Two-dimensional Spatial Domain Research Articles

The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime

The existence and stability of localized patterns of criminal activityare studied for the reaction-diffusion model of urban crime that wasintroduced by Short et. al. [Math. Models. Meth. Appl. Sci., 18, Suppl. (2008), pp. 1249--1267]. Such patterns,characterized by the concentration of criminal activity in localizedspatial regions, are referred to as hot-spot patterns and they occurin a parameter regime far from the Turing point associated with thebifurcation of spatially uniform solutions. Singular perturbationtechniques are used to construct steady-state hot-spot patterns in oneand two-dimensional spatial domains, and new types of nonlocaleigenvalue problems are derived that determine the stability of thesehot-spot patterns to ${\mathcal O}(1)$ time-scale instabilities. Froman analysis of these nonlocal eigenvalue problems, a criticalthreshold $K_c$ is determined such that a pattern consisting of $K$hot-spots is unstable to a competition instability if $K>K_c$. Thisinstability, due to a positive real eigenvalue, triggers the collapseof some of the hot-spots in the pattern. Furthermore, in contrast tothe well-known stability results for spike patterns of theGierer-Meinhardt reaction-diffusion model, it is shown for the crimemodel that there is only a relatively narrow parameter range whereoscillatory instabilities in the hot-spot amplitudes occur. Such aninstability, due to a Hopf bifurcation, is studied explicitly for asingle hot-spot in the shadow system limit, for which the diffusivityof criminals is asymptotically large. Finally, the parameter regimewhere localized hot-spots occur is compared with the parameter regime,studied in previous works, where Turing instabilities from a spatiallyuniform steady-state occur.

SINGULARLY PERTURBED PERIODIC PARABOLIC EQUATIONS WITH ALTERNATING BOUNDARY LAYER TYPE SOLUTIONS IN SPATIALLY TWO-DIMENSIONAL DOMAINS

In this article, we continue the analysis of a class of singularly perturbed parabolic equations with alternating boundary layer type solutions. For such problems, the degenerate (reduced) equations obtained by setting a small parameter equal to zero correspond to algebraic equations that have several isolated roots. As time increases, solutions of these equations periodically go through two comparatively long lasting stages with fast transitions between these stages. During one of these stages, the solution outside the boundary layer (i.e. the regular part of the asymptotic solution) is close to one of the roots of the degenerate equation. During the other stage, the regular part of the asymptotic solution is close to the other root. Here we discuss some specific features of the solutions' behavior for such problems in certain two-dimensional spatial domains.

Developing signal processing method for recognizing defects in metal samples based on heat diffusion properties in sonic infrared image sequences

In sonic infrared (SonicIR) imaging, heat is generated in defect areas during the sonic pulse; the heat appears bright in SonicIR images as the indication of a defect. However, in practical applications of SonicIR, there are lots of disturbing bright areas in infrared images, such as heat reflection and paint problem. When crack size is small, the generated heat appears not bright enough to be recognizable. Based on heat diffusion properties in the one-dimensional temporal and two-dimensional spatial domain, a method is developed to automatically recognize defect signals from SonicIR image sequences. The algorithm is verified with the SonicIR image sequences of 100 metal plates which may have different thickness, materials, or crack sizes.

An Alternate Approach to Optimal L 2-Error Analysis of Semidiscrete Galerkin Methods for Linear Parabolic Problems with Nonsmooth Initial Data

In this article, we propose and analyze an alternate proof of a priori error estimates for semidiscrete Galerkin approximations to a general second order linear parabolic initial and boundary value problem with rough initial data. Our analysis is based on energy arguments without using parabolic duality. Further, it follows the spirit of the proof technique used for deriving optimal error estimates for finite element approximations to parabolic problems with smooth initial data and hence, it unifies both theories, that is, one for smooth initial data and other for nonsmooth data. Moreover, the proposed technique is also extended to a semidiscrete mixed method for linear parabolic problems. In both cases, optimal L 2-error estimates are derived, when the initial data is in L 2. A superconvergence phenomenon is also observed, which is then used to prove L ∞-estimates for linear parabolic problems defined on two-dimensional spatial domain again with rough initial data.

