Structured population on two patches: modeling dispersal and delay

We derive from the age-structured model a system of delay differential equations to describe the interaction of spatial dispersal (over two patches) and time delay (arising from the maturation period). Our model analysis shows that varying the immature death rate can alter the behavior of the homogeneous equilibria, leading to transient oscillations around an intermediate equilibrium and complicated dynamics (in the form of the coexistence of possibly stable synchronized periodic oscillations and unstable phase-locked oscillations) near the largest equilibrium.     

A mathematical model of the stress induced during avascular tumour growth

In this paper a mathematical model is developed to describe the effect of nonuniform growth on the mechanical stress experienced by cells within an avascular tumour. The constitutive law combines the stress-strain relation of linear elasticity with a growth term that is derived by analogy with thermal expansion. To accommodate the continuous nature of the growth process, the law relates the rate of change of the stress tensor to the rate of change of the strain (rather than relating the stress to the strain directly). By studying three model problems which differ in detail, certain characteristic features are identified. First, cells near the tumour boundary, where nutrient levels and cell proliferation rates are high, are under compression. By contrast, cells towards the centre of the tumour, where nutrient levels are low and cell death dominant, are under tension. The implications of these results and possible model developments are also discussed. Received: 15 November 1999 / Published online: 5 May 2000     

Structured and Unstructured Continuous Models for <Emphasis Type="Italic">Wolbachia</Emphasis> Infections

We introduce and investigate a series of models for an infection of a diplodiploid host species by the bacterial endosymbiont Wolbachia. The continuous models are characterized by partial vertical transmission, cytoplasmic incompatibility and fitness costs associated with the infection. A particular aspect of interest is competitions between mutually incompatible strains. We further introduce an age-structured model that takes into account different fertility and mortality rates at different stages of the life cycle of the individuals. With only a few parameters, the ordinary differential equation models exhibit already interesting dynamics and can be used to predict criteria under which a strain of bacteria is able to invade a population. Interestingly, but not surprisingly, the age-structured model shows significant differences concerning the existence and stability of equilibrium solutions compared to the unstructured model.     

COMPARATIVE POPULATION DYNAMICS OF FUR SEALS

The population sizes, trends, exploitation, and life history parameters for the ten fur seal species and subspecies are summarized. The largest population is that of Arctocephalus pusillus pusillus with approximately two million seals, and the smallest is A. townsendi with approximately 7,000 individuals. Most populations are legally protected, although controlled harvesting may occur. None of the fur seal populations is currently known to be decreasing. Data are presented for parameters related to the survival of pups, juveniles, adults, and territorial males, and to reproduction, including the age of attainment of territorial status, aggregation sizes, age of first parturition, pregnancy rates, sex ratios of young animals, and information on the birth seasons of the different species. Since pinnipeds are often of concern in fisheries management, their daily consumption rates are of importance, and consequently data on body masses are summarized and the paucity of data on consumption rates as a function of body mass noted. A simplified age-structured model is developed, and the results of this model are compared with results from more detailed models based on two published life tables for Callorhinus ursinus. This comparison shows that the use of the simplified age-structured model is justified to explore changes in population growth rate. However, the simplified model does show exaggerated age structure effects compared to the more detailed models. This model is used to compare the population dynamics of those species for which sufficient data are available. Areas in which limited, or no, data are available for the different fur seal species are highlighted.     

An explicit representation of the Luria–Delbrück distribution

The probability distribution of the number of mutant cells in a growing single-cell population is presented in explicit form. We use a discrete model for mutation and population growth which in the limit of large cell numbers and small mutation rates reduces to certain classical models of the Luria-Delbrück distribution. Our results hold for arbitrarily large values of the mutation rate and for cell populations of arbitrary size. We discuss the influence of cell death on fluctuation experiments and investigate a version of our model that accounts for the possibility that both daughter cells of a non-mutant cell might be mutants. An algorithm is presented for the quick calculation of the distribution. Then, we focus on the derivation of two essentially different limit laws, the first of which applies if the population size tends to infinity while the mutation rate tends to zero such that the product of mutation rate times population size converges. The second limit law emerges after a suitable rescaling of the distribution of non-mutant cells in the population and applies if the product of mutation rate times population size tends to infinity. We discuss the distribution of mutation events for arbitrary values of the mutation rate and cell populations of arbitrary size, and, finally, consider limit laws for this distribution with respect to the behavior of the product of mutation rate times population size. Thus, the present paper substantially extends results due to Lea and Coulson (1949), Bartlett (1955), Stewart et al. (1990), and others.     

