System Of Reaction-diffusion Equations Research Articles

Non-polynomial spline method for computational study of reaction diffusion system

This work addresses an efficient and new numerical technique utilizing non-polynomial splines to solve system of reaction diffusion equations (RDS). These system of equations arise in pattern formation of some special biological and chemical reactions. Different types of RDS are in the form of spirals, hexagons, stripes, and dissipative solitons. Chemical concentrations can travel as waves in reaction-diffusion systems, where wave like behaviour can be seen. The purpose of this research is to develop a stable, highly accurate and convergent scheme for the solution of aforementioned model. The method proposed in this paper utilizes forward difference for time discretization whereas for spatial discretization cubic non-polynomial spline is used to get approximate solution of the system under consideration. Furthermore, stability of the scheme is discussed via Von-Neumann criteria. Different orders of convergence is achieved for the scheme during a theoretical convergence test. Suggested method is tested for performance on various well known models such as, Brusselator, Schnakenberg, isothermal as well as linear models. Accuracy and efficiency of the scheme is checked in terms of relative error (E R ) and L ∞ norms for different time and space step sizes. The newly obtained results are analyzed and compared with those available in literature.

Incorporating different enzyme kinetics in amperometric biosensor for the steady-state conditions: A complete theoretical and numerical approach

The well-defined physical, chemical, and biological interactions that define the behaviour of biosensors are described by nonlinear differential equations. Because these biosensors have so many uses in many domains, it is desirable to represent them mathematically. The enzyme kinetic and the diffusion rate across the enzymatic layer determine the distinct enzymatic processes. The biosensors are modelled by a system of reaction-diffusion equations involving a nonlinear part associated with the reaction rate. The nonlinear equations in this model have been solved analytically (Rajendran-Joy method) and numerically (MATLAB® v2016b software). Approximate steady-state analytical solutions for the substrate and reaction product concentrations and the output current are presented for specific cases of first-order, Michaelis-Menten and ping-pong kinetics. The influence of various parameters on concentrations and currents is discussed. Further, the sensitivity of parameters in current was also analysed. The analytical results are compared with numerically simulated results for various kinetics.

Regulatory disturbances in the dynamical signaling systems of and NO in fibroblasts cause fibrotic disorders.

Studying the calcium dynamics within a fibroblast cell individually has provided only a restricted understanding of its functions. However, research efforts focusing on systems biology approaches for such investigations have been largely neglected by researchers until now. Fibroblast cells rely on signaling from calcium and nitric oxide (NO) to maintain their physiological functions and structural stability. Various studies have demonstrated the correlation between NO and the control of dynamics in cells. However, there is currently no existing model to assess the disruptions caused by various factors in regulatory dynamics, potentially resulting in diverse fibrotic disorders. A mathematical model has been developed to investigate the effects of changes in parameters such as buffer, receptor, sarcoplasmic endoplasmic reticulum -ATPase (SERCA) pump, and source influx on the regulation and dysregulation of spatiotemporal calcium and NO dynamics in fibroblast cells. This model is based on a system of reaction-diffusion equations, and numerical simulations are conducted using the finite element method. Disturbances in key processes related to calcium and nitric oxide, including source influx, buffer mechanism, SERCA pump, and inositol trisphosphate receptor, may contribute to deregulation in the calcium and NO dynamics within fibroblasts. The findings also provide new insights into the extent and severity of disorders resulting from alterations in various parameters, potentially leading to deregulation and the development of fibrotic disease.

Modeling of reaction–desorption process by core–shell particles dispersed in continuously stirred tank reactor (CSTR)

