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  • Springer  (314,923)
  • American Physical Society  (55,992)
  • 1995-1999  (232,868)
  • 1990-1994  (138,047)
  • 1999  (79,891)
  • 1998  (78,960)
  • 1997  (74,017)
  • 1991  (70,552)
  • 1990  (67,495)
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  • 1995-1999  (232,868)
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  • 1
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    Bulletin of mathematical biology 52 (1990), S. 153-197 
    ISSN: 1522-9602
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    Topics: Biology , Mathematics
    Notes: Abstract It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns onHydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.
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  • 2
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    Bulletin of mathematical biology 52 (1990), S. 319-334 
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    Bulletin of mathematical biology 52 (1990), S. 335-337 
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  • 4
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    Bulletin of mathematical biology 52 (1990), S. I 
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    Bulletin of mathematical biology 52 (1990), S. 335-348 
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    Notes: Abstract The classical metaphor of the genetic program written in the DNA nucleotidic sequences is reconsidered. Recent works on algorithmic complexity and logical properties of computer programs and data are used to question the explanatory value of that metaphor. Structural properties of strings are looked for which would be necessary to apply to DNA sequences if the metaphor is to be taken literally. The notion of sophistication is used to quantify meaningful complexity and to distinguish it from classical computational complexity. In this context, the distinction between program and data becomes relevant and an alternative metaphor of DNA as data to a parallel computing network embedded in the global geometrical and biochemical structure of the cell is discussed. An intermediate picture of an evolving network emerges as the most likely where the output of the cellular computing network can produce, at a different time scale, changes in the structure of the network itself by means of changes in the DNA activity patterns.
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  • 6
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    Bulletin of mathematical biology 52 (1990), S. 349-358 
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    Notes: Abstract When two strings of symbols are aligned it is important to know whether the observed number of matches is better than that expected between two independent sequences with the same frequency of symbols. When strings are of different lengths, nulls need to be inserted in order to align the sequences. One approach is to use simple approximations of sampling for replacement. We describe an algorithm for exactly determining the frequencies of given numbers of matches, sampling without replacement. This does not lead to a simple closed form expression. However we show examples where sampling with, or without, replacement give very similar results and the simple approach may be adequate for all but the smallest cases.
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  • 7
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    Bulletin of mathematical biology 52 (1990), S. 509-525 
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    Notes: Abstract Pairwise optimal alignments between three or more sequences are not necessarily consistent as a whole, but consistent and inconsistent residues are usually distributed in clusters. An efficient method has been developed for locating consistent regions when each pairwise alignment is given in the form of a “skeletal representation” (Bull. math. Biol. 52, 359–373). This method is further extended so that the combination of pairwise alignments that gives the greatest consistency is found when possibly many alignments are equally optimal for each pairwise comparison. A method for acceleration of simultaneous multiple sequence alignment is proposed in which consistent regions serve as “anchor points” limiting application of direct multi-way alignment to the rest of “inconsistent” regions.
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  • 8
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    Bulletin of mathematical biology 52 (1990), S. 527-534 
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    Notes: Abstract Sufficient conditions are given for the unlimited growth or otherwise in multitype population size dependent Galton-Watson processes. These conditions are given in terms of moments of offspring distributions and extend known conditions for processes with one type.
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  • 9
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    Bulletin of mathematical biology 52 (1990), S. 535-547 
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    Notes: Abstract The variability of the duration of the cell cycle is explained by the phenomenon of sensitive dependence upon initial conditions; as may occur in deterministic non-linear systems. Chaotic dynamics of a system is the result of this sensitive dependence. First a deterministic system is formulated that is equivalent to the Smith-Martin transition probability model of the cell cycle. Next the model is extended to a dynamic process that ranges over the cell generations. A deterministic non-linear relationship between the cycle time of the mother and daughter cell is established. It clarifies the variability of mother-daughter correlation for the different cell types. The model is fitted to two different cell cultures; it shows that the graph of the non-linear relation has the same shape for different cell types.
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    Bulletin of mathematical biology 52 (1990), S. 583-596 
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    Bulletin of mathematical biology 52 (1990), S. I 
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    Bulletin of mathematical biology 52 (1990), S. 549-582 
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    Notes: Abstract Recently a mathematical model of the prevascular phases of tumor growth by diffusion has been investigated (S. A. Maggelakis and J. A. Adam,Math. Comput. Modeling, in press). In this paper we examine in detail the results and implications of that mathematical model, particularly in the light of recent experimental work carried out on multicellular spheroids. The overall growth characteristics are determined in the present model by four parameters:Q, γ, b, andδ, which depend on information about inhibitor production rates, oxygen consumption rates, volume loss and cell proliferation rates, and measures of the degree of non-uniformity of the various diffusion processes that take place. The integro-differential growth equation is solved for the outer spheroid radiusR 0(t) and three related inner radii subject to the solution of the governing time-independent diffusion equations (under conditions of diffusive equilibrium) and the appropriate boundary conditions. Hopefully, future experimental work will enable reasonable bounds to be placed on parameter values referred to in this model: meanwhile, specific experimentally-provided initial data can be used to predict subsequent growth characteristics ofin vitro multicellular spheroids. This will be one objective of future studies.
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  • 13
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    Notes: Abstract Workers of the speciesLeptothorax acervorum show age-polyethism, they start their life as broodworkers and later on they become nestworkers and foragers. Nestworkers and foragers of this ant species are inactive for 72% and 15% of the total time respectively. The short bursts of activity within the nest do not occur randomly but are synchronized so that the whole nest population exhibits nonperiodic pulses of activity: the ants were seen to wake each other actively. In addition starvation experiments were done to assess whether ants react upon food availability. In appeared that during a longlasting period of starvation the proportion of active ants in the nest is at a higher approximately constant level.
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  • 14
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    Bulletin of mathematical biology 53 (1991), S. 281-312 
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    Bulletin of mathematical biology 53 (1991), S. 253-279 
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    Bulletin of mathematical biology 53 (1991), S. 313-326 
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    Bulletin of mathematical biology 53 (1991), S. I 
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  • 18
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    Bulletin of mathematical biology 53 (1991), S. 345-353 
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    Notes: Abstract The statistical correlation of nucleotides in a DNA sequence is described by a set of redundanciesD 1,D 2,D 3,... By calculation of {D n} of 2341 coding regions of nucleic acid sequences it is demonstrated that about 2/3 of sequences has correlation length ≤2, 10% of sequences—correlation with 3-periodicity and others—long range aperiodic correlations. The implications of the results from the interactions of random mutation and natural selection are discussed briefly.
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    Bulletin of mathematical biology 53 (1991), S. 327-343 
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    Notes: Abstract A basic characteristic of biological memory is that it has a graded duration, which, even for socalled short-term memory, can vary from minutes to days (i.e. over about three orders of magnitude), depending on the training protocol, which one can think of as determining the “strength” of the memory. Furthermore, the molecular analysis of simple learning in invertebrates has revealed many examples where “learning” is produced by adecrease in an appropriate membrane conductance. This paper provides a quantitative analysis of a simple kinetic scheme where by a conductance decrease can be produced by repetitive nerve impulses, with a duration that varies with stimulus frequency. The simplest model considered is based on the actual kinetics of the naturally-occurring ionophore Monazomycin. This model yields durations ranging only over a factor of about 10, for reasonable parameter values. However, a simple modification of the model yields memory durations ranging over three or more orders of magnitude. We also show that Monazomycin-like kinetics can appear as the result of a combination of simple uni- and bi-molecular reactions, thus making more plausible the possibility that the effects described here may operate in actual biological systems.
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    Bulletin of mathematical biology 53 (1991), S. 579-589 
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    Notes: Abstract A mathematical model (Kliemann, W. 1987.Bull. math. Biol. 49, 135–152.) that predicts the quantitative branching pattern of dendritic tree was evaluated using the apical and basal dendrites of rat hippocampal neurons. The Wald statistics for χ2-test was developed for the branching pattern of dendritic trees and for the distribution of the maximal order of the tree. Using this statistic, we obtained a reasonable, but not excellent, fit of the mathematical model for the dendritic data. The model's predictability of branching patterns was greatly enhanced by replacing one of the assumptions used for the original model “splitting of branches for all dendritic orders is stochastically independent”, with a new assumption “branches are more likely to split in areas where there is already a high density of branches”. The modified model delivered an excellent fit for basal dendrites and for the apical dendrites of hippocampal neurons from young rats (30–34 days postpartum). This indicates that for these cells the development of dendritic patterns is the result of a purely random and a systematic component, where the latter one depends on the density of dendritic branches in the brain area considered. For apical dendrites there is a trend towards decreasing pattern predictability with increasing age. This appears to reflect the late arrival of afferents and subsequent synaptogenesis proximal on the apical dendritic tree of hippocampal neurons.
