Reaction-diffusion Model Research Articles

Spatiotemporal analysis of a modified Leslie–Gower model with cross-diffusion and harvesting

This paper considers a modified Leslie–Gower prey-predator reaction–diffusion model introducing harvesting of both species. Both the temporal and spatiotemporal dynamics of the model have been examined. We have found the stability regions and drawn bifurcation diagrams to determine the harvesting effect on the model, revealing that the harvesting has a stabilizing effect. Local bifurcations, such as transcritical and Hopf bifurcations, appear in the temporal system. For the spatiotemporal model, Turing instability conditions have been determined. The amplitude equation for the critical modes has been derived using multiple time scale analyses by taking the harvesting effort as the bifurcating parameter. Also, we have verified the theoretical results by plotting several kinds of stationary patterns, including stripes, spots, and a mix of stripes and spots. This study’s critical observation is that as harvesting effort rises, the patterns steadily turn into spots, i.e., harvesting influences pattern creation strongly. This fosters a dynamic equilibrium, allowing competitors to maintain distance, optimize resource use and survive.

Two-phase flow effect on methane conversion in pyrolysis reactors embedding molten salts or metals

In this work, we numerically investigate the impact of flowrate, sparger and reactor geometry on methane conversion in lab-scale cylindrical pyrolysis reactors embedding molten salts or metals. A multiscale approach is enforced, where a diffusion-reaction model yielding the conversion vs time within a single rising bubble is combined with the average residence time of gas bubbles obtained through the two-phase turbulent bubbly flow model. We find that the relative size of the sparger and the height-to-diameter ratio of the molten medium are key parameters for controlling the average residence time of the bubbles, which always results lower than that predicted by the terminal velocity of a single bubble rising in an overall static melt. The highest relative discrepancy (order 300%) between the single-bubble estimate and the CFD-based residence time is found in the case of molten salts in low-aspect ratio reactors fed by small spargers. The physical mechanism underpinning the influence of geometric parameters on the overall conversion is addressed in detail, together with the impact of two-phase flow effects on the estimation of bubble size in laboratory scale bubble reactors.

Bifurcation analysis of a single population reaction–diffusion model with discrete memory delay and distributed competition delay

Bifurcation analysis of a single population reaction–diffusion model with discrete memory delay and distributed competition delay

Speed and Shape of Population Fronts with Density-Dependent Diffusion

There is growing empirical evidence that animal movement patterns depend on population density. We investigate travelling wave solutions in reaction-diffusion models of animal range expansion in the case that population diffusion is density-dependent. We find that the speed of the selected wave depends critically on the strength of diffusion at low density. For sufficiently large low-density diffusion, the wave propagates at a speed predicted by a simple linear analysis. For small or zero low-density diffusion, the linear analysis is not sufficient, but a variational approach yields exact or approximate expressions for the speed and shape of population fronts.

Oxygen diffusion effects in thermo-oxidative degradation of typical tire rubber

Thermo-oxidative degradation of tire rubber has been demonstrated as a green method for upcycling waste tire rubber. However, the complicated tire compositions present challenges to achieving the homogeneity and efficiency of the reclaimed products, which restricts their widespread industrial adoption. To address this challenge, natural rubber(NR)and natural rubber/butadiene rubber(NR/BR)were innovatively designed to simulate complex tire compositions and investigate the influence of oxygen diffusion on thermo-oxidative degradation at 150–240 °C. The evolution of chemical structural changes and mechanical properties during degradation was traced by FTIR, UV–vis, and nanoindentation test. A basic reactive-diffusion model based on Fickian oxygen diffusion was used to simulate the diffusion profiles. It was found that recrosslinking decreases the oxygen permeability coefficient during NR/BR degradation as the temperature increases, making it difficult for oxygen to diffuse into the inner layer, and therefore tire rubber degrades unevenly. Lower temperatures and prolonged treatment times were recommended to enhance degradation. These findings provide substantial guidance for optimizing the recycling process of tire rubber and its sustainable utilization.

Dynamical systems analysis of a reaction-diffusion SIRS model with optimal control for the COVID-19 spread

We examine an SIRS reaction-diffusion model with local dispersal and spatial heterogeneity to study COVID-19 dynamics. Using the operator semigroup approach, we establish the existence of disease-free equilibrium (DFE) and endemic equilibrium (EE), and derive the basic reproduction number, R 0 . Simulations show that without dispersal, reinfection and limited medical resources problems can cause a plateau in cases. Dispersal and spatial heterogeneity intensify localised outbreaks, while integrated control strategies (vaccination and treatment) effectively reduce infection numbers and epidemic duration. The possibility of reinfection demonstrates the need for adaptable health measures. These insights can guide optimised control strategies for enhanced public health preparedness.