Stone Age to recent land-use history at Pegrema, northern Lake Onega, Russian Karelia

The village of Pegrema in Karelia may be regarded as a Stone Age innovation centre in the large Lake Onega area. Two pollen and plant macrofossil diagrams are presented which represent the first contribution to the study of human impact in the area using anthropogenic pollen indicators. A continuous but sporadic human presence from the Mesolithic onwards is demonstrated. While there is no archaeological evidence relating to the period 4200–3000 B.P., the pollen data suggest continuous, though rather sparse human presence. The data do not support any natural catastrophes in Pegrema as has been suggested elsewhere. Cerealia pollen is recorded earlier than expected (c. 5000 B.P.=. In the Bronce Age and Iron Age, the settlement of the Zaonezhye peninsula is reflected by a slight increase in herb pollen representation, sporadic Cerealia pollen and several periods of regression in Picea. The long introductory period of agriculture to the area, as well as the similarities and discrepancies between different sources of evidence (palaeoecological, archaeological and historical) are discussed at some length. The start of land clearance for permanent cultivation in the profile Pegrema S was dated to the late 13^th century. The beginning of more intensive field cultivation in the 15^th century is clearly seen in the pollen succession at both localities. The fluctuation in anthropogenic indicators can be related to population density based on historical data. The village of Pegrema was depopulated in 1956 which is reflected in a distinct decline in settlement indicators. Received February 17 / Accepted May 19, 2000     

Traveling waves and pulses in a one-dimensional network of excitable integrate-and-fire neurons

 We study the existence and stability of traveling waves and pulses in a one-dimensional network of integrate-and-fire neurons with synaptic coupling. This provides a simple model of excitable neural tissue. We first derive a self-consistency condition for the existence of traveling waves, which generates a dispersion relation between velocity and wavelength. We use this to investigate how wave-propagation depends on various parameters that characterize neuronal interactions such as synaptic and axonal delays, and the passive membrane properties of dendritic cables. We also establish that excitable networks support the propagation of solitary pulses in the long-wavelength limit. We then derive a general condition for the (local) asymptotic stability of traveling waves in terms of the characteristic equation of the linearized firing time map, which takes the form of an integro-difference equation of infinite order. We use this to analyze the stability of solitary pulses in the long-wavelength limit. Solitary wave solutions are shown to come in pairs with the faster (slower) solution stable (unstable) in the case of zero axonal delays; for non-zero delays and fast synapses the stable wave can itself destabilize via a Hopf bifurcation. Received: 27 October 1998     

Simulation of ‘hitch-hiking’ genealogies

An ancestral influence graph is derived, an analogue of the coalescent and a composite of Griffiths' (1991) two-locus ancestral graph and Krone and Neuhauser's (1997) ancestral selection graph. This generalizes their use of branching-coalescing random graphs so as to incorporate both selection and recombination into gene genealogies. Qualitative understanding of a ‘hitch-hiking’ effect on genealogies is pursued via diagrammatic representation of the genealogical process in a two-locus, two-allele haploid model. Extending the simulation technique of Griffiths and Tavaré (1996), computational estimation of expected times to the most recent common ancestor of samples of n genes under recombination and selection in two-locus, two-allele haploid and diploid models are presented. Such times are conditional on sample configuration. Monte Carlo simulations show that ‘hitch-hiking’ is a subtle effect that alters the conditional expected depth of the genealogy at the linked neutral locus depending on a mutation-selection-recombination balance. Received: 21 July 2000 / Published online: 5 December 2000     

Comparative statics and stochastic dynamics of age-structured populations

Arguments from the comparative statics of populations with fixed vital rates are of limited use in studying age-structured populations subject to stochastically varying vital rates. In an age-structured population that experiences a sequence of independently and identically distributed Leslie matrices, the expectation of the Malthusian parameters of the Leslie matrices has no exact interpretation either as the ensemble average of the long-run rate of growth of each sample path of the population (Eq. (3)) or as the long-run rate of growth of the ensemble average of total population size (Eq. (4)). On the other hand, the Malthusian parameter of the expectation of a sequence of Leslie matrices is exactly the logarithm of the finite growth rate of the ensemble average of total population size when Leslie matrices are independently and identically distributed (though not in general when Leslie matrices are sequentially dependent). These observations appear to contradict the claims of a recent study using computer simulation of age-structured populations with stochastically varying vital rates.     