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  • 22
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    Notes: Abstract In order for immune cells to carry out many of their functions, including clearance of infectious agents from tissue, they must first encounter their targets in the tissue. This encounter process is often the rate-limiting step in the overall function. Most immune cells exhibit chemotactic ability, and previous continuum models for encounter rates and dynamics have shown that chemotaxis can be a great advantage to cells by greatly increasing encounter rates relative to those for randomly moving cells. This paper describes computer simulations of discrete cell-target encounter events in two dimensions, for the two cases considered by the continuum models: where only a single cell and a single target are present, and where many cells and targets are present. The results of these simulations verify our previous model predictions that a small amount of chemotactic bias dramatically decreases the encounter time, while further increases in the amount of bias have a much smaller effect. Chemotactic ability is shown to be an important determinant of the kinetics of target clearance, and its effects depend on the initial cell-target ratio and the initial distributions of cells and targets. To the best of our knowledge, this work provides the first computer simulations of particle-target encounter in which there is biased motion of particles toward their targets, and is therefore of general interest beyond specific application to immune cell function.
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    Bulletin of mathematical biology 53 (1991), S. 657-663 
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    Bulletin of mathematical biology 53 (1991), S. 881-899 
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    Notes: Abstract The concept of a one-way block, arising from a region of depressed tissue, has remained central to theories for cardiac arrhythmias. We show that both the geometry of a depressed region and spatial heterogeneities in depression are key factors for inducing such a block. By using an asymptotic approximation, known as the eikonal equation, to model qualitatively the movement of a depolarization wave-front down a Purkinje fibre bundle, we show how a one-way block in conduction may result from asymmetric constriction in the width of a depressed bundle. We demonstrate that this theory is valid for biologically relevant parameters and simulate a one-way block by numerically solving the eikonal approximation. We consider the case of non-uniform depression, where the planar travelling wave speed is spatially dependent. Here, numerical simulations indicate that such a spatial dependency may, in itself, be sufficient to produce a one-way block.
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    Bulletin of mathematical biology 53 (1991), S. 911-940 
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    Notes: Abstract The MacArthur-Wilson equilibrium theory of island biogeography has been one of the more influential concepts in modern biogeography and ecology. In this paper, we synthesize the theory and examine effects of different immigration/extinction rate-species diversity curves on original predictions from the theory by using the System Dynamics simulation modeling approach. Moreover, we develop a comprehensive and generic System Dynamics model to incorporate a variety of recent modifications and extensions of the theory, including area effect, distance effect, competition effect, habitat diversity effect, target effect, and rescue effect. Through computer simulation with STELLA, a more profound understanding of the theory of island biogeography can be gained. The System Dynamics modeling approach is especially appropriate for such a study because it maximizes the utilization of the ecological data by incorporating qualitative information so that a complex, imprecisely-defined ecological system can be studied quantitatively, effectively, and comprehensively. Our simulation results show that different monotonic rate-species diversity curves do not affect the essence of the theory of island biogeography, while the magnitude of equilibrium species diversity may be greatly affected. Non-monotonic rate-species diversity curves may result in potential multiple equilibria of species diversity. In addition, our model suggests that a non-monotonic relationship may exist between the equilibrium turnover rate and island area and between the equilibrium turnover rate and distance.
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    Bulletin of mathematical biology 53 (1991), S. 665-678 
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    Notes: Abstract A method to estimate a lower bound of the Kolmogorov entropy—the so calledK 2-entropy—from a time series is presented which avoids use of the generalized correlation integral. The influence of the norm is studied. The method is demonstrated on some standard examples. The entropy of the attractor apparent in the EEG of the foetal sheep is estimated and the results are compared with results obtained from synthesized data featuring some basic properties of EEG. This gives an insight into the limitations of the procedure.
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    Bulletin of mathematical biology 53 (1991), S. 685-700 
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    Notes: Abstract The autonomous oscillations in yeast continuous cultures are investigated analytically and related to the behaviour of the single cell by means of a suitable modified version of Monod’s classical chemostat model. Two main cell phases or states are considered to account for the experimentally observed changes occurring in the cell growth course: the budded phase and the unbudded one. Thus, a sort of two compartment structure is given to the total biomass. The model so far obtained allows one to analyse the local properties of the predicted steady states under various assumptions, both on the yield coefficients and the specific growth rates. Necessary conditions for the local instability are derived and the existence of stable limit cycles is shown by computer simulation. With respect to the qualitative changes in the metabolic parameters, this analysis agrees with the results obtained by simulation of complex structured and segregated models. However, the oscillation period is too long compared with the experimental one and this fact may be mainly due to the strong simplifying assumptions on the dynamic evolution of the transfer rates between the two compartments. The model’s usefulness seems until now restricted to the identification of the relationships between the cell cycle regulation and the oscillation triggering.
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  • 28
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    Notes: Abstract The mathematical model developed by Riveroet al. (1989,Chem. Engng Sci. 44, 2881–2897) is applied to literature data measuring chemotactic bacterial population distributions in response to steep as well as shallow attractant gradients. This model is based on a fundamental picture of the sensing and response mechanisms of individual bacterial cells, and thus relates individual cell properties such as swimming speed and tumbling frequency to population parameters such as the random motility coefficient and the chemotactic sensitivity coefficient. Numerical solution of the model equations generates predicted bacterial density and attractant concentration profiles for any given experimental assay. We have previously validated the mathematical model from experimental work involving a step-change in the attractant gradient (Fordet al., 1991Biotechnol. Bioengng.37, 647–660; For and Lauffenburger, 1991,Biotechnol. Bioengng,37, 661–672). Within the context of this experimental assay, effects of attractant diffusion and consumption, random motility, and chemotactic sensitivity on the shape of the profiles are explored to enhance our understanding of this complex phenomenon. We have applied this model to various other types of gradients with successful intepretation of data reported by Dalquistet al. (1972,Nature New Biol. 236, 120–123) forSalmonella typhimurum validating the mathematical model and supportin the involvement of high and low affinity receptors for serine chemotaxis by these cells.
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    Bulletin of mathematical biology 53 (1991), S. 797-804 
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    Bulletin of mathematical biology 53 (1991), S. 845-851 
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    Notes: Abstract Disconnected recurrences of the stop signal, serine and arginine appear in the original representation of the genetic code, and of the stop signal, arginine, serine and leucine in the codon ring representation. To achieve connectedness along with structural continuity, arook’s tour representation is presented here. On the basis of structural similarities and disparities in their side groups, each of the 20 amino acids is associated with a domain comprised of from one to six contiguous squares on the chess board. As the rook moves on the chess board, it reaches all 64 squares in the ordering of the codon numbers, which prescribe the codons by a simple formula based on the position and size of the nucleotides in a triplet. Recurrences of the stop signal, arginine and serine occur naturally on the tour as the rook enters each of the latter domains for the second time. A mathematical equivalent of the rook’s tour may enter as a programming device in the implementation of the code by the RNAs.
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    Bulletin of mathematical biology 59 (1997), S. 23-41 
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    Notes: Abstract We consider a host-solitary parasitoid system with three categories of individuals: parasitoids, healthy hosts and parasitized hosts. Parasitoids are assumed to discriminate perfectly between the two kinds of hosts and they can reject those which are already parasitized. If parasitoids systematically accept or reject superparasitism or behave randomly, the system is always unstable. Using an optimal foraging model, we determine the behavior of parasitoids which leads to maximization of the instantaneous reproductive rate. When following this adaptive decision rule, parasitoids accept or refuse superparasitism according to the densities of both healthy and parasitized hosts. We study the dynamics of the system when parasitoids follow the optimal rule and show that under certain conditions it possesses a locally stable equilibrium point. In addition, our model predicts that at equilibrium parasitoids show partial preferences for superparasitism.
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    Bulletin of mathematical biology 59 (1997), S. 205-232 
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    Notes: Abstract A system of differential equations for the control of tumor growth cells in a cycle nonspecific chemotherapy is analyzed. Spontaneously acquired drug resistance is taken into account, and a criterion for the selection of chemotherapeutic treatment is used. This criterion purports to describe the possibility of improvement of the patient's health when treatment is discontinued. Contrary to our early results which also take drug resistance into account, in this context strategies of continuous chemotherapy in which rest periods take part may be better than maximum drug concentration throughout the treatment (which appears to be in accordance with clinical practice). This bears out our previous conjecture that when drug resistance is accounted for, the imperfections in the usual modelling of treatment criteria, which in general do not allow for patient recuperation, ruled out the possibility of rest periods in optimal continuous chemotherapy.
