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1.
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
2.
J. M. Mahaffy 《Journal of mathematical biology》1985,22(2):137-144
A technique is discussed for locating the Hopf bifurcation of an n-dimensional system of delay differential equations which arises from a model for control of protein biosynthesis. Certain parameter values are shown to allow a Hopf bifurcation to periodic orbits. At the Hopf bifurcation the periodic orbits are shown to be stable either analytically or numerically depending on the parameter values.On leave from North Carolina State University.Supported in part by N.S.F. Grant # MCS 81-02828 相似文献
3.
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
4.
Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example 总被引:1,自引:0,他引:1
Odo Diekmann Mats Gyllenberg J. A. J. Metz Shinji Nakaoka Andre M. de Roos 《Journal of mathematical biology》2010,61(2):277-318
We consider the interaction between a general size-structured consumer population and an unstructured resource. We show that
stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations
(a renewal equation for the consumer population birth rate coupled to a delay differential equation for the resource concentration).
As many results for such systems are available (Diekmann et al. in SIAM J Math Anal 39:1023–1069, 2007), we can draw rigorous
conclusions concerning dynamical behaviour from an analysis of a characteristic equation. We derive the characteristic equation
for a fairly general class of population models, including those based on the Kooijman–Metz Daphnia model (Kooijman and Metz in Ecotox Env Saf 8:254–274, 1984; de Roos et al. in J Math Biol 28:609–643, 1990) and a model introduced
by Gurney–Nisbet (Theor Popul Biol 28:150–180, 1985) and Jones et al. (J Math Anal Appl 135:354–368, 1988), and next obtain
various ecological insights by analytical or numerical studies of special cases. 相似文献
5.
To understand joint effects of logistic growth in target cells and intracellular delay on viral dynamics in vivo, we carry
out two-parameter bifurcation analysis of an in-host model that describes infections of many viruses including HIV-I, HBV
and HTLV-I. The bifurcation parameters are the mitosis rate r of the target cells and an intracellular delay τ in the incidence of viral infection. We describe the stability region of the chronic-infection equilibrium E* in the two-dimensional (r, τ) parameter space, as well as the global Hopf bifurcation curves as each of τ and r varies. Our analysis shows that, while both τ and r can destabilize E* and cause Hopf bifurcations, they do behave differently. The intracellular delay τ can cause Hopf bifurcations only when r is positive and sufficiently large, while r can cause Hopf bifurcations even when τ = 0. Intracellular delay τ can cause stability switches in E* while r does not. 相似文献
6.
研究了一类由两个神经元构成的时滞神经网络模型的稳定性和局部Hopf分支,并结合一般泛函微分方程的全局Hopf分支定理,利用度理论研究了全局Hopf分支的存在性. 相似文献
7.
Lindström T 《Journal of mathematical biology》2002,45(5):396-418
In this paper we derive and analyze a discrete version of Rosenzweig's (Am. Nat. 1973) food-chain model. We provide substantial
analytical and numerical evidence for the general dynamical patterns of food chains predicted by De Feo and Rinaldi (Am. Nat.
1997) remaining largely unaffected by this discretization. Our theoretical analysis gives rise to a classification of the
parameter space into various regions describing distinct governing dynamical behaviors. Predator abundance has a local optimum
at the edge of chaos.
Received: 13 August 1999 / Revised version: 12 March 2002 / Published online: 17 October 2002
Mathematics Subject Classification (1991): 92D40
Keywords or phrases: Discrete food-chain – Discrete Hopf (Neimark-Sacker) bifurcation – Pulsewise birth processes – Mean yield maximization –
Nicholson-Bailey model 相似文献
8.
S. A. van Gils S. G. Janssens Yu. A. Kuznetsov S. Visser 《Journal of mathematical biology》2013,66(4-5):837-887
Neural field models with transmission delays may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results. 相似文献
9.
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
10.
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
11.
The well-known neural mass model described by Lopes da Silva et al. (1976) and Zetterberg et al. (1978) is fitted to actual
EEG data. This is achieved by reformulating the original set of integral equations as a continuous-discrete state space model.
