General Reaction-diffusion Model Research Articles

Analysis of a general reaction–diffusion model using Lie symmetries and conservation laws

Analysis of a general reaction–diffusion model using Lie symmetries and conservation laws

Asymptotic stability results of generalized discrete time reaction diffusion system applied to Lengyel-Epstein and Dagn Harrison models

In this research paper, we delve into the analysis of a generalized discrete reaction-diffusion system. Our study begins with the discretization of a generalized reaction-diffusion model, achieved through second-order and L1-difference approximations. We explore the local stability of its unique solution, both in the absence and presence of the diffusion term. To determine the conditions for global asymptotic stability of the steady-state solution, we employ suitable techniques including the direct Lyapunov method. To illustrate the practical application of this theoretical framework, we provide several numerical simulations that examine both the Lengyel-Epstein reaction-diffusion model and the discrete Degn-Harrison reaction-diffusion model. These simulations serve to validate the predictions of asymptotic stability.

Persistence of a competition model of plankton allelopathy in time–space periodic environment

This work is devoted to the study of a competition model of plankton allelopathy imposed in time-space periodic environment. We prove that the system admits positive periodic solutions under certain conditions. We further obtain some sufficient conditions for the uniqueness and global stability of the positive periodic solution, which shows that the model is persistent. The main tools for our arguments are comparison theorems based on the maximum principle, sub- and supersolutions method, and an iteration method, which also permit the treatment of some more general reaction–diffusion models in periodic environment.

Discrete kinetic analysis of a general reaction–diffusion model constructed by Euler discretization and coupled map lattices

In this paper, we provide a framework for the Turing instability in a general two-dimensional discrete reaction–diffusion model that utilizes Euler discretization and coupled map lattices. We obtained explicit criterions for the normal forms of Neimark–Sacker bifurcation and flip bifurcation in the absence of diffusion. We derive the general conditions that govern the emergence of pure Turing instability, Neimark–Sacker–Turing instability, Flip–Turing instability, spatially homogeneous stable states, spatially homogeneous periodic oscillation states, and spatially homogeneous period-doubling oscillatory states under spatially inhomogeneous conditions. As an example, we employ a ratio-dependent predation model and verify the derived general conditions through a series of numerical simulations. Additionally, we apply the calculation of Maximum Lyapunov Exponent(MLE) to simulate and demonstrate the path from flip bifurcation to chaos.

Global stability of a PDE-ODE model for acid-mediated tumor invasion

In this paper, we study the global dynamics of a general reaction-diffusion model based on acid-mediated invasion hypothesis, which is a candidate explanation for the Warburg effect. Our theoretical results characterize the effects of acid resistance and mutual competition of healthy cells and tumor cells on tumor progression in the long term, i.e., whether the healthy cells and tumor cells coexist or the tumor cells prevail after tumor invasion. A key feature of this model is the density-limited tumor diffusion term for tumor cells, which might give rise to the degeneracy of the parabolic equation. To overcome this difficulty, we combine the construction of suitable Lyapunov functionals and upper/lower solutions. This paper continues and improves the work begun in [15].

Correspondences Between Parameters in a Reaction-Diffusion Model and Connexin Functions During Zebrafish Stripe Formation

Different diffusivities among interacting substances actualize the potential instability of a system. When these elicited instabilities manifest as forms of spatial periodicity, they are called Turing patterns. Simulations using general reaction-diffusion (RD) models demonstrate that pigment patterns on the body trunk of growing fish follow a Turing pattern. Laser ablation experiments performed on zebrafish reveal apparent interactions among pigment cells, which allow for a three-component RD model to be derived. However, the underlying molecular mechanisms responsible for Turing pattern formation in this system remain unknown. A zebrafish mutant with a spotted pattern was found to have a defect in Connexin41.8 (Cx41.8) which, together with Cx39.4, exists in pigment cells and controls pattern formation. Here, molecular-level evidence derived from connexin analyses is linked to the interactions among pigment cells described in previous RD modeling. Channels on pigment cells are generalized as “gates,” and the effects of respective gates were deduced. The model uses partial differential equations (PDEs) to enable numerical and mathematical analyses of characteristics observed in the experiments. Furthermore, the improved PDE model, including nonlinear reaction terms, enables the consideration of the behavior of components realistically.

