate, that integrated over the i-state distribution together produce the output of the population model. When the environmental condition is treated as a given function of time (input), the population model becomes linear in the state. Density dependence and interaction with other populations is captured by feedback via a shared environment, i.e., by letting the environmental condition be influenced by the populations' outputs. This yields a systematic methodology for formulating community models by coupling nonlinear input-output relations defined by state-linear population models. For some combinations of submodels an (infinite dimensional) PSPM can without loss of relevant information be replaced by a finite dimensional ODE. We then call the model ODE-reducible. The present paper provides (a) a test for checking whether a PSPM is ODE reducible, and (b) a catalogue of all possible ODE-reducible models given certain restrictions, to wit: (i) the i-state dynamics is deterministic, (ii) the i-state space is one-dimensional, (iii) the birth rate can be written as a finite sum of environment-dependent distributions over the birth states weighted by environment independent 'population outputs'. So under these restrictions our conditions for ODE-reducibility are not only sufficient but in fact necessary. Restriction (iii) has the desirable effect that it guarantees that the population trajectories are after a while fully determined by the solution of the ODE so that the latter gives a complete picture of the dynamics of the population and not just of its outputs.

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). We also show how to derive an ODE system describing the asymptotic large time behaviour of the population when growth, death and reproduction all depend on the environmental condition through a common factor (so for a very strict form of physiological age).

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. We show that the principle of linearised stability holds for this model, and develop some analytical methods for the investigation of characteristic equations for fixed delays. For a general maturation rate we resort to numerical methods and we extend the pseudospectral discretisation technique to approximate the state-dependent delay equation with a system of ordinary differential equations. This is the first application of the technique to nonlinear state-dependent delay equations, and currently the only method available for studying the stability of equilibria by means of established software packages for bifurcation analysis. The numerical method is validated on some cases when the maturation rate is independent of maturity and the model can be reformulated as a fixed-delay equation via a suitable time transformation. We exploit the analytical and numerical methods to investigate the stability boundary in parameter planes. Our study shows some drastic qualitative changes in the stability boundary under assumptions on the model parameters, which may have important biological implications.

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. With the methods of adaptive dynamics, we derive both analytical and numerical results for the evolution of immigration when life-history traits are patch-type dependent. The model exhibits evolutionary branching in a wide parameter range and the subsequent coevolution can lead to a stable coexistence of a generalist, settling in patches of any type, and a specialist that only immigrates into patches of high reward. We find that individuals always settle in the patches of high reward, in which survival until maturation, relative fecundity and emigration probability are high. We investigate how the probability to immigrate into patches of low reward changes with model parameters. For example, we show that immigration into patches of low reward increases when the emigration probability in these patches increases. Further, immigration into patches of low reward decreases when the patches of high reward become less safe during the dispersal season.

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. In our model, new phenotypes arise in the middle of the environmental gradient, while newly appearing environments are filled by existing phenotypes shifting their adaptive optima. Slow environmental change leads to loss of phenotypes in the middle of the extant environmental range, whereas fast change causes extinction at one extreme of the environmental range. We compared our model predictions against a well-known yet partially unexplained pattern of intense hoofed mammal diversification associated with grassland expansion during the Late Miocene. We additionally used the model outcomes to cast new insight into Cope's law of the unspecialized. Our general finding is that the rate of environmental change determines where generation and loss of diversity occur in the phenotypic and physical spaces.

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. We further assume a third timescale, which is even slower than the evolutionary timescale. We call this the geological timescale and we assume that slow climatic change occurs within this timescale. We study the evolution of our model population over this very slow geological timescale with bifurcation plots of the standard adaptive dynamics framework. The bifurcation parameter being varied describes the abiotic environment that changes over the geological timescale. We construct evolutionary trees over the geological timescale and observe both gradual phenotypic evolution and punctuated branching events. We concur with the established notion that branching of a monomorphic population on an environmental gradient only happens when the gradient is not too shallow and not too steep. However, we show that evolution within the habitat can produce polymorphic populations that inhabit steep gradients. What is necessary is that the environmental gradient at some point in time is such that the initial branching of the monomorphic population can occur. We also find that phenotypes adapted to environments in the middle of the existing environmental range are more likely to branch than phenotypes adapted to extreme environments.

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. When the benefits are indirect (genes for the offspring) and the population is under frequency-dependent ecological selection, the benefits depend on the mating strategy of other females as well. This case is particularly relevant to speciation models that seek to explore the stability of reproductive isolation by assortative mating under frequency-dependent ecological selection. We show that the indirect benefits are to be quantified by the reproductive values of couples, and describe how the evolutionarily stable time-thresholds can be found. We conclude with an example based on the Levene model, in which we analyze the evolutionarily stable assortative mating strategies and the strength of reproductive isolation provided by them.

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. Using the theory of adaptive dynamics, we find that a predator-prey population at equilibrium always favours less timidity. Low levels of timidity, however, may destabilise the population and lead to cycles. Large-amplitude cycles favour a positive level of timidity, but if such cycles do not occur, timidity will evolve all the way to zero, where the prey no longer responds to the predator by seeking refuge, and in which case the DeAngelis-Beddington functional response has become identical to the Holling type-II 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.pneumoniae is necessary, for example, for making successful predictions of changes in the composition of the strain community under intervention policies. Here we analyze data on the strains of S. pneumoniae carried in attendees of day care units in the metropolitan area of Oslo, Norway. We introduce a variant of approximate Bayesian computation methods, which is suitable for estimating the parameters governing the transmission dynamics in a setting where small local populations of hosts are subject to epidemics of different pathogenic strains due to infections independently acquired from the community. We find evidence for strong between-strain competition, as the acquisition of other strains in the already colonized hosts is estimated to have a relative rate of 0.09 (95% credibility interval [0.06, 0.14]). We also predict the frequency and size distributions for epidemics within the day care unit, as well as other epidemiologically relevant features. The assumption of ecological neutrality between the strains is observed to be compatible with the data. Model validation checks and the consistency of our results with previous research support the validity of our conclusions.

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. Survival during dispersal may depend on body condition, and competitive ability increases with increasing body condition. The evolutionarily stable strategy predicts that families abandon patches which are too unsafe or do not produce enough successful dispersers. From families that invest in retaining their natal patches, individuals stay in the patch that are less suitable for dispersal whereas the better dispersers disperse. However, this clear within-family pattern is often not reflected in the population-wide body condition distribution of dispersers or non-dispersers. This may be an explanation why empirical data do not show any general relationship between body condition and dispersal. When all individuals are equally good dispersers, then there exist equivalence classes defined by the competitive weight that remains in a patch. An equivalence class consists of infinitely many dispersal strategies that are selectively neutral. This provides an explanation why very diverse patterns found in body condition dependent dispersal data can all be equally evolutionarily stable.

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.