Reaction-diffusion Model Research Articles

Numerical Simulation of Polyacrylamide Hydrogel Prepared via Thermally Initiated Frontal Polymerization.

Traditional polymer curing techniques present challenges such as a slow processing speed, high energy consumption, and considerable initial investment. Frontal polymerization (FP), a novel approach, transforms monomers into fully cured polymers through a self-sustaining exothermic reaction, which enhances speed, efficiency, and safety. This study focuses on acrylamide hydrogels, synthesized via FP, which hold significant potential for biomedical applications and 3D printing. Heat conduction is critical in FP, particularly due to its influence on the temperature distribution and reaction rate mechanisms, which affect the final properties of polymers. Therefore, a comprehensive analysis of heat conduction and chemical reactions during FP is presented through the establishment of mathematical models and numerical methods. Existing research on FP hydrogel synthesis primarily explores chemical modifications, with limited studies on numerical modeling. By utilizing Differential Scanning Calorimetry (DSC) data on the curing kinetics of polymerizable deep eutectic solvents (DES), this paper employs Malek's model selection method to establish an autocatalytic reaction model for FP synthesis. In addition, the finite element method is used to solve the reaction-diffusion model, examining the temperature evolution and curing degree during synthesis. The results affirm the nth-order autocatalytic model's accuracy in studying acrylamide monomer curing kinetics. Additionally, factors such as trigger temperature and solution initial temperature were found to influence the FP reaction's frontal propagation speed. The model's predictions on acrylamide hydrogel synthesis align with experimental data, filling the gap in numerical modeling for hydrogel FP synthesis and offering insights for future research on numerical models and temperature control in the FP synthesis of high-performance hydrogels.

Self-organization of PIP3 waves is controlled by the topology and curvature of cell membranes

Phosphatidylinositol (3,4,5)-trisphosphate (PIP3) is a signaling lipid on the plasma membrane that plays a fundamental role in cell signaling with a strong impact on cell physiology and diseases. It is responsible for the protruding edge formation, cell polarization, macropinocytosis, and other membrane remodeling dynamics in cells. It has been shown that the membrane confinement and curvature affects the wave formation of PIP3 and F-actin. But, even in the absence of F-actin, a complex self-organization of the spatiotemporal PIP3 waves is observed. In recent findings, we have shown that these waves can be guided and pinned on strongly bended Dictyostelium membranes caused by molecular crowding and curvature-limited diffusion. Based on these experimental findings, we investigate the spatiotemporal PIP3 wave dynamics on realistic three-dimensional cell-like membranes to explore the effect of curvature-limited diffusion, as observed experimentally. We use an established stochastic reaction-diffusion model with enzymatic Michaelis-Menten-type reactions that mimics the dynamics of Dictyostelium cells. As these cells mimic the three-dimensional shape and size observed experimentally, we found that the PIP3 wave directionality can be explained by a Hopf-like and a reverse periodic-doubling bifurcation for uniform diffusion and curvature-limited diffusion properties. Finally, we compare the results with recent experimental findings and discuss the discrepancy between the biological and numerical results.

Complex pattern evolution of a two-dimensional space diffusion model of malware spread

In order to investigate the propagation mechanism of malware in cyber-physical systems (CPSs), the cross-diffusion in two-dimensional space is attempted to be introduced into a class of susceptible-infected (SI) malware propagation model depicted by partial differential equations (PDEs). Most of the traditional reaction-diffusion models of malware propagation only take into account the self-diffusion in one-dimensional space, but take less consideration of the cross-diffusion in two-dimensional space. This paper investigates the spatial diffusion behaviour of malware nodes spreading through physical devices. The formations of Turing patterns after homogeneous stationary instability triggered by Turing bifurcation are investigated by linear stability analysis and multiscale analysis methods. The conditions under the occurence of Hopf bifurcation and Turing bifurcation in the malware model are obtained. The amplitude equations are derived in the vicinity of the bifurcation point to explore the conditions for the formation of Turing patterns in two-dimensional space. And the corresponding patterns are obtained by varying the control parameters. It is shown that malicious virus nodes spread in different forms including hexagons, stripes and a mixture of the two. This paper will extend a new direction for the study of system security theory.