An Accelerating Numerical Computation of the Diffusion Term in a Nonlocal Reaction-Diffusion Equation

In this paper we propose and compare two methods to optimize the numerical computations for the diffusion term in a nonlocal formulation for a reaction-diffusion equation. The diffusion term is particularly computationally intensive due to the integral formulation, and thus finding a better way of computing its numerical approximation could be of interest, given that the numerical analysis usually takes place on large input domains having more than one dimension. After introducing the general reaction-diffusion model, we discuss a numerical approximation scheme for the diffusion term, based on a finite difference method. In the next sections we propose two algorithms to solve the numerical approximation scheme, focusing on finding a way to improve the time performance. While the first algorithm (sequential) is used as a baseline for performance measurement, the second algorithm (parallel) is implemented using two different memory-sharing parallelization technologies: Open Multi-Processing (OpenMP) and CUDA. All the results were obtained by using the model in image processing applications such as image restoration and segmentation.

Spatiotemporal dynamics of a diffusive predator-prey model with nonlocal effect and delay

In this paper, we discuss a diffusive predator-prey model with nonlocality and delay. Stability and bifurcation analysis suggest that the joint impacts of the nonlocal term and delay result in instability of the positive constant steady state. Moreover, steady state, Hopf and steady state-Hopf bifurcations and interactions of these bifurcations may occur under certain conditions. Normal forms of steady state, Hopf and steady state-Hopf bifurcations for a general reaction-diffusion model with nonlocal effects and delay are derived. In numerical simulations, spatially inhomogeneous steady states and periodic solutions and heteroclinic connections between these solutions are obtained.

Sharp relaxation rates for plane waves of general reaction-diffusion models

It is a well-known and classical result that spectrally stable traveling waves of a general reaction-diffusion system in one spatial dimension are asymptotically stable with exponential relaxation rates. In a series of works in the 1990s, Goodman [Trans. Am. Math. Soc. 311, 683 (1989)], Kapitula [Trans. Am. Math. Soc. 349, 1901 (1997)], and Xin [Commun. Partial Differ. Equations 17, 1889 (1992)] considered plane traveling waves for such systems and they have succeeded in showing asymptotic stability for such objects. Interestingly, the (estimates for the) relaxation rates that they have exhibited are all algebraic and dimension dependent. It was heuristically argued that as the spectral gap closes in dimensions n ≥ 2, algebraic rates are the best possible. In this paper, we revisit this issue. We rigorously calculate the sharp relaxation rates in L∞ based spaces, both for the asymptotic phase and the radiation terms. They are, indeed, algebraic, but about twice better than the best ones obtained in these early works although this can be mostly attributed to the inefficiencies of using Sobolev embeddings to control L∞ norms by high order L2 based Sobolev spaces. Finally, we explicitly construct the leading order profiles, both for the phase and for the radiation terms. Our approach relies on the method of scaling variables, as introduced in the work of Gallay and Wayne [Arch. Ration. Mech. Anal. 163, 209 (2002) and Gallay and Wayne, Philos. Trans. R. Soc. A 360, 2155 (2002)] and, in fact, provides sharp relaxation rates in a class of weighted L2 spaces as well.

Dynamic analysis of the anomalous diffusion in catalyst particles considering chemical reactions with non-linear kinetics

In this work, a generalized reaction-diffusion model useful to study the interactions between anomalous diffusion and chemical reactions in catalyst particles is presented. The fractional reaction-diffusion model considers the linearization of the non-linear reaction kinetic terms, which allows the solution of the mathematical model through the Fourier transform. The proposed methodology can be used for the dynamic analysis of the concentration profiles in the catalytic pellets, including internal and external mass transfer resistances, as well as for the estimation of effectiveness factor (EF). These findings revealed that the analysis of dynamic reaction-diffusion linearized models where the mass transfer is described by the anomalous transport can be analysed in the frequency domain. Two kinetic models, a general power-law and Langmuir-Hinshelwood, were considered. The results demonstrated the effect of use the traditional Fick’s law and the anomalous transport (i.e., subdiffusion or superdiffusion phenomena) in a porous media system, particularly on the space-time concentration profile predictions and the influence on the effectiveness factor of the catalytic particle.