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    Bulletin of mathematical biology 59 (1997), S. 255-262 
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    Notes: Abstract A logistic density-dependent matrix model is developed in which the matrices contain only parameters and recruitment is a function of adult population density. The model was applied to simulate introductions of white-tailed deer into an area; the fitted model predicted a carrying capacity of 215 deer, which was close to the observed carrying capacity of 220 deer. The rate of population increase depends on the dominant eigenvalue of the Leslie matrix, and the age structure of the simulated population approaches a stable age distribution at the carrying capacity, which was similar to that generated by the Leslie matrix. The logistic equation has been applied to study many phenomena, and the matrix model can be applied to these same processes. For example, random variation can be added to life history parameters, and population abundances generated with random effects on fecundity show both the affect of annual variation in fecundity and a longer-term pattern resulting from the age structure.
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    Bulletin of mathematical biology 59 (1997), S. 399-406 
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    Bulletin of mathematical biology 59 (1997), S. 707-724 
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    Notes: Abstract A system of differential equations for the control of tumor cells growth in a cycle nonspecific chemotherapy is presented. Spontaneously acquired drug resistance is accounted for, as well as the evolution in time of normal cells. In addition, optimization of conflicting objectives forms the aim of the chemotherapeutic treatment. For general cell growth, some results are given, whereas for the special case of Malthusian (exponential) growth of tumor cells and rather general growth rate for normal cells, the optimal strategy is worked out. The latter, from the clinical standpoint, corresponds to maximum drug concentration throughout the treatment.
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    Bulletin of mathematical biology 59 (1997), S. 787-807 
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    Bulletin of mathematical biology 59 (1997), S. 809-831 
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    Notes: Abstract This study examines the influence of various host-feeding patterns on host-parasitoid population dynamics. The following types of host-feeding patterns are considered: concurrent and non-destructive, non-concurrent and non-destructive, and non-concurrent and destructive. The host-parasitoid population dynamics is described by the Lotka-Volterra continuous-time model. This study shows that when parasitoids behave optimally, i.e. they maximize their fitness measured by the instantaneous per capita growth rate, the non-destructive type of host feeding stabilizes host-parasitoid dynamics. Other types of host feeding, i.e. destructive, concurrent, or non-concurrent, do not qualitatively change the neutral stability of the Lotka-Volterra model. Moreover, it is shown that the pattern of host feeding which maximizes parasitoid fitness is either non-concurrent and destructive, or concurrent and non-destructive host feeding, depending on the host abundance and parameters of the model. The effects of the adaptive choice of host-feeding patterns on host-parasitoid population dynamics are discussed.
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    Bulletin of mathematical biology 59 (1997), S. 931-952 
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    Notes: Abstract Game theory has had remarkable success as a framework for the discussion of animal behaviour and evolution. It suggested new interpretations and prompted new observational studies. Most of this work has been done with 2-player games. That is the individuals of a population compete in pairwise interactions. While this is often the case in nature, it is not exclusively so. Here we introduce a class of models for situations in which more than two (possibly very many) individuals compete simultaneously. It is shown that the solutions (i.e. the behaviour which may be expected to be observable for long periods) are more complex than for 2-player games. The concluding section lists some of the new phenomena which can occur.
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    Notes: Abstract A method allowing to measure the inhomogeneous distribution of purines/pyrimidines in nucleotide sequences is developed. We show that this measure relates to the coding or non-coding character of the considered sequence. Coding sequences present a near to the random Pu or Py distribution. This property is shared by both protein-coding DNA and functional RNA-coding DNA. Non-coding sequences present a highly clustered inhomogeneity. We propose the hypothesis, corroborated with appropriate computer simulations, that this is due to the action of various transposition events accumulated for long time periods.
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    Bulletin of mathematical biology 59 (1997), S. 1047-1075 
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    Notes: Abstract The potential generated in the smooth muscle of the vas deferens on release of a quantum of transmitter from a varicosity was analyzed using a three-dimensional bidomain continuum model. Current was injected at the origin of the bidomain; this current had the temporal characteristics of the junctional current. The membrane potential, intracellular potential, and extracellular potential, as well as the extracellular current, were then calculated throughout the bidomain at different times. Calculations were performed to show the effect of changing the anisotropy ratios of the intracellular and extracellular conductivities on the spread of current and potential in each of the three dimensions. These results provide a theoretical framework for ascertaining the time course of transmitter interaction at a varicosity following the secretion of a quantum of transmitter.
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    Bulletin of mathematical biology 59 (1997), S. 1145-1154 
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    Notes: Abstract Parabolic growth invariably results in the survival of all competing types. Under the constraint of constant total concentration, there is a unique equilibrium in the simplex interior, which is asymptotically stable inside the whole simplex. The appropriate Lyapunov function is obtained in terms of the excess productivity which is shown to be maximized for the competitive system with fractional order kinetics. Claims to the contrary are refuted.
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    Bulletin of mathematical biology 59 (1997), S. 1191-1201 
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    Bulletin of mathematical biology 60 (1998), S. 195-196 
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    Bulletin of mathematical biology 60 (1998), S. 101-129 
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    Notes: Abstract Due to the increasing importance of the extracellular matrix in many biological problems, in this paper we develop a model for fibroblast and collagen orientation with the ultimate objective of understanding how fibroblasts form and remodel the extracellular matrix, in particular its collagen component. The model uses integrodifferential equations to describe the interaction between the cells and fibers at a point in space with various orientations. The equations are studied both analytically and numerically to discover different types of solutions and their behavior. In particular we examine solutions where all the fibroblasts and collagen have discrete orientations, a localized continuum of orientations and a continuous distribution of orientations with several maxima. The effect of altering the parameters in the system is explored, including the angular diffusion coefficient for the fibroblasts, as well as the strength and range of the interaction between fibroblasts and collagen. We find the initial conditions and the range of influence between the collagen and the fibroblasts are the two factors which determine the behavior of the solutions. The implications of this for wound healing and cancer are discussed including the conclusion that the major factor in determining the degree of scarring is the initial deposition of collagen.
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    Bulletin of mathematical biology 60 (1998), S. 215-230 
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    Notes: Abstract This paper considers the time to extinction for a stochastic epidemic model of SEIR form without replacement of susceptibles. It first shows how previous rigorous results can be heuristically explained in terms of the more transparent dynamics of an approximating deterministic system. The model is then extended to include a host population structured into patches, with weak nearest-neighbour mixing of infection. It is shown, by considering the approximating deterministic system, that the expected time to extinction in a population of n + 1 patches each of size N is of the form a log N + bn, provided that N 〉 N c where N c is a critical patch size below which transits are unlikely to occur. This corresponds to the simple decomposition of the time of an epidemic into the time it takes to spread through one patch plus the time it takes to transit to each of n successive patches. Expressions for this threshold and the coefficients of the time to extinction are given in terms of the transmission parameters of infection and the coupling strength between patches. These expressions are compared with numerical results using parameters relevant to a study of phocine distemper virus in North Sea seals, and the agreement is found to be good for large and small N. In the region when N ≈ N c , where transits may or may not occur, interesting transitional behaviour is seen, leading to a non-monotonicity of the extinction time as a function of N.
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    Bulletin of mathematical biology 60 (1998), S. 409-415 
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    Bulletin of mathematical biology 60 (1998), S. 355-372 
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    Notes: Abstract When directly transmitted infectious diseases are modeled assuming an everlasting induced immunity (and constant contact rate), there are well-established formulas to deal with, which is not true if we include the loss of induced immunity. In general, the immunity induced by the disease is everlasting. We propose a model considering the loss of immunity and present methods for the estimation of two epidemiological parameters: the force of infection and the basic reproduction ratio. We also analyze the effects of the loss of immunity on these parameters. Based on these results, we conclude that reinfection can play an important role in highly vaccinated populations.
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    Bulletin of mathematical biology 60 (1998), S. 449-475 
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    Notes: Abstract We studied mathematical models for the length distributions of actin filaments under the effects of polymerization/depolymerization, and fragmentation. In this paper, we emphasize the effects of these two processes acting alone. In this case, simple discrete and continuous models can be derived and solved explicitly (in several special cases), making the problem interesting from a modeling and pedagogical point of view. In a companion paper (Ermentrout and Edelstein-Keshet, 1998, Bull. Math. Biol. 60, 477–503) we investigate what happens when the processes act together, with particular attention to fragmentation by gelsolin, and with a greater level of biological detail.