Abstract Transient responses of reaction–desorption process were predicted from mathematical solutions of modeling equations for CSTR (continuously stirred tank reactor) containing core–shell adsorbent particles. Analytical solutions on the core–shell particles were derived for core–shell spherical, cylindrical, and slab-type morphologies, assuming inert-cores. Unlike continuous adsorber, CSTRs for reaction–desorption process containing spherical particles exhibited the slowest reduction rate of concentration of adsorbate, because the amount of adsorbed component on the particles is the largest among three kinds of particle shapes. Factors affecting the transient concentration in bulk medium of reaction–desorption process were investigated by adjusting inert-core thickness, inlet flow rate, initial concentration of reactant in inflow stream, amount of adsorbent, and Thiele modulus. Concentration profile inside the particles as well as average intra-particle concentration could be also predicted for comparison with bulk concentration of CSTR. For non-linear isotherm and non-linear reaction kinetics, concentration of active component could be predicted by solving non-linear coupled differential equation using finite element method. For connected CSTRs in series, systems of reaction-diffusion equations were solved by finite element method to study the effect of number of connected reactors. When the number of reactors was sufficiently large, the reactor system could be approximated to fixed bed reactor for reaction–desorption process.

Stability analysis and simulations of tumor growth model based on system of reaction-diffusion equation in two-dimensions

<abstract><p>In this study, we introduce a novel framework for exploring the dynamics of tumor growth and an evolution model for two-stage carcinogenic mutations in two-dimensions based on a system of reaction-diffusion equations. It is shown theoretically that the system is globally stable in the absence of both delay and diffusion. The inclusion of diffusion does not destabilize the system, while including delay does capture the key elements of how normal cells convert into cancer cells. To further validate these results, several numerical experiments are performed for different parameter values involved in the model equation. These parameter values are chosen in the sense that they have some biological meanings using the steady states of the equilibrium points. For the purpose of simulation, a stable Euler scheme is used for temporal discretization, while a Fourier spectral method is used for space variables, which is a natural choice due to the periodic boundary conditions in the model equation. The numerical simulation results further confirm our theoretical justification.</p></abstract>

Turing Patterns in Networks

The first pattern formation model was proposed by the mathematician Alan M. Turing. This model consists of a system of reaction-diffusion equations that produces stationary patterns by means of the so-called “Turing instability.” In this paper, we found the conditions that the network and the parameters need to fulfill in order to achieve the Turing instability in a particular reaction-diffusion system called the Mimura–Murray model on different network topologies, including some simulations on an innovative kind of network, based on the Wolfram model, that evolves over time, generating interesting topologies that exhibit lattice-like topology. In addition, the equations are solved and simulated in Wolfram Language, and some examples of applications in biology and sociology are presented.

Computation of Turing Bifurcation Normal Form for n -Component Reaction-Diffusion Systems

General expressions are derived for the amplitude equation valid at a Turing bifurcation of a system of reaction-diffusion equations in one spatial dimension, with an arbitrary number of components. The normal form is computed up to fifth order, which enables the detection and analysis of codimension-two points where the criticality of the bifurcation changes. The expressions are implemented within a Python package, in which the user needs to specify only expressions for the reaction kinetics and the values of diffusion constants. The code is augmented with a Mathematica routine to compute curves of Turing bifurcations in a parameter plane and automatically detect codimension-two points. The software is illustrated with examples that show the versatility of the method including a case with cross-diffusion, a higher-order scalar equation and a four-component system.

An amperometric biosensor and its steady state current in the case of substrate and product inhibition: Taylors series method and Adomian decomposition method

In this paper, an amperometric biosensor model with substrate and product inhibition kinetics is analysed. This model is a steady-state system of reaction-diffusion equations with non-linear terms related to non-Michaelis-Menten kinetics of an enzymatic reaction. We present the approximate analytical expression of the substrate and product concentrations using well-founded methods, namely the Taylors series method (TSM) and the Adomian decomposition method (ADM). These methods proved that they fit for all values of parameters in this model. The steady-state biosensor current, biosensor substrate sensitivity and resistance are also discussed. We also present the numerical solution of the described model using MATLAB programming, and it is noted that there is satisfactory agreement in comparing the analytical solution with numerical results for all possible values of parameters. The effects of the parameters, such as inhibition constants, diffusion parameters, bulk concentration and Michaelis-Menten constant on the sensitivity and the resistance of the biosensor are analysed.

