Showing papers in "Siam Journal on Applied Mathematics in 1998"

Global attractivity in delayed Hopfield neural network models

Abstract: Two different approaches are employed to investigate the global attractivity of delayed Hopfield neural network models. Without assuming the monotonicity and differentiability of the activation functions, Liapunov functionals and functions (combined with the Razumikhin technique) are constructed and employed to establish sufficient conditions for global asymptotic stability independent of the delays. In the case of monotone and smooth activation functions, the theory of monotone dynamical systems is applied to obtain criteria for global attractivity of the delayed model. Such criteria depend on the magnitude of delays and show that self-inhibitory connections can contribute to the global convergence.

Computable elastic distances between shapes

Abstract: We dene distances between geometric curves by the square root of the minimal energy required to transform one curve into the other. The energy is formally dened from a left invariant Riemannian distance on an innite dimensional group acting on the curves, which can be explicitly computed. The obtained distance boils down to a variational problem for which an optimal matching between the curves has to be computed. An analysis of the distance when the curves are polygonal leads to a numerical procedure for the solution of the variational problem, which can eciently be implemented, as illustrated by experiments. 1.1. Generalities. The problem of matching two objects together is very impor- tant in computer vision and shape recognition. In many cases, recognition is based on shapes (outlines), with the help of some suitably designed distance. A general principle is to associate with any pair (O1;O 2) of objects to be compared a measure of discrepancy d(O1;O 2 ):The recognition of an observed object O may be done by nding, from a dictionary of \templates," the previously recorded object Otemp, for which d(O;Otemp) is minimal. Clearly, the denition of the distance is the crucial step of the method, and much research has been done in this direction. We shall not try here to provide a review of the huge literature existing on the subject (see, for example, (17)) but rather focus on methods related to deformable templates, with which we are directly concerned. Instead of basing recognition on a nite collection of points of interest (primitives) taken from the outline of an object (corners, inflexion points, etc.), which is a popular way of handling the problem, our purpose is to base the comparison on the whole outline, considered as a plane curve. The distance we shall dene incorporates some deformation energy between the curves. The approach, as we will see, turns out to be intrinsic and robust to usual Euclidean transformations. The method is related to the wide literature on \snakes" (14), (7), (21) etc. in the way that our distance corresponds to some continuous process of deformation of one curve into another. It is also related to papers on elastic matching, such as (8); however, we provide an elastic matching algorithm which is based on a true distance between intrinsic properties of the shapes, taking into account possible invariance to scaling or Euclidean transformations in the case they are required. From this point of view, our results are indebted to the seminal work of Grenander on group theory applied to pattern recognition (cf. (10) and (11), in particular; see also (1), (2), 12). Another source of inspiration may come from mathematical physics, since we are going to look to the path (process) of lowest energy which deforms one object into

Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice

Abstract: We consider innite systems of ODEs on the two-dimensional integer lattice, given by a bistable scalar ODE at each point, with a nearest neighbor coupling between lattice points. For a class of ideal nonlinearities, we obtain traveling wave solutions in each direction e i , and we explore the relation between the wave speed c, the angle , and the detuning parameter a of the nonlinearity. Of particular interest is the phenomenon of \propagation failure," and we study how the critical value a = a() depends on , where a() is dened as the value of the parameter a at which propagation failure (that is, wave speed c = 0) occurs. We show that a :R!R is continuous at each point where tan is irrational, and is discontinuous where tan is rational or innite.

Coexistence region and global dynamics of a harvested predator-prey system

Abstract: The study of populational dynamics with harvesting is related to the optimal man- agement of renewable resources. In this paper we consider a predator-prey model in which two ecologically interacting species are harvested independently with constant rates. The main purpose of the present work is to offer a complete mathematical analysis for the model and to describe some of the significant qualitative results that may be expected to arise from the interplay of biological forces. Our study shows that the maximum safe harvest may be far less than what would be assumed from a local analysis for the equilibria. It is also proved that the harvested predator-prey system may exhibit very complicated dynamics such as a spontaneous appearance of a homoclinic orbit and multiple limit cycles.