Two for tau: Automated model discovery reveals two-stage tau aggregation dynamics in Alzheimer’s disease

Alzheimer’s disease is a neurodegenerative disorder characterized by the presence of amyloid-β plaques and the accumulation of misfolded tau proteins and neurofibrillary tangles in the brain. A thorough understanding of the local accumulation of tau is critical to develop effective therapeutic strategies. Tau pathology has traditionally been described using reaction–diffusion models, which succeed in capturing the global spread, but fail to accurately describe the local aggregation dynamics. Current mathematical models enforce a single-peak behavior in tau aggregation, which does not align well with clinical observations. Here we identify a more accurate description of tau aggregation that reflects the complex patterns observed in patients. We propose an innovative approach that uses constitutive neural networks to autonomously discover bell-shaped aggregation functions with multiple peaks from clinical positron emission tomography (PET) data of misfolded tau protein. Our method reveals previously overlooked two-stage aggregation dynamics by uncovering a two-term ordinary differential equation that links the local accumulation rate to the tau concentration. When trained on data from amyloid-β positive and negative subjects, the neural network clearly distinguishes between both groups and uncovers a more subtle relationship between amyloid-β and tau than previously postulated. In line with the amyloid-tau dual pathway hypothesis, our results show that the presence of toxic amyloid-β influences the accumulation of tau, particularly in the earlier disease stages. We expect that our approach to autonomously discover the accumulation dynamics of pathological proteins will improve simulations of tau dynamics in Alzheimer’s disease and provide new insights into disease progression.Significance StatementIn Alzheimer’s disease, understanding the local dynamics of tau protein aggregation is crucial for developing effective treatments. Traditional models for tau protein dynamics use reaction-diffusion models that fail to accurately capture these local patterns. Our study introduces a novel approach that leverages constitutive neural networks to autonomously discover the complex, multi-peak aggregation dynamics from clinical PET data. This method reveals a previously overlooked two-stage tau accumulation process and a nuanced relationship between amyloid-β and tau. By distinguishing between amyloid-β positive and negative subjects, our model supports the amyloid-tau dual pathway hypothesis and offers novel insights into tau protein aggregation that have the potent to advance our understanding of Alzheimer’s disease progression.

Experimental and kinetic studies on steam gasification of oxidized carbon-rich fraction from coal gasification fine slag

Gasification is an important method to realize the resource utilization of coal gasification fine slag (CGFS). In this work, the carbon-rich fraction (CF) of CGFS was modified by oxidation, and the gasification reactivities, structures, and kinetics of samples were studied. The results demonstrate that compared to CF, the gasification characteristic temperatures of oxidized CF are lower, the weight loss rates are faster, and the gasification reactivities are significantly enhanced. These improvements are attributed to the structural evolutions of oxidized CF. Compared with CF, the specific surface area and pore volume of oxidized CF increase by at least 2.5 and 1.5 times, the proportions of oxygen-containing groups rise, and the carbon microstructure of different oxidized CF varies significantly. CF oxidized by steam exhibits the most developed porosity and the most disordered carbon microstructure, corresponding to the highest gasification reactivity of all the samples. The kinetics results showcase that the diffusion reaction model is suitable for describing the gasification process of samples, so porosity emerged as the primary factor determining the gasification reactivity. However, the influence of carbon microstructure and active groups on the gasification reactivity of oxidized CF is enhanced when the porosity meets the diffusion requirements of the gasification reaction.

Step-by-step time discrete Physics-Informed Neural Networks with application to a sustainability PDE model

The use of Artificial Neural Networks (ANNs) has spread massively in several research fields. Among the various applications, ANNs have been exploited for the solution of Partial Differential Equations (PDEs). In this context, the so-called Physics-Informed Neural Networks (PINNs) are considered, i.e. neural networks generally constructed in such a way as to compute a continuous approximation in time and space of the exact solution of a PDE.In this manuscript, we propose a new step-by-step approach that allows to define PINNs capable of providing numerical solutions of PDEs that are discrete in time and continuous in space. This is done by establishing connections between the network outputs and the numerical approximations computed by a classical one-stage method for stiff Initial Value Problems (IVPs). Links are also highlighted between the step-by-step PINNs derived here, and the time discrete models based on Runge–Kutta (RK) methods proposed so far in literature. To evaluate the efficiency of the new approach, we build such PINNs to solve a nonlinear diffusion-reaction PDE model describing the process of production of renewable energy through dye-sensitized solar cells. The numerical experiments show that not only the new step-by-step PINNs are able to well reproduce the model solution, but also highlight that the proposed approach can constitute an improvement over existing continuous and time discrete models.

