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Discrete & Continuous Dynamical Systems - B

2004 , Volume 4 , Issue 4

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A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type
C.B. Muratov
2004, 4(4): 867-892 doi: 10.3934/dcdsb.2004.4.867 +[Abstract](73) +[PDF](273.3KB)
We identified a variational structure associated with traveling waves for systems of reaction-diffusion equations of gradient type with equal diffusion coefficients defined inside an infinite cylinder, with either Neumann or Dirichlet boundary conditions. We show that the traveling wave solutions that decay sufficiently rapidly exponentially at one end of the cylinder are critical points of certain functionals. We obtain a global upper bound on the speed of these solutions. We also show that for a wide class of solutions of the initial value problem an appropriately defined instantaneous propagation speed approaches a limit at long times. Furthermore, under certain assumptions on the shape of the solution, there exists a reference frame in which the solution of the initial value problem converges to the traveling wave solution with this speed at least on a sequence of times. In addition, for a class of solutions we establish bounds on the shape of the solution in the reference frame associated with its leading edge and determine accessible limiting traveling wave solutions. For this class of solutions we find the upper and lower bounds for the speed of the leading edge.
A reaction-diffusion system modeling direct and indirect transmission of diseases
W. E. Fitzgibbon , M. Langlais and  J.J. Morgan
2004, 4(4): 893-910 doi: 10.3934/dcdsb.2004.4.893 +[Abstract](109) +[PDF](249.6KB)
We study the global existence and approximation of the solutions to a reaction diffusion system coupled with an ordinary differential equation modeling direct transmission between individuals and indirect transmission via a contaminated environment of an epidemic disease.
A criterion for non-persistence of travelling breathers for perturbations of the Ablowitz--Ladik lattice
A. Berger , R.S. MacKay and  Vassilis Rothos
2004, 4(4): 911-920 doi: 10.3934/dcdsb.2004.4.911 +[Abstract](25) +[PDF](204.1KB)
The Ablowitz-Ladik lattice has a two-parameter family of travelling breathers. We derive a necessary condition for their persistence under perturbations of the system. From this we deduce non-persistence for a variety of examples of perturbations. In particular, we show that travelling breathers do not persist under many reversible perturbations unless an additional symmetry is preserved, and we address the case of Hamiltonian perturbations.
The algorithmic information content for randomly perturbed systems
C. Bonanno
2004, 4(4): 921-934 doi: 10.3934/dcdsb.2004.4.921 +[Abstract](42) +[PDF](219.3KB)
In this paper we prove estimates on the behaviour of the Kolmogorov-Sinai entropy relative to a partition for randomly perturbed dynamical systems. Our estimates use the entropy for the unperturbed system and are obtained using the notion of Algorithmic Information Content. The main result is an extension of known results to study time series obtained by the observation of real systems.
Dynamical systems and computable information
Vieri Benci , C. Bonanno , Stefano Galatolo , G. Menconi and  M. Virgilio
2004, 4(4): 935-960 doi: 10.3934/dcdsb.2004.4.935 +[Abstract](46) +[PDF](745.4KB)
We present some new results that relate information to chaotic dynamics. In our approach the quantity of information is measured by the Algorithmic Information Content (Kolmogorov complexity) or by a sort of computable version of it (Computable Information Content) in which the information is measured by using a suitable universal data compression algorithm. We apply these notions to the study of dynamical systems by considering the asymptotic behavior of the quantity of information necessary to describe their orbits. When a system is ergodic, this method provides an indicator that equals the Kolmogorov-Sinai entropy almost everywhere. Moreover, if the entropy is null, our method gives new indicators that measure the unpredictability of the system and allows various kind of weak chaos to be classified. Actually, this is the main motivation of this work. The behavior of a 0-entropy dynamical system is far to be completely predictable except that in particular cases. In fact there are 0-entropy systems that exhibit a sort of weak chaos, where the information necessary to describe the orbit behavior increases with time more than logarithmically (periodic case) even if less than linearly (positive entropy case). Also, we believe that the above method is useful to classify 0-entropy time series. To support this point of view, we show some theoretical and experimental results in specific cases.