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    Bulletin of mathematical biology 60 (1998), S. 197-213 
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    Notes: Abstract A possible experimental design for combination experiments is to compare the doseresponse curve of a single agent with the corresponding curve of the same agent using either a fixed amount of a second one or a fixed dose ratio. No interaction is then often defined by a parallel shift of these curves. We have performed a systematic study for various types of doseresponse relations both for the dose-additivity (Loewe additivity) and for the independence (Bliss independence) criteria for defining zero interaction. Parallelism between doseresponse curves of a single agent and those of the same agent in the presence of a fixed amount of another one is found for the Loewe-additivity criterion for linear doseresponse relations. For nonlinear relations, one has to differentiate between effect parallelism (parallel shift on the effect scale) and dose parallelism (parallel shift on the dose scale). In the case of Loewe additivity, zero-interaction dose parallelism is found for power, Weibull, median-effect and logistic doseresponse relations, given that special parameter relationships are fulfilled. The mechanistic model of competitive interaction exhibits dose parallelism but not effect parallelism for Loewe additivity. Bliss independence and Loewe additivity lead to identical results for exponential doseresponse curves. This is the only case for which dose parallelism was found for Bliss independence. Parallelism between single-agent doseresponse relations and Loewe additivity mixture relations is found for examples with a fixed doseratio design. However, this is again not a general property of the design adopted but holds only if special conditions are fulfilled. The comparison of combination doseresponse curves with single-agent relations has to be performed taking into account both potency and shape parameters. The results of this analysis lead to the conclusion that parallelism between zero interaction combination and single-agent doseresponse relations is found only for special cases and cannot be used as a general criterion for defining zero-interaction in combined-action assessment even if the correct potency shift is taken into account.
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    Bulletin of mathematical biology 59 (1997), S. 763-785 
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    Notes: Abstract The purpose of this study was to investigate strategies in the monotherapy treatment of HIV infection in the presence of drug-resistant (mutant) strains. A mathematical system is developed to model resistance in HIV chemotherapy. It includes the key players in the immune response to HIV infection: virus and both uninfected CD4+ and infected CD4+ T-cell populations. We model the latent and progressive stages of the disease, and then introduce monotherapy treatment. The model is a system of differential equations describing the interaction of two distinct classes of HIV—drug-sensitive (wild type) and drug-resistant (mutant)—with lymphocytes in the peripheral blood. We then introduce chemotherapy effects. In the absence of treatment, the model produces the three types of qualitative clinical behavior—anuninfected steady state, andinfected steady state (latency), andprogression to AIDS. Simulation of treatment is provided for monotherapy, during theprogression to AIDS state, in the consideration of resistance effects. Treatment benefit is based on an increase or retention in CD4+ T-cell counts together with a low viral titer. We explore the following treatment approaches: an antiviral drug which reduces viral infectivity that is administered early—when the CD4+ T-cell count is ≥300/mm3, and late—when the CD4+ T-cell count is less than 300/mm3. We compare all results with data. When treatment is initiated during the progression to AIDS state, treatment prevents T-cell collapse, but gradually loses effectiveness due to drug resistance. We hypothesize that it is the careful balance of mutant and wild-type HIV strains which provides the greatest prolonged benefit from treatment. This is best achieved when treatment is initiated when the CD4+ T-cell counts are greater than 250/mm3, but less than 400/mm3 in this model (i.e. not too early, not too late). These results are supported by clinical data. The work is novel in that it is the first model to accurately simultate data before, during and after monotherapy treatment. Our model also provides insight into recent clinical results, as well as suggests plausible guidelines for clinical testing in the monotherapy of HIV infection.
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    Bulletin of mathematical biology 59 (1997), S. 833-856 
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    Notes: Abstract A mathematical model which describes adhesion of bacteria to host cell lines is presented. The model is flexible enough to account for the following situations: extracellular bacteria are either in exponential or in stationary phase. Adhesion is described as a reversible binding process in which the bacteria attach to or detach from specific receptors uniformly distributed on the cell surface. In turn, attached bacteria can either replicate or, conversely, they are restrained to remain in stationary phase. In the first case, however, we must consider the problem of whether the decrease of unoccupied receptors as adhesion progresses imposes a limit to the replicating capacity of the attached bacteria. The effect exerted by the multiplicity of infection (MOI), i.e. the ratio of the number of bacteria to the number of host cells, on the process of adhesion is also contemplated by the model. This has revealed that experiments performed at the same values of MOI can show completely different levels of adhered bacteria, depending on the number of host cells in the assays. This finding demonstrates that the report of the MOI values is insufficient to characterize comparative studies of bacterial adhesion since it could lead to a misunderstanding of the corresponding data. Simplified models based on the steady-state approximation and in equilibrium analysis by means of a Lagmuir adsorption isotherm for the attached bacteria are also discussed. This allows us to define the adhesion coefficient (β) in a given bacterium-cell system so that, with the exception of those systems where these coefficients cannot be defined, larger values of β are related to a greater adhesion capacity. An overview of the procedures to perform quantitative adhesion data analysis is outlined. Finally, theoretical predictions are compared with experimental results from the literature.
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    Bulletin of mathematical biology 59 (1997), S. 897-910 
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    Notes: Abstract A new measure of toxicity based on stochastic modelling of single photon-counting processes, representing time-resolved phagocyte luminescence of xenobiotic-perturbed human neutrophils, has been constructed. The stochastic measure of toxicity has been verified by the QSAR method, and then compared and contrasted with the traditional toxicity measure used in bio- and chemiluminescent research. Phenol and benzene homologues were chosen as perturbers due to their importance from the viewpoint of ecotoxicology and occupational medicine.
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    Bulletin of mathematical biology 59 (1997), S. 953-973 
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    Notes: Abstract We describe a mathematical model of the flow and deformation in a human teat. Our aim is to compare the theoretical milk yield during infant breast feeding with that obtained through the use of a breast pump. Infants use a peristaltic motion of the tongue, along with some suction, to extract milk, whereas breast pumps use a cyclic pattern of suction only. Our model is based on quasi-linear poroelasticity whereby the teat is modelled as a cylindrical porous elastic material saturated with fluid. We impose a cyclic axial suction pressure difference across the teat and impose a radial compressive force moving along the teat which mimics infant suckling. This is compared to the case of cyclic and steady pumping only which models the action of breast pumps. The results illustrate that there is an optimal time to apply the compressive force during the suction cycle that will increase the flow rate in our theoretical teat. The model and results may be of use in the future design of effective breast pumps.
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    Bulletin of mathematical biology 59 (1997), S. 993-1012 
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    Notes: Abstract In the present work, we study the propagation of solitary waves in a prestressed thick walled elastic tube filled with an incompressible inviscid fluid. In order to include the geometric dispersion in the analysis the wall inertia and shear deformation effects are taken into account for the inner pressure-cross-sectional area relation. Using the reductive perturbation technique, the propagation of weakly non-linear waves in the long-wave approximation is examined. It is shown that, contrary to thin tube theories, the present approach makes it possible to have solitary waves even for a Mooney-Rivlin (M-R) material. Due to dependence of the coefficients of the governing Korteweg-deVries equation on initial deformation, the solution profile changes with inner pressure and the axial stretch. The variation of wave profiles for a class of elastic materals are depicted in graphical forms. As might be seen from these illustrations, with increasing thickness ratio, the profile of solitary wave is steepened for a M-R material but it is broadened for biological tissues.
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    Bulletin of mathematical biology 59 (1997), S. 1077-1100 
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    Notes: Abstract Adult dermal wounds, in contrast to fetal wounds, heal with the formation of scar tissue. A crucial factor in determining the degree of scarring is the ratio of types I and III collagen, which regulates the diameter of the combined fibers. We developed a reaction-diffusion model which focuses on the control of collagen synthesis by different isoforms of the polypeptide transforming growth factor-β (TGFβ). We used the model to investigate the current controversy as to whether the fibroblasts migrate into the wound from the surrounding unwounded dermis or from the underlying subcutaneous tissue. Numerical simulations of a spatially independent, temporal model led to a value of the collagen ratio consistent with that of healthy tissue for the fetus, but corresponding to scarring in the adult. We investigated the effect of topical application of TGFβ and show that addition of isoform 3 reduces scar tissue formation, in agreement with the experiment. However, numerical solutions of the reaction-diffusion system do not exhibit this sensitivity to growth factor application. Mathematically, this corresponds to the observation that behind healing wavefront solutions, a particular healed state is always selected independent of transients, even though there is a continuum of possible positive steady states. We explain this phenomenon using a caricature system of equations, which reflects the key qualitative features of the full model but has a much simpler mathematical form. Biologically, our results suggest that the migration into a wound of fibroblasts and TGFβ from the surrounding dermis alone cannot account for the essential features of the healing process, and that fibroblasts entering from the underlying subcutaneous tissue are crucial to the healing process.