The local linearization approach is then used to discretize the state equation and to construct a nonlinear Kalman filter.
On this basis, a maximum likelihood procedure is used for estimating the model parameters for several EEG recordings. The
analysis of the noise-free differential equations of the estimated models suggests that there are two different types of alpha
rhythms: those with a point attractor and others with a limit cycle attractor. These attractors are also found by means of
a nonlinear time series analysis of the EEG recordings. We conclude that the Hopf bifurcation described by Zetterberg et al.
(1978) is present in actual brain dynamics.
Received: 11 August 1997 / Accepted in revised form: 20 April 1999 相似文献
12.
Malaria fever is highly periodic and is associated with the parasite replication cycles in red blood cells. The existence
of periodicity in malaria infection demonstrates that parasite replication in different red blood cells is synchronized. In
this article, rigorous mathematical analysis of an age-structured human malaria model of infected red blood cells (Rouzine
and McKenzie, Proc Natl Acad Sci USA 100:3473–3478, 2003) is provided and the synchronization of Plasmodium falciparum erythrocytic stages is investigated. By using the replication rate as the bifurcation parameter, the existence of Hopf bifurcation
in the age-structured malaria infection model is obtained. Numerical simulations indicate that synchronization with regular
periodic oscillations (of period 48 h) occurs when the replication rate increases. Therefore, Kwiatkowski and Nowak’s observation
(Proc Natl Acad Sci USA 88:5111–5113, 1991) that synchronization could be generated at modest replication rates is confirmed. 相似文献
13.
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
14.
In this paper, we consider an interaction of prey and predator species where prey species have the ability of group defence.
Thresholds, equilibria and stabilities are determined for the system of ordinary differential equations. Taking carrying capacity
as a bifurcation parameter, it is shown that a Hopf bifurcation can occur implying that if the carrying capacity is made sufficiently
large by enrichment of the environment, the model predicts the eventual extinction of the predator providing strong support
for the so-called ‘paradox of enrichment’. 相似文献
15.
Aminoglycoside–arginine conjugates (AAC and APAC) are multi-target inhibitors of human immunodeficiency virus type-1 (HIV-1).
Here, we predict new conjugates of neomycin with two arginine peptide chains binding at specific sites on neomycin [poly-arginine-neomycin-poly-arginine
(PA-Neo-PA)]. The rationale for the design of such compounds is to separate two short arginine peptides with neomycin, which
may extend the binding region of the CXC chemokine receptor type 4 (CXCR4). We used homology models of CXCR4 and unliganded
envelope glycoprotein 120 (HIV-1IIIB gp120) and docked PA-Neo-PAs and APACs to these using a multistep docking procedure. The results indicate that PA-Neo-PAs
spread over two negatively charged patches of CXCR4. PA-Neo-PA–CXCR4 complexes are energetically more favorable than AACs/APAC–CXCR4
complexes. Notably, our CXCR4 model and docking procedure can be applied to predict new compounds that are either inhibitors
of gp120–CXCR4 binding without affecting stromal cell-derived factor 1α (SDF-1α) chemotaxis activity, or inhibitors of SDF-1α–CXCR4
binding resulting in an anti-metastasis effect. We also predict that PA-Neo-PAs and APACs can interfere with CD4–gp120 binding
in unliganded conformation.
Figure The r5-Neo-r5-CXCR4 complex. CXCR4 is shown in CPK representation. The negatively charged residues are shown in red and positively charged residues in blue. The r5-Neo-r5 is shown in stick representation, neomycin core is colored yellow and arginine moieties are colored magenta. Two negatively charged patches separated by neutral and positively charged residues are visible. 相似文献
16.