A hierarchy of models for multilane vehicular traffic I: modeling

Abstract: In the present paper multilane models for vehicular traffic are considered. A microscopic multilane model based on reaction thresholds is developed. Based on this model an Enskog- like kinetic model is developed. In particular, care is taken to incorporate the correlations between the vehicles. From the kinetic model a fluid dynamic model is derived. The macroscopic coefficients are deduced from the underlying kinetic model. Numerical simulations are presented for all three levels of description in [A. Klar and R. Wegener, SIAM J. Appl. Math., 59 (1999), pp. 1002--1011]. Moreover, a comparison of the results is given there.

Phase segregation dynamics in particle systems with long range interactions II: interface motion

Abstract: We study properties of the solutions of a family of second-order integrodifferential equations, which describe the large scale dynamics of a class of microscopic phase segregation models with particle conserving dynamics. We first establish existence and uniqueness as well as some properties of the instantonic solutions. Then we concentrate on formal asymptotic (sharp interface) limits. We argue that the obtained interface evolution laws (a Stefan-like problem and the Mullins--Sekerka solidification model) coincide with the ones which can be obtained in the analogous limits from the Cahn--Hilliard equation, the fourth-order PDE which is the standard macroscopic model for phase segregation with one conservation law.

Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping

Abstract: In this paper, we study the mathematical properties of a variational second order evolution equation, which includes the equations modelling vibrations of the Euler--Bernoulli and Rayleigh beams with the global or local Kelvin--Voigt (K--V) damping. In particular, our results describe the semigroup setting, the strong asymptotic stability and exponential stability of the semigroup, the analyticity of the semigroup, as well as characteristics of the spectrum of the semigroup generator under various conditions on the damping. We also give an example to show that the energy of a vibrating string does not decay exponentially when the K--V damping is distributed only on a subinterval which has one end coincident with one end of the string.

Large time behavior of the solutions to a hydrodynamic model for semiconductors

Abstract: We establish the global existence of smooth solutions to the Cauchy problem for the one-dimensional isentropic Euler--Poisson (or hydrodynamic) model for semiconductors for small initial data. In particular we show that, as $t\to\infty$, these solutions converge to the stationary solutions of the drift-diffusion equations. The existence and uniqueness of stationary solutions to the drift-diffusion equations are proved without the smallness assumption.

The Gaussian moment closure for gas dynamics

Abstract: The moment closure method of Levermore applied to the Boltzmann equation for rarefied gas dynamics leads to a hierarchy of symmetric hyperbolic systems of partial differential equations. The Euler system is the first member of this hierarchy of closures. In this paper we investigate the next member, the 10 moment Gaussian closure. We first reduce the collision term to an integral which may be explicitly evaluated for the special case of Maxwell molecular interaction. The resulting collision term for this case is shown to be equivalent to the term obtained by replacing the Boltzmann collision operator with the Bhatnagar, Gross, and Krook (BGK) approximation. We then analyze the Gaussian system applied to the canonical flow problem of a stationary planar shock. An analytic shock profile for the Gaussian closure is derived and compared with the numerical solutions of the Boltzmann and Navier--Stokes equations. The results show reasonable agreement for weak shocks and close agreement between the downstream Ga...

Weak existence for the semigeostrophic equations formulated as a coupled Monge-Ampère/transport problem

Abstract: Hoskins's semigeostrophic equations are reformulated as a coupled Monge--Ampere/ transport problem [B. J. Hoskins, Quart. J. Royal Met. Soc., 97 (1971), pp. 139--153]. Existence of global weak solu...

