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1.
We introduce inhomogeneous, substrate dependent cell division in a time discrete, nonlinear matrix model of size-structured
population growth in the chemostat, first introduced by Gage et al. [8] and later analysed by Smith [13]. We show that mass conservation is verified, and conclude that our system admits one
non zero globally stable equilibrium, which we express explicitly. Then we run numerical simulations of the system, and compare
the predictions of the model to data related to phytoplankton growth, whose obtention we discuss. We end with the identification
of several parameters of the system.
Received: 9 February 2000 / Revised version: 10 October 2001 / Published online: 23 August 2002
RID="*"
ID="*" Present address: Department of Mathematics and Statistics, University of Victoria, B.C., Canada. e-mail: jarino@math.uvic.ca
Key words or phrases: Chemostat – Structured population models – Discrete model – Inhomogeneous division size 相似文献
2.
Haslach HW 《Journal of mathematical biology》2002,45(5):441-460
Intracranial saccular aneurysms have been clinically observed to emit a transient sound, a bruit, on each heartbeat. The
mechanism causing the bruits has been a matter of contention. A qualitative analysis of the nonlinear dynamical properties
of the Shah-Humphrey model for periodic pressure forcing of a thin-necked saccular aneurysm, using the Fung nonlinear constitutive
model for the aneurysm material, shows that a small blood pressure jump on each beat, whether the pressure is weakly aperiodic
or periodic, induces transients in the radial deformation response of the aneurysmal wall on each heartbeat. These transient
vibrations, which have a component with frequency near the natural frequency of the system but are not resonant phenomena
and which decay rapidly to a limit cycle during each distinct forcing pressure cycle, can generate the bruits.
Received: 21 November 2000 / Revised version: 9 August 2001 / Published online: 23 August 2002
Mathematics Subject Classification (2000): 92B99, 70K40, 70K05
Key words or phrases: Intracranial saccular aneurysm – Bruit – Spectrum – Nonlinear dynamics – Transients – Vortex shedding – Fung model 相似文献
3.
We start from a stochastic SIS model for the spread of epidemics among a population partitioned into M sites, each containing
N individuals; epidemic spread occurs through within-site (`local') contacts and global contacts. We analyse the limit behaviour
of the system as M and N increase to ∞. Two limit procedures are considered, according to the order in which M and N go to ∞; independently of the order, the limiting distribution of infected individuals across sites is a probability measure,
whose evolution in time is governed by the weak form of a PDE. Existence and uniqueness of the solutions to this problem is
shown. Finally, it is shown that the infected distribution converges, as time goes to infinity, to a Dirac measure at the
value x
*
, the equilibrium of a single-patch SIS model with contact rate equal to the sum of local and global contacts.
Received: 18 July 2001 / Revised version: 16 March 2002 / Published online: 26 September 2002
Mathematics Subject Classification (2000): 92D30, 60F99
Key words or phrases: SIS epidemic – Metapopulation – Markov population processes – Weak convergence of measures 相似文献
4.
We consider a partially coupled diffusive population model in which the state variables represent the densities of the immature
and mature population of a single species. The equation for the mature population can be considered on its own, and is a delay
differential equation with a delay-dependent coefficient. For the case when the immatures are immobile, we prove that travelling
wavefront solutions exist connecting the zero solution of the equation for the matures with the delay-dependent positive equilibrium
state. As a perturbation of this case we then consider the case of low immature diffusivity showing that the travelling front
solutions continue to persist. Our findings are contrasted with recent studies of the delayed Fisher equation. Travelling
fronts of the latter are known to lose monotonicity for sufficiently large delays. In contrast, travelling fronts of our equation
appear to remain monotone for all values of the delay.
Received: 1 November 2001 / Revised version: 10 May 2002 / Published online: 23 August 2002
Mathematics Subject Classification (2000): 35K57, 92D25
Key words or phrases: Age-structure – Time-delay – Travelling Fronts – Reaction-diffusion 相似文献
5.