Protected polymorphism in the two-locus haploid model with unpredictable fitnesses

 In an unpredictable environment, the distributions of alleles from which polymorphism can be maintained forever belong to a certain set, the C-viability kernel. Such a set is calculated in the two-locus haploid model, as well as the corresponding fitnesses at any time which make this maintenance possible. The dependence of the C-viability kernel on the set U of admissible fitnesses and on the recombination rate r is studied. Notably, the C-viability kernel varies rapidly in the neighborhood of equal fitness of AB and ab; it becomes empty when ab has a fitness below a certain function, which is delineated, of the recombination rate. The properties of the two-locus model under constraints, out of equilibrium and with unpredictable selection are thus presented. Received: 20 May 1999     

Introducing inter-individual growth variability in the assessment of a cephalopod population: application to the English Channel squid Loligo forbesi

A new approach is presented here to better take into account inter-individual growth variability in age-structured models used for stock assessment. Cohort analysis requires knowledge of the age structure of the catch, generally derived from an age–length key and length-structure information. Age distribution at length is estimated by applying conditional quantile regression to a data set of lengths and ages estimated from calcareous parts. A “stochastic” age–length key that describes the probability of age-at-length is applied to the English Channel squid Loligo forbesi. Age distribution at length from quantile regression proved to be considerably less biased than that resulting from both polymodal decomposition (PD) and two separate slicing methods. Both catch-at-age and stock size were underestimated using classical methods. Estimations of fishing mortalities from classical methods were higher causing underestimation in yield simulations. Quantile regression offers a more complete statistical analysis of the stochastic relationships among random variables than mean regression and PD.     

The Rhodococcus fascians-plant interaction: morphological traits and biotechnological applications

Rhodococcus fascians is a Gram-positive bacterium that infects dicotyledonous and monocotyledonous plants, leading to an alteration in the normal growth process of the host. The disease results from the modulation of the plant hormone balances, and cytokinins are thought to play an important role in the induction of symptoms. Generally, on the aerial parts of the plants, existing meristems were found to be most sensitive to the action of R. fascians, but, depending on the infection procedure, differentiated tissues as well gave rise to shoots. Similarly, in roots not only actively dividing cells, but also cells with a high competence to divide were strongly affected by R. fascians. The observed symptoms, together with the determined hormone levels in infected plant tissue, suggest that auxins and molecules of bacterial origin are also involved in leafy gall formation. The complexity of symptom development is furthermore illustrated by the necessary and continuous presence of the bacteria for symptom persistence. Indeed, elimination of the bacteria from a leafy gall results in the further development of the multiple embryonic buds of which it consists. This interesting characteristic offers novel biotechnological applications: a leafy gall can be used for germplasm storage and for plant propagation. The presented procedure proves to be routinely applicable to a very wide range of plants, encompassing several recalcitrant species. Received: 14 January 1999 / Accepted: 19 June 1999     

A generalized transport model for biased cell migration in an anisotropic environment

 A generalized transport model is derived for cell migration in an anisotropic environment and is applied to the specific cases of biased cell migration in a gradient of a stimulus (taxis; e.g., chemotaxis or haptotaxis) or along an axis of anisotropy (e.g., contact guidance). The model accounts for spatial or directional dependence of cell speed and cell turning behavior to predict a constitutive cell flux equation with drift velocity and diffusivity tensor (termed random motility tensor) that are explicit functions of the parameters of the underlying random walk model. This model provides the connection between cell locomotion and the resulting persistent random walk behavior to the observed cell migration on longer time scales, thus it provides a framework for interpreting cell migration data in terms of underlying motility mechanisms. Received: 8 April 1999     