Effects of random motility on microbial growth and competition in a flow reactor

Abstract: The authors investigate the effects of random motility on the ability of a microbial population to survive in pure culture and to be a good competitor for scarce nutrient in mixed culture in a flow reactor model consisting of a nonlinear parabolic system of partial differential equations. For pure culture (*a single population), a sharp condition is derived which distinguishes between the two outcomes: (1) washout of the population from the reactor or (2) persistence of the population and the existence of a unique single-population steady state. The simulations suggest that this steady state is globally attracting. For the case of two populations competing for scarce nutrient, they obtain sufficient conditions for the uniform persistence of the two populations, for the existence of a coexistence steady state, and for the ability of one population to competitively exclude a rival. Extensive simulations are reported which suggest that (1) all solutions approach some steady state solution, (2) all possible outcomes exhibited by the classical competitive Lotka-Volterra ODE model can occur in the model, and (3) the outcome of competition between two bacterial strains can depend rather subtly on their respective random motility coefficients.

Three-dimensional competitive Lotka-Volterra systems with no periodic orbits

Abstract: The following conjecture of M. L. Zeeman is proved. If three interacting species modeled by a competitive Lotka--Volterra system can each resist invasion at carrying capacity, then there can be no coexistence of the species. Indeed, two of the species are driven to extinction. It is also proved that in the other extreme, if none of the species can resist invasion from either of the others, then there is stable coexistence of at least two of the species. In this case, if the system has a fixed point in the interior of the positive cone in R3 , then that fixed point is globally asymptotically stable, representing stable coexistence of all three species. Otherwise, there is a globally asymptotically stable fixed point in one of the coordinate planes of R3 , representing stable coexistence of two of the species.

Spectral properties of classical waves in high-contrast periodic media

Abstract: We introduce and investigate the band gap structure of the frequency spectrum for classical electromagnetic and acoustic waves in a high-contrast, two-component periodic medium. The asymptotics with respect to the high-contrast is considered. The limit medium is described in terms of appropriate self-adjoint operators and the convergence to the limit is proven. These limit operators give an idea of the spectral structure and suggest new numerical approaches as well. The results are obtained in arbitrary dimension and for rather general geometry of the medium. In particular, two-dimensional (2D) photonic band gap structures and their acoustic analogues are covered.

Global asymptotic behavior of the chemostat: general response functions and different removal rates

Abstract: In this paper, we consider a competition model between n species in a chemostat that incorporates both monotone and nonmonotone general response functions and distinct removal rates. We show that only the species with the lowest break-even concentration survives, provided that the variation of distinct removal rates relative to the flow rate of the chemostat can be controlled by either the difference between the two lowest break-even concentrations or by a parameter based on the structure of response functions. LaSalle's extension theorem of the Lyapunov stability theory and fluctuation lemma are the main tools.

Global and exploding solutions for nonlocal quadratic evolution problems

Abstract: Nonlinear nonlocal parabolic equations modeling the evolution of density of mutually interacting particles are considered. The inertial type nonlinearity is quadratic and nonlocal while the diffusive term, also nonlocal, is anomalous and fractal, i.e., represented by a fractional power of the Laplacian. Conditions for global in time existence versus finite time blow-up are studied. Self-similar solutions are constructed for certain homogeneous initial data.

A uniqueness result for scattering by infinite rough surfaces

Abstract: Consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane with an unbounded, piecewise Lyapunov boundary. This problem models time-harmonic electromagnetic scattering in transverse magnetic polarization by one-dimensional rough, perfectly conducting surfaces. A radiation condition is introduced for the problem, which is a generalization of the usual one used in the study of diffraction by gratings when the solution is quasi-periodic, and allows a variety of incident fields including an incident plane wave to be included in the results obtained. We show in this paper that the boundary value problem for the scattered field has at most one solution. For the case when the whole boundary is Lyapunov and is a small perturbation of a flat boundary we also prove existence of solution and show a limiting absorption principle.