THE SPATIAL PATTERNS THROUGH DIFFUSION-DRIVEN INSTABILITY IN MODIFIED LESLIE-GOWER AND HOLLING-TYPE II PREDATOR-PREY MODEL

Formation of spatial patterns in prey-predator system is a central issue in ecology. In this paper Turing structure through diffusion driven instability in a modified Leslie-Gower and Holling-type II predator-prey model has been investigated. The parametric space for which Turing spatial structure takes place has been found out. Extensive numerical experiments have been performed to show the role of diffusion coefficients and other important parameters of the system in Turing instability that produces some elegant patterns that have not been observed in the earlier findings. Finally it is concluded that the diffusion can lead the prey population to become isolated in the two-dimensional spatial domain.

Morphological spatial patterns in a reaction diffusion model for metal growth

In this paper a reaction-diffusion system modelling metal growth processes is considered, to investigate - within the electrodeposition context- the formation of morphological patterns in a finite two-dimensional spatial domain. Nonlinear dynamics of the system is studied from both the analytical and numerical points of view. Phase-space analysis is provided and initiation of spatial patterns induced by diffusion is shown to occur in a suitable region of the parameter space. Investigations aimed at establishing the role of some relevant chemical parameters on stability and selection of solutions are also provided. By the numerical approximation of the equations, simulations are presented which turn out to be in good agreement with experiments for the electrodeposition of Au-Cu and Au-Cu-Cd alloys.

Stochastic dynamics and mean field approach in a system of three interacting species

Abstract The spatio-temporal dynamics of three interacting species, two preys and one predator, in the presence of two different kinds of noise sources is studied, by using Lotka-Volterra equations. A correlated dichotomous noise acts on the interaction parameter between the two preys, and a multiplicative white noise affects directly the dynamics of the three species. After analyzing the time behaviour of the three species in a single site, we consider a two-dimensional spatial domain, applying a mean field approach and obtaining the time behaviour of the first and second order moments for different multiplicative noise intensities. We find noise-induced oscillations of the three species with an anticorrelated behaviour of the two preys. Finally, we compare our results with those obtained by using a coupled map lattice (CML) model, finding a good qualitative agreement. However, some quantitative discrepancies appear, that can be explained as follows: i) different stationary values occur in the two approaches; ii) in the mean field formalism the interaction between sites is extended to the whole spatial domain, conversely in the CML model the species interaction is restricted to the nearest neighbors; iii) the dynamics of the CML model is faster since an unitary time step is considered.

Finite element methods for partial Volterra integro-differential equations on two-dimensional unbounded spatial domains

The precise and effective way for solving a partial Volterra integro-differential equation on unbounded spatial domain is to derive artificial boundary conditions for some feasible and prescribed computational domain. Thus the unsolvable problem on unbounded domain is converted to the computational problem on bounded domain, which is often called reduced problem. A feasible numerical method will then be applied to solve the reduced problem with the artificial boundary conditions which often exhibit “nonlocal” properties. In this paper, the finite element method for the spatial space and the recently developed discontinuous Galerkin time-stepping method for the temporal space are considered to solve the reduced problem. The convergence rate, explicitly with those interesting parameters: spatial grid-size h, temporal mesh-size k, the location of the artificial boundary d, and the truncation number M, is obtained. This paper is the first theoretical analysis of the numerical methods for Volterra integro-differential equations on unbounded spatial domains.

Receptive Field Properties of Neurons in the Early Visual Cortex Revealed by Local Spectral Reverse Correlation

We introduce a novel class of white-noise analyses, named local spectral reverse correlation (LSRC), which is capable of revealing various aspects of visual receptive field profiles that were undetectable previously in a single simple measurement. The method is based on spectral analyses in a two-dimensional spatial frequency domain for spatially localized areas within and around their receptive fields. Extracellular single-unit recordings were performed for area 17 and 18 neurons in anesthetized cats. A dynamic dense noise pattern was presented in which the pattern covered an area two to three times larger than the classical receptive field. Spike trains were then cross-correlated with frequency spectra of localized noise pattern to obtain spatially localized selectivity maps in the two-dimensional frequency domain. Our findings are as follows. (1) The new LSRC method allows measurements of two-dimensional frequency tunings and their spatial extent even for cells with substantial nonlinearity. (2) A small subset of neurons shows spatial inhomogeneity in the two-dimensional frequency tunings. (3) In addition to facilitatory response profiles, we can also visualize suppressive profiles localized both in space and spatial frequency domains. Our results suggest that the new analysis technique can be a powerful tool for measuring visual response profiles that contain inhomogeneity in space, as well as for studying neurons with substantial nonlinearities. These features make the method particularly suitable for studying response profiles of neurons in early as well as intermediate extrastriate visual areas.