Manipulation of rotation for triangular plasma photonic crystals in dielectric barrier discharge

Rotation manipulation in the fields of metamaterials and metasurfaces has led to a variety of striking properties. Here, we propose an efficient scheme for realizing rotation-controllable plasma metamaterials in dielectric barrier discharge. Rotating triangular plasma photonic crystals (RTPPCs) are obtained by self-organization of filaments in simply ambient air. Independent control of the angular velocity and the lattice constant of RTPPCs is realized. A phenomenological reaction–diffusion model with two coupled layers is established to reveal the underlying mechanism of RTPPCs. Moreover, the changes in the bandgaps with angular reorientation of RTPPCs are demonstrated by using microwave diagnosis. Experimental observations and numerical simulations are in good agreement. Our method provides an additional degree of freedom to tailor plasma metamaterials, which may find potential applications, such as integrated optical components, wireless communications, precision radiolocation, time-resolved imaging, and sensing.

Biofilm-NOM-ferrihydrite synergy facilitate the dissimilatory iron bioreduction

Dissimilatory microbial iron reduction, a key biogeochemical process, influences the cycling of iron, hazardous metals such as arsenic, cadmium and chromium, and natural organic matter (NOM) across diverse environments like soils and aquatic systems. Despite its ecological significance, the interaction between ferrihydrite, NOM, and biofilms, particularly the synergy within these conditions, remains unclear. This study explored the bioreduction of ferrihydrite by Shewanella oneidensis MR-1 biofilms, analyzing how varying concentrations of NOM and electron donor affects this process. We developed a kinetic model to quantify iron transformation, revealing that NOM performs dual functions: as a carbon source and an electron shuttle. In carbon-rich environments, NOM acts as an electron shuttle, while insufficient carbon sources, biofilms metabolize NOM for electron transfer to ferrihydrite. The biofilm's extensive surface area facilitates the coprecipitation of NOM and ferrihydrite, enhancing electron transfer efficiency. The study noted a significant conversion of ferrihydrite into goethite and lepidocrocite, particularly at high C/Fe ratios. Additionally, integrating a coupled reactive-diffusion model, we aimed to predict iron transformation and distribution in biofilm systems. This approach emphasizes the importance of the biofilm-NOM-ferrihydrite synergy in iron biogeochemical cycles and the need for incorporating biofilm dynamics in realistic biogeochemical process models. The findings provide a new insight into the biofilm function in mediating the ferrihydrite bioreduction process in different carbon resources in real scenarios.

A reaction–diffusion vector-borne disease model with incubation period in almost periodic environments

To study the multiple effects of seasonal fluctuations, spatial heterogeneity and extrinsic incubation period on vector-borne disease transmission, a nonlocal almost periodic reaction–diffusion model of vector-borne diseases is proposed and studied. We first give a characterization of the upper Lyapunov exponent λ∗ for a class of linear almost periodic reaction–diffusion systems with time delay, and provide a numerical scheme to compute it. Then we show that λ∗ is a threshold value determining the uniform persistence and extinction of our model. Specifically, the disease will die out when λ∗<0, while the disease is uniformly persistent when λ∗>0. Some numerical simulations finally are presented to verify our theoretical results, and to investigate the effects of diffusion rates, extrinsic incubation period and spatial heterogeneity on disease transmission.

Variational techniques for a one-dimensional energy balance model

Abstract. A one-dimensional climate energy balance model (1D EBM) is a simplified climate model for the zonally averaged global temperature profile, based on the Earth's energy budget. We examine a class of 1D EBMs which emerges as the parabolic equation corresponding to the Euler–Lagrange equations of an associated variational problem, covering spatially inhomogeneous models such as with latitude-dependent albedo. Sufficient conditions are provided for the existence of at least three steady-state solutions in the form of two local minima and one saddle, that is, of coexisting “cold”, “warm” and unstable “intermediate” climates. We also give an interpretation of minimizers as “typical” or “likely” solutions of time-dependent and stochastic 1D EBMs. We then examine connections between the value function, which represents the minimum value (across all temperature profiles) of the objective functional, regarded as a function of greenhouse gas concentration, and the global mean temperature (also as a function of greenhouse gas concentration, i.e. the bifurcation diagram). Specifically, the global mean temperature varies continuously as long as there is a unique minimizing temperature profile, but coexisting minimizers must have different global mean temperatures. Furthermore, global mean temperature is non-decreasing with respect to greenhouse gas concentration, and its jumps must necessarily be upward. Applicability of our findings to more general spatially heterogeneous reaction–diffusion models is also discussed, as are physical interpretations of our results.