Faedo-Galerkin method for the time-fractional reaction-diffusion model

This paper investigates a time-fractional reaction-diffusion equation characterized by a Caputo derivative of fractional order , incorporates non-local and memory effects that are essential for accurately modeling complex diffusion processes in such environments. These effects arise from the fractal nature of the media, which leads to anomalous diffusion behavior that cannot be adequately described by classical diffusion models. We start by giving a definition of a weak solution for this problem, ensuring that the solution satisfies the equation in a generalized sense. Subsequently, the Faedo-Galerkin method combined with compactness arguments is employed to establish the existence and uniqueness of the weak solution. This approach involves approximating the solution using a sequence of finite-dimensional functions and then passing to the limit to obtain the desired solution . Moreover, we establish the stability of these solutions, which is fundamental for understanding their behavior under varying conditions and ensuring the reliability of the model . Our findings contribute to a deeper understanding of anomalous diffusion phenomena in fractal media and provide a solid foundation for further research in this area.

An in-silico study of conventional and FLASH radiotherapy iso-effectiveness: potential impact of radiolytic oxygen depletion on tumor growth curves and tumor control probability

Objective. This work aims to investigate the iso-effectiveness of conventional and FLASH radiotherapy on tumors through in-silico mathematical models. We focused on the role of radiolytic oxygen depletion (ROD), which has been argued as a possible factor to explain the FLASH effect.Approach. We used a spatiotemporal reaction-diffusion model, including ROD, to simulate tumor oxygenation and response. From those oxygen distributions we obtained surviving fractions (SFs) using the linear-quadratic (LQ) model with the oxygen enhancement ratios (OERs). We then employed the calculated SFs to describe the evolution of preclinical tumor volumes through a mathematical model of tumor response, and we also extrapolated those results to calculate tumor control probabilities (TCPs) using the Poisson-LQ approach.Main results. Our study suggests that the ROD effect may cause differences in SF between FLASH and conventional radiotherapy, especially in lowα/βandpoorly oxygenatedcells. However, a statistical analysis showed that these changes in SF generally do not result in significant differences in the evolution of preclinical tumor growth curves when the sample size is small, because such differences in SF may not be noticeable in the heterogeneity of the population of animals. Nonetheless, when extrapolating this effect to TCP curves, we observed important differences between both techniques (TCP is lower in FLASH radiotherapy). When analyzing the response of tumors with heterogeneous oxygenations, differences in TCP are more important forwell oxygenatedtumors. This apparent contradiction with the results obtained for homogeneously oxygenated cells is explained by the complex interplay between the heterogeneity of tumor oxygenation, the OER effect, and the ROD effect.Significance. This study supports the experimentally observed iso-effectiveness of FLASH and conventional radiotherapy when analyzing the volume evolution of preclinical tumors (that are far from control). However, this study also hints that tumor growth curves may be less sensitive to small variations in SF than tumor control probability: ROD may lead to increased SF in FLASH radiotherapy, which while not large enough to cause significant differences in tumor growth curves, could lead to important differences in clinical TCPs. Nonetheless, it cannot be discarded that other effects not modeled in this work, like radiation-induced immune effects, can contribute to tumor control and maintain the iso-effectiveness of FLASH radiotherapy. The study of tumor growth curves may not be the ideal experiment to test the iso-effectiveness of FLASH, and experiments reporting TCP orD50may be preferred.

Investigating the impact of stochasticity on HIV infection dynamics in CD4+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{+}$$\\end{document} T cells using a reaction-diffusion model