Artificial boundary conditions for parabolic Volterra integro-differential equations on unbounded two-dimensional domains

In this paper we study the numerical solution of parabolic Volterra integro-differential equations on certain unbounded two-dimensional spatial domains. The method is based on the introduction of a feasible artificial boundary and the derivation of corresponding artificial (fully transparent) boundary conditions. Two examples illustrate the application and numerical performance of the method.

Solvation of Coumarin 314 at Water/Air Interfaces Containing Anionic Surfactants. I. Low Coverage

Through the use of molecular dynamics techniques, we analyze equilibrium and dynamical aspects of the solvation of Coumarin 314 adsorbed at water/air interfaces in the presence of sodium dodecyl sulfate surfactant molecules. Three different coverages in the submonolayer regime were considered, 500, 250, and 100 A(2)/SDS molecule. The surfactant promotes two well-differentiated solvation environments, which can be clearly distinguished in terms of their structures for the largest surfactant coverage considered. The first one is characterized by the probe lying adjacent or exterior to two-dimensional spatial domains formed by clustered surfactant molecules. A second type of solvation environment is found in which the coumarin appears embedded within compact surfactant domains. Equilibrium and dynamical aspects of the interfacial orientation of the probe are investigated. Our results show a gradual transition from parallel to perpendicular dipolar alignment of the probe with respect to the interface as the concentration of surfactant rho(s) increases. The presence of the surfactant leads to an increase in the roughness and in the characteristic width of the water/air interface. These modifications are also manifested by the decorrelation times for the probe reorientational dynamics, which become progressively slower with rho(s) in both solvation states, although much more pronounced for the embedded ones. The dynamical characteristics of the solvation responses of the charged interfaces are also analyzed, and the implications of our findings to the interpretation of available experimental measurements are discussed.

Traveling wave solutions of a reaction diffusion model for competing pioneer and climax species

Presented is a reaction-diffusion model for the interaction of pioneer and climax species. For certain parameters the system exhibits bistability and traveling wave solutions. Specifically, we show that when the climax species diffuses at a slow rate there are traveling wave solutions which correspond to extinction waves of either the pioneer or climax species. A leading order analysis is used in the one-dimensional spatial case to estimate the wave speed sign that determines which species becomes extinct. Results of these analyses are then compared to numerical simulations of wave front propagation for the model on one and two-dimensional spatial domains. A simple mechanism for harvesting is also introduced.

Modelling Population Growth with Delayed Nonlocal Reaction in 2-Dimensions

In this paper, we consider the population growth of a single species living in a two-dimensional spatial domain. New reaction-difusion equation models with delayed nonlocal reaction are developed in two-dimensional bounded domains combining diferent boundary conditions. The important feature of the models is the reflection of the joint efect of the difusion dynamics and the nonlocal maturation delayed efect. We consider and ana- lyze numerical solutions of the mature population dynamics with some wellknown birth functions. In particular, we observe and study the occurrences of asymptotically stable steady state solutions and periodic waves for the two-dimensional problems with nonlocal delayed reaction. We also investigate numerically the efects of various parameters on the period, the peak and the shape of the periodic wave as well as the shape of the asymptotically stable steady state solution.

BIFURCATIONS IN A HUMAN MIGRATION MODEL OF SCHEURLE–SEYDEL TYPE-I: TURING BIFURCATION

In this paper we consider a model for the behavior of students in graduate programs at neighboring universities which is a modified form of the model proposed by [Scheurle & Seydel, 2000], and observe that the stationary solution of this two-component system becomes unstable in the presence of diffusion. We assume that both types of individuals are continuously distributed throughout a bounded two-dimensional spatial domain of two types (regular hexagon and rhombus), across whose boundaries there is no migration, and which simultaneously undergo simple (Fickian) diffusion but the spatial flow is influenced not only by its own but also by the other's density (cross diffusion). We will show that at a critical value of a parameter a Turing bifurcation takes place: a spatially nonhomogenous solution (pattern) arises.