Higher-order galerkin finite element method for nonlinear coupled reaction-diffusion models

In this article, generalized nonlinear coupled reaction-diffusion (NCRD) models are analyzed using the higher-order Galerkin finite element method (FEM). The authors present finite element analysis for one- and two-dimensional NCRD models by applying quadratic Lagrange basis functions. Optimal order convergence in space is proved by applying quadratic FEM in spatial discretization and utilizing the Crank-Nicolson (CN) scheme for time discretization. To address the nonlinearity of NCRD models, CN extrapolation and predictorcorrector schemes are employed. The stability analysis for generalized NCRD models and the applied CN scheme are discussed by utilizing the eigenvalues method. Finally, to assess the accuracy and efficiency of the proposed scheme and to capture different patterns of NCRD models, various one- and two-dimensional models are considered. The obtained results are numerically and graphically compared with existing literature. It is observed that quadratic basis functions in FEM improve accuracy and efficiency of the results.

Bifurcations and pattern formation in a host–parasitoid model with nonlocal effect

In this paper, we analyse Turing instability and bifurcations in a host–parasitoid model with nonlocal effect. For a ordinary differential equation model, we provide some preliminary analysis on Hopf bifurcation. For a reaction–diffusion model with local intraspecific prey competition, we first explore the Turing instability of spatially homogeneous steady states. Next, we show that the model can undergo Hopf bifurcation and Turing–Hopf bifurcation, and find that a pair of spatially nonhomogeneous periodic solutions is stable for a (8,0)-mode Turing–Hopf bifurcation and unstable for a (3,0)-mode Turing–Hopf bifurcation. For a reaction–diffusion model with nonlocal intraspecific prey competition, we study the existence of the Hopf bifurcation, double-Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation successively, and find that a spatially nonhomogeneous quasi-periodic solution is unstable for a (0,1)-mode double-Hopf bifurcation. Our results indicate that the model exhibits complex pattern formations, including transient states, monostability, bistability, and tristability. Finally, numerical simulations are provided to illustrate complex dynamics and verify our theoretical results.

Multiscale modeling of an entrained flow solar calciner for thermochemical energy storage via calcium looping

Calcium looping has been proposed as an alternative route addressing the intermittent nature of solar energy in terms of thermochemical energy storage, with the added benefit of CO2 capture. In this work, a multiscale approach aims to predict and optimize the performance of a pilot-scale solar calciner operating at realistic conditions, incorporating a diffusion–reaction model with multiphase flow considerations in a downer reactor. The model has been successfully validated with experimental data from the literature studying the effect of particle size on the observed rate, and with limestone decomposition experiments carried out in a lab -scale downer reactor. It was revealed that particles larger than 100 μm exhibit internal mass transport resistances, due to CO2 accumulation within the particle, whereas heat transport limitations are not expected in the operating conditions of interest. A reference scenario demonstrated 92 % limestone conversion at realistic operating conditions, feasible in concentrated solar power plants. The solar flux density on the reactor wall influences the efficiency of the system critically since heat loss to the environment leads to greater solar input. To tackle the low efficiency, a partially insulated, indirectly irradiated reactor was investigated: the efficiency increased from 8 % to 26 %, while integration in a Calcium-Looping system can further boost the efficiency to 34 %. This approach constitutes a basis for future scale-up considerations.

Parameter Identifiability in PDE Models of Fluorescence Recovery After Photobleaching.

Identifying unique parameters for mathematical models describing biological data can be challenging and often impossible. Parameter identifiability for partial differential equations models in cell biology is especially difficult given that many established in vivo measurements of protein dynamics average out the spatial dimensions. Here, we are motivated by recent experiments on the binding dynamics of the RNA-binding protein PTBP3 in RNP granules of frog oocytes based on fluorescence recovery after photobleaching (FRAP) measurements. FRAP is a widely-used experimental technique for probing protein dynamics in living cells, and is often modeled using simple reaction-diffusion models of the protein dynamics. We show that current methods of structural and practical parameter identifiability provide limited insights into identifiability of kinetic parameters for these PDE models and spatially-averaged FRAP data. We thus propose a pipeline for assessing parameter identifiability and for learning parameter combinations based on re-parametrization and profile likelihoods analysis. We show that this method is able to recover parameter combinations for synthetic FRAP datasets and investigate its application to real experimental data.

