Oscillatory reaction-diffusion equations on rings

We study the behavior of traveling waves in - systems on both homogeneous and inhomogeneous rings. The stability regions in parameter space of - waves were previously known [15, 19]; the results are extended here. We show the existence of Hopf bifurcations of traveling waves and the stability of the limit cycles born at the Hopf bifurcation for some parameter ranges. Using a Lindstedt-type perturbation scheme, we formally construct periodic solutions of the - system near a Hopf bifurcation and show that the periodic solutions superimposed on the original traveling wave have the effect of altering its overall frequency and amplitude. We also study the - system on an annulus ofvariable width, which does not possess reflection symmetry about any axis. We formally construct traveling waves on this variable-width annulus by a perturbation scheme, and find that perturbing the width of the annulus alters the amplitude and frequency of traveling waves on the domain by a small (order ²) amount. For typical parameter values, we find that the speed, frequency, and stability are unaffected by the direction of travel of the wave on the annulus, despite the rotationally asymmetric inhomogeneity. This indicates that the - system on a variable-width domain cannot account for directional preferences of traveling waves in biological systems.     

Three-dimensional waves in excitable reaction-diffusion systems

The eikonal equation [5] for excitable media is generalised to three dimensional systems. The main result of the investigation is the demonstration of the existence of toroidal and twisted toroidal scroll waves in the limit of large values of the major radius of the torus. The existence of a helical wave near the z-axis follows from the eidonal equation but its connection with the twisted toroidal scroll remains to be demonstrated. The eikonal equation also predicts a non-uniform rate of rotation of the cross-sectional spiral wave near the toroidal axis. The notion of geometrical stability is introduced for the case of an expanding sphere; in particular it is shown that a discussion of stability of solutions of the eikonal equation must take into account the possible shift in the origin of the coordinate systems with respect to which patterns are defined. On leave of absence from: The Department of Mathematics, Glasgow College of Technology, Cowcaddens Road, Glasgow G4 0BA, UK     

A collocation approach to the numerical calculation of simple gradients in reaction-diffusion systems

Reaction diffusion equations are frequently used to model pattern formation problems in biology, but numerical experiments in two or three space dimensions can be expensive in computing time. We show that the spectral method with collocation is a particularly efficient method for the numerical study of the evolution of simple patterns in such models. In many cases of interest, the scheme is sufficiently simple and efficient for calculations to be carried out on a micro-computer.     

Parameter domains for instability of uniform states in systems with many delays

 We present a computational method for determining regions in parameter space corresponding to linear instability of a spatially uniform steady state solution of any system of two coupled reaction-diffusion equations containing up to four delay terms. At each point in parameter space the required stability properties of the linearised system are found using mainly the Principle of the Argument. The method is first developed for perturbations of a particular wavenumber, and then generalised to allow arbitrary perturbations. Each delay term in the system may be of either a fixed or a distributed type, and spatio-temporal delays are also allowed. Received 19 September 1995; received in revised form 4 September 1996     

A delay reaction-diffusion model of the dynamics of botulinum in fish

A model of interaction between fish and a bacterium (Clostridium botulinum) responsible for avian botulism is introduced, considering diffusion of both fish and bacterium in water. The fish population moves randomly in water. Death fish disintegrate in water, at different locations, causing bacteria to diffuse through water and infect other fish. Existence of uniform steady states is investigated and the linearized stability of the positive uniform steady state is analyzed. A Hopf bifurcation is proved to occur from the uniform steady state when the bifurcation parameter, here the time delay, passes through a critical value and diffusion coefficients satisfy some conditions, that induces time oscillations of the populations. Comments on diffusion-driven instability are provided, and numerical simulations are carried out to illustrate the results.     

Galerkin-Ritz procedures for approximate solutions to systems of reaction-diffusion equations

For any essentially nonlinear system of reaction-diffusion equations of the generic form ∂c_i/∂t=D_i∇²c_i+Q_i(c,x,t) supplemented with Robin type boundary conditions over the surface of a closed bounded three-dimensional region, it is demonstrated that all solutions for the concentration distributionn-tuple function c=(c ₁(x,t),...,c _n (x,t)) satisfy a differential variational condition. Approximate solutions to the reaction-diffusion intial-value boundary-value problem are obtainable by employing this variational condition in conjunction with a Galerkin-Ritz procedure. It is shown that the dynamical evolution from a prescribed initial concentrationn-tuple function to a final steady-state solution can be determined to desired accuracy by such an approximation method. The variational condition also admits a systematic Galerkin-Ritz procedure for obtaining approximate solutions to the multi-equation elliptic boundary-value problem for steady-state distributions c=−c(x). Other systems of phenomenological (non-Lagrangian) field equations can be treated by Galerkin-Ritz procedures based on analogues of the differential variational condition presented here. The method is applied to derive approximate nonconstant steady-state solutions for ann-species symbiosis model.     