Transition curves for the quasi-periodic Mathieu equation

Abstract: In this work we investigate an extension of Mathieu's equation, the quasi-periodic (QP) Mathieu equation given by \[ \ddot{\psi} + [\delta + \eps \,( \cos t + \cos \omega t)]\, \psi = 0 \] for small $\eps$ and irrational $\omega$. Of interest is the generation of stability diagrams that identify the points or regions in the $\delta$-$\omega$ parameter plane (for fixed $\eps$) for which all solutions of the QP Mathieu equation are bounded. Numerical integration is employed to produce approximations to the true stability diagrams both directly and through contour plots ofLyapunov exponents. In addition, we derive approximate analytic expressions for transition curves using two distinct techniques: (1) a regular perturbation method under which transition curves $\delta = \delta(\omega; \eps)$ are each expanded in powers of $\eps$, and (2) the method of harmonic balance utilizing Hill's determinants. Both analytic methods deliver results in good agreement with those generated numerically in the sense that pre...

A linear sampling method for the detection of leukemia using microwaves

Abstract: This paper continues our study of the use of the linear sampling method for using microwaves to detect leukemia in the lower leg [SIAM J. Appl. Math., 58 (1998), pp. 926--941], [Computational Radiology and Imaging: Therapy and Diagnostics, C. Borgers and F. Natterer, eds., IMA Vol. Math. Appl. 110, Springer-Verlag, Berlin, 1999, pp. 137--156]. In the present work we no longer ignore the presence of skin and assume that the sources are placed in a thin sheath that is wrapped around the leg. This requires us to consider a resistive boundary value problem and to extend the linear sampling method to treat problems of this type. Numerical examples are given, showing the viability of this new model.

Optimal harvesting for periodic age-dependent population dynamics

Abstract: Here we investigate the optimal harvesting problem for some periodic age-dependent population dynamics; namely, we consider the linear Lotka--McKendrick model with periodic vital rates and a periodic forcing term that sustains oscillations. Existence and uniqueness of a positive periodic solution are demonstrated and the existence and uniqueness of the optimal control are established. We also state necessary optimality conditions. A numerical algorithm is developed to approximate the optimal control and the optimal harvest. Some numerical results are presented.

Midgap defect modes in dielectric and acoustic media

Abstract: We consider three dimensional lossless periodic dielectric (photonic crystals) and acoustic media having a gap in the spectrum If such a periodic medium is perturbed by a strong enough defect, defect eigenmodes arise, localized exponentially around the defect, with the corre- sponding eigenvalues in the gap We use a modied Birman-Schwinger method to derive equations for these eigenmodes and corresponding eigenvalues in the gap, in terms of the spectral attributes of an auxiliary Hilbert-Schmidt operator We prove that in three dimensions, under some natural conditions on the periodic background, the number of eigenvalues generated in a gap of the periodic operator is nite, and give an estimate on the number of these midgap eigenvalues In particular, we show that if the defect is weak there are no midgap eigenvalues

Mathematical analysis of dynamic models of suspension bridges

Abstract: In this paper we present some mathematical analysis of dynamic (PDE) models of suspension bridges as proposed by Lazer and McKenna. Our results are illustrated by numerical simulation with physical interpretation.

The inverse conductivity problem with one measurement: bounds on the size of the unknown object

Abstract: We consider the problem of determining, within an electrically conducting body $\Omega$ of conductivity $\sigma\equiv 1$, an unknown inclusion D, whose conductivity is $\sigma\equiv k eq 1$, when one pair of current density and voltage measurements {}from the exterior of $\Omega$ is available. We prove upper and lower bounds on the size of the unknown inclusion D. We also consider the case of nonuniform and nonisotropic conductivities in $\Omega$ and in D.

Traveling waves in buffered systems: applications to calcium waves

Abstract: Traveling waves of calcium are widely observed under conditions where the free calcium is heavily buffered. It is thus of considerable physiological interest to determine the precise effects that buffers have on the properties of traveling waves. Since calcium waves are widely believed to be the result of the reaction and diffusion of calcium, we study the properties of traveling waves in simple reaction-diffusion equations in which the diffusing species is buffered. For the buffered bistable equation we derive a constraint on the model parameters in order for a traveling wave of excitation to exist, and we show that stationary buffers cannot eliminate traveling waves. When the buffer is of low affinity, a constant effective diffusion coefficient may be defined, but no effective diffusion coefficient can be defined for high affinity buffers. We derive an approximate expression for the wave speed, show numerically that this approximation applies for both high and low affinity buffers, and hence derive an a...