How Obstacles Perturb Population Fronts and Alter Their Genetic Structure.

As populations spread into new territory, environmental heterogeneities can shape the population front and genetic composition. We focus here on the effects of an important building block of heterogeneous environments, isolated obstacles. With a combination of experiments, theory, and simulation, we show how isolated obstacles both create long-lived distortions of the front shape and amplify the effect of genetic drift. A system of bacteriophage T7 spreading on a spatially heterogeneous Escherichia coli lawn serves as an experimental model system to study population expansions. Using an inkjet printer, we create well-defined replicates of the lawn and quantitatively study the population expansion of phage T7. The transient perturbations of the population front found in the experiments are well described by a model in which the front moves with constant speed. Independent of the precise details of the expansion, we show that obstacles create a kink in the front that persists over large distances and is insensitive to the details of the obstacle’s shape. The small deviations between experimental findings and the predictions of the constant speed model can be understood with a more general reaction-diffusion model, which reduces to the constant speed model when the obstacle size is large compared to the front width. Using this framework, we demonstrate that frontier genotypes just grazing the side of an isolated obstacle increase in abundance, a phenomenon we call ‘geometry-enhanced genetic drift’, complementary to the founder effect associated with spatial bottlenecks. Bacterial range expansions around nutrient-poor barriers and stochastic simulations confirm this prediction. The effect of the obstacle on the genealogy of individuals at the front is characterized by simulations and rationalized using the constant speed model. Lastly, we consider the effect of two obstacles on front shape and genetic composition of the population illuminating the effects expected from complex environments with many obstacles.

Nonlinear stability of a heterogeneous state in a PDE-ODE model for acid-mediated tumor invasion.

This work studies a general reaction-diffusion model for acid-mediated tumor invasion, where tumor cells produce excess acid that primarily kills healthy cells, and thereby invade the microenvironment. The acid diffuses and could be cleared by vasculature, and the healthy and tumor cells are viewed as two species following logistic growth with mutual competition. A key feature of this model is the density-limited diffusion for tumor cells, reflecting that a healthy tissue will spatially constrain a tumor unless shrunk. Under appropriate assumptions on model parameters and on initial data, it is shown that the unique heterogeneous state is nonlinearly stable, which implies a long-term coexistence of the healthy and tumor cells in certain parameter space. Our theoretical result suggests that acidity may play a significant role in heterogeneous tumor progression.

Exactly solvable reaction diffusion models on a Bethe Lattice through the empty-interval method

The most general reaction-diffusion model on a Bethe Lattice with nearest-neighbor interactions is introduced, which can be solved exactly through the empty-interval method. The stationary solutions of such models are discussed. For some special choice of reaction rates the dynamics of the system is also studied.

A general reaction–diffusion model of acidity in cancer invasion

We model the metabolism and behaviour of a developing cancer tumour in the context of its microenvironment, with the aim of elucidating the consequences of altered energy metabolism. Of particular interest is the Warburg Effect, a widespread preference in tumours for cytosolic glycolysis rather than oxidative phosphorylation for glucose breakdown, as yet incompletely understood. We examine a candidate explanation for the prevalence of the Warburg Effect in tumours, the acid-mediated invasion hypothesis, by generalising a canonical non-linear reaction-diffusion model of acid-mediated tumour invasion to consider additional biological features of potential importance. We apply both numerical methods and a non-standard asymptotic analysis in a travelling wave framework to obtain an explicit understanding of the range of tumour behaviours produced by the model and how fundamental parameters govern the speed and shape of invading tumour waves. Comparison with conclusions drawn under the original system--a special case of our generalised system--allows us to comment on the structural stability and predictive power of the modelling framework.