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    Bulletin of mathematical biology 59 (1997), S. 1125-1144 
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    Notes: Abstract Oscillations in cytosolic Ca2+ concentrations in living cells are often a manifestation of propagating waves of Ca2+. Numerical simulations with a realistic model of inositol 1, 4, 5-trisphosphate (IP3)-induced Ca2+ wave trains lead to wave speeds that increase linearly at long times when (a) IP3 levels are in the range for Ca2+ oscillations, (b) a gradient of phase is established by either an initial ramp or pulse of IP3, and (c) IP3 concentrations asymptotically become uniform. We explore this phenomenon with analytical and numerical methods using a simple two-variable reduction of the De Young-Keizer model of the IP3 receptor that includes the influence of Ca2+ buffers. For concentrations of IP3 in the oscillatory regime, numerical solution of the resulting reaction diffusion equations produces nonlinear wave trains that shows the same asymptotic growth of wave speed. Due to buffering, diffusion of Ca2+ is quite slow and, as previously noted, these waves occur without appreciable bulk movement of Ca2+. Thus, following Neu and Murray, we explore the behavior of these waves using an asymptotic expansion based on the small size of the buffered diffusion constant for Ca2+. We find that the gradient in phase of the wave obeys Burgers' equation asymptotically in time. This result is used to explain the linear increase of the wave speed observed in the simulations.
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    Bulletin of mathematical biology 59 (1997), S. 1183-1189 
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    Notes: Abstract The robustness of patterning events in development is a key feature that must be accounted for in proposed models of these events. When considering explicitly cellular systems, robustness can be exhibited at different levels of organization. Consideration of two widespread patterning mechanisms suggests that robustness at the level of cell communities can result from variable development at the level of individual cells; models of these mechanisms show how interactions between participating cells guarantee community-level robustness. Cooperative interactions enhance homogeneity within communities of like cells and the sharpness of boundaries between communities of distinct cells, while competitive interactions amplify small inhomogeneities within communities of initially equivalent cells, resulting in fine-grained patterns of cell specialization.
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    Bulletin of mathematical biology 60 (1998), S. 1-26 
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    Notes: Abstract Many models have been proposed for spatial pattern formation in embryology and analyzed for the standard case of zero-flux boundary conditions. However, relatively little attention has been paid to the role of boundary conditions on the form of the final pattern. Here we investigate, numerically, the effect of nonstandard boundary conditions on a model pattern generator, which we choose to be of a cell-chemotactic type. We specifically focus on the role of boundary conditions and the effects of scale and aspect ratio, and study the spatiotemporal dynamics of pattern formation. We illustrate the properties of the model by application to the spatiotemporal sequence of skeletal development.
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    Bulletin of mathematical biology 60 (1998), S. 79-100 
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    Notes: Abstract A model, based on the principles of continuum mechanics, is presented for the analysis of cell-velocity fields within wool follicles. The model requires specification of three follicle characteristics in the form of spatially varying fields: viscosity, cell density and cell production rate. The viscosity is introduced as an attempt to model both complex intercellular interactions and individual cell deformation as the cells move. It is demonstrated that the distribution of cell production is more important than axial variation in viscosity in determining the overall flow pattern.
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    Bulletin of mathematical biology 60 (1998), S. 131-150 
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    Notes: Abstract A microbial trichome extracts nutrients from its immediate surroundings. It may also oxidize electron donors, reduce electron acceptors, and exude the ‘waste’ products of endogenous redox metabolism. Finally, it may effect light harvesting. These exchange fluxes are summed up in a generic model, which covers photoautotrophs as well as chemoheterotrophs. The focus is on endogenous metabolism and the cellular homeostasis of both reducing and phosphorylating equivalents. A novel result is the formulation of four ‘rules’, akin to the Pasteur effect, which govern the compatibility of endogenous metabolism with various assimilatory processes.
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    Bulletin of mathematical biology 60 (1998), S. 49-65 
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    Notes: Abstract In this paper we present a deterministic, discrete-time model for a two-patch predator-prey metapopulation. We study optimal harvesting for the metapopulation using dynamic programming. Some rules are established as generalizations of rules for a single-species metapopulation harvesting theory. We also establish rules to harvest relatively more (or less) vulnerable prey subpopulations and more (or less) efficient predator subpopulations.
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    Bulletin of mathematical biology 61 (1999), S. 1-17 
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    Notes: Abstract An equivalent electrical circuit is given for a branch of an amphibian motor-nerve terminal in a volume conductor. The circuit allows for longitudinal current flow inside the axon as well as between the axon and its Schwann cell sheath, and also for the radial leakage of current through the Schwann cell sheath. Analytical and numerical solutions are found for the spatial and time dependence of the membrane potential resulting from the injection of depolarizing current pulses by external electrodes at one or two separate locations on the terminal. These solutions show that the depolarization at an injection site can cause a hyperpolarization at sites a short distance away. This effect becomes more pronounced in a short terminal with sealed-end boundary conditions. The hyperpolarization provides a possible explanation for recent experimental results, which show that the average quantal release due to a test depolarizing current pulse delivered by an electrode at one site on a nerve terminal is reduced by the application of an identical conditioning pulse at a neighbouring site.
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    Bulletin of mathematical biology 61 (1999), S. 113-140 
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    Notes: Abstract Synthetic barriers such as gloves, condoms and masks are widely used in efforts to prevent disease transmission. Due to manufacturing defects, tears arising during use, or material porosity, there is inevitably a risk associated with use of these barriers. An understanding of virus transport through the relevant passageways would be valuable in quantifying the risk. However, experimental investigations involving such passageways are difficult to perform, owing to the small dimensions involved. This paper presents a mathematical model for analyzing and predicting virus transport through barriers. The model incorporates a mathematical description of the mechanisms of virus transport, which include carrier-fluid flow, Brownian motion, and attraction or repulsion via virus-barrier interaction forces. The critical element of the model is the empirically determined rate constant characterizing the interaction force between the virus and the barrier. Once the model has been calibrated through specification of the rate constant, it can predict virus concentration under a wide variety of conditions. The experiments used to calibrate the model are described, and the rate constants are given for four bacterial viruses interacting with a latex membrane in saline. Rate constants were also determined for different carrier-fluid salinities, and the salt concentration was found to have a pronounced effect. Validation experiments employing laser-drilled pores in condoms were also performed to test the calibrated model. Model predictions of amount of transmitted virus through the drilled holes agreed well with measured values. Calculations using determined rate constants show that the model can help identify situations where barrier-integrity tests could significantly underestimate the risk associated with barrier use.
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    Bulletin of mathematical biology 61 (1999), S. 221-238 
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    Notes: Abstract Evaluation of the fluid flow pattern in a non-pregnant uterus is important for understanding embryo transport in the uterus. Fertilization occurs in the fallopian tube and the embryo (fertilized ovum) enters the uterine cavity within 3 days of ovulation. In the uterus, the embryo is conveyed by the uterine fluid for another 3 to 4 days to a successful implantation site at the upper part of the uterus. Fluid movements within the uterus may be induced by several mechanisms, but they seem to be dominated by myometrial contractions. Intra-uterine fluid transport in a sagittal cross-section of the uterus was simulated by a model of wall-induced fluid motion within a two-dimensional channel. The time-dependent fluid pattern was studied by employing the lubrication theory. A comprehensive analysis of peristaltic transport resulting from symmetric and asymmetric contractions is presented for various displacement waves on the channel walls. The results provide information on the flow field and possible trajectories by which an embryo may be transported before implantation at the uterine wall.
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    Bulletin of mathematical biology 61 (1999), S. 379-398 
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    Notes: Abstract A mechanistically based mathematical model is used to investigate some of the important factors in priming hepatocytes to enter the G1 phase of the cell cycle. The model considers all of the relevant biochemical mechanisms from signal-receptor binding to the elevation of AP-1(activation protein transcription factor) levels. Focus is centered on the chain of biochemical events governing the sequential activation of protein kinase C (PKC), mitogen-activated protein kinase (MAPK) and AP-1. Factors such as amplitude and duration of growth factors signals, the kinetics of guanosine diphosphate (GDP) to guanosine triphosphate (GTP) conversion, and the negative feedback control mechanisms governing initial steps in cellular replication were theoretically examined. The results of our theoretical assessments support the finding that specific mutations along the PKC-AP1 pathways can have a critical effect on the rate at which cells enter the division cycle.
