**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**12

# Search results for: reaction-diffusion

##### 12 A Study of Numerical Reaction-Diffusion Systems on Closed Surfaces

**Authors:**
Mei-Hsiu Chi,
Jyh-Yang Wu,
Sheng-Gwo Chen

**Abstract:**

**Keywords:**
Close surfaces,
high-order approach,
numerical solutions,
reaction-diffusion systems.

##### 11 A Qualitative Description of the Dynamics in the Interactions between Three Populations: Pollinators, Plants, and Herbivores

**Authors:**
Miriam Sosa-Díaz,
Faustino Sánchez-Garduño

**Abstract:**

**Keywords:**
Bifurcation,
heteroclinic orbits,
steady state,
traveling
wave.

##### 10 Analysis of a Spatiotemporal Phytoplankton Dynamics: Higher Order Stability and Pattern Formation

**Authors:**
Randhir Singh Baghel,
Joydip Dhar,
Renu Jain

**Abstract:**

**Keywords:**
Phytoplankton dynamics,
Reaction-diffusion system,
Local stability,
Hopf-bifurcation,
Global stability,
Chaos,
Pattern Formation,
Higher-order stability analysis.

##### 9 A Model to Study the Effect of Excess Buffers and Na+ Ions on Ca2+ Diffusion in Neuron Cell

**Authors:**
Vikas Tewari,
Shivendra Tewari,
K. R. Pardasani

**Abstract:**

Calcium is a vital second messenger used in signal transduction. Calcium controls secretion, cell movement, muscular contraction, cell differentiation, ciliary beating and so on. Two theories have been used to simplify the system of reaction-diffusion equations of calcium into a single equation. One is excess buffer approximation (EBA) which assumes that mobile buffer is present in excess and cannot be saturated. The other is rapid buffer approximation (RBA), which assumes that calcium binding to buffer is rapid compared to calcium diffusion rate. In the present work, attempt has been made to develop a model for calcium diffusion under excess buffer approximation in neuron cells. This model incorporates the effect of [Na+] influx on [Ca2+] diffusion,variable calcium and sodium sources, sodium-calcium exchange protein, Sarcolemmal Calcium ATPase pump, sodium and calcium channels. The proposed mathematical model leads to a system of partial differential equations which have been solved numerically using Forward Time Centered Space (FTCS) approach. The numerical results have been used to study the relationships among different types of parameters such as buffer concentration, association rate, calcium permeability.

**Keywords:**
Excess buffer approximation,
Na+ influx,
sodium calcium exchange protein,
sarcolemmal calcium atpase pump,
forward time centred space.

##### 8 Stability Analysis of Impulsive BAM Fuzzy Cellular Neural Networks with Distributed Delays and Reaction-diffusion Terms

**Authors:**
Xinhua Zhang,
Kelin Li

**Abstract:**

In this paper, a class of impulsive BAM fuzzy cellular neural networks with distributed delays and reaction-diffusion terms is formulated and investigated. By employing the delay differential inequality and inequality technique developed by Xu et al., some sufficient conditions ensuring the existence, uniqueness and global exponential stability of equilibrium point for impulsive BAM fuzzy cellular neural networks with distributed delays and reaction-diffusion terms are obtained. In particular, the estimate of the exponential convergence rate is also provided, which depends on system parameters, diffusion effect and impulsive disturbed intention. It is believed that these results are significant and useful for the design and applications of BAM fuzzy cellular neural networks. An example is given to show the effectiveness of the results obtained here.

**Keywords:**
Bi-directional associative memory,
fuzzy cellular neuralnetworks,
reaction-diffusion,
delays,
impulses,
global exponentialstability.

##### 7 Haar Wavelet Method for Solving Fitz Hugh-Nagumo Equation

**Authors:**
G.Hariharan,
K.Kannan

**Abstract:**

In this paper, we develop an accurate and efficient Haar wavelet method for well-known FitzHugh-Nagumo equation. The proposed scheme can be used to a wide class of nonlinear reaction-diffusion equations. The power of this manageable method is confirmed. Moreover the use of Haar wavelets is found to be accurate, simple, fast, flexible, convenient, small computation costs and computationally attractive.

**Keywords:**
FitzHugh-Nagumo equation,
Haar wavelet method,
adomain decomposition method,
computationally attractive.

##### 6 Species Spreading due to Environmental Hostility, Dispersal Adaptation and Allee Effects

**Authors:**
Sanjeeva Balasuriya

**Abstract:**

A phenomenological model for species spreading which incorporates the Allee effect, a species- maximum attainable growth rate, collective dispersal rate and dispersal adaptability is presented. This builds on a well-established reaction-diffusion model for spatial spreading of invading organisms. The model is phrased in terms of the “hostility" (which quantifies the Allee threshold in relation to environmental sustainability) and dispersal adaptability (which measures how a species is able to adapt its migratory response to environmental conditions). The species- invading/retreating speed and the sharpness of the invading boundary are explicitly characterised in terms of the fundamental parameters, and analysed in detail.

**Keywords:**
Allee effect,
dispersal,
migration speed,
diffusion,
invasion.

##### 5 Solution of Density Dependent Nonlinear Reaction-Diffusion Equation Using Differential Quadrature Method

**Authors:**
Gülnihal Meral

**Abstract:**

**Keywords:**
Density Dependent Nonlinear Reaction-Diffusion Equation,
Differential Quadrature Method,
Implicit Euler Method.