On the $L^2$-moment closure of transport equations: The Cattaneo approximation
T. Hillen
2004, 4(4): 961-982 doi: 10.3934/dcdsb.2004.4.961 +[Abstract](38) +[PDF](979.1KB)
We consider the moment-closure approach to transport equations which arise in Mathematical Biology. We show that the negative $L^2$-norm is an entropy in the sense of thermodynamics, and it satisfies an $H$-theorem. With an $L^2$-norm minimization procedure we formally close the moment hierarchy for the first two moments. The closure leads to semilinear Cattaneo systems, which are closely related to damped wave equations. In the linear case we derive estimates for the accuracy of this moment approximation. The method is used to study reaction-transport models and transport models for chemosensitive movement. With this method also order one perturbations of the turning kernel can be treated - in extension of an earlier theory on the parabolic limit of transport equations (Hillen and Othmer 2000). Moreover, this closure procedure allows us to derive appropriate boundary conditions for the Cattaneo approximation. Finally, we illustrate that the Cattaneo system is the gradient flow of a weighted Dirichlet integral and we show simulations.
The moment closure for higher order moments and for general transport models will be studied in a second paper.
Hölder continuous solutions of an obstacle thermistor problem
Walter Allegretto , Yanping Lin and  Shuqing Ma
2004, 4(4): 983-997 doi: 10.3934/dcdsb.2004.4.983 +[Abstract](47) +[PDF](227.2KB)
In this paper we consider a thermistor problem with a current source, i.e., a nonlocal boundary condition. The electric potential is unknown on part of the boundary, but the current through it is known. We apply a decomposition technique and transform the equation satisfied by the potential into two elliptic problems with usual boundary conditions. The unique solvability of the initial boundary value problem is achieved.
Modelling the effect of imperfect vaccines on disease epidemiology
S.M. Moghadas
2004, 4(4): 999-1012 doi: 10.3934/dcdsb.2004.4.999 +[Abstract](41) +[PDF](343.7KB)
We develop a mathematical model to monitor the effect of imperfect vaccines on the transmission dynamics of infectious diseases. It is assumed that the vaccine efficacy is not $100\%$ and may wane with time. The model will be analyzed using a new technique based on some results related to the Poincaré index of a piecewise smooth Jordan curve defined as the boundary of a positively invariant region for the model. Using global analysis of the model, it is shown that reducing the basic reproductive number ($\mathcal{R}_0$) to values less than one no longer guarantees disease eradication. This analysis is extended to determine the threshold level of vaccination coverage that guarantees disease eradication.
Analytical and numerical solutions for a class of optimization problems in elasticity
G. Machado and  L. Trabucho
2004, 4(4): 1013-1032 doi: 10.3934/dcdsb.2004.4.1013 +[Abstract](32) +[PDF](1982.6KB)
The subject of topology optimization methods in structural design has increased rapidly since the publication of [5], where some ideas from homogenization theory were put into practice. Since then, several engineering applications have been developed successfully. However, in the literature, there is a lack of analytical solutions, even for simple cases, which might help in the validation of the numerical results. In this work, we develop analytical solutions for simple minimum compliance problems, in the framework of elasticity theory. We compare these analytical solutions with numerical results obtained via an algorithm proposed in [4].
Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model
Theodore Kolokolnikov and  Michael J. Ward
2004, 4(4): 1033-1064 doi: 10.3934/dcdsb.2004.4.1033 +[Abstract](34) +[PDF](499.7KB)
In the limit of small activator diffusivity $\varepsilon$, and in a bounded domain in $\mathbb{R}^{N}$ with $N=1$ or $N=2$ under homogeneous Neumann boundary conditions, the bifurcation behavior of an equilibrium one-spike solution to the Gierer-Meinhardt activator-inhibitor system is analyzed for different ranges of the inhibitor diffusivity $D$. When $D=\infty$, it is well-known that a one-spike solution for the resulting shadow Gierer-Meinhardt system is unstable, and the locations of unstable equilibria coincide with the points in the domain that are furthest away from the boundary. For a unit disk domain it is shown that as $D$ is decreased below a critical bifurcation value $D_{c}$, with $D_{c}=O(\varepsilon^2 e^{2/\varepsilon})$, the spike at the origin becomes stable, and unstable spike solutions bifurcate from the origin. The locations of these bifurcating spikes tend to the boundary of the domain as $D$ is decreased further. Similar bifurcation behavior is studied in a one-parameter family of dumbbell-shaped domains. This motivates a further analysis of the existence of certain near-boundary spikes. Their location and stability is given in terms of the modified Green's function. Finally, for the dumbbell-shaped domain, an intricate bifurcation structure is observed numerically as $D$ is decreased below some $O(1)$ critical value.
A mathematical model for the propagation of a hantavirus in structured populations
Cédric Wolf
2004, 4(4): 1065-1089 doi: 10.3934/dcdsb.2004.4.1065 +[Abstract](41) +[PDF](284.6KB)
We analysed a mathematical model for the propagation of Puumala hantavirus (PUU), within a population of bank voles (Clethrionomys glareolus). This model includes the chronological age of individuals and the time elapsed since an individual is infected. The hantavirus propagates via direct transmission (contacts between individuals) and indirect transmission (through the environment). Demographic parameters are population density dependent and the maturation rate is adult density dependent. This leads to a weakly coupled system of hyperbolic equations featuring nonlocal nonlinearities. We give a global existence and uniqueness result.
Asymptotic theory for disc-like crystal growth (I) --- Basic state solutions
Jian-Jun Xu and  Junichiro Shimizu
2004, 4(4): 1091-1116 doi: 10.3934/dcdsb.2004.4.1091 +[Abstract](30) +[PDF](470.3KB)
The present paper is concerned with disc-like crystal growth from a pure undercooled melt. We obtained the uniformly valid asymptotic solution for the basic state in the limit of the aspect ratio $\delta = b/L$ << $1$ , in terms of the matched asymptotic expansion method. The solution obtained under the present model shows that the growth of the top/bottom interface of the disc is very slow, dominated by the kinetic effect, while the growth of its side-interface is much faster and is dominated by heat diffusion mechanism, with negligible effects by the surface tension and the kinetic attachment. Furthermore, we performed the linear stability analysis for the basic state at the early stage of growth. It is found that the system allows two discrete sets of unstable eigen-modes over the side-interface: the axi-symmetric eigen-modes and non-axi-symmetric eigen-modes. The onset of the instability is when the thickness of the disc reaches at a critical value $b_c$. The axi-symmetric eigen-modes are found can be further distinguished as the A-modes, anti-symmetric and the S-modes, symmetric about the central plane. The most dangerous axi-symmetric modes is a well isolated base mode $A_0$ with the index ($n = 0$). This mode is responsible for the formation of anti-symmetric pattern about the central plane of the disc, which is observed at the early stage of growth. We have compared the theoretical predictions with the available experimental data of ice-disc growth obtained by Shimada and Furukawa. It is found that both are in good agreements.
Fisher waves in an epidemic model
Xiao-Qiang Zhao and  Wendi Wang
2004, 4(4): 1117-1128 doi: 10.3934/dcdsb.2004.4.1117 +[Abstract](87) +[PDF](197.7KB)
The existence of Fisher type monotone traveling waves and the minimal wave speed are established for a reaction-diffusion system modeling man-environment-man epidemics via the method of upper and lower solutions as applied to a reduced second order ordinary differential equation with infinite time delay.
Diffusion approximation for the one dimensional Boltzmann-Poisson system
N. Ben Abdallah and  M. Lazhar Tayeb
2004, 4(4): 1129-1142 doi: 10.3934/dcdsb.2004.4.1129 +[Abstract](129) +[PDF](237.2KB)
The diffusion limit of the initial-boundary value problem for the Boltzmann-Poisson system is studied in one dimension. By carefully analyzing entropy production terms due to the boundary, $L^p$ estimates are established for the solution of the scaled Boltzmann equation (coupled to Poisson) with well prepared initial and boundary conditions. A hybrid Hilbert expansion taking advantage of the regularity of the limiting system allows to prove the convergence of the solution towards the solution of the Drift-Diffusion-Poisson system and to exhibit a convergence rate.