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    Bulletin of mathematical biology 61 (1999), S. 273-301 
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    Notes: Abstract Normal cardiac muscle contraction occurs in response to a rapid rise followed by a slower decay in intracellular calcium concentration. When cardiac muscle cells are loaded with calcium, an intracellular store releases calcium into the cytosol by the process of calcium-induced calcium release (CICR). This release contributes to the rise in intracellular calcium which in turn triggers contraction. We use two qualitative piecewise linear reaction-diffusion models of this behaviour to investigate the speed, stability and waveform of plane waves using singular perturbation techniques.
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    Bulletin of mathematical biology 61 (1999), S. 365-377 
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    Notes: Abstract Properties of two of the stochastic circulatory models theoretically introduced by Smith et al., 1997, Bull. Math. Biol. 59, 1–22 were investigated. The models assumed the gamma distribution of the cycle time under either the geometric or Poisson elimination scheme. The reason for selecting these models was the fact that the probability density functions of the residence time of these models are formally similar to those of the Bateman and gamma-like function models, i.e., the two common deterministic models. Using published data, the analytical forms of the probability density functions of the residence time and the distributions of the simulated values of the residence time were determined on the basis of the deterministic models and the stochastic circulatory models, respectively. The Kolmogorov-Smirnov test revealed that even for 1000 xenobiotic particles, i.e., a relatively small number if the particles imply drug molecules, the probability density functions of the residence time based on the deterministic models closely matched the distributions of the simulated values of the residence time obtained on the basis of the stochastic circulatory models, provided that parameters of the latter models fulfilled selected conditions.
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    Bulletin of mathematical biology 60 (1998), S. 919-935 
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    Notes: Abstract To analyze signals measured from human blood flow in the time-frequency domain, we used the wavelet transform which gives good time resolution for high-frequency components and good frequency resolution for low-frequency components. Five characteristic frequency peaks, corresponding to five almost periodic rhythmic activities, were found on the time scale of minutes. These oscillations were characterized by time and spatial invariant measures. The potential of this approach in studying the blood-flow dynamics was illustrated by revealing differences between the groups of control subjects and athletes.
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    Bulletin of mathematical biology 60 (1998), S. 997-998 
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    Bulletin of mathematical biology 60 (1998), S. 1167-1200 
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    Notes: Abstract The interaction of a pair of weakly coupled biological bursters is examined. Bursting refers to oscillations in which an observable slowly alternates between phases of relative quiescence and rapid oscillatory behavior. The motivation for this work is to understand the role of electrical coupling in promoting the synchronization of bursting electrical activity (BEA) observed in the β-cells of the islet of Langerhans, which secrete insulin in response to glucose. By studying the coupled fast subsystem of a model of BEA, we focus on the interaction that occurs during the rapid oscillatory phase. Coupling is weak, diffusive and non-scalar. In addition, non-identical oscillators are permitted. Using perturbation methods with the assumption that the uncoupled oscillators are near a Hopf bifurcation, a reduced system of equations is obtained. A detailed bifurcation study of this reduced system reveals a variety of patterns but suggests that asymmetrically phase-locked solutions are the most typical. Finally, the results are applied to the unreduced full bursting system and used to predict the burst pattern for a pair of cells with a given coupling strength and degree of heterogeneity.
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    Bulletin of mathematical biology 60 (1998), S. 1017-1037 
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    Notes: Abstract Hemodynamic forces affect endothelial cell morphology and function. In particular, circumferential cyclic stretch of blood vessels, due to pressure changes during the cardiac cycle, is known to affect the endothelial cell shape, mediating the alignment of the cells in the direction perpendicular to stretch. This change in cell shape proceeds a drastic reorganization at the internal level. The cellular scaffolding, mainly composed of actin filaments, reorganize in the direction which later becomes the cell’s long axis. How this external mechanical stimulus is ’sensed’ and transduced into the cell is still unknown. Here, we develop a mathematical model depicting the dynamics of actin filaments, and the influence of the cyclic stretch of the substratum based on the experimental evidence that external stimuli may be transduced inside the cell via transmembrane proteins which are coupled with actin filaments on the cytoplasmic side. Based on this view, we investigate two approaches describing the formulation of the transduction mechanisms involving the coupling between filaments and the membrane proteins. As a result, we find that the mechanical stimulus could cause the experimentally observed reorganization of the entire cytoskeleton simply by altering the dynamics of the filaments connected with the integral membrane proteins, as described in our model. Comparison of our results with previous studies of cytoskeletal dynamics reveals that the cytoskeleton, which, in the absence of the effect of stretch would maintain its isotropic distribution, slowly aligns with the precise direction set by the external stimulus. It is found that even a feeble stimulus, coupled with a strong internal dynamics, is sufficient to align actin filaments perpendicular to the direction of stretch.
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    Bulletin of mathematical biology 60 (1998), S. 1149-1166 
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    Notes: Abstract We study a general predator—prey system in a spatially heterogeneous environment. The predation process, which occurs on a behavioural time-scale, is much faster than the other processes (reproduction, natural mortality and migrations) occurring on the population dynamics time-scale. We show that, taking account of this difference in time-scales, and assuming that the prey have a refuge, the dynamics of the system on a slow time-scale become donor-controlled. Even though predators may control the prey density locally and on a behavioural fast time-scale, nevertheless, both globally and on a slow time-scale, the prey dynamics are independent of predator density: the presence of predators generates a constant prey mortality. In other words, in heterogeneous environments, the prey population dynamics depend in a switch-like manner on the presence or absence of predators, not on their actual density.
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    Bulletin of mathematical biology 61 (1999), S. 19-32 
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    Notes: Abstract Ratio-dependent predator-prey models set up a challenging issue regarding their dynamics near the origin. This is due to the fact that such models are undefined at (0, 0). We study the analytical behavior at (0, 0) for a common ratio-dependent model and demonstrate that this equilibrium can be either a saddle point or an attractor for certain trajectories. This fact has important implications concerning the global behavior of the model, for example regarding the existence of stable limit cycles. Then, we prove formally, for a general class of ratio-dependent models, that (0, 0) has its own basin of attraction in phase space, even when there exists a non-trivial stable or unstable equilibrium. Therefore, these models have no pathological dynamics on the axes and at the origin, contrary to what has been stated by some authors. Finally, we relate these findings to some published empirical results.
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    Bulletin of mathematical biology 61 (1999), S. 157-177 
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    Notes: Abstract We explore the behavior of richly connected inhibitory neural networks under parameter changes that correspond to weakening of synaptic efficacies between network units, and show that transitions from irregular to periodic dynamics are common in such systems. The weakening of these connections leads to a reduction in the number of units that effectively drive the dynamics and thus to simpler behavior. We hypothesize that the multiple interconnecting loops of the brain’s motor circuitry, which involve many inhibitory connections, exhibit such transitions. Normal physiological tremor is irregular while other forms of tremor show more regular oscillations. Tremor in Parkinson’s disease, for example, stems from weakened synaptic efficacies of dopaminergic neurons in the nigro-striatal pathway, as in our general model. The multiplicity of structures involved in the production of symptoms in Parkinson’s disease and the reversibility of symptoms by pharmacological and surgical manipulation of connection parameters suggest that such a neural network model is appropriate. Furthermore, fixed points that can occur in the network models are suggestive of akinesia in Parkinson’s disease. This model is consistent with the view that normal physiological systems can be regulated by robust and richly connected feedback networks with complex dynamics, and that loss of complexity in the feedback structure due to disease leads to more orderly behavior.
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    Bulletin of mathematical biology 61 (1999), S. 987-1008 
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    Notes: Abstract Determining molecular structure from interatomic distances is an important and challenging problem. Given a molecule with n atoms, lower and upper bounds on interatomic distances can usually be obtained only for a small subset of the $$\frac{{n(n - 1)}}{2}$$ atom pairs, using NMR. Given the bounds so obtained on the distances between some of the atom pairs, it is often useful to compute tighter bounds on all the $$\frac{{n(n - 1)}}{2}$$ pairwise distances. This process is referred to as bound smoothing. The initial lower and upper bounds for the pairwise distances not measured are usually assumed to be 0 and ∞. One method for bound smoothing is to use the limits imposed by the triangle inequality. The distance bounds so obtained can often be tightened further by applying the tetrangle inequality—the limits imposed on the six pairwise distances among a set of four atoms (instead of three for the triangle inequalities). The tetrangle inequality is expressed by the Cayley—Menger determinants. For every quadruple of atoms, each pass of the tetrangle inequality bound smoothing procedure finds upper and lower limits on each of the six distances in the quadruple. Applying the tetrangle inequalities to each of the ( 4 n ) quadruples requires O(n 4) time. Here, we propose a parallel algorithm for bound smoothing employing the tetrangle inequality. Each pass of our algorithm requires O(n 3 log n) time on a CREW PRAM (Concurrent Read Exclusive Write Parallel Random Access Machine) with $$O\left( {\frac{n}{{\log n}}} \right)$$ processors. An implementation of this parallel algorithm on the Intel Paragon XP/S and its performance are also discussed.