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 dynamics of a time-delayed nonlocal reaction-diffusion model of within-host viral infections.

In this paper, we study a time-delayed nonlocal reaction-diffusion model of within-host viral infections. We introduce the basic reproduction number and show that the infection-free steady state is globally asymptotically stable when , while the disease is uniformly persistent when . In the case where all coefficients and reaction terms are spatially homogeneous, we obtain an explicit formula of and the global attractivity of the positive constant steady state. Numerically, we illustrate the analytical results, conduct sensitivity analysis, and investigate the impact of drugs on curtailing the spread of the viruses.

Dynamics of an EIS spatially heterogeneous rabies model

This paper is devoted to the study of population dynamics of an EIS epidemic reaction-diffusion model in spatially heterogeneous environments, where the rates of disease-induced mortality and disease transmission are space dependent. For such a class of systems, we provide clear pictures on the global dynamics in terms of the basic reproduction number R0, which improves the local dynamics proposed in Wang and Zhao (2012) [20]. When R0≤1, the disease-free steady state is globally asymptotically stable. Different from earlier works for many epidemic models, we obtain the positive steady states by using bifurcation theory for R0>1, which can be easily generalized to the case where the reaction-diffusion system involves partial diffusion coefficients of zero.

Spontaneous signal generation by an excitable system for cell migration.

Eukaryotic cells exhibit random migration in the absence of extracellular directional cues. This random migration acts as basal motility for various migratory responses such as chemotaxis. The self-organization of random motility requires the internal signals that determine the anterior side of the migrating cell be generated spontaneously from the intrinsic activities of intracellular signaling networks. Recent studies have identified an excitable system as the mechanism of the spontaneous signal generation. Here, we discuss how the excitable system of Ras, a small G protein, regulates signaling networks in Dictyostelium discoideum as a model organism. The excitability produces a domain where an active form of Ras is enriched on the cell membrane without extracellular directional cues, such that Ras serves as the anterior signal. The typical spatiotemporal characteristics are mathematically explained by reaction-diffusion models. These models further enable a quantitative analysis of the dynamics that depends on the internal cellular states and surrounding environments. Downstream of the Ras excitable system, a phosphoinositide metabolic network composed of PI3K, PTEN, PI(3,4,5)P3 and PI(4,5)P2 exhibits bistability to discretize the anterior and posterior regions of the cell membrane. Upstream, a local excitation and global inhibition local excitation global inhibition network, which works for gradient sensing in the presence of chemoattractant gradients, spatiotemporally biases the excitability of Ras for chemotaxis. In parallel with the Ras excitable system, the cGMP signaling pathway constitutes another excitable system of its own periodicity to ensure flexible migratory dynamics. In addition to these intracellular signaling networks, an intercellular signaling network activated by secreted cAMP is coupled with the Ras excitable system for collective cell migration. Finally, we discuss how the excitable system of Ras operates as a platform of information integration by receiving multiple intrinsic and extrinsic signals to ensure spontaneous cellular activity and robust responses in eukaryotic cell migration under natural complex environments.

On propagation in networks, promising models beyond network diffusion to describe degenerative brain diseases and traumatic brain injuries.

Introduction: Connections among neurons form one of the most amazing and effective network in nature. At higher level, also the functional structures of the brain is organized as a network. It is therefore natural to use modern techniques of network analysis to describe the structures of networks in the brain. Many studies have been conducted in this area, showing that the structure of the neuronal network is complex, with a small-world topology, modularity and the presence of hubs. Other studies have been conducted to investigate the dynamical processes occurring in brain networks, analyzing local and large-scale network dynamics. Recently, network diffusion dynamics have been proposed as a model for the progression of brain degenerative diseases and for traumatic brain injuries. Methods: In this paper, the dynamics of network diffusion is re-examined and reaction-diffusion models on networks is introduced in order to better describe the degenerative dynamics in the brain. Results: Numerical simulations of the dynamics of injuries in the brain connectome are presented. Different choices of reaction term and initial condition provide very different phenomenologies, showing how network propagation models are highly flexible. Discussion: The uniqueness of this research lies in the fact that it is the first time that reaction-diffusion dynamics have been applied to the connectome to model the evolution of neurodegenerative diseases or traumatic brain injury. In addition, the generality of these models allows the introduction of non-constant diffusion and different reaction terms with non-constant parameters, allowing a more precise definition of the pathology to be studied.