Reaction kinetics of amperometric enzyme electrode in various geometries using the Akbari-Ganji method

A kinetic model of amperometric enzyme electrodes of planar, cylindrical and spherical shape factors is discussed. This paper considers the mathematical formulation of the kinetics mechanism in the electrode’s diffusion layer. This model comprises a non-linear system of reaction diffusion equations. Simple and closed-form analytical expressions for the dimensionless concentration of mediator, enzyme, substrate and resultant current are gained using the Akbari-Ganji method. Comparing the analytical and numerical results reveals a satisfactory level of agreement. The influence of the parameter and electrode’s geometry on mediator concentration, enzyme concentration and current are analysed.

Learning system parameters from turing patterns.

The Turing mechanism describes the emergence of spatial patterns due to spontaneous symmetry breaking in reaction-diffusion processes and underlies many developmental processes. Identifying Turing mechanisms in biological systems defines a challenging problem. This paper introduces an approach to the prediction of Turing parameter values from observed Turing patterns. The parameter values correspond to a parametrized system of reaction-diffusion equations that generate Turing patterns as steady state. The Gierer-Meinhardt model with four parameters is chosen as a case study. A novel invariant pattern representation based on resistance distance histograms is employed, along with Wasserstein kernels, in order to cope with the highly variable arrangement of local pattern structure that depends on the initial conditions which are assumed to be unknown. This enables us to compute physically plausible distances between patterns, to compute clusters of patterns and, above all, model parameter prediction based on training data that can be generated by numerical model evaluation with random initial data: for small training sets, classical state-of-the-art methods including operator-valued kernels outperform neural networks that are applied to raw pattern data, whereas for large training sets the latter are more accurate. A prominent property of our approach is that only a single pattern is required as input data for model parameter predicion. Excellent predictions are obtained for single parameter values and reasonably accurate results for jointly predicting all four parameter values.

Mathematical modelling of a mono-enzyme dual amperometric biosensor for enzyme-catalyzed reactions using homotopy analysis and Akbari-Ganji methods

The mathematical model is described in this article as a mono-enzyme dual biosensor. This model involves coupling the enzyme-catalyzed reactions under steady-state conditions for a reaction-diffusion equation describing the substrate and product. Pseudo-experimental responses to chemical mixtures were produced using a computer model of a biocatalytic amperometric biosensor that used a mono-enzyme-catalyzed (nonspecific) competitive conversion of two substrates.The non-linear system of reaction-diffusion equations into an approximate analytical expression is solved by employing the Homotopy Analysis Method (HAM) and Akbari-Ganji Method (AGM) are used. These expressions are compared with numerical simulations carried out using MATLAB. Further, the biosensor's resistance, sensitivity, and current density were analytically expressed.

Theoretical analysis of amperometric response towards PPO-based rotating disk bioelectrodes: Taylors series and hyperbolic function method

A mathematical model for the electroenzymatic process of a rotating disk electrode in which polyphenol oxidase occurs is discussed. The model is related to a system of reaction-diffusion equations that includes a nonlinear factor relating to the enzymatic reaction's Michaelis-Menten kinetics. The approximate analytical techniques (Taylor's series and Hyperbolic function) are used to solve the nonlinear differential equations describing the diffusion-coupled enzymatic reactions. For all parameter values γE(=γL2) and α1(αS∞). Closed analytical equations for the concentration of the substrate, the concentration of the product, and the related current response have been obtained. These analytical results are in good agreement when compared to simulation results ( Matlab program). The findings are reliable over the whole solution domain.

Linking chemical signaling to mechanical components to determine tissue shape in the Drosophila wing imaginal disc

During organogenesis in the Drosophila wing imaginal disc, the regulation of epithelial cell height and curvature is crucial in developing correct tissue shapes. This requires the interplay between mechanical forces and morphogen-mechanogen pathways, at both the cell and tissue levels. Morphogens, such as Decapentaplegic (Dpp), regulate cell growth and division, as well as mechanogen activity. Mechanogens, such as RhoGTPases, are small diffusible molecules that regulate mechanical components, such as actin and myosin, to coordinate cell shape and tissue geometry. Even though the effect of morphogens in regulating mechanogens is critical for proper tissue formation, insufficient work has been done to understand this in the context of epithelial organogenesis. In this study, a combination of experimental and mathematical modeling approaches are used to study the linkage between Dpp and Rho1, Cdc42 in the wing imaginal disc. By using experiments, a new regulation between Dpp/pMad and Cdc42 has been identified, as well as the interaction between Cdc42 and Rho1. A mathematical model has been developed by using a system of reaction-diffusion equations to model Dpp, Rho1, and Cdc42 dynamics, as well as the newly identified regulations. In particular, the signaling model uses a discrete node structure from a multi-scale subcellular element model. We apply this model to investigate the robustness of the proposed networks, in addition to the effect of tissue geometry on the concentration profiles of both the morphogens and mechanogens.