The disease dynamics affect the human life. When one person is affected with a disease and if it is not treated well, it can weaken the immune system of the body. Human Immunodeficiency Virus (HIV) is a virus that attacks the immune system, of the body which is the defense line against diseases. If it is not treated well then HIV progresses to its advanced stages and it is known as Acquired Immunodeficiency Syndrome (AIDS). HIV is typically a disease that can transferred from one person to another in several ways such as through blood, breastfeeding, sharing needles or syringes, and many others. So, the need of the hour is to consider such important disease dynamics and that will help mankind to save them from such severe disease. For the said purpose the reaction-diffusion HIV CD4+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{+}$$\\end{document} T cell model with drug therapy under the stochastic environment is considered. The underlying model is numerically investigated with two time-efficient schemes and the effects of various parameters used in the model are analyzed and explained in a real-life scenario. Additionally, the obtained results will help the decision-makers to avoid such diseases. The random version of the HIV model is numerically investigated under the influence of time noise in Ito^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\hat{o}}$$\\end{document} sense. The proposed stochastic backward Euler (SBE) scheme and proposed stochastic Implicit finite difference (SIFD) scheme are developed for the computational study of the underlying model. The consistency of the schemes is proven in the mean square sense and the given system of equations is compatible with both schemes. The stability analysis proves that both schemes and schemes are unconditionally stable. The given system of equations has two equilibria, one is disease-free equilibrium (DFE) and the other is endemic equilibrium. The simulations are drawn for the different values of the parameters. The proposed SBE scheme showed the convergent behavior towards the equilibria for the given values of the parameters but also showed negative behavior that is not biological. The proposed SIFD scheme showed better results as compared with the stochastic SBE scheme. This scheme has convergent and positive behavior towards the equilibria points for the given values of the parameters. The effect of various parameters is also analyzed. Simulations are drawn to evaluate the efficacy of the schemes.

Diffusion of Ge Donors in β‐Ga2O3

Diffusion of Ge donors in β‐Ga2O3 is studied using a combination of secondary‐ion mass spectrometry, diffusion simulations, and first‐principles calculations, and compared to previous studies on Sn diffusion. Ge is implanted into (01)‐oriented samples and annealed at temperatures from 900 to 1050 °C for a total of 8 h. From previous first‐principles calculations, Sn is predicted to diffuse via the formation of a mobile complex with VGa that migrates through a sequence of exchange and rotation jumps. Herein, it is similarly predicted that Ge diffusion is mediated by VGa. However, the microscopic mechanism differs, as Ge can diffuse more easily through exchange combined with complex dissociation, rather than rotational jumps. This is explained by the difference in Ga‐site preference of Ge compared to Sn, and the three‐split mechanism that enables low migration barriers for VGa. The dissociation mechanism leads to a considerably faster transport for Ge as compared to Sn. The experimentally obtained Ge diffusion profiles are successfully fitted using a reaction–diffusion model based on the predicted diffusion mechanism, yielding a migration barrier of 2.5 ± 0.2 eV for the complex. The 2.72 eV obtained from first‐principles calculations is in good agreement with this value.

Interpreting the demic diffusion of early farming in Europe with a three-population model

In 1971, Ammerman and Cavalli-Sforza demonstrated that reaction-diffusion equations could be usefully applied to the archaeological question of the spread of early farming in Europe. Their basic premise was demic diffusion, i.e., the iterative short-range colonization of virgin land by the descendants of the original Near Eastern farmers. This hypothesis has been vindicated by ancient DNA studies, which show limited acculturation of the autochthonous hunter-gatherers, who when converted to farming were apparently assimilated into preexisting farming communities. In this brief report, I describe a reaction-diffusion model incorporating various interactions among the Near Eastern farmers, converted farmers, and hunter-gatherers. Predictions, derived in terms of the model parameters, are examined vis-à-vis the ancient DNA and archaeological evidence. Of particular interest is the theoretical requirement that the hunter-gatherers behaved more competitively toward the converted farmers than the Near Eastern (specifically Anatolian) famers. Based on “Ammerman AJ, Cavalli-Sforza LL. Measuring the rate of spread of early farming in Europe. Man 1971; 6: 674-688.”

Pore Network Modeling of Intraparticle Transport Phenomena Accompanied by Chemical Reactions.

In this work, a 3D pore network model (PNM) is introduced for modeling reaction-diffusion phenomena, with and without coupled heat transfer, in a spherical porous catalyst particle. The particle geometry is generated by packing thousands of microspheres inside a large sphere to represent the 3D geometry, porosity, and tortuosity of a spherical catalyst particle. A pore-network representation is extracted from this geometry, and a PNM for diffusion-reaction and heat conduction is constructed. This newly proposed particle-scale PNM allows for the application of realistic 3D nonuniform boundary conditions on the particle's surface, which is commonly encountered in slender packed-bed reactors. Concentration profiles inside the particle, and effectiveness of the reactions, is analyzed.

Pattern dynamics of networked epidemic model with higher-order infections.