On cognitive epidemic models: spatial segregation versus nonpharmaceutical interventions.

Fick's law and the Fokker-Planck law of diffusion are applied to manifest the cognitive dispersal of individuals in two reaction-diffusion SEIR epidemic models, where the disease transmission is illustrated by nonlocal infection mechanisms in heterogeneous environments. Building upon the well-posedness of solutions, threshold dynamics are discussed in terms of the basic reproduction numbers for the two cognitive epidemic models. The numerical investigation reveals that the Fokker-Planck law can better describe the diffusion of individuals by taking different dispersal strategies of exposed individuals in our cognitive epidemic models, and provides some insights on spatial segregation and nonpharmaceutical interventions: (i) spatial segregation occurs in the random diffusion model when the nonlocal infection radius is small, while it appears in the symmetric diffusion model when the radius is large; (ii) nonpharmaceutical interventions on restricting the dispersal of exposed and infected individuals do not contribute to reducing the infection proportion, but rather eliminate the disease in a region, which expands as the nonlocal infection radius increases. We additionally find that the final infection size in the random diffusion model is significantly smaller than that in the symmetric diffusion model and decreases as the nonlocal infection radius increases.

Global dynamics of two-species reaction–diffusion competition model with Gompertz growth

ABSTRACT In this paper, we investigate a two-species reaction–diffusion competition model with Gompertz growth, where the intrinsic growth rates and carrying capacities of environments are heterogeneous. At firstly, assuming two competing species only admit different diffusive rates, we show that ‘slower diffuser prevails’, which is consistent with the well-known result in Dockery J, Hutson V, Mischaikow K, Pernarowski M. [The evolution of slow dispersal rates: a reaction–diffusion model. J Math Biol. 1998;37(1):61–83; Hastings A. Can spatial variation alone lead to selection for dispersal? Theor Popul Biol. 1983;24:244–251]. Then, for the “weak competition” case, we establish a prior estimate, which combined with the theory of monotone dynamical system and spectral analysis implies that the model admits a unique coexistence steady state, which is globally asymptotically stable. Finally, for the “strong–weak competition” case, we give the expression of critical competition intensity and the weak competitor will be wiped out.

On the factors enhancing hydrogen trapping in spherical cavities in metals

Using the reaction–diffusion model for hydrogen (H) trapping in spherical cavities in metals (Zibrov and Schmid, 2022), we theoretically analyze the case of H trapping only in chemisorption sites at the cavity surface. We show that the model can be reduced to a form that is similar to the conventional model of H trapping in point defects. The reduced model includes analytical expressions for correction factors to account for correlated H retrapping by the same cavity and inhomogeneous H potential energy landscape in the vicinity of the cavity surface. We show that these correction factors play a significant role in thermal desorption (TDS) simulations only in the case of very low cavity number densities (detrapping-limited regime), which is rarely encountered in experiments.

Modeling and Simulating an Epidemic in Two Dimensions with an Application Regarding COVID-19

We derive a reaction–diffusion model with time-delayed nonlocal effects to study an epidemic’s spatial spread numerically. The model describes infected individuals in the latent period using a structured model with diffusion. The epidemic model assumes that infectious individuals are subject to containment measures. To simulate the model in two-dimensional space, we use the continuous Runge–Kutta method of the fourth order and the discrete Runge–Kutta method of the third order with six stages. The numerical results admit the existence of traveling wave solutions for the proposed model. We use the COVID-19 epidemic to conduct numerical experiments and investigate the minimal speed of spread of the traveling wave front. The minimal spreading speeds of COVID-19 are found and discussed. Also, we assess the power of containment measures to contain the epidemic. The results depict a clear drop in the spreading speed of the traveling wave front after applying containment measures to at-risk populations.