Weinberger HF 《Journal of mathematical biology》2002,45(6):511-548
It is shown that the methods previously used by the author [Wei82] and by R. Lui [Lui89] to obtain asymptotic spreading results
and sometimes the existence of traveling waves for a discrete-time recursion with a translation invariant order preserving
operator can be extended to a recursion with a periodic order preserving operator. The operator can be taken to be the time-one
map of a continuous time reaction-diffusion model, or it can be a more general model of time evolution in population genetics
or population ecology in a periodic habitat. Methods of estimating the speeds of spreading in various directions will also
be presented.
Received: 12 July 2001 / Revised version: 19 July 2002 / Published online: 17 October 2002
Mathematics Subject Classification (2000): 92D40, 92D25, 35K55, 35K57, 35B40
Keywords or phrases: Periodic – Spreading speed – Traveling wave 相似文献
6.
The urine concentrating mechanism of mammals and birds depends on a counterflow configuration of thousands of nearly parallel
tubules in the medulla of the kidney. Along the course of a renal tubule, cell type may change abruptly, resulting in abrupt
changes in the physical characteristics and transmural transport properties of the tubule. A mathematical model that faithfully
represents these abrupt changes will have jump discontinuities in model parameters. Without proper treatment, such discontinuities
may cause unrealistic transmural fluxes and introduce suboptimal spatial convergence in the numerical solution to the model
equations. In this study, we show how to treat discontinuous parameters in the context of a previously developed numerical
method that is based on the semi-Lagrangian semi-implicit method and Newton's method. The numerical solutions have physically
plausible fluxes at the discontinuities and the solutions converge at second order, as is appropriate for the method.
Received: 13 November 2001 / Revised version: 28 June 2002 / Published online: 26 September 2002
This work was supported in part by the National Institutes of Health (National Institute of Diabetes and Digestive and Kidney
Diseases, grant DK-42091.)
Mathematics Subject Classification (2000): 65-04, 65M12, 65M25, 92-04, 92C35, 35-04, 35L45
Keywords or phrases: Mathematical models – Differential equations – Mathematical biology – Kidney – Renal medulla – Semi-Lagrangian semi-implicit 相似文献
7.
We examine a generalised SIR model for the infection dynamics of four competing disease strains. This model contains four
previously-studied models as special cases. The different strains interact indirectly by the mechanism of cross-immunity;
individuals in the host population may become immune to infection by a particular strain even if they have only been infected
with different but closely related strains. Several different models of cross-immunity are compared in the limit where the
death rate is much smaller than the rate of recovery from infection. In this limit an asymptotic analysis of the dynamics
of the models is possible, and we are able to compute the location and nature of the Takens–Bogdanov bifurcation associated
with the presence of oscillatory dynamics observed by previous authors.
Received: 5 December 2001 / Revised version: 5 May 2002 / Published online: 17 October 2002
Keywords or phrases: Infection – Pathogen – Epidemiology – Multiple strains – Cross-immunity – Oscillations – Dynamics – Bifurcations 相似文献
8.
We establish the existence of traveling front solutions and small amplitude traveling wave train solutions for a reaction-diffusion
system based on a predator-prey model with Holling type-II functional response. The traveling front solutions are equivalent
to heteroclinic orbits in R
4
and the small amplitude traveling wave train solutions are equivalent to small amplitude periodic orbits in R
4
. The methods used to prove the results are the shooting argument and the Hopf bifurcation theorem.
Received: 25 May 2001 / Revised version: 5 August 2002 / Published online: 19 November 2002
RID="*"
ID="*" Research was supported by the National Natural Science Foundations (NNSF) of China.
RID="*"
ID="*" Research was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. On leave
from the Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada.
Mathematics Subject Classification (2000): 34C35, 35K57
Key words or phrases: Traveling wave solution – Wazewski set – Shooting argument – Hopf bifurcation
Acknowledgements. We would like to thank the two referees for their careful reading and helpful comments. 相似文献
9.
We derive models for chemosensitive movement based on Cattaneo's law of heat propagation with finite speed. We apply the
model to pattern formation as observed in experiments with Dictyostelium discoideum, with Salmonella typhimurium and with Escherichia coli. For Salmonella typhimurium we make predictions on pattern formation which can be tested in experiments. We discuss the relations of the Cattaneo models
to classical models and we develop an effective numerical scheme.
