Reaction-diffusion Research Articles

Existence of a Global Attractor for the Reaction–Diffusion Equation with Memory and Lower Regularity Terms

This paper investigates the large time behavior for the reaction–diffusion equation with memory and the forcing term g∈H−1(Ω). We prove the existence of a global attractor in L2(Ω)×Lμ2(R;H01(Ω)). Due to the lower regularity of g, one can hardly use the traditional energy estimates to derive the existence of a bounded absorbing set in the higher regularity space and then the compactness of the semigroup. Here, we utilize the contractive function method to establish the asymptotic smoothness of the semigroup.

Bifurcation and stability analysis of a Leslie–Gower diffusion predator–prey model with prey refuge and Beddington–DeAngelis functional response

In this paper, we consider the spatiotemporal dynamics behaviors of a Leslie–Gower diffusion predator–prey system with prey refuge and Beddington–DeAnglis (B‐D) functional response. By using the Poincaré inequality and topological degree theory, we first investigate the Turing instability of the reaction–diffusion system and prove the existence of nonconstant positive steady‐state solutions. Then we discuss the steady‐state bifurcation and the direction to Hopf bifurcation of the PDE model by the local bifurcation theorem and center manifold theory. Finally, some numerical simulations are presented to supplement the analytic results in one dimension which indicates that changes in prey refuge and diffusion coefficient can increase the complexity of the system.

Effects of Anisotropy, Convection, and Relaxation on Nonlinear Reaction-Diffusion Systems

The effects of relaxation, convection, and anisotropy on a two-dimensional, two-equation system of nonlinearly coupled, second-order hyperbolic, advection–reaction–diffusion equations are studied numerically by means of a three-time-level linearized finite difference method. The formulation utilizes a frame-indifferent constitutive equation for the heat and mass diffusion fluxes, taking into account the tensorial character of the thermal diffusivity of heat and mass diffusion. This approach results in a large system of linear algebraic equations at each time level. It is shown that the effects of relaxation are small although they may be noticeable initially if the relaxation times are smaller than the characteristic residence, diffusion, and reaction times. It is also shown that the anisotropy associated with one of the dependent variables does not have an important role in the reaction wave dynamics, whereas the anisotropy of the other dependent variable results in transitions from spiral waves to either large or small curvature reaction fronts. Convection is found to play an important role in the reaction front dynamics depending on the vortex circulation and radius and the anisotropy of the two dependent variables. For clockwise-rotating vortices of large diameter, patterns similar to those observed in planar mixing layers have been found for anisotropic diffusion tensors.

Galerkin approximation of Burgers-Huxley equation with fractional order arising in reaction mechanism and diffusion transport

Abstract Burgers-Huxley model depicts a prototype model of the interaction of convection effect, reaction mechanisms and diffusion transport, used to study the liquid crystal and nerve fibers. This study introduces Galerkin approximation for time-fractional Burgers-Huxley equation (TFBHE) . The Caputo derivative is used to evaluate the temporal part using the $L_1$ formula. The Galerkin approach employs cubic B-spline as a shape and test function, resulting in a symmetric matrix that is easily convergent. In addition, the three-point quadrature rule is implemented to evaluate the integration of complex function . The Von Neumann analysis is used to discuss stability of the scheme. The performance and robustness of the technique is measured using various error norms. The results are compared with the exact solution, demonstrating effectiveness of the proposed method.

The Kinetics of Polymer Brush Growth in the Frame of the Reaction Diffusion Front Formalism

We studied the properties of a reaction front that forms in irreversible reaction–diffusion systems with concentration-dependent diffusivities during the synthesis of polymer brushes. A coarse-grained model of the polymerization process during the formation of polymer brushes was designed and investigated for this purpose. In this model, a certain amount of initiator was placed on an impenetrable surface, and the “grafted from” procedure of polymerization was carried out. The system consisted of monomer molecules and growing chains. The obtained brush consisted of linear chains embedded in nodes of a face-centered cubic lattice with excluded volume interactions only. The simulations were carried out for high rafting densities of 0.1, 0.3, and 0.6 and for reaction probabilities of 0.02, 0.002, and 0.0002. Simulations were performed by means of the Monte Carlo method while employing the Dynamic Lattice Liquid model. Some universal behavior was found, i.e., irrespective of reaction rate and grafting density, the width of the reaction front as well as the height of the front show for long times the same scaling with respect to time. During the formation of the polymer layer despite the observed difference in dispersion of chain lengths for different grafting densities and reaction rates at a given layer height, the quality of the polymer layer does not seem to depend on these parameters.

A Space Distributed Model and Its Application for Modeling the COVID-19 Pandemic in Ukraine

A space distributed model based on reaction–diffusion equations, which was previously developed, is generalized and applied to COVID-19 pandemic modeling in Ukraine. Theoretical analysis and a wide range of numerical simulations demonstrate that the model adequately describes the second wave of the COVID-19 pandemic in Ukraine. In particular, comparison of the numerical results obtained with the official data shows that the model produces very plausible total numbers of the COVID-19 cases and deaths. An extensive analysis of the impact of the parameters arising from the model is presented as well. It is shown that a well-founded choice of parameters plays a crucial role in the applicability of the model.