Stability of traveling wave solutions for a nonlocal Lotka-Volterra model.

In this paper, we studied the stability of traveling wave solutions of a two-species Lotka-Volterra competition model in the form of a coupled system of reaction diffusion equations with nonlocal intraspecific and interspecific competitions in space at times. First, the uniform upper bounds for the solutions of the model was proved. By using the anti-weighted method and the energy estimates, the asymptotic stability of traveling waves with large wave speeds of the system was established.

A control strategy for the sterile insect technique using exponentially decreasing releases to avoid the hair-trigger effect

In this paper, we introduce a control strategy for applying the Sterile Insect Technique (SIT) to eliminate the population ofAedesmosquitoes which are vectors of various deadly diseases like dengue, zika, chikungunya… in a wide area. We use a system of reaction-diffusion equations to model the mosquito population and study the effect of releasing sterile males. Without any human intervention, and due to the so-calledhair-trigger effect, the introduction of only a few individuals (eggs or fertilized females) can lead to the invasion of mosquitoes in the whole region after some time. To avoid this phenomenon, our strategy is to keep releasing a small number of sterile males in the treated zone and move this release forward with a negative forcing speedcto push back the invasive front of wild mosquitoes. By using traveling wave analysis, we show in the present paper that the strategy succeeds in repulsing the population while consuming a finite amount of mosquitoes in any finite time interval even though we treat a moving half-space {x>ct}. Moreover, we succeed in constructing a ‘forced’ traveling wave for our system moving at the same speed as the releases. We also provide some numerical illustrations for our results.

A robust nonlinear model of steam reformers with applications to energy efficiency

We have developed and implemented a computational methodology to solve for the heat flow and the gas concentrations in a class of industrial catalytic steam reformers. The problem under consideration consists of a nonlinearly coupled system composed of a boundary-value problem and reaction-diffusion partial differential equations. Such reformers have broad industrial applications and consume a large amount of energy and natural resources globally. The methodology is applied to tackle two outstanding questions regarding catalytic processes, for which our methodology delivers efficient answers to the following research questions: (i) How to solve the energy/catalyst investment dichotomy, i.e., locations with relatively inexpensive energy require less catalyst and in the opposite case of high energy costs, more catalyst is needed. (ii) How to express the energy input, which is difficult to ascertain for large burner designs, in terms of easily measurable parameters in real time, such as the input temperatures. These questions are answered by our approach leading to a powerful design tool for catalytic processes in general, with huge potential to mitigate environmental impacts. The complex nonlinear system of reaction-diffusion equations is solved by means of an implicit-explicit method and the nonlinear boundary value problem is solved by an iterative method. The coupling between the two systems is also performed through an iterative method. The convergence of the latter is confirmed by extensive numerical investigations, which are consistent with observed measurements in industrial installations. Furthermore, an analysis of the syngas production with different catalyst density, reactor inlet temperature, furnace energy flame and catalyst length are performed, which confirm that the variation of these parameters can result in a more optimized process. A typical consequence is the following: Fixing all parameters, except the catalyst density and the reactor inlet temperature, we show that an increase of 100 K in the latter allows us to use 3 times less catalyst density to achieve the same target CO concentration. In other words, we have a quantifiable and substantial trade off between energy and catalyst investment.