Received: 8 October 2001 / Revised version: 2 August 2002 / Published online: 19 November 2002
Key words or phrases: Chemotaxis – Aggregation – Cattaneo model – Numerical schemes
Acknowledgements. We are very grateful for comments of S. Noelle concerning the numerical scheme. We thank K.P. Hadeler and C. Schmeiser for
helpful remarks. The research was supported by the Deutsche Forschungsgemeinschaft, research project ANumE and the Austrian Science Foundation, grant no. W008. 相似文献
10.
We consider a discrete time model of semelparous biennial population dynamics. Interactions between individuals are modelled
with the aid of an ``environmental' variable I. The impact on and the sensitivity to the environmental condition is age specific. The main result is that competitive exclusion
between the year classes is possible as is their coexistence. For moderate values of the basic reproduction ratio R
0
there is a strict dichotomy: depending on the other parameters we either find competitive exclusion or coexistence. We characterize
rather precisely the patterns of age specific impact and sensitivity that lead to either of these outcomes.
Received: 13 July 2001 / Revised version: 26 June 2002 / Published online: 19 November 2002
Key words or phrases: Competitive exclusion – Semelparous species – Periodical insects 相似文献
11.
Andreasen V 《Journal of mathematical biology》2003,46(6):504-536
The persistence of Influenza A in the human population relies on continual changes in the viral surface antigens allowing
the virus to reinfect the same hosts every few years. The epidemiology of such a drifting virus is modeled by a discrete season-to-season
map. During the epidemic season only one strain is present and its transmission dynamics follows a standard epidemic model.
After the season, cross-immunity to next year's virus is determined from the proportion of hosts that were infected during
the season. A partial analysis of this map shows the existence of oscillations where epidemics occur at regular or irregular
intervals.
Received: 16 February 2001 / Revised version: 11 June 2002 /
Published online: 28 February 2003
Key words or phrases: Infectious disease – Influenza drift – Cross-immunity – Seasonal epidemics – Iterated map 相似文献
12.
A mathematical model is developed with a highly controlled birth and death process for precursor cells. This model is both
biologically- and statistically-based. The controlled growth and differentiation (CGD) model limits the number of replications
allowed in the development of a tissue or organ and thus, more closely reflects the presence of a true stem cell population.
Leroux et al. (1996) presented a biologically-based dose-response model for developmental toxicology that was derived from a partial differential
equation for the generating function. This formulation limits further expansion into more realistic models of mammalian development.
The same formulae for the probability of a defect (a system of ordinary differential equations) can be derived through the
Kolmogorov forward equations due to the nature of this Markov process. This modified approach is easily amenable to the expansion
of more complicated models of the developmental process such as the one presented here. Comparisons between the Leroux et al. (1996) model and the controlled growth and differentiation (CGD) model as developed in this paper are also discussed.
Received: 8 June 2001 / Revised version: 15 June 2002 / Published online: 26 September 2002
Keywords or phrases: Teratology – Multistate process – Cellular kinetics – Numerical simulation 相似文献
13.
Exact formulas for the mean and variance of the proportion of different types in a fixed generation of a multi-type Galton-Watson
process are derived. The formulas are given in terms of iterates of the probability generating function of the offspring distribution.
It is also shown that the sequence of types backwards from a randomly sampled particle in a fixed generation is a non-homogeneous
Markov chain where the transition probabilities can be given explicitly, again in terms of probability generating functions.
Two biological applications are considered: mutations in mitochondrial DNA and the polymerase chain reaction.
Received: 10 June 2001 / Revised version: 21 November 2001 / Published online: 23 August 2002
Mathematics Subject Classification (2000): Primary 60J80, Secondary 92D10, 92D25
Key words or phrases: Multi-type Galton-Watson process – sampling formula – PCR – mitochondrial DNA 相似文献
14.
Waxman D 《Journal of mathematical biology》2003,46(3):225-240
Two results are presented for problems involving alleles with a continuous range of effects. The first result is a simple
yet highly accurate numerical method that determines the equilibrium distribution of allelic effects, moments of this distribution,
and the mutational load. The numerical method is explicitly applied to the mutation-selection balance problem of stabilising
selection. The second result is an exact solution for the distribution of allelic effects under weak stabilising selection
for a particular distribution of mutant effects. The exact solution is shown to yield a distribution of allelic effects that,
depending on the mutation rate, interpolates between the ``House of Cards' approximation and the Gaussian approximation.