Mechanisms and rate equations for dissociating systems.

The results presented in the previous paper (Boeker, E.A. (1978), Biochemistry 17 (preceding paper in this issue) indicate that the dissociation of the decamer of arginine decarboxylase of Escherichia coli B is enhanced by Na+ and retarded by H+. In this system, substances which increase the rate of dissociation can be treated kinetically either as substrates or activators, and substances which retard dissociation can be treated as products or inhibitors. In addition, the events needed for dissociation can occur in an ordered or a random sequence, and the dissociation itself, from a decamer to five dimers, can be a sequential or a concerted process. In order to provide a framework for the experimental results, mechanisms for the dissociation of arginine decarboxylase that take all of these factors into account are described. In addition, it is shown that the usual methods of steady-state kinetics can be applied to these systems when true initial rates are measured; rate equations are presented for each mechanism. The results can be used for any dissociating of three or more subunits and will describe the dissociation of a dimer under certain conditions.     

The asymptotic transectional/circumferential homogeneity of the solutions of reaction-diffusion systems in/on cylinder-like domains

It is often reported that an animal with spotty coat markings on its body has a tail with stripe-shaped pattern. In other various biological and chemical phenomena in/on cylinder-like domains, longitudinally periodic band patterns are observed much more often than the other non-uniform patterns. This paper mathematically explains these observations by proving that, in/on a long and narrow cylinder-like domain, any solution of reaction-diffusion system asymptotically loses its spatial dependence in the transectional/circumferential direction.     

Coexistence for systems governed by difference equations of Lotka-Volterra type

The question of the long term survival of species in models governed by Lotka-Volterra difference equations is considered. The criterion used is the biologically realistic one of permanence, that is populations with all initial values positive must eventually all become greater than some fixed positive number. We show that in spite of the complex dynamics associated even with the simplest of such systems, it is possible to obtain readily applicable criteria for permanence in a wide range of cases.     

Plankton blooms and patchiness generated by heterogeneous physical environments

Microscopic turbulent motions of water have been shown to influence the dynamics of microscopic species living in that habitat. The number, stability, and excitability of stationary states in a predator–prey model of plankton species can therefore change when the strength of turbulent motions varies. In a spatial system these microscopic turbulent motions are naturally of different strength and form a heterogeneous physical environment. Spatially neighboring plankton communities with different physical conditions can impact each other due to diffusive coupling. We show that local variations in the physical conditions can influence the global system in form of propagating pulses of high population densities. For this we consider three different local predator–prey models with different local responses to variations in the physical environment. The degree of spatial heterogeneity can, depending on the model, promote or reduce the number of propagating pulses, which can be interpreted as patchy plankton distributions and recurrent blooms.     

Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations

In this paper we use a dynamical systems approach to prove the existence of a unique critical value c ^* of the speed c for which the degenerate density-dependent diffusion equation u _ct = [D(u)u _x ] _x + g(u) has: 1. no travelling wave solutions for 0 < c < c ^*, 2. a travelling wave solution u(x, t) = (x - c ^* t) of sharp type satisfying (– ) = 1, () = 0 ^*; '(^*–) = – c ^*/D'(0), '(^*+) = 0 and 3. a continuum of travelling wave solutions of monotone decreasing front type for each c > c ^*. These fronts satisfy the boundary conditions (– ) = 1, '(– ) = (+ ) = '(+ ) = 0. We illustrate our analytical results with some numerical solutions.     

Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases

A reaction-diffusion system which describes the spatial spread of bacterial diseases is studied. It consists of two nonlinear parabolic equations which concern the evolution of the bacteria population and of the human infective population in an urban community, respectively. Different boundary conditions of the third type are considered, for the two variables. This model is suitable to study oro-faecal transmitted diseases in the European Mediterranean regions. A threshold parameter is introduced such that for suitable values of it the epidemic eventually tends to extinction, otherwise a globally asymptotically stable spatially inhomogeneous stationary endemic state appears. The case in which the bacteria diffuse but the human population does not, has also been considered.Work performed under the auspices of the G.N.A.F.A. [L. M.] and the G.N.F.M. [V. C.] C.N.R. in the context of the Program of Preventive Medicine (Project MPP1), C.N.R., Italy     

Mechanisms for stabilisation and the maintenance of solubility in proteins from thermophiles

Background  

The database of protein structures contains representatives from organisms with a range of growth temperatures. Various properties have been studied in a search for the molecular basis of protein adaptation to higher growth temperature. Charged groups have emerged as key distinguishing factors for proteins from thermophiles and mesophiles.     

Activation waves in a model of platelet aggregation: existence of solutions and stability of travelling fronts

Platelets cohere to one another to form platelet aggregates as part of the blood's clotting response. The ability of a platelet to participate in this process depends on its prior activation by chemicals released into the blood plasma by other activated platelets. We study the piecewise-linear system of reaction-diffusion equations which, in one spatial dimension, describe the chemically-mediated spread of platelet activation. We establish the existence of classical solutions to this system of equations, and show that these solutions do not blow up in finite time. We also explicitly construct travelling front solutions and discuss their stability. Finally, we present numerical evidence which suggests that for a broad range of initial data with the correct limiting values at ± , the solution to the initial value problem rapidly evolves into the travelling front solution provided the front is linearly stable.     

The geometry and motion of reaction-diffusion waves on closed two-dimensional manifolds

Chemical or biological systems modelled by reaction diffusion (R.D.) equations which support simple one-dimensional travelling waves (oscillatory or otherwise) may be expected to produce intricate two or three-dimensional spatial patterns, either stationary or subject to certain motion. Such structures have been observed experimentally. Asymptotic considerations applied to a general class of such systems lead to fundamental restrictions on the existence and geometrical form of possible structures. As a consequence of the geometrical setting, it is a straightforward matter to consider the propagation of waves on closed two-dimensional manifolds. We derive a fundamental equation for R.D. wave propagation on surfaces and discuss its significance. We consider the existence and propagation of rotationally symmetric and double spiral waves on the sphere and on the torus. On leave of absence from: Department of Mathematics, Glasgow College of Technology, Cowcaddens Road, Glasgow G4 0BA, Scotland, UK     

Empirical validation of gradient field models for an accurate ADC measured on clinical 3T MR systems in body oncologic applications

PurposeTo empirically corroborate vendor-provided gradient nonlinearity (GNL) characteristics and demonstrate efficient GNL bias correction for human brain apparent diffusion coefficient (ADC) across 3T MR systems and spatial locations.MethodsSpatial distortion vector fields (DVF) were mapped in 3D using a surface fiducial array phantom for individual gradient channels on three 3T MR platforms from different vendors. Measured DVF were converted into empirical 3D GNL tensors and compared with their theoretical counterparts derived from vendor-provided spherical harmonic (SPH) coefficients. To illustrate spatial impact of GNL on ADC, diffusion weighted imaging using three orthogonal gradient directions was performed on a volunteer brain positioned at isocenter (as a reference) and offset superiorly by 10–17 cm (>10% predicted GNL bias). The SPH tensor-based GNL correction was applied to individual DWI gradient directions, and derived ADC was compared with low-bias reference for human brain white matter (WM) ROIs.ResultsEmpiric and predicted GNL errors were comparable for all three studied 3T MR systems, with <1.0% differences in the median and width of spatial histograms for individual GNL tensor elements. Median (±width) of ADC (10⁻³mm²/s) histograms measured at isocenter in WM reference ROIs from three MR systems were: 0.73 ± 0.11, 0.71 ± 0.14, 0.74 ± 0.17, and at off-isocenters (before versus after GNL correction) were respectively 0.63 ± 0.14 versus 0.72 ± 0.11, 0.53 ± 0.16 versus 0.74 ± 0.18, and 0.65 ± 0.16 versus 0.76 ± 0.18.ConclusionThe phantom-based spatial distortion measurements validated vendor-provided gradient fields, and accurate WM ADC was recovered regardless of spatial locations and clinical MR platforms using system-specific tensor-based GNL correction for routine DWI.     

崔-Lawson和Logistic方程参数的优化估计方法

本文提出用单形寻优与微分方程数值解法的联合方法,进行生态学中一些微分动力系统的参数的优化估计。用这种方法来估计崔-Lawson和Logistic方程的各参数效果极好。