##### 4 Stability Analysis of Impulsive Stochastic Fuzzy Cellular Neural Networks with Time-varying Delays and Reaction-diffusion Terms

**Authors:**
Xinhua Zhang,
Kelin Li

**Abstract:**

In this paper, the problem of stability analysis for a class of impulsive stochastic fuzzy neural networks with timevarying delays and reaction-diffusion is considered. By utilizing suitable Lyapunov-Krasovskii funcational, the inequality technique and stochastic analysis technique, some sufficient conditions ensuring global exponential stability of equilibrium point for impulsive stochastic fuzzy cellular neural networks with time-varying delays and diffusion are obtained. In particular, the estimate of the exponential convergence rate is also provided, which depends on system parameters, diffusion effect and impulsive disturbed intention. It is believed that these results are significant and useful for the design and applications of fuzzy neural networks. An example is given to show the effectiveness of the obtained results.

**Keywords:**
Exponential stability,
stochastic fuzzy cellular neural networks,
time-varying delays,
impulses,
reaction-diffusion terms.

##### 3 Symmetry Breaking and the Emergence of Branching Structures in Morphogenesis: Minimal Conditions and Mechanical Interactions between Cells

**Authors:**
M. Margarida Costa,
Jorge Simão

**Abstract:**

The minimal condition for symmetry breaking in morphogenesis of cellular population was investigated using cellular automata based on reaction-diffusion dynamics. In particular, the study looked for the possibility of the emergence of branching structures due to mechanical interactions. The model used two types of cells an external gradient. The results showed that the external gradient influenced movement of cell type-I, also revealed that clusters formed by cells type-II worked as barrier to movement of cells type-I.

**Keywords:**
Morphogenesis,
branching structures,
symmetrybreaking.

##### 2 Modeling and Simulating Reaction-Diffusion Systems with State-Dependent Diffusion Coefficients

**Authors:**
Paola Lecca,
Lorenzo Dematte,
Corrado Priami

**Abstract:**

The present models and simulation algorithms of intracellular stochastic kinetics are usually based on the premise that diffusion is so fast that the concentrations of all the involved species are homogeneous in space. However, recents experimental measurements of intracellular diffusion constants indicate that the assumption of a homogeneous well-stirred cytosol is not necessarily valid even for small prokaryotic cells. In this work a mathematical treatment of diffusion that can be incorporated in a stochastic algorithm simulating the dynamics of a reaction-diffusion system is presented. The movement of a molecule A from a region i to a region j of the space is represented as a first order reaction Ai k- ! Aj , where the rate constant k depends on the diffusion coefficient. The diffusion coefficients are modeled as function of the local concentration of the solutes, their intrinsic viscosities, their frictional coefficients and the temperature of the system. The stochastic time evolution of the system is given by the occurrence of diffusion events and chemical reaction events. At each time step an event (reaction or diffusion) is selected from a probability distribution of waiting times determined by the intrinsic reaction kinetics and diffusion dynamics. To demonstrate the method the simulation results of the reaction-diffusion system of chaperoneassisted protein folding in cytoplasm are shown.

**Keywords:**
Reaction-diffusion systems,
diffusion coefficient,
stochastic simulation algorithm.

##### 1 Stochastic Simulation of Reaction-Diffusion Systems

**Authors:**
Paola Lecca,
Lorenzo Dematte

**Abstract:**

Reactiondiffusion systems are mathematical models that describe how the concentration of one or more substances distributed in space changes under the influence of local chemical reactions in which the substances are converted into each other, and diffusion which causes the substances to spread out in space. The classical representation of a reaction-diffusion system is given by semi-linear parabolic partial differential equations, whose general form is ÔêétX(x, t) = DΔX(x, t), where X(x, t) is the state vector, D is the matrix of the diffusion coefficients and Δ is the Laplace operator. If the solute move in an homogeneous system in thermal equilibrium, the diffusion coefficients are constants that do not depend on the local concentration of solvent and of solutes and on local temperature of the medium. In this paper a new stochastic reaction-diffusion model in which the diffusion coefficients are function of the local concentration, viscosity and frictional forces of solvent and solute is presented. Such a model provides a more realistic description of the molecular kinetics in non-homogenoeus and highly structured media as the intra- and inter-cellular spaces. The movement of a molecule A from a region i to a region j of the space is described as a first order reaction Ai k- → Aj , where the rate constant k depends on the diffusion coefficient. Representing the diffusional motion as a chemical reaction allows to assimilate a reaction-diffusion system to a pure reaction system and to simulate it with Gillespie-inspired stochastic simulation algorithms. The stochastic time evolution of the system is given by the occurrence of diffusion events and chemical reaction events. At each time step an event (reaction or diffusion) is selected from a probability distribution of waiting times determined by the specific speed of reaction and diffusion events. Redi is the software tool, developed to implement the model of reaction-diffusion kinetics and dynamics. It is a free software, that can be downloaded from http://www.cosbi.eu. To demonstrate the validity of the new reaction-diffusion model, the simulation results of the chaperone-assisted protein folding in cytoplasm obtained with Redi are reported. This case study is redrawing the attention of the scientific community due to current interests on protein aggregation as a potential cause for neurodegenerative diseases.

**Keywords:**
Reaction-diffusion systems,
Fick's law,
stochastic simulation algorithm.