Reaction-diffusion Equation Model Research Articles

Reaction–diffusion equation model in the electrical double layer for exploring the response signal in electrochemical sensing

Electron transfer in the electrode–solution interface is responsible for the signal in the electrochemical sensor. However, the Electrical Double-Layer (EDL) characteristic is usually ignored in signal modeling. Here, we constructed a model based on the reaction–diffusion equation to describe the dynamics in the EDL during sensing. With this model, many common experimental phenomena, such as transient/steady-state current variation or detection nonlinearization, can be unraveled by exploring the effects of position, kinetics proportionality coefficient (k), and reaction order ([Formula: see text]). We also put forward some suggestions to improve the stability and linear detection range for future sensor design.

Dynamical behavior of a temporally discrete non-local reaction-diffusion equation on bounded domain

This paper focuses on the study of global dynamics of a class of temporally discrete non-local reaction-diffusion equations on bounded domains. Similar to classical reaction diffusion equations and integro-difference equations, temporally discrete reaction-diffusion equations can also be used to describe the dispersal phenomena in population dynamics. In this paper, we first derived a temporally discrete reaction diffusion equation model with time delay and nonlocal effects to model the evolution of a single species population with age-structured located in a bounded domain. By establishing a new maximum principle and applying the monotone iteration method, the global stabilities of the trivial solution and the positive steady state solution are obtained respectively under some appropriate assumptions.

Threshold dynamics of a reaction–diffusion equation model for cholera transmission with waning vaccine-induced immunity and seasonality

Threshold dynamics of a reaction–diffusion equation model for cholera transmission with waning vaccine-induced immunity and seasonality

Adaptive Extraction of Oil Painting Texture Features Based on Reaction Diffusion Equation

The oil painting retrieval technology based on the reaction diffusion equation has attracted widespread attention in the fields of oil painting processing and pattern recognition. The description and extraction of oil painting information and the classification method of oil paintings are two important processes in content-based oil painting retrieval. Inspired by the restoration and decomposition functional model of equal oil painting, we propose a reaction diffusion equation model. The new model contains two reaction diffusion equations with different principal parts. One principal part is total variation diffusion, which is used to remove noise. The other main part is thermal diffusion, which is used to modify the source term of the denoising reaction-diffusion equation to achieve the effect of protecting the texture of the oil painting. The interaction of the two reaction-diffusion equations finally achieves denoising while maintaining the boundaries and textures. Under the framework of the above reaction diffusion equation model, we introduce Laplace flow to replace the original total variation flow, so that the new denoising reaction diffusion equation combines the isotropic diffusion and total variation flow of the thermal reaction diffusion equation to achieve the effect of adaptive theoretical research. Using regularization methods and methods, we, respectively, get the well-posedness of the two model solutions, which provides the necessary preparation for numerical calculations. Based on the statistical theory and classification principles of support vector machines, combined with the characteristics of oil painting classification, the research and analysis are carried out from the three important aspects of kernel function, training algorithm, and multiclass classifier algorithm that affect the classification effect and speed. Numerical experiments show that the given filter model has a better processing effect on images with different types and different degrees of noise pollution. On this basis, an oil painting classification system based on texture features is designed, combined with an improved gray-level cooccurrence matrix algorithm and a multiclass support vector machine classification model, to extract, train, and classify oil paintings. Experiments with three types of oil paintings prove that the system can achieve a good oil painting classification effect. Different from the original model, the new model is based on the framework of reaction-diffusion equations. In addition, the new model has good effects in removing step effects, maintaining boundaries and denoising, especially in maintaining texture.

Traveling Waves Solutions for Delayed Temporally Discrete Non-Local Reaction-Diffusion Equation

This paper deals with the existence of traveling wave solutions to a delayed temporally discrete non-local reaction diffusion equation model, which has been derived recently for a single species with age structure. When the birth function satisfies monotonic condition, we obtained the traveling wavefront by using upper and lower solution methods together with monotonic iteration techniques. Otherwise, without the monotonicity assumption for birth function, we constructed two auxiliary equations. By means of the traveling wavefronts of the auxiliary equations, using the Schauder’ fixed point theorem, we proved the existence of a traveling wave solution to the equation under consideration with speed c>c*, where c*>0 is some constant. We found that the delayed temporally discrete non-local reaction diffusion equation possesses the dynamical consistency with its time continuous counterpart at least in the sense of the existence of traveling wave solutions.