Convergence analysis of the numerical method for the primitive equations formulated in mean vorticity on a Cartesian grid
Cheng Wang
2004, 4(4): 1143-1172 doi: 10.3934/dcdsb.2004.4.1143 +[Abstract](39) +[PDF](298.6KB)
A second order numerical method for the primitive equations (PEs) of large-scale oceanic flow formulated in mean vorticity is proposed and analyzed, and the full convergence in $L^2$ is established. In the reformulation of the PEs, the prognostic equation for the horizontal velocity is replaced by evolutionary equations for the mean vorticity field and the vertical derivative of the horizontal velocity. The total velocity field (both horizontal and vertical) is statically determined by differential equations at each fixed horizontal point. The standard centered difference approximation is applied to the prognostic equations and the determination of numerical values for the total velocity field is implemented by FFT-based solvers. Stability of such solvers are established and the convergence analysis for the whole scheme is provided in detail.
Modelling the dynamics of endemic malaria in growing populations
G.A. Ngwa
2004, 4(4): 1173-1202 doi: 10.3934/dcdsb.2004.4.1173 +[Abstract](57) +[PDF](343.1KB)
A mathematical model for endemic malaria involving variable human and mosquito populations is analysed. A threshold parameter $R_0$ exists and the disease can persist if and only if $R_0$ exceeds $1$. $R_0$ is seen to be a generalisation of the basic reproduction ratio associated with the Ross-Macdonald model for malaria transmission. The disease free equilibrium always exist and is globally stable when $R_0$ is below $1$. A perturbation analysis is used to approximate the endemic equilibrium in the important case where the disease related death rate is nonzero, small but significant. A diffusion approximation is used to approximate the quasi-stationary distribution of the associated stochastic model. Numerical simulations show that when $R_0$ is distinctly greater than $1$, the endemic deterministic equilibrium is globally stable. Furthermore, in quasi-stationarity, the stochastic process undergoes oscillations about a mean population whose size can be approximated by the stable endemic deterministic equilibrium.
Size-structured population dynamics models and their numerical solutions
L. M. Abia , O. Angulo and  J.C. López-Marcos
2004, 4(4): 1203-1222 doi: 10.3934/dcdsb.2004.4.1203 +[Abstract](48) +[PDF](245.2KB)
This paper presents a review of the numerical methods for the solution of the size-structured population balance models. The methods are compared with regards to accuracy, efficiency, generality and mathematical methodology.
Inexact Levenberg-Marquardt method for nonlinear equations
Jinyan Fan and  Jianyu Pan
2004, 4(4): 1223-1232 doi: 10.3934/dcdsb.2004.4.1223 +[Abstract](55) +[PDF](202.0KB)
In this paper, we present an inexact Levenberg-Marquardt (LM) method for singular system of nonlinear equations, where the LM parameter is chosen as the norm of the function and the trial step is computed approximately. Under the local error bound condition which is weaker than the non- singularity, we show that the new inexact LM method preserves the quadratic convergence of the traditional LM method where the parameter is chosen to be larger than a positive constant and the Jacobi at the solution is nonsingular.
Age-dependent population dynamics diffusive systems
Bedr'Eddine Ainseba
2004, 4(4): 1233-1247 doi: 10.3934/dcdsb.2004.4.1233 +[Abstract](38) +[PDF](207.1KB)
A nonlinear and nonlocal reaction-diffusion system of population growth is investigated which allows for consideration of both age-size and spatial effects. The mortality and fertility processes of the population are assumed to be linear to simplify the exposition. Local existence, continuation property, positivity, and global existence are obtained. This theory is applied to some specific reaction-diffusion epidemic model including the SI system, the SIS system with vertical transmission, and the SIR system.

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