A nonlinear structured population model of tumor growth with quiescence

A nonlinear structured cell population model of tumor growth is considered. The model distinguishes between two types of cells within the tumor: proliferating and quiescent. Within each class the behavior of individual cells depends on cell size, whereas the probabilities of becoming quiescent and returning to the proliferative cycle are in addition controlled by total tumor size. The asymptotic behavior of solutions of the full nonlinear model, as well as some linear special cases, is investigated using spectral theory of positive simigroup of operators. Supported in part by the National Science Foundation under Grant No. DMS-8722947     

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.     

Model-Based Analysis of Mechanisms Responsible for Sleep-Induced Carbon Dioxide Differences

This work describes a comprehensive mathematical model of the human respiratory control system which incorporates the central mechanisms for predicting sleep-induced changes in chemical regulation of ventilation. The model integrates four individual compartments for gas storage and exchange, namely alveolar air, pulmonary blood, tissue capillary blood, body tissues, and gas transport between them. An essential mechanism in the carbon dioxide transport is its dissociation into bicarbonate and acid, where a buffering mechanism through hemoglobin is used to prevent harmfully low pH levels. In the current model, we assume high oxygen levels and consider intracellular hydrogen ion concentration as the principal respiratory control variable. The resulting system of delayed differential equations is solved numerically. With an appropriate choice of key parameters, such as velocity of blood flow and gain of a non-linear controller function, the model provides steady-state results consistent with our experimental observations measured in subjects across sleep onset. Dynamic predictions from the model give new insights into the behaviour of the system in subjects with different buffering capacities and suggest novel hypotheses for future experimental and clinical studies.     

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. in SIAM J Math Anal 39:1023–1069, 2007), we can draw rigorous conclusions concerning dynamical behaviour from an analysis of a characteristic equation. We derive the characteristic equation for a fairly general class of population models, including those based on the Kooijman–Metz Daphnia model (Kooijman and Metz in Ecotox Env Saf 8:254–274, 1984; de Roos et al. in J Math Biol 28:609–643, 1990) and a model introduced by Gurney–Nisbet (Theor Popul Biol 28:150–180, 1985) and Jones et al. (J Math Anal Appl 135:354–368, 1988), and next obtain various ecological insights by analytical or numerical studies of special cases.     

Evolution of condition-dependent dispersal under kin competition

Dispersers often differ in body condition from non-dispersers. The social dominance hypothesis explains dispersal of weak individuals, but it is not yet well understood why strong individuals, which could easily retain their natal site, are sometimes exposed to risky dispersal. Based on the model for dispersal under kin competition by Hamilton and May, we construct a model where dispersal propensity depends on body condition. We consider an annual species that inhabits a patchy environment with varying patch qualities. Offspring body condition corresponds to the quality of the natal patch and competitive ability increases with body condition. Our main general result balances the fitness benefit from not dispersing and retaining the natal patch and the benefit from dispersing and establishing somewhere else. We present four different examples for competition, which all hint that dispersal of strong individuals may be a common outcome under the assumptions of the present model. In three of the examples, the evolutionarily stable dispersal probability is an increasing function of body condition. However, we found an example where, counterintuitively, the evolutionarily stable dispersal probability is a non-monotone function of body condition such that both very weak and very strong individuals disperse with high probability but individuals of intermediate body condition do not disperse at all.     

How should we define fitness in structured metapopulation models? Including an application to the calculation of evolutionarily stable dispersal strategies

We define a fitness concept applicable to structured metapopulations consisting of infinitely many equally coupled patches. In addition, we introduce a more easily calculated quantity Rm that relates to fitness in the same manner as R0 relates to fitness in ordinary population dynamics: the Rm of a mutant is only defined when the resident population dynamics converges to a point equilibrium and Rm is larger (smaller) than 1 if and only if mutant fitness is positive (negative). Rm corresponds to the average number of newborn dispersers resulting from the (on average less than one) local colony founded by a newborn disperser. Efficient algorithms for calculating its numerical value are provided. As an example of the usefulness of these concepts we calculate the evolutionarily stable conditional dispersal strategy for individuals that can account for the local population density in their dispersal decisions. Below a threshold density x, at which staying and leaving are equality profitable, everybody should stay and above x everybody should leave, where profitability is measured as the mean number of dispersers produced through lines of descent consisting of non-dispersers.     

Adjustment of the human respiratory system to increased upper airway resistance during sleep

A cardiorespiratory model incorporating control of the human upper airway during sleep is described. Most previous models have not considered the possibility that the upper airway could be a limiting factor for gas exchange. Our model was developed to also predict certain pathophysiological phenomena in the cardiorespiratory system that characterize heavy snoring or sleep apnea. We started by adapting our collapsible upper airway model to include the impact of nasal passage and larynx, and extended the model with equations for gas exchange in the lungs. A feedback loop both to the respiratory pump and the upper airway dilator muscles was included. The model enabled successful breath-by-breath simulations of obstructive events of the upper airway. Although the model incorporates several physiologically relevant components of the system, the simulation results suggest that only few parameters suffice to predict the key adjustments that the cardiorespiratory system is known to make in patients with heavy snoring.