EDGE DETECTION ALGORITHM FOR COLOR IMAGES BASED ON THE REACTION-DIFFUSION EQUATION AND THE CELLULAR NEURAL NETWORK MODEL

EDGE DETECTION ALGORITHM FOR COLOR IMAGES BASED ON THE REACTION-DIFFUSION EQUATION AND THE CELLULAR NEURAL NETWORK MODEL

Time periodic reaction–diffusion equations for modeling 2-LTR dynamics in HIV-infected patients

Considering multiple effects of spatial factors (spatial heterogeneity and mobility) and temporal heterogeneity (the periodic administration of the drug) in HIV infection and treatment, we formulate a time periodic reaction–diffusion equation model to study the HIV viral dynamics. The basic reproduction number R0 is derived and the global threshold dynamics is explored in terms of R0. When R0<1, the infection-free periodic solution is globally attractive. When R0>1, the model system is uniformly persistent. The global stability analysis is also performed in the critical case of R0=1 with spatial heterogeneity but without time periodicity. The global asymptotic stability of the infection steady state for the spatial homogeneous model is discussed. Furthermore, numerical simulations reveal that (i) the value of R0 depends on spatial heterogeneity, which indicates that different human individuals or different organs in one individual may have different infection states even if the drug efficacy is the same; and (ii) the 2-LTR level at the steady state may increase and the steady-state viral load may decrease dramatically if raltegravir is administered from the beginning of thetreatment.

Complex dynamics of a time periodic nonlocal and time-delayed model of reaction–diffusion equations for modeling CD4[formula omitted] T cells decline

Pyroptosis, an intensively inflammatory form of programmed cell death triggered during abortive HIV infection, is associated with the release of inflammatory cytokines. The inflammatory cytokines can attract more CD4 + T cells to be infected. Based on the new biological perspective, a time periodic reaction–diffusion equation model with spatial heterogeneity and latent period is developed to investigate whether or not pyroptosis can explain CD4+ T cells decline during HIV infection. Threshold dynamics is explored in terms of the basic reproduction number ℛ0. It is shown that the infection-free periodic solution is globally attractive if ℛ0<1, while there exists positive infection periodic solution and the virus is uniformly persistent if ℛ0>1, which might be a new finding for viral infection dynamical models. Theoretical analyses and simulations for the spatially homogeneous model demonstrate rich dynamics, including the occurrence of Turing instability, which may result in Hopf bifurcation and spatially inhomogeneous pattern formations. It turns out that the new model issues some new challenges due to the enhancement infection term in the global stability problem and Turing instability analysis for the high dimensional system. Our results reveal that the inflammatory cytokines can make the CTLs level increase, which is a new phenomenon not presented in the existing literature.

On Impulsive Reaction-Diffusion Models in Higher Dimensions

We formulate a general impulsive reaction-diffusion equation model to describe the population dynamics of species with distinct reproductive and dispersal stages. The seasonal reproduction is modeled by a discrete-time map, while the dispersal is modeled by a reaction-diffusion partial differential equation. Study of this model requires a simultaneous analysis of the differential equation and the recurrence relation. When boundary conditions are hostile we provide critical domain results showing how extinction versus persistence of the species arises, depending on the size and geometry of the domain. We show that there exists an extreme volume size such that if $|\Omega|$ falls below this size the species is driven extinct, regardless of the geometry of the domain. To construct such extreme volume sizes and critical domain sizes, we apply Schwarz symmetrization rearrangement arguments, the classical Rayleigh--Faber--Krahn inequality, and the spectrum of uniformly elliptic operators. The critical domain re...

Repulsion effect on superinfecting virions by infected cells.