Dynamics behaviours of N-kink solitons in conformable Fisher–Kolmogorov–Petrovskii–Piskunov equation

PurposeThis manuscript is related to compute $N$-kink soliton solutions for conformable Fisher–Kolmogorov equation (CFKE) by using the generalized extended direct algebraic method (EDAM). The considered problem has important applications in mathematical biology and reaction diffusion processes. Also, the mentioned problem has significant applications in population dynamics. The fractional order conformable derivative has many features as compared to the other fractional order differential operators. For instance, the chain, product and quotient procedures do not satisfy by other fractional differential operators, but conformable operators obey the mentioned rules. Hence, we compute the soliton solutions for the mentioned problem and present its various dynamical behaviours graphically.Design/methodology/approachThe generalized EDAM is used in this article to examine the calculation of N-kink soliton solutions for the CFKE. In mathematical biology and reaction-diffusion processes, the topic under consideration holds great significance, especially when considering population dynamics.FindingsThe results highlight the benefits of utilising conformable derivatives in mathematical modelling and further our understanding of fractional differential equations and their applications.Research limitations/implicationsThe work focuses primarily on N-kink soliton solutions, which may limit the examination of alternative types of solutions (e.g., multi-soliton or periodic solutions) that might give new insights into the dynamics of the CFKE.Practical implicationsThe generated N N-kink soliton solutions can enhance mathematical models in biological contexts, notably in modelling population dynamics, disease propagation and ecological interactions, leading to better forecasts and interventions.Social implicationsPublic health initiatives can benefit from the understanding of disease transmission and intervention efficacy that comes from modelling population dynamics and reaction-diffusion processes.Originality/valueThe use of the generalized EDAM to obtain solutions for N-kink soliton problems is an innovative method for solving the conformable Fisher–Kolmogorov equation, demonstrating the power of this mathematical tool.

Publisher Correction: Numerical Study of the Reaction Diffusion Prey–Predator Model Having Holling II Increasing Function in the Predator Under Noisy Environment

Publisher Correction: Numerical Study of the Reaction Diffusion Prey–Predator Model Having Holling II Increasing Function in the Predator Under Noisy Environment

Geometric Ergodicity of a Stochastic Reaction–Diffusion Tuberculosis Model with Varying Immunity Period

Geometric Ergodicity of a Stochastic Reaction–Diffusion Tuberculosis Model with Varying Immunity Period

WELL-POSEDNESS RESULTS FOR GENERAL REACTION–DIFFUSION TRANSPORT OF OXYGEN IN ENCAPSULATED CELLS

Abstract We provide well-posedness results for nonlinear parabolic partial differential equations (PDEs) given by reaction–diffusion equations describing the concentration of oxygen in encapsulated cells. The cells are described in terms of a core and a shell, which introduces a discontinuous diffusion coefficient as the material properties of the core and shell differ. In addition, the cells are subject to general nonlinear consumption of oxygen. As no monotonicity condition is imposed on the consumption, monotone operator theory cannot be used. Moreover, the discontinuity in the diffusion coefficient bars us from applying classical results on strong solutions. However, by directly applying a Galerkin method, we obtain uniqueness and existence of the strong form solution. These results provide the basis to study the dynamics of cells in critical states.

Synchronization of fuzzy reaction–diffusion neural networks via semi-intermittent hybrid control

Synchronization of fuzzy reaction–diffusion neural networks via semi-intermittent hybrid control

Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment

Abstract This study proposed and analyzed a vector-borne reaction–diffusion–advection model with vector-bias mechanism and heterogeneous parameters in one-dimensional habitat. The basic reproduction number R 0 {{\mathfrak{R}}}_{0} in connection with principal eigenvalue of elliptic eigenvalue problem is characterized as the role of determining the threshold dynamics of the system. The main objective of this study is to investigate the asymptotic profiles and monotonicity of R 0 {{\mathfrak{R}}}_{0} with respect to diffusion rates and advection rates under certain conditions. Through exploring the level set of R 0 {{\mathfrak{R}}}_{0} , we also find that there exists a unique surface separating the dynamics. Our results also reveal that the infected hosts and vectors will aggregate at the downstream end if the ratio of advection rates and diffusion rates is sufficiently large.

A novel efficient analytical technique and its application to fractional Cauchy reaction–diffusion equations

In this research paper, we suggest a novel analytical method to obtain the approximate solutions of linear and nonlinear differential equations of arbitrary order. The proposed technique is a good combination of Kharrat–Toma transform and q-homotopy analysis method. Additionally, we also find out the solution of Cauchy reaction–diffusion equation associated with regularized version of Hilfer–Prabhakar fractional derivative. The Cauchy reaction–diffusion equation is broadly used to describe the dynamical processes in physics, geology, biology, ecology, etc. Some characteristic properties of the considered model and existence as well as the uniqueness of the solutions are also discussed. The outcomes of the considered model are exhibited graphically to show the accuracy and efficiency of the suggested method.