Are Assumptions about the Model Type Necessary in Reaction-Diffusion Modeling? A FRAP Application

At present, fluorescence recovery after photobleaching (FRAP) data are interpreted using various types of reaction-diffusion (RD) models: the model type is usually fixed first, and corresponding model parameters are inferred subsequently. In this article, we describe what we believe to be a novel approach for RD modeling without using any assumptions of model type or parameters. To the best of our knowledge, this is the first attempt to address both model-type and parameter uncertainties in inverting FRAP data. We start from the most general RD model, which accounts for a flexible number of molecular fractions, all mobile, with different diffusion coefficients. The maximal number of possible binding partners is identified and optimal parameter sets for these models are determined in a global search of the parameter-space using the Simulated Annealing strategy. The numerical performance of the described techniques was assessed using artificial and experimental FRAP data. Our general RD model outperformed the standard RD models used previously in modeling FRAP measurements and showed that intracellular molecular mobility can only be described adequately by allowing for multiple RD processes. Therefore, it is important to search not only for the optimal parameter set but also for the optimal model type.

Nonuniform autonomous one-dimensional exclusion nearest-neighbor reaction-diffusion models

The most general nonuniform reaction-diffusion models on a one-dimensional lattice with boundaries, for which the time evolution equations of corre- lation functions are closed, are considered. A transfer matrix method is used to find the static solution. It is seen that this transfer matrix can be obtained in a closed form, if the reaction rates satisfy certain conditions. We call such models superautonomous. Possible static phase transitions of such models are investigated. At the end, as an example of superau- tonomous models, a nonuniform voter model is introduced, and solved explicitly.

On the Solutions of the Generalized Reaction-Diffusion Model for Bacterial Colony

In this paper, the Adomian’s decomposition method has been developed to yield approximate solution of the reaction-diffusion model of fractional order which describe the evolution of the bacterium Bacillus subtilis, which grows on the surface of thin agar plates. The fractional derivatives are described in the Caputo sense. The method introduces a promising tool for solving many linear and nonlinear fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the approach is easy to implement and accurate when applied to partial differential equations of fractional order.

Compact Modeling and Simulation of Circuit Reliability for 65-nm CMOS Technology

Negative bias temperature instability (NBTI) and channel hot carrier (CHC) are the leading reliability concerns for nanoscale transistors. The de facto modeling method to analyze CHC is based on substrate current Isub, which becomes increasingly problematic with technology scaling as various leakage components dominate Isub. In this paper, we present a unified approach that directly predicts the change of key transistor parameters under various process and design conditions for both NBTI and CHC effects. Using the general reaction-diffusion model and the concept of surface potential, the proposed method continuously captures the performance degradation across subthreshold and strong inversion regions. Models are comprehensively verified with an industrial 65-nm technology. By benchmarking the prediction of circuit performance degradation with the measured ring oscillator data and simulations of an amplifier, we demonstrate that the proposed method very well predicts the degradation. For 65-nm technology, NBTI is the dominant reliability concern, and the impact of CHC on circuit performance is relatively small.

Scalability and Reliability Characteristics of CVD HfO2 Gate Dielectrics with HfN Electrodes for Advanced CMOS Applications

Metal-oxide-semiconductor (MOS) devices using a thermally robust gate stack were fabricated. The equivalent oxide thickness of gate stack has been aggressively scaled down to 0.75 and for MOS capacitors and metal-oxide-semiconductor field effect transistors, respectively, after a thermal budget required by the conventional complementary metal-oxide-semiconductor gate-first process. The reliability issues such as time-dependent dielectric breakdown (TDDB) and bias temperature instability (BTI) of the devices are studied. The stress electric-field-dependent TDDB characteristics are demonstrated and explained by a model taking into account the high energetic carrier trapping in the and at the interfacial layer. The polarity dependent BTI characteristics are observed which can be explained by a generalized reaction-diffusion model. These intrinsic reliability characteristics are correlated with the low pre-existing charge traps in gate stack resulting from a high temperature postdeposition annealing of the gate stack.

Exactly solvable reaction diffusion models on a Cayley tree

The most general reaction-diffusion model on a Cayley tree with nearest-neighbor interactions is introduced, which can be solved exactly through the empty-interval method. The stationary solutions of such models, as well as their dynamics, are discussed. Concerning the dynamics, the spectrum of the evolution Hamiltonian is found and shown to be discrete, hence there is a finite relaxation time in the evolution of the system towards its stationary state.