The exact solution is also used to test the accuracy of the numerical method.
Received: 7 November 2001 / Revised version: 5 September 2002 / Published online: 18 December 2002
Key words or phrases: Continuum of alleles – Numerical solution – Exact solution – Mutation selection balance – Stabilising selection 相似文献
15.
Early development and quorum sensing in bacterial biofilms 总被引:3,自引:0,他引:3
Ward JP King JR Koerber AJ Croft JM Sockett RE Williams P 《Journal of mathematical biology》2003,47(1):23-55
We develop mathematical models to examine the formation, growth and quorum sensing activity of bacterial biofilms. The growth
aspects of the model are based on the assumption of a continuum of bacterial cells whose growth generates movement, within
the developing biofilm, described by a velocity field. A model proposed in Ward et al. (2001) to describe quorum sensing, a process by which bacteria monitor their own population density by the use of quorum
sensing molecules (QSMs), is coupled with the growth model. The resulting system of nonlinear partial differential equations
is solved numerically, revealing results which are qualitatively consistent with experimental ones. Analytical solutions derived
by assuming uniform initial conditions demonstrate that, for large time, a biofilm grows algebraically with time; criteria
for linear growth of the biofilm biomass, consistent with experimental data, are established. The analysis reveals, for a
biologically realistic limit, the existence of a bifurcation between non-active and active quorum sensing in the biofilm.
The model also predicts that travelling waves of quorum sensing behaviour can occur within a certain time frame; while the
travelling wave analysis reveals a range of possible travelling wave speeds, numerical solutions suggest that the minimum
wave speed, determined by linearisation, is realised for a wide class of initial conditions.
Received: 10 February 2002 / Revised version: 29 October 2002 /
Published online: 19 March 2003
Key words or phrases: Bacterial biofilm – Quorum sensing – Mathematical modelling – Numerical solution – Asymptotic analysis – Travelling wave
analysis 相似文献
16.
The purpose of this paper is to present a mathematical model for the tumor vascularization theory of tumor growth proposed
by Judah Folkman in the early 1970s and subsequently established experimentally by him and his coworkers [Ausprunk, D. H.
and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during
tumor angiogenesis, Microvasc Res., 14, 53–65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract
derived from vitreous Am. J. Opthalmol., 84, 323–328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58–64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: an experimental
model using the rabbit cornea, J. Nat. Cancer Inst., 52, 413–419]. In the simplest version of this model, an avascular tumor secretes a tumor growth factor (TGF) which is transported
across an extracellular matrix (ECM) to a neighboring vasculature where it stimulates endothelial cells to produce a protease
that acts as a catalyst to degrade the fibronectin of the capillary wall and the ECM. The endothelial cells then move up the
TGF gradient back to the tumor, proliferating and forming a new capillary network. In the model presented here, we include
two mechanisms for the action of angiostatin. In the first mechanism, substantiated experimentally, the angiostatin acts as
a protease inhibitor. A second mechanism for the production of protease inhibitor from angiostatin by endothelial cells is
proposed to be of Michaelis-Menten type. Mathematically, this mechanism includes the former as a subcase.
Our model is different from other attempts to model the process of tumor angiogenesis in that it focuses (1) on the biochemistry
of the process at the level of the cell; (2) the movement of the cells is based on the theory of reinforced random walks;
(3) standard transport equations for the diffusion of molecular species in porous media.
One consequence of our numerical simulations is that we obtain very good computational agreement with the time of the onset
of vascularization and the rate of capillary tip growth observed in rabbit cornea experiments [Ausprunk, D. H. and J. Folkman
(1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis,
Microvasc Res., 14, 73–65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract
derived from vitreous Am. J. Opthalmol., 84, 323–328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58–64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: An experimental
model using the rabbit cornea, J. Nat. Cancer Inst., 52, 413–419]. Furthermore, our numerical experiments agree with the observation that the tip of a growing capillary accelerates
as it approaches the tumor [Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58–64].