A nonlinear stability analysis of pattern formation in thin liquid films

The development of spontaneous stationary equilibrium patterns in thin liquid films is investigated by means of a hexagonal-planform weakly nonlinear stability analysis applied to the appropriate governing evolution equation for this phenomenon. In the long-wavelength limit the mathematical system modeling the liquid film can be reduced to such a single nonlinear partial differential time-evolution equation describing the layer thickness on an unbounded two-dimensional spatial domain and including the effects of gravity, intermolecular forces, and temperature-dependent surface tension. The main result of this analysis is that supercritical equilibrium patterns can occur for an interval of mean layer thickness with subcritical rupture occurring outside that interval. These patterns consist of surface ridges and hexagonal network-like cells or close-packed configurations of nanodroplets separated by relatively flat ultra thin films. In particular those morphological phase separation patterns are generated by the coupling between the long-range attractive and short-range repulsive intermolecular forces with cells being stable for the thicker layers; nanodroplets, for the thinner ones; and ridges, for layers of intermediate thickness. These theoretical predictions are in accord with both relevant experimental evidence involving thin liquid polymer, crystal, and metal films coating a solid substrate and numerical simulations of similar model equations as well as being consistent with dewetting-type rupture occurring for such situations by hole formation in relatively thick layers but by drop formation in thinner ones.

Identification of material damage in two-dimensional domains using theSQUID-based nondestructive evaluation system

Problems on the identification of two-dimensional spatial domains arising in thedetection and characterization of material damage are considered. Forelectromagnetic nondestructive evaluation systems, observations of the magneticflux from the front surface are used in a output least-squares approach. Parameterestimation techniques based on the method of mappings are discussed andapproximation schemes are developed applying a finite-element Galerkinapproach. Theoretical convergence results for computational techniques are givenand results are applied to numerical experiments to demonstrate the efficacy ofthe proposed schemes.

Competing spatial and temporal instabilities in a globally coupled bistable semiconductor system near a codimension-two bifurcation

We study complex spatiotemporal dynamics in a globally coupled bistable reaction-diffusion model on a two-dimensional spatial domain. It is demonstrated that complex behavior appears near a codimension-two bifurcation point due to the competition of spatial and temporal instabilities. We derive sufficient conditions for the appearance of mixed spatiotemporal modes, and clarify the origin of a menagery of complex dynamics, such as periodic and chaotic oscillations of current filaments, low-dimensional spatiotemporal chaos including a Shil'nikov attractor, and periodic back-and-forth motion of current density fronts. Such dynamics is found in a wide range of domain sizes for square and rectangular domains. The type of dynamics is sensitive to small variations in the domain shape. We discuss and explain the differences between spatiotemporal dynamics on one-dimensional and two-dimensional domains.

Optical Pattern Recognition Experiments of Walsh Spatial Frequency Domain Filtering Method

An optical parallel processor for an image recognition system using microoptical devices has been developed. This system expands input images to coefficients of a two-dimensional Walsh spatial frequency domain and discriminates them by performing the inner product to the reference filter. In this paper, we demonstrate parallel optical discrimination of Arabic numerals using this method. First, the mechanism of this recognition method is presented. Next, the reference filter is designed. In order to implement the reference filter by using the microoptical devices, the fabrication problems were solved. Finally, the recognition system is constructed and effects of deformation on the input image are experimentally evaluated.

Optimal sensor location for parameter estimation of distributed processes

The problem under consideration is to plan sensor movements in prescribed feasible regions in such a way as to maximize the accuracy of parameter identification of a distributed system defined in a two-dimensional spatial domain. A general functional defined on the Fisher information matrix is used as the design criterion. Central to the approach is the formulation of the problem as an optimal control one, possibly with state inequality constraints. Its solution is obtained numerically with the use of a method of successive linearizations which is capable of handling various constraints imposed on the sensors' motions. Simple numerical examples are included which clearly demonstrate the ideas and trends presented in the paper.

Unravelling the Turing bifurcation using spatially varying diffusion coefficients

. The Turing bifurcation is the basic bifurcation generating spatial pattern, and lies at the heart of almost all mathematical models for patterning in biology and chemistry. In this paper the authors determine the structure of this bifurcation for two coupled reaction diffusion equations on a two-dimensional square spatial domain when the diffusion coefficients have a small explicit variation in space across the domain. In the case of homogeneous diffusivities, the Turing bifurcation is highly degenerate. Using a two variable perturbation method, the authors show that the small explicit spatial inhomogeneity splits the bifurcation into two separate primary and two separate secondary bifurcations, with all solution branches distinct. This splitting of the bifurcation is more effective than that given by making the domain slightly rectangular, and shows clearly the structure of the Turing bifurcation and the way in which the! var ious solution branches collapse together as the spatial variation is reduced. The authors determine the stability of the solution branches, which indicates that several new phenomena are introduced by the spatial variation, including stable subcritical striped patterns, and the possibility that stable stripes lose stability supercritically to give stable spotted patterns..