Existence and stability of a stable stationary solution with a boundary layer for a system of reaction-diffusion equations with Neumann boundary conditions

Рассматривается начально-краевая задача для сингулярно возмущенной параболической системы двух уравнений типа реакция-диффузия с условиями Неймана с коэффициентами диффузии различной степени малости без требования квазимонотонности правых частей. Получено асимптотическое приближение стационарного решения с пограничным слоем и доказаны теоремы существования, асимптотической устойчивости по Ляпунову и локальной единственности такого решения. Полученный результат применен к классу задач химической кинетики.

The gradient discretisation method for the chemical reactions of biochemical systems

PurposeThe main purpose of this paper is to introduce the gradient discretisation method (GDM) to a system of reaction diffusion equations subject to non-homogeneous Dirichlet boundary conditions. Then, the authors show that the GDM provides a comprehensive convergence analysis of several numerical methods for the considered model. The convergence is established without non-physical regularity assumptions on the solutions.Design/methodology/approachIn this paper, the authors use the GDM to discretise a system of reaction diffusion equations with non-homogeneous Dirichlet boundary conditions.FindingsThe authors provide a generic convergence analysis of a system of reaction diffusion equations. The authors introduce a specific example of numerical scheme that fits in the gradient discretisation method. The authors conduct a numerical test to measure the efficiency of the proposed method.Originality/valueThis work provides a unified convergence analysis of several numerical methods for a system of reaction diffusion equations. The generic convergence is proved under the classical assumptions on the solutions.

The Stochastic Gierer–Meinhardt System

The Gierer–Meinhardt system occurs in morphogenesis, where the development of an organism from a single cell is modelled. One of the steps in the development is the formation of spatial patterns of the cell structure, starting from an almost homogeneous cell distribution. Turing proposed different activator–inhibitor systems with varying diffusion rates in his pioneering work, which could trigger the emergence of such cell structures. Mathematically, one describes these activator–inhibitor systems as coupled systems of reaction-diffusion equations with different diffusion coefficients and highly nonlinear interaction. One famous example of these systems is the Gierer–Meinhardt system. These systems usually are not of monotone type, such that one has to apply other techniques. The purpose of this article is to study the stochastic reaction-diffusion Gierer–Meinhardt system with homogeneous Neumann boundary conditions on a one or two-dimensional bounded spatial domain. To be more precise, we perturb the original Gierer–Meinhardt system by an infinite-dimensional Wiener process and show under which conditions on the Wiener process and the initial conditions, a solution exists. In dimension one, we even show the pathwise uniqueness. In dimension two, uniqueness is still an open question.

Switching Interacting Particle Systems: Scaling Limits, Uphill Diffusion and Boundary Layer

This paper considers three classes of interacting particle systems on {{mathbb {Z}}}: independent random walks, the exclusion process, and the inclusion process. Particles are allowed to switch their jump rate (the rate identifies the type of particle) between 1 (fast particles) and epsilon in [0,1] (slow particles). The switch between the two jump rates happens at rate gamma in (0,infty ). In the exclusion process, the interaction is such that each site can be occupied by at most one particle of each type. In the inclusion process, the interaction takes places between particles of the same type at different sites and between particles of different type at the same site. We derive the macroscopic limit equations for the three systems, obtained after scaling space by N^{-1}, time by N^2, the switching rate by N^{-2}, and letting Nrightarrow infty . The limit equations for the macroscopic densities associated to the fast and slow particles is the well-studied double diffusivity model. This system of reaction-diffusion equations was introduced to model polycrystal diffusion and dislocation pipe diffusion, with the goal to overcome the limitations imposed by Fick’s law. In order to investigate the microscopic out-of-equilibrium properties, we analyse the system on [N]={1,ldots ,N}, adding boundary reservoirs at sites 1 and N of fast and slow particles, respectively. Inside [N] particles move as before, but now particles are injected and absorbed at sites 1 and N with prescribed rates that depend on the particle type. We compute the steady-state density profile and the steady-state current. It turns out that uphill diffusion is possible, i.e., the total flow can be in the direction of increasing total density. This phenomenon, which cannot occur in a single-type particle system, is a violation of Fick’s law made possible by the switching between types. We rescale the microscopic steady-state density profile and steady-state current and obtain the steady-state solution of a boundary-value problem for the double diffusivity model.