An erratum to this article is available at . 相似文献
17.
The nature of the association between two species may vary depending on population abundances, age or size of individuals,
or environmental conditions. Interactions may switch between beneficial and detrimental depending on the net balance of costs
and benefits involved for each species. We study the repercussion of the ecological setting on the outcomes of conditional
or variable interactions by means of a model that incorporates density-dependent interaction coefficients; that is, interaction α-functions. These characterize the responsiveness and sensitivity of the association to changes in partner's abundance, and can take
positive and negative values. Variable outcomes – and transitions between them – are categorized as homeo- or allo-environmental, that is, occurring under the same ecological setting, or not, respectively. Bifurcation analyses show that these dynamics
are moulded by ecological factors that are: intrinsic to the nature of the association (concerning the sensitivity of the interaction), and extrinsic to the association itself (the quality of the environment referred to each species alone). The influence of these factors
may be conflicting; consequently, the dynamics involve catastrophic events. In a facultative variable association, stable coexistence is expected when environmental conditions are adverse; otherwise, the exclusion of one species is the likely outcome. Remarkable situations as the switching of victim-exploiter roles illustrate the theoretical perspective.
Received: 15 December 2001 / Revised version: 18 November 2002 /
Published online: 28 February 2003
Key words or phrases: Variable population interactions – Conditional interactions – Costs and benefits – Density dependent interaction coefficient
– Hysteresis – Symbiosis – Mutualism – Parasitism 相似文献
18.
Magarshak et al. represented an RNA molecule as a complex vector and an RNA secondary structure Γ as a complex matrix S
Γ in such a way that the molecule represented by was compatible with the secondary structure Γ if and only if . They only considered Watson-Crick base pairs and their representation cannot be extended to allow for G⋅U pairs. In this paper we study a generalization of Magarshak's representation that allows for these pairs, and in particular
we provide a family of algebraic structures where that generalization can be carried out. We also show that this representation
can be used to compare secondary structures, through transfer matrices which transform the representation of one secondary
structure into the representation of the other.
Received: 10 December 2001 / Revised version: 7 May 2002 /
Published online: 28 February 2003
Key words or phrases: RNA secondary structure – Algebra – Finite field 相似文献
19.
We deduce and prove a general formula to approximate the change in frequency of a mutant allele under weak selection, when
this allele is introduced in small frequency into a population which was previously at a fixation state. We apply the formula
to autosomal genes in partial selfing models and to autosomal as well as sex-linked genes in partial sib mating models. It
is shown that the fate of a rare mutant allele depends not only on the selection parameters, the inbreeding coefficient and
the reproductive values of the sexes in sex-differentiated populations, but also on coefficients of relatedness between mates.
This is interpreted as a kin selection effect caused by inbreeding per se.
Received: 3 December 2001 / Revised version: 10 April 2002 / Published online: 19 November 2002
Research supported in part by NSERC of Canada and FCAR of Québec.
Mathematics Subject Classification (2000): Primary 60J80, Secondary 92D10, 92D25
Keywords or phrases: Adaptive topography – Partial selfing – Partial sib mating – Kin selection 相似文献
20.
Progression age enhanced backward bifurcation in an epidemic model with super-infection 总被引:10,自引:0,他引:10
We consider a model for a disease with a progressing and a quiescent exposed class and variable susceptibility to super-infection.
The model exhibits backward bifurcations under certain conditions, which allow for both stable and unstable endemic states
when the basic reproduction number is smaller than one.
Received: 11 October 2001 / Revised version: 17 September 2002 /
Published online: 17 January 2003
Present address: Department of Biological Statistics and Computational Biology, 434 Warren Hall, Cornell University, Ithaca, NY 14853-7801
This author was visiting Arizona State University when most of the research was done.
Research partially supported by NSF grant DMS-0137687.
This author's research was partially supported by NSF grant DMS-9706787.
Key words or phrases: Backward bifurcation – Multiple endemic equilibria – Alternating stability – Break-point density – Super-infection – Dose-dependent
latent period – Progressive and quiescent latent stages – Progression age structure – Threshold type disease activation –
Operator semigroups – Hille-Yosida operators – Dynamical systems – Persistence – Global compact attractor 相似文献