Mutation and recombination with tight linkage

An exact solution of the mutation-recombination equation in continuous time is presented, with linear ordering of the sites and at most one mutation or crossover event taking place at every instant of time. The differential equation may be obtained from a mutation-recombination model with discrete generations, in the limit of short generations, or weak mutation and recombination. The solution relies on the multilinear structure of the dynamical system, and on the commuting properties of the mutation and recombination operators. It is obtained through diagonalization of the mutation term, followed by a transformation to certain measures of linkage disequilibrium that simultaneously linearize and diagonalize the recombination dynamics. The collection of linkage disequilibria, as well as their decay rates, are given in closed form. Received: 26 January 1999 / Revised version: 20 October 2000 / Published online: 10 April 2001     

The interaction of thin-film flow, bacterial swarming and cell differentiation in colonies of Serratia liquefaciens

 The rate of expansion of bacterial colonies of S. liquefaciens is investigated in terms of a mathematical model that combines biological as well as hydrodynamic processes. The relative importance of cell differentiation and production of an extracellular wetting agent to bacterial swarming is explored using a continuum representation. The model incorporates aspects of thin film flow with variable suspension viscosity, wetting, and cell differentiation. Experimental evidence suggests that the bacterial colony is highly sensitive to its environment and that a variety of mechanisms are exploited in order to proliferate on a variety of surfaces. It is found that a combination of effects are required to reproduce the variation of bacterial colony motility over a large range of nutrient availability and medium hardness. Received: 29 April 1999     

Regulation of platelet production: the normal response to perturbation and cyclical platelet disease

An age-structured model for the regulation of platelet production is developed, and compared with both normal and pathological platelet production. We consider the role of thrombopoietin (TPO) in this process, how TPO affects the transition between megakaryocytes of various ploidy classes, and their individual contributions to platelet production. After the estimation of the relevant parameters of the model from both in vivo and in vitro data, we use the model to numerically reproduce the normal human response to a bolus injection of TPO. We further show that our model reproduces the dynamic characteristics of autoimmune cyclical thromobocytopenia if the rate of platelet destruction in the circulation is elevated to more than twice the normal value.     

Extinction in relation to demographic and environmental stochasticity in age-structured models

The demographic variance of an age-structured population is defined. This parameter is further split into components generated by demographic stochasticity in each vital rate. The applicability of these parameters are investigated by checking how an age-structured population process can be approximated by a diffusion with only three parameters. These are the deterministic growth rate computed from the expected projection matrix and the environmental and demographic variances. We also consider age-structured populations where the fecundity at any stage is either zero or one, and there is neither environmental stochasticity nor dependence between individual fecundity and survival. In this case the demographic variance is uniquely determined by the vital rates defining the projection matrix. The demographic variance for a long-lived bird species, the wandering albatross in the southwestern part of the Indian Ocean, is estimated. We also compute estimates of the age-specific contributions to the total demographic variance from survival, fecundity and the covariance between survival and fecundity.     

Differential responses to anxiogenic drugs in a mouse model of panic disorder as revealed by Fos immunocytochemistry in specific areas of the fear circuitry

Summary. Sensitivity to pharmacological challenges has been reported in patients with panic disorder. We have previously validated transgenic mice overexpressing the neurotrophin-3 (NT-3) receptor, TrkC (TgNTRK3), as an engineered murine model of panic disorder. We could determine that TgNTRK3 mice presented increased cellularity in brain regions, such as the locus ceruleus, that are important neural substrates for the expression of anxiety in severe anxiety states. Here, we investigated the sensitivity to induce anxiety and panic-related symptoms by sodium lactate and the effects of various drugs (the α2-adrenoceptor antagonist, yohimbine and the adenosine antagonist, caffeine), in TgNTRK3 mice. We found enhanced panicogenic sensitivity to sodium lactate and an increased intensity and a differential pattern of Fos expression after the administration of yohimbine or caffeine in TgNTRK3. Our findings validate the relevance of the NT-3/TrkC system to pathological anxiety and raise the possibility that a specific set of fear-related pathways involved in the processing of anxiety-related information may be differentially activated in panic disorder.     

A continuous, age-structured insect population model

 A continuous age-structured model of cannibalistic insect populations is constructed and analyzed. The model is a continuous analog of the model used in the recent work of Costantino et al. in which discrete modeling, mathematical analysis, statistical techniques, and laboratory experiments were used to demonstrate the presence of nonlinear dynamics, including chaos, in laboratory Tribolium cultures. A special case of the continuous model (no larva-on-egg cannibalism) is analyzed and the results are compared to the analogous special case of the discrete model. Received: 6 January 1999