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    Notes: Abstract We observed that amphiphile-induced microexovesicles may be spherical or cylindrical, depending on the species of the added amphiphile. The spherical microexovesicle corresponds to an extreme local difference between the two monolayer areas of the membrane segment with a fixed area, while the cylindrical microexovesicle corresponds to an extreme local area difference if the area of the budding segment is increased due to lateral influx of anisotropic membrane constituents. Protein analysis showed that both types of vesicles are highly depleted in the membrane skeleton. It is suggested that a partial detachment of the skeleton in the budding region is favoured due to accumulated skeleton shear deformations in this region.
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    Bulletin of mathematical biology 61 (1999), S. 1209-1210 
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    Bulletin of mathematical biology 61 (1999), S. 1187-1207 
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    Notes: Abstract The possibility of chaos control in biological systems has been stimulated by recent advances in the study of heart and brain tissue dynamics. More recently, some authors have conjectured that such a method might be applied to population dynamics and even play a nontrivial evolutionary role in ecology. In this paper we explore this idea by means of both mathematical and individual-based simulation models. Because of the intrinsic noise linked to individual behavior, controlling a noisy system becomes more difficult but, as shown here, it is a feasible task allowed to be experimentally tested.
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    Bulletin of mathematical biology 61 (1999), S. 573-595 
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    Notes: Abstract In an attempt to improve the understanding of complex metabolic dynamic phenomena, we have analysed several ‘metabolic networks’, dynamical systems which, under a single formulation, take into account the activity of several catalytic dissipative structures, interconnected by substrate fluxes and regulatory signals. These metabolic networks exhibit a rich variety of self-organized dynamic patterns, with e.g., phase transitions emerging in the whole activity of each network. We apply Hurst’s R/S analysis to several time series generated by these metabolic networks, and measure Hurst exponents H 〈 0.5 in most cases. This value of H, indicative of antipersistent processes, is detected at very high significance levels, estimated with detailed Monte Carlo simulations. These results show clearly the considered type of metabolic networks exhibit long-term memory phenomena.
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    Bulletin of mathematical biology 61 (1999), S. 597-600 
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    Bulletin of mathematical biology 61 (1999), S. 437-467 
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    Notes: Abstract The secondary structures of nucleic acids form a particularly important class of contact structures. Many important RNA molecules, however, contain pseudo-knots, a structural feature that is excluded explicitly from the conventional definition of secondary structures. We propose here a generalization of secondary structures incorporating ‘non-nested’ pseudo-knots, which we call bi-secondary structures, and discuss measures for the complexity of more general contact structures based on their graph-theoretical properties. Bi-secondary structures are planar trivalent graphs that are characterized by special embedding properties. We derive exact upper bounds on their number (as a function of the chain length n) implying that there are fewer different structures than sequences. Computational results show that the number of bi-secondary structures grows approximately like 2.35n. Numerical studies based on kinetic folding and a simple extension of the standard energy model show that the global features of the sequence-structure map of RNA do not change when pseudo-knots are introduced into the secondary structure picture. We find a large fraction of neutral mutations and, in particular, networks of sequences that fold into the same shape. These neutral networks percolate through the entire sequence space.
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    Bulletin of mathematical biology 61 (1999), S. 683-700 
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    Notes: Abstract A braced framework of tubular struts, in the walls and air spaces of frog lungs, suspends the respiratory surface and holds the lung open at zero transmural pressure withstanding imploding forces created by abdominal viscera, much as would the supports of a bell tent. The struts are tubes, having a larger second moment of area than do solid struts of the same cross-sectional area, and so are stronger, and contain pulmonary vessels within a flexible wall. The orthogonal arrangement of the struts in the framework, explained in part by Maxwell’s Lemma and Michell’s Theorem, strengthens the framework and minimizes its weight; orthogonality is maintained as the lungs change size. A model is presented, in which a frog might control pre-and post-pulmonary vascular resistances and, hence, blood volume in the struts, without compromising pulmonary perfusion. Such adjustments could vary the area of lung and the extent of perfused capillaries exposed to pulmonary gas, helping match the lung’s surface area, weight and metabolic load to activity.
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    Notes: Abstract A molecular-level theory is constructed for the control of fast neurotransmitter release, based on recent experimental findings that depolarization shifts presynaptic autoreceptors to a low affinity state and that an autoreceptor must be bound to a transmitter before it can become associated with the exocytotic apparatus. It is assumed that such an association blocks release; experimental support for this assumption is cited. The theory provides mechanisms for key experimental results concerning the essence of the matter, what controls the time course of evoked release? The same general model can account for both evoked and spontaneous release. The new theory can be regarded as a molecular implementation of the (phenomenological) calcium-voltage hypothesis that was suggested earlier.
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    Bulletin of mathematical biology 61 (1999), S. 799-805 
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    Bulletin of mathematical biology 61 (1999), S. 625-649 
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    Notes: Abstract We have developed cellular automaton models for two species competing in a patchy environment. We have modeled three common types of competition: facilitation (in which the winning species can colonize only after the losing species has arrived) inhibition (in which either species is able to prevent the other from colonizing) and tolerance (in which the species most tolerant of reduced resource levels wins). The state of a patch is defined by the presence or absence of each species. State transition probabilities are determined by rates of disturbance, competitive exclusion, and colonization. Colonization is restricted to neighboring patches. In all three models, disturbance permits regional persistence of species that are excluded by competition locally. Persistence, and hence diversity, is maximized at intermediate disturbance frequencies. If disturbance and dispersal rates are sufficiently high, the inferior competitor need not have a dispersal advantage to persist. Using a new method for measuring the spatial patterns of nominal data, we show that none of these competition models generates patchiness at equilibrium. In the inhibition model, however, transient patchiness decays very slowly. We compare the cellular automaton models to the corresponding mean-field patch-occupancy models, in which colonization is not restricted to neighboring patches and depends on spatially averaged species frequencies. The patch-occupancy model does an excellent job of predicting the equilibrium frequencies of the species and the conditions required for coexistence, but not of predicting transient behavior.
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    Bulletin of mathematical biology 61 (1999), S. 1093-1120 
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    Notes: Abstract We investigate the sequence of patterns generated by a reaction—diffusion system on a growing domain. We derive a general evolution equation to incorporate domain growth in reaction—diffusion models and consider the case of slow and isotropic domain growth in one spatial dimension. We use a self-similarity argument to predict a frequency-doubling sequence of patterns for exponential domain growth and we find numerically that frequency-doubling is realized for a finite range of exponential growth rate. We consider pattern formation under different forms for the growth and show that in one dimension domain growth may be a mechanism for increased robustness of pattern formation.
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    Bulletin of mathematical biology 61 (1999), S. 1151-1186 
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    Notes: Abstract The persistence of linear dominance hierarchies is often attributed to higher probabilities of a win after a win or a loss after a loss in agonistic interactions, yet there has been no theory on the evolution of such prior-experience effects. Here an analytic model, based on the idea that contests are determined by subjective perceptions of resource-holding potential (RHP) which animals may revise in the light of experience, demonstrates that winner and loser effects can evolve through round-robin competition among triads of animals drawn randomly from their population, and that the probability of a hierarchy increases with the strength of the combined effect. The effects are pure, in the sense that a contestant observes neither its own RHP nor its opponent’s RHP or RHP perception or win—loss record; and so the strength of an effect is unmodified by the RHPs of particular individuals, but depends on the distribution of RHP among the population at large. The greater the difference between an individual’s and its opponent’s RHP perception, the more likely it is to win a contest; however, if it overestimates its RHP, then the cost of fighting increases with the overestimate. A winner or loser effect exists only if the fitness gain of the beta individual in a hierarchy, relative to that of the alpha, is less than 0.5. Then a loser effect can exist alone, or it can coexist with a winner effect; however, there cannot exist a winner effect without a loser effect.
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    Bulletin of mathematical biology 61 (1999), S. 1121-1149 
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    Notes: Abstract Mathematical models predict that a population which oscillates in the absence of time-dependent factors can develop multiple attracting final states in the advent of periodic forcing. A periodically-forced, stage-structured mathematical model predicted the transient and asymptotic behaviors of Tribolium (flour beetle) populations cultured in periodic habitats of fluctuating flour volume. Predictions included multiple (2-cycle) attractors, resonance and attenuation phenomena, and saddle influences. Stochasticity, combined with the deterministic effects of an unstable ’saddle cycle’ separating the two stable cycles, is used to explain the observed transients and final states of the experimental cultures. In experimental regimes containing multiple attractors, the presence of unstable invariant sets, as well as stochasticity and the nature, location, and size of basins of attraction, are all central to the interpretation of data.
