Abstract: The diffusion-reaction equations are important Partial Differential Equations in mathematical biology, material science, physics, and so on. However, finding efficient numerical methods for diffusion-reaction systems on curved surfaces is still an important and difficult problem. The purpose of this paper is to present a convergent geometric method for solving the reaction-diffusion equations on closed surfaces by an O(r)-LTL configuration method. The O(r)-LTL configuration method combining the local tangential lifting technique and configuration equations is an effective method to estimate differential quantities on curved surfaces. Since estimating the Laplace-Beltrami operator is an important task for solving the reaction-diffusion equations on surfaces, we use the local tangential lifting method and a generalized finite difference method to approximate the Laplace-Beltrami operators and we solve this reaction-diffusion system on closed surfaces. Our method is not only conceptually simple, but also easy to implement.
Abstract: In population dynamics the study of both, the
abundance and the spatial distribution of the populations in a
given habitat, is a fundamental issue a From ecological point of
view, the determination of the factors influencing such changes
involves important problems. In this paper a mathematical model to
describe the temporal dynamic and the spatiotemporal dynamic of the
interaction of three populations (pollinators, plants and herbivores) is
presented. The study we present is carried out by stages: 1. The
temporal dynamics and 2. The spatio-temporal dynamics. In turn,
each of these stages is developed by considering three cases which
correspond to the dynamics of each type of interaction. For instance,
for stage 1, we consider three ODE nonlinear systems describing
the pollinator-plant, plant-herbivore and plant-pollinator-herbivore,
interactions, respectively. In each of these systems different types of
dynamical behaviors are reported. Namely, transcritical and pitchfork
bifurcations, existence of a limit cycle, existence of a heteroclinic
orbit, etc. For the spatiotemporal dynamics of the two mathematical
models a novel factor are introduced. This consists in considering
that both, the pollinators and the herbivores, move towards those
places of the habitat where the plant population density is high.
In mathematical terms, this means that the diffusive part of the
pollinators and herbivores equations depend on the plant population
density. The analysis of this part is presented by considering pairs of
populations, i. e., the pollinator-plant and plant-herbivore interactions
and at the end the two mathematical model is presented, these models
consist of two coupled nonlinear partial differential equations of
reaction-diffusion type. These are defined on a rectangular domain
with the homogeneous Neumann boundary conditions. We focused
in the role played by the density dependent diffusion term into
the coexistence of the populations. For both, the temporal and
spatio-temporal dynamics, a several of numerical simulations are
included.
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.
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.
Abstract: In this paper, for the understanding of the phytoplankton dynamics in marine ecosystem, a susceptible and an infected class of phytoplankton population is considered in spatiotemporal domain.
Here, the susceptible phytoplankton is growing logistically and the
growth of infected phytoplankton is due to the instantaneous Holling
type-II infection response function. The dynamics are studied in terms of the local and global stabilities for the system and further
explore the possibility of Hopf -bifurcation, taking the half saturation period as (i.e., ) the bifurcation parameter in temporal domain.
It is also observe that the reaction diffusion system exhibits spatiotemporal
chaos and pattern formation in phytoplankton dynamics,
which is particularly important role play for the spatially extended phytoplankton system. Also the effect of the diffusion coefficient
on the spatial system for both one and two dimensional case is obtained. Furthermore, we explore the higher-order stability analysis
of the spatial phytoplankton system for both linear and no-linear system. Finally, few numerical simulations are carried out for pattern
formation.
Abstract: In this study, the density dependent nonlinear reactiondiffusion
equation, which arises in the insect dispersal models, is
solved using the combined application of differential quadrature
method(DQM) and implicit Euler method. The polynomial based
DQM is used to discretize the spatial derivatives of the problem. The
resulting time-dependent nonlinear system of ordinary differential
equations(ODE-s) is solved by using implicit Euler method. The
computations are carried out for a Cauchy problem defined by a onedimensional
density dependent nonlinear reaction-diffusion equation
which has an exact solution. The DQM solution is found to be in a
very good agreement with the exact solution in terms of maximum
absolute error. The DQM solution exhibits superior accuracy at large
time levels tending to steady-state. Furthermore, using an implicit
method in the solution procedure leads to stable solutions and larger
time steps could be used.
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.
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.
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.
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.
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.
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.