《Journal of biological dynamics》2013,7(4):387-409
In this paper, we rigorously analyse an ordinary differential equation system that models fighting the HIV-1 virus with a genetically modified virus. We show that when the basic reproduction ratio ?0<1, then the infection-free equilibrium E 0 is globally asymptotically stable; when ?0>1, E 0 loses its stability and there is the single-infection equilibrium E s. If ?0∈(1, 1+δ) where δ is a positive constant explicitly depending on system parameters, then the single-infection equilibrium E s that is globally asymptotically stable, while when ?0>1+δ, E s becomes unstable and the double-infection equilibrium E d comes into existence. When ?0 is slightly larger than 1+δ, E d is stable and it loses its stability via Hopf bifurcation when ?0 is further increased in some ways. Through a numerical example and by applying a normal form theory, we demonstrate how to determine the bifurcation direction and stability, as well as the estimates of the amplitudes and the periods of the bifurcated periodic solutions. We also perform numerical simulations which agree with the theoretical results. The approaches we use here are a combination of analysis of characteristic equations, fluctuation lemma, Lyapunov function and normal form theory. 相似文献
17.
Vielle B 《Journal of mathematical biology》2000,41(6):546-558
The aim of this paper is to carry out a stability analysis for periodic breathing in humans that incorporates the dynamic
characteristics of ventilation control. A simple CO2 model that takes into account the main elements of the respiratory system, i.e. the lungs and the ventilatory controller
with its dynamic properties, is presented. This model results in a three-dimensional non-linear delay differential system
for which there exists a unique equilibrium point. Our stability analysis of this equilibrium point leads to the definition
of a new explicit stability criterion and to the demonstration of the existence of a Hopf bifurcation. Numerical simulations
illustrate the influence of physiological parameters on the stability of ventilation, and particularly the major role of the
dynamic characteristics of the respiratory controller.
Received: 2 February 1999 / Revised version: 18 June 1999 / Published online: 23 October 2000 相似文献
18.
Understanding bursting oscillations as periodic slow passages through bifurcation and limit points 总被引:3,自引:0,他引:3
We consider a biochemical system consisting of two allosteric enzyme reactions coupled in series. The system has been modeled by Decroly and Goldbeter (J. Theor. Biol. 124, 219 (1987)) and is described by three coupled, first-order, nonlinear, differential equations. Bursting oscillations correspond to a succession of alternating active and silent phases. The active phase is characterized by rapid oscillations while the silent phase is a period of quiescence. We propose an asymptotic analysis of the differential equations which is based on the limit of large allosteric constants. This analysis allows us to construct a time-periodic bursting solution. This solution is jumping periodically between a slowly varying steady state and a slowly varying oscillatory state. Each jump follows a slow passage through a bifurcation or limit point which we analyze in detail. Of particular interest is the slow passage through a supercritical Hopf bifurcation. The transition is from an oscillatory solution to a steady state solution. We show that the transition is delayed considerably and characterize this delay by estimating the amplitude of the oscillations at the Hopf bifurcation point. 相似文献
19.
This paper gives an explanation for the experimentally observed onset latencies of the inhibitory responses that vary from
a few milliseconds to hundreds of milliseconds in systems where the conduction delays are only several milliseconds in the
feedback pathways. To do this we use a simple mathematical model. The model consists of two delay differential equations (DDE)
where the nonlinear relation between the postsynaptic potential and the firing frequency of the neuron population arises from
the stoichiometry of the transmitter-receptor kinetics. The parameters of the model refer to the hippocampal feedback system,
and the modeling results are compared with corresponding experiments.
Received: 31 May 2002 / Accepted: 5 February 2003 /
Published online: 20 May 2003
Correspondence to: C. Hauptmann (e-mail: chauptma@cnd.mcgill.ca)
Acknowledgements. We thank Prof. Krnjevic and Prof. Glavinovic for helpful and extensive discussions about this problem. This work was supported
by MITACS (Canada), the Natural Sciences and Engineering Research Council (NSERC grant OGP-0036920, Canada), the Alexander
von Humboldt Stiftung, Le Fonds pour la Formation de Chercheurs et l'Aide à la Recherche (FCAR grant 98ER1057, Québec), and
the Leverhulme Trust (U.K.). 相似文献