A new chemical networked system: spatial-temporal evolution and control

Abstract This paper constructs a scale-free chemical network based on the Gierer-Meinhardt (GM) system and investigates its Turing instability. We establish a fractional-order single-node GM system with delay and design a fractional-order proportional derivative (PD) control strategy for the issue of bifurcation control. Using delay as bifurcation parameter, the existence of Hopf bifurcation is proven, and control over bifurcation threshold points is achieved through a fractional-order PD control strategy. For the scale-free chemical network based on the GM system, we obtain the condition of how the Turing instability occurs. We derive how the number of edges for the new nodes changes the stability of the network-organized system and investigate the relationship between degrees of nodes and eigenvalues of the network matrix. We give the instability condition caused by diffusion in the network-organized system. Finally, the numerical simulations verify analytical results.

Traveling waves in a delayed reaction–diffusion SIR epidemic model with a generalized incidence function

Traveling waves in a delayed reaction–diffusion SIR epidemic model with a generalized incidence function

Study on the Acidic Modification of Mesoporous HZSM-5 Zeolite and Its Catalytic Cracking Performance

Mesoporous HZSM-5 zeolites with nanocrystal stacking morphology were directly synthesized via hydrothermal methods without mesoporous templates. The synthesized mesoporous HZSM-5 was subjected to hydrothermal–citric acid washing treatment. The structural and acidic properties of the samples before and after modification were characterized using various techniques. The catalytic performance for butene conversion to propylene was investigated under atmospheric pressure, 500 °C, and a butene weight hourly space velocity (WHSV) of 10 h−1 in a continuous-flow micro-fixed bed reactor. The results show that propylene selectivity increased significantly from 24.7% before modification to 44%, and propylene yield increased from 22% to 38%. After 2 h of hydrothermal–citric acid washing modification, the catalyst maintained a butene conversion rate of 76% and a selectivity of 47% at 525 °C and a WHSV of 10 h−1 after 130 h of continuous reaction, with a propylene yield of 37%. The results indicate that moderate hydrothermal–citric acid washing modification leads to the removal of aluminum from the zeolite framework, reducing the amount and strength of acid but increasing the mesopore quantity. This helps control the reaction pathways and diffusion of intermediate products, suppresses some side reactions, and improves the selectivity and yield of the desired product, propylene, while significantly enhancing catalytic stability.

Note on the diffusive prey-predator model with variable coefficients and degenerate diffusion

It is of interest to understand effects of variable coefficients and degenerate diffusion on the longtime behaviors of solutions of reaction diffusion equations. Recently, Yang and Yao (2024) studied a classical prey-predator model and proved that the degradation of the diffusion coefficient of the prey and variable coefficients satisfying the appropriate conditions will not affect dynamical properties. In this note we shall simplify the proof of Yang and Yao (2024) and delete the condition (4).

An Algorithm for Creating a Synaptic Cleft Digital Phantom Suitable for Further Numerical Modeling

One of the most significant applications of mathematical numerical methods in biology is the theoretical description of the convectional reaction–diffusion of chemical compounds. Initial biological objects must be appropriately mimicked by digital domains that are suitable for further use in computational modeling. In the present study, an algorithm for the creation of a digital phantom describing a local part of nervous tissue—namely, a synaptic contact—is established. All essential elements of the synapse are determined using a set of consistent Boolean operations within the COMSOL Multiphysics software 6.1. The formalization of the algorithm involves a sequence of procedures and logical operations applied to a combination of 3D Voronoi diagrams, an experimentally defined inner synapse area, and a simple ellipsoid under different sets of biological parameters. The obtained digital phantom is universal and may be applied to different types of neuronal synapses. The clear separation of the designed domains reveals that the boundary’s conditions and internal flux dysconnectivity functions can be set up explicitly. Digital domains corresponding to the parts of a synapse are appropriate for further application of the derived numeric meshes, with various capacities of the included elements. Thus, the obtained digital phantom can be effectively used for further modeling of the convectional reaction–diffusion of chemical compounds in nervous tissue.

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.

Spatiotemporal fluctuation induces Turing pattern formation in the chemical Brusselator

Chemical reactions are embedded in spatiotemporal fluctuations instead of a constant environment. Here, we aimed to assess reaction–diffusion (RD) with dichotomous noise‐controlling system parameters in the Brusselator and examine the effect of these fluctuations on the dynamic behavior of chemical reactions. By performing a multiscale perturbation analysis, we demonstrated that the correlated noise can broaden the Turing region even if molecular memory (autocorrelation time) exists. However, for small noise, short‐term memory promotes Turing instability. The instability of the Brusselator is determined by the noise strength, which belongs to the optimal region if the diffusion coefficient is fixed. Turing pattern selection and stability are also governed by the dynamic character of the amplitude equation, and the entire Turing instability region shifts to the right in the phase space with noise perturbation. Finally, numerical simulations validate the theoretical derivation that correlated noise can amplify Turing pattern formation to maintain distinct patterns.