Multidimensional reaction diffusion equations with nonlinear boundary conditions

Abstract: We consider diffusion models in which distributed nonlinear absorption mechanisms compete with nonlinear boundary sources. Assuming that the nonlinearities are weak, formal asymptotic approximation...

Convergence of a branching particle method to the solution of the Zakai equation

Abstract: We construct a sequence of branching particle systems Un convergent in distribution to the solution of the Zakai equation. The algorithm based on this result can be used to solve numerically the filtering problem. The result is an improvement of the one presented in a recent paper [Crisan and T. Lyons, Prob. Theory Related Fields, 109 (1997), pp. 217--244], because it eliminates the extra degree of randomness introduced there.

Moment Lyapunov exponent and stability index for linear conservative system with small random perturbation

Abstract: Asymptotic expansion series for the moment Lyapunov exponent and stability index are constructed and justified for the two-dimensional linear stochastic system close to a harmonic oscillator. As an example, a one-degree-of-freedom mechanical system parametrically excited in stiffness and damping is considered and several terms of the expansion are obtained.

On the asymmetric May-Leonard model of three competing species

Abstract: In this paper we analyze the global asymptotic behavior of the asymmetric May--Leonard model of three competing species:\, $\frac{dx_i}{dt}=x_i(1-x_i-\beta_ix_{i-1}-\alpha_ix_{i+1})$, $x_i(0) > 0$, $i=1,2,3$ with $x_{0}=x_3$, $x_4=x_1$ under the assumption \, $0 B1B2B3 and P is a saddle point with one-dimensional stable manifold $\Gamma$ if A1A2A3 B1B2B3 then P is global asymptotically stable in Int(R3+ ), (ii) if A1A2A3 < B1B2B3 then for each initial condition $x_0 ot\in\Gamma$, the solution $\v...

Optimal vaccination patterns in age-structured populations

Abstract: Within the framework of homogeneous age-structured SIR-models, optimal vaccination strategies are considered. There are two concepts for a vaccination strategy: the effort (the number of vaccinatio...

Acoustic scattering by an inhomogeneous layer on a rigid plate

Abstract: The problem of scattering of time-harmonic acoustic waves by an inhomogeneous fluid layer on a rigid plate in R 2 is considered. The density is assumed to be unity in the media: within the layer the sound speed is assumed to be an arbitrary bounded measurable function. The problem is modelled by the reduced wave equation with variable wavenumber in the layer and a Neumann condition on the plate. To formulate the problem and prove uniqueness of solution a radiation condition appropriate for scattering by infinite rough surfaces is introduced, a generalization of the Rayleigh expansion condition for diffraction gratings. With the help of the radiation condition the problem is reformulated as a system of two second kind integral equations over the layer and the plate. Under additional assumptions on the wavenumber in the layer, uniqueness of solution is proved and the nonexistence of guided wave solutions of the homogeneous problem established. General results on the solvability of systems of integral equations on unbounded domains are used to establish existence and continuous dependence in a weighted norm of the solution on the given data.

Fast and slow compaction in sedimentary basins

Abstract: A mathematical model of compaction in sedimentary basins is presented and analyzed. Compaction occurs when accumulating sediments compact under their own weight, expelling pore water in the process. If sedimentation is rapid or the permeability is low, then high pore pressures can result, a phenomenon which is of importance in oil drilling operations. Here we show that one-dimensional compaction can be described in its simplest form by a nonlinear diffusion equation, controlled principally by a dimensionless parameter $\lambda$, which is the ratio of the hydraulic conductivity to the sedimentation rate. Large $\lambda$ corresponds to very permeable sediments, or slow sedimentation, a situation which we term "fast compaction," since the rapid pore water expulsion allows the pore water pressure to equilibrate to a hydrostatic value. On the other hand, small $\lambda$ corresponds to "slow compaction," and the pore pressure is in excess above the hydrostatic value and more nearly equal to the overburden value...