In this paper, the repulsion effect of superinfecting virion by infected cells is studied by a reaction diffusion equation model for virus infection dynamics. In this model, the diffusion of virus depends not only on its concentration gradient but also on the concentration of infected cells. The basic reproduction number, linear stability of steady states, spreading speed and existence of traveling wave solutions for the model are discussed. It is shown that viruses spread more rapidly with the repulsion effect of infected cells on superinfecting virions, than with random diffusion only. For our model, the spreading speed of free virus is not consistent with the minimal traveling wave speed. With our general model, numerical computations of the spreading speed show that the repulsion of superinfecting virion promotes the spread of virus, which confirms, not only qualitatively but also quantitatively, the experimental result of Doceul et al. (Science 327:873-876, 2010).

MODELING FLUCTUATIONS IN A MINIMAL PLANKTON MODEL: ROLE OF SPATIAL HETEROGENEITY AND STOCHASTICITY

The present paper studies the naturally observed phenomenon of population fluctuation in the context of a minimal plankton model. The analysis of the basic model reveals asymptotic stable behavior that is unable to explain any kind of population outburst. We introduce the nonuniform spatial distribution of plankton by considering physical diffusion of the species concerned. The resulting reaction-diffusion equation model is first analyzed with constant diffusivity and then with variable diffusivity for the zooplankton. The model with variable diffusivity is analyzed by using Floquet theory. In both cases, the model is seen to exhibit stable behavior. Finally, we study another characteristic feature of any ecological population, namely, environmental fluctuations. We achieve this by perturbing the growth rate of phytoplankton and death rate of zooplankton by using colored noise. The resulting model is analyzed by evaluating the spectral density functions. It is observed that the stochastic model can generate a large fluctuation in population concentration for high amplitude driving forces.

Pattern Dynamics in a Two-Dimensional Gas Discharge System

Reaction-diffusion equation model for a 2-dimensional gas discharge system is introduced in close relationship with the recent experiment by Nasuno. The model shows formation of spots, molecule-like organization of a cluster of spots, moving of spots, and intermittent collapse and recreation of spots, in agreement with experiments. The pattern dynamics are classified into distributed, localized, and moving spots, and periodic wave pattern. The phase diagram displayed against the current and pressure has some agreement with that observed in the experiment. Dynamic change of spot numbers as well as the structure of the spot cluster is studied in terms of the collective flow of charge, and discussed as chaotic itinerancy.

Merging traveling waves for the porous-Fisher's equation

We study a reaction-diffusion equation model for population dynamics. By focusing on the diffusive behavior expected in a population that seeks to avoid over-crowding, we derive a nonlinear-diffusion porous-Fisher's equation. Using explicit traveling wave solutions, initially-separated, expanding populations are studied as they first coalesce. The nonlinear interactions of the merging populations are examined using perturbation theory and the method of matched asymptotic expansions. Results are also extended to the axisymmetric case.

Formation of band-type electric patterns in Characean cells

It is observed by experiments that band patterns of alternating acid and alkaline zones are formed on the surface of Characean cells under illumination. In order to understand theoretically such pattern formation, we employ a reaction-diffusion equation model with activator-inhibitor interaction. We study the existence problem of band patterns and hysteresis phenomena such as patterns appearing and disappearing with changing light intensity by using singular perturbation methods.

Inertial manifold for a reaction diffusion equation model of competition in a chemostat

AbstractThe existence of an inertial manifold for a reaction-diffusion equation model of the chemostat is established.

Higher-dimensional localized patterns in excitable media

The excitable reaction-diffusion equation model of the form ετu t=ε 2▿ 2u+ƒ(u)–v , v t=▿ 2v+u−γv is considered. When ƒ(u) is assumed to be of McKean's piecewise linear type, the interfacial approach can be applied to the stability of various localized patterns in higher-dimensional spaces. It is shown that a band-shaped localized pattern is destabilized into a zig-zag mode or a varicose mode and that a disk-shaped localized pattern is destabilized into static modes and, when τ is small, into an oscillatory mode like a “breather motion”. Numerical simulations are performed to confirm such destabilizations for more general nonlinear functions ƒ(u).