Current research on pattern formations in networked reaction-diffusion (RD) systems predominantly focuses on the impacts of diffusion heterogeneity between nodes, often overlooking the contact heterogeneity between individuals within nodes in the reaction terms. In this paper, we establish a networked RD model incorporating infection through higher-order interaction in simplicial complexes in the reaction terms. Through theoretical and numerical analysis, we find that these higher-order interactions may induce Turing instability in the system. Notably, the relationship between the size of the Turing instability range and the average 2-simplices degree within nodes can be approximated by a quadratic function. Additionally, as the average 2-simplices degree increases, the amplitude of the patterns exhibits three distinct trends: increasing, decreasing, and initially increasing then decreasing, while the average infection density increases consistently. We then provide a possible explanation for these observations. Our findings offer new insights into the effects of contact heterogeneity within nodes on networked pattern formations, thereby informing the development of epidemic prevention and control measures.

Existence of global attractor in reaction–diffusion model of obesity-induced Alzheimer’s disease and its control strategies

Evidence suggests that obesity, diabetes, and aging notably increase susceptibility to dementia-related conditions such as Alzheimer’s disease (AD). This article explores the correlations between obesity, diabetes, and AD. It introduces a diffusion-driven model encompassing variables like glucose dynamics, insulin levels, beta cells, microglia, cytokines, amyloid-β plaques, neurofibrillary tangles (τ plaques), neurodegeneration, and cognitive decline. The study includes stability analysis (local and global), examining boundedness and long-time behavior via showing the existence of a global attractor for the diffusion-driven model. A global sensitivity analysis, utilizing the Partial Rank Correlation Coefficient (PRCC), identifies factors sensitively impacting Aβ plaque growth, τ plaques, and neurodegeneration. The deterministic model solution illustrates spatiotemporal dynamics, revealing a link between obesity and Alzheimer’s, which is characterized by distinct patchy patterns. While Alzheimer’s has no cure, employing optimal control techniques can help alleviate its effects and enhance affected individuals’ quality of life. An optimal control problem for AD management is developed, optimizing multiple aspects of disease management. The study highlights the efficacy of long-term healthy lifestyle practices and customized anti-amyloid therapy in significantly delaying obesity-induced AD progression. This research sheds light on the connection between obesity and Alzheimer’s, underscoring the negative impact of pro-inflammatory microglia on cognitive decline while proposing control strategies.

Spatial patterns of the Brusselator model with asymmetric Lévy diffusion

Abstract The formation of spatial patterns is an important issue in reaction–diffusion systems. Previous studies have mainly focused on the spatial patterns in reaction–diffusion models equipped with symmetric diffusion (such as normal or fractional Laplace diffusion), namely, assuming that spatial environments of the systems are homogeneous. However, the complexity and heterogeneity of spatial environments of biochemical reactions in vivo can lead to asymmetric diffusion of reactants. Naturally, there arises an open question of how the asymmetric diffusion affects dynamical behaviors of biochemical reaction systems. To answer this, we build a general asymmetric Lévy diffusion model based on the theory of a continuous time random walk. In addition, we investigate the two-species Brusselator model with asymmetric Lévy diffusion, and obtain a general condition for the formation of Turing and wave patterns. More interestingly, we find that even though the Brusselator model with symmetric diffusion cannot produce steady spatial patterns for some parameters, the asymmetry of Lévy diffusion for this model can produce wave patterns. This is different from the previous result that wave instability requires at least a three-species model. In addition, the asymmetry of Lévy diffusion can significantly affect the amplitude and frequency of the spatial patterns. Our results enrich our knowledge of the mechanisms of pattern formation.

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

Prediction of spatiotemporal dynamics using deep learning: Coupled neural networks of long short-terms memory, auto-encoder and physics-informed neural networks

Several classic reaction-diffusion models using partial differential equations (PDEs) have been established to elucidate the formation mechanism of vegetation patterns. However, predictive modeling of complex spatiotemporal dynamics using traditional numerical methods can be significantly challenging in many practical scenarios. Physics-Informed Neural Networks (PINNs) provide a new approach to predict the solution of PDEs. However, the generalization of PINNs is not satisfactory when pretrained PINNs is directly used in non-trained space (defined as explorations). This may be attributed to the lack of training in the time dimension. Therefore, a framework (LA-PINNs) is proposed to predict the evolutionary solution of the non-dimensional vegetation-sand model. The framework couples neural networks of Long-Short Terms Memory, Auto-Encoder and Physics-Informed Neural Networks. The predictions of LA-PINNs are much better than those of PINNs. Then we studied the effects of hyperparameters on the accuracy of predictions. Based on training in time dimension by LSTM module and pretraining for quick-training strategy, LA-PINNs can improve the accuracy of explorations.