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    The journal of Fourier analysis and applications 4 (1998), S. 283-297 
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    Keywords: Primary 42B25 ; 30D10 ; Secondary 26A16 ; 46B45 ; 47B10 ; 47B35 ; Mean oscillation ; Paley-Wiener space ; Besov spaces ; wavelets ; commutators ; Hankel operators ; Schatten-von Neumann ideals
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    Notes: Abstract The oscillatory behavior of functions with compactly supported Fourier transform is characterized in a quantified way using various function spaces. In particular, the results in this article show that the oscillations of a function at large scale are comparable to the oscillations of its samples on an appropriate discrete set of points. Several open questions about spaces of sequences are answered and applications in the study of commutator operators on the Paley-Wiener space are shown.
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    The journal of Fourier analysis and applications 4 (1998), S. 329-340 
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    Keywords: 42C15 ; 46E35 ; Wavelets ; function spaces ; fractals
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    Notes: Abstract Wavelets on self-similar fractals are introduced. It is shown that for certain totally disconnected fractals, function spaces may be characterized by means of the magnitude of the wavelet coefficients of the functions.
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    The journal of Fourier analysis and applications 5 (1999), S. 1-19 
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    Keywords: Primary: 42A20 ; Secondary 42C20 ; divergence of Fourier series ; rearrangement of Fourier series
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    Notes: Abstract There exists a continuous function whose Fourier sum, when taken in decreasing order of magnitude of the coefficients, diverges unboundedly almost everywhere.
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    The journal of Fourier analysis and applications 5 (1999), S. 73-85 
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    Keywords: 42C10 ; 46B15 ; 46E30 ; Wavelet ; unimodular wavelet ; unconditional basis
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    Notes: Abstract We present weak sufficient conditions for decay of a wavelet so that the wavelet basis is an unconditional basis in Lp(ℝ), 1 〈p 〈 ∞. We also prove that some unimodular wavelets yield unconditional bases in Lp(ℝ).
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    The journal of Fourier analysis and applications 5 (1999), S. 87-104 
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    Keywords: 42C15 ; 46E35 ; 42B30 ; refinable distribution ; Triebel-Lizorkin space ; Besov space ; multiresolution ; wavelet ; joint spectral radius
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    Notes: Abstract The aim of this article is to characterize compactly supported refinable distributions in Triebel-Lizorkin spaces and Besov spaces by projection operators on certain wavelet space and by some operators on a finitely dimensional space.
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    The journal of Fourier analysis and applications 4 (1998), S. 357-375 
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    Keywords: 42C15 ; 94A12 ; wavelets ; interpolation ; orthogonal expansions
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    Notes: Abstract Gibbs' phenomenon occurs for most orthogonal wavelet expansions. It is also shown to occur with many wavelet interpolating series, and a characterization is given. By introducing modifications in such a series, it can be avoided. However, some series that exhibit Gibbs' phenomenon for orthogonal series do not for the associated sampling series.
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    The journal of Fourier analysis and applications 5 (1999), S. 21-44 
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    Keywords: 42B99 ; 47B35 ; 15A54 ; 60G35 ; Positive extensions ; Toeplitz operators ; matrix functions on bitorus ; Wiener algebra ; band method ; entropy ; almost periodic functions ; ARMA processes
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    Notes: Abstract Let S be a band in Z2 bordered by two parallel lines that are of equal distance to the origin. Given a positive definite ℓ1 sequence of matrices {cj}j∈S we prove that there is a positive definite matrix function f in the Wiener algebra on the bitorus such that the Fourier coefficients $$\widehat{f(k)}$$ equal ck for k ∈ S. A parameterization is obtained for the set of all positive extensions f of {cj}j∈S. We also prove that among all matrix functions with these properties, there exists a distinguished one that maximizes the entropy. A formula is given for this distinguished matrix function. The results are interpreted in the context of spectral estimation of ARMA processes.
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    The journal of Fourier analysis and applications 5 (1999), S. 67-71 
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    Keywords: 42C15 ; Frame ; Frame sequence ; Fourier frame
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Given a real sequence {λn}n∈ℤ. Suppose that $$\left\{ {e^{i\lambda _n x} } \right\}_{n \in \mathbb{Z}}$$ is a frame for L2[−π, π] with bounds A, B. The problem is to find a positive constant L such that for any real sequence {μn}n∈ℤ with ¦μn −λn¦ ≤δ 〈L, $$\left\{ {e^{i\mu _n x} } \right\}_{n \in \mathbb{Z}}$$ is also a frame for L2[−π, π]. Balan [1] obtained $$L_R = \tfrac{1}{4} - \tfrac{1}{\pi }$$ arcsin $$\left( {\tfrac{1}{{\sqrt 2 }}\left( {1 - \sqrt {\tfrac{A}{B}} } \right)} \right)$$ . This value is a good stability bound of Fourier frames because it covers Kadec's 1/4-theorem $$\left( {L_R = \tfrac{1}{4}ifA = B} \right)$$ and is better than $$L_{DS} = \tfrac{1}{\pi }\ln \left( {1 + \sqrt {\tfrac{A}{B}} } \right)$$ (see Duffin and Schaefer [3]). In this paper, a sharper estimate is given.
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  • 97
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    The journal of Fourier analysis and applications 5 (1999), S. 105-125 
    ISSN: 1531-5851
    Keywords: 26B05 ; 42B10 ; 42C99 ; frame ; Gabor system ; Riesz basis ; stability ; wavelet
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract If the sequence of functions ϕj, k is a wavelet frame (Riesz basis) or Gabor frame (Riesz basis), we obtain its perturbation system ψj,k which is still a frame (Riesz basis) under very mild conditions. For example, we do not need to know that the support of ϕ or ψ $$(\hat \phi or\hat \psi )$$ is compact as in [14]. We also discuss the stability of irregular sampling problems. In order to arrive at some of our results, we set up a general multivariate version of Littlewood-Paley type inequality which was originally considered by Lemarié and Meyer [17], then by Chui and Shi [9], and Long [16].
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  • 98
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    The journal of Fourier analysis and applications 5 (1999), S. 185-192 
    ISSN: 1531-5851
    Keywords: 42C15 ; 30A10 ; 94A12 ; lower bound ; exponential frame ; sine-type-function ; irregular sampling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Lower frame bounds for sequences of exponentials are obtained in a special version of Avdonin's theorem on “1/4 in the mean” [1] and in a theorem of Duffin and Schaeffer [4].
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  • 99
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    The journal of Fourier analysis and applications 5 (1999), S. 303-308 
    ISSN: 1531-5851
    Keywords: 42B20 ; 42B30 ; Hardy spaces ; Calderon-Zygmund singular integral operator ; multipliers
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Calderón-Zygmund singular integral operators have been extensively studied for almost half a century. This paper provides a context for and proof of the following result: If a Calderón-Zygmund convolution singular integral operator is bounded on the Hardy space H1 (Rn), then the homogeneous of degree zero kernel is in the Hardy space H1(Sn−1) on the sphere.
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  • 100
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    The journal of Fourier analysis and applications 5 (1999), S. 285-302 
    ISSN: 1531-5851
    Keywords: 42C05 ; 22D25 ; 46L55 ; 47C05 ; spectral pair ; translations ; tilings ; Fourier basis ; operator extensions ; induced representations ; spectral resolution ; Hilbert space
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let Ω ⊂ℝd have finite positive Lebesgue measure, and let $$\mathcal{L}^2$$ (Ω) be the corresponding Hilbert space of $$\mathcal{L}^2$$ -functions on Ω. We shall consider the exponential functionse λ on Ω given bye λ(x)=e i2πλ·x . If these functions form an orthogonal basis for $$\mathcal{L}^2$$ (Ω), when λ ranges over some subset Λ in ℝ d , then we say that (Ω, Λ) is a spectral pair, and that Λ is a spectrum. We conjecture that (Ω, Λ) is a spectral pair if and only if the translates of some set Ω′ by the vectors of Λ tile ℝd. In the special case of Ω=Id, the d-dimensional unit cube, we prove this conjecture, with Ω′=Id, for d≤3, describing all the tilings by Id, and for all d when Λ is a discrete periodic set. In an appendix we generalize the notion of spectral pair to measures on a locally compact abelian group and its dual.
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