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The stability analysis of time-delay neural networks with reaction-diffusion terms in the sense of Riemann–Liouville derivative is still an open problem, which will be considered in this paper. We first extend a new inequality on Riemann–Liouville fractional-order derivative, which plays an important role in the subsequent proof. Using the Lyapunov direct method, Jensen's integral inequality, and the linear matrix inequality (LMI) method, several easy-to-test criteria expressed by system parameters and given parameters are given to ensure the stability of the system under consideration. The advantage of our method is that we can directly calculate the integral-order derivative of the Lyapunov function, which can be very convenient to test the stability of practical system. Finally, the validity and conciseness of the results are verified by numerical simulation. [ABSTRACT FROM AUTHOR]

The MinDE proteins from E. coli have received great attention as a paradigmatic biological pattern‐forming system. Recently, it has surfaced that these proteins do not only generate oscillating concentration gradients driven by ATP hydrolysis, but that they can reversibly deform giant vesicles. In order to explore the potential of Min proteins to actually perform mechanical work, we introduce a new model membrane system, flat vesicle stacks on top of a supported lipid bilayer. MinDE oscillations can repeatedly deform these flat vesicles into tubules and promote progressive membrane spreading through membrane adhesion. Dependent on membrane and buffer compositions, Min oscillations further induce robust bud formation. Altogether, we demonstrate that under specific conditions, MinDE self‐organization can result in work performed on biomimetic systems and achieve a straightforward mechanochemical coupling between the MinDE biochemical reaction cycle and membrane transformation. [ABSTRACT FROM AUTHOR]

A space spectral interpolation collocation method is proposed to study non-linear reaction-diffusion systems with complex dynamics characters. A detailed solution process is elucidated, and some pattern formations are given. The numerical results have a good agreement with theoretical ones. The method can be extended to fractional calculus and fractal calculus. [ABSTRACT FROM AUTHOR]

In this paper, a delayed HIV/AIDS epidemic model with treatment and spatial diffusion is introduced. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state is discussed. By using the cross-iteration method and Schauder's fixed point theorem, we reduce the existence of traveling waves to the existence of a pair of upper–lower solutions. By constructing a pair of upper–lower solutions, we derive the existence of a traveling wave solution connecting the disease-free steady state and the endemic steady state. It is shown that the existence of traveling waves of the proposed HIV/AIDS epidemic model is fully determined by the basic reproduction number and the minimal wave speed. Finally, numerical simulations are performed to show the feasibility and effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a reaction–diffusion model with delay effect and Dirichlet boundary condition is considered. Firstly, the existence, multiplicity, and patterns of spatially nonhomogeneous steady-state solution are obtained by using the Lyapunov–Schmidt reduction. Secondly, by means of space decomposition, we subtly discuss the distribution of eigenvalues of the infinitesimal generator associated with the linearized system at a spatially nonhomogeneous synchronous steady-state solution, and then we derive some sufficient conditions to ensure that the nontrivial synchronous steady-state solution is asymptotically stable. By using the symmetric bifurcation theory of differential equations together with the representation theory of standard dihedral groups, we not only investigate the effect of time delay on the pattern formation, but also obtain some important results on the spontaneous bifurcation of multiple branches of nonlinear wave solutions and their spatiotemporal patterns. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a reaction–diffusion malaria model which incorporates vector-bias, spatial heterogeneity, sensitive and resistant strains. The main question that we study is the threshold dynamics of the model, in particular, whether the existence of spatial structure would allow two strains to coexist. In order to achieve this goal, we define the basic reproduction number R i and introduce the invasion reproduction number R ^ i for strain i (i = 1 , 2) . A quantitative analysis shows that if R i < 1 , then disease-free steady state is globally asymptotically stable, while competitive exclusion, where strain i persists and strain j dies out, is a possible outcome when R i > 1 > R j (i ≠ j , i , j = 1 , 2) , and a unique solution with two strains coexist to the model is globally asymptotically stable if R i > 1 , R ^ i > 1 . Numerical simulations reinforce these analytical results and demonstrate epidemiological interaction between two strains, discuss the influence of resistant strains and study the effects of vector-bias on the transmission of malaria. [ABSTRACT FROM AUTHOR]

It is well known that movement strategies in ecology and in economics can make the difference between extinction and persistence. We present a unifying model for the dynamics of ecological populations and street vendors, which are an important part of many informal economies. We analyze this model to study the effects of directed movement of populations subject to strong Allee effect. We begin with the study of the existence of equilibrium solutions subject to homogeneous Dirichlet or no-flux boundary conditions. Next, we study the evolution problem and show that if the directed movement effect is small, the solutions behave like those of the classical reaction-diffusion equation with bistable growth pattern. We present numerical simulations, which show that directed movement can help overcome a strong Allee effect and provide some partial analytical results in this direction. We conclude by making a connection to the ideal free distribution and analyze what happens under competition, finding that an ideal free distribution strategy is a local neighborhood invader. [ABSTRACT FROM AUTHOR]

[Display omitted] In the past years, a significant amount of effort has been directed at the observation and characterisation of caries using experimental techniques. Nevertheless, relatively little progress has been made in numerical modelling of the underlying demineralisation process. The present study is the first attempt to provide a simplified calculation framework for the numerical simulation of the demineralisation process at the length scale of enamel rods and its validation by comparing the data with statistical analysis of experimental results. FEM model was employed to simulate a time-dependent reaction-diffusion equation process in which H ions diffuse and cause demineralisation of the enamel. The local orientation of the hydroxyapatite crystals was taken into account. Experimental analysis of the demineralising front was performed using advanced high-resolution synchrotron X-ray micro-Computed Tomography. Further experimental investigations were conducted by means of SEM and STEM imaging techniques. Besides establishing and validating the new modelling framework, insights into the role of the etchant solution pH level were obtained. Additionally, some light was shed on the origin of different types of etching patterns by simulating the demineralisation process at different etching angles of attack. The implications of this study pave the way for simulations of enamel demineralisation within different complex scenarios and across the range of length scales. Indeed, the framework proposed can incorporate the presence of chemical species other than H ions and their diffusion and reaction leading to dissolution and re-precipitation of hydroxyapatite. It is the authors' hope and aspiration that ultimately this work will help identify new ways of controlling and preventing caries. [ABSTRACT FROM AUTHOR]

This paper focuses on the synchronization of reaction-diffusion complex dynamical networks with coupling delay. In order to reflect the uncertainties of the controller, the nonfragile problem is considered. Furthermore, we also take into account the dissipativity analysis problem, which contains the H ∞ performance and passivity performance in a unified framework. By utilizing the Lyapunov functional method, two sufficient delay-dependent conditions, which ensure the considered system is globally asymptotically synchronized onto the unforced node and strictly dissipative, are established in terms of linear matrix inequality. Finally, three numerical examples are employed to demonstrate the effectiveness of the design methods. [ABSTRACT FROM AUTHOR]

A reaction-diffusion predator-prey system with prey-taxis and predator-taxis describes the spatial interaction and random movement of predator and prey species, as well as the spatial movement of predators pursuing prey and prey evading predators. The spatial pattern formation induced by the prey-taxis and predator-taxis is characterized by the Turing type linear instability of homogeneous state and bifurcation theory. It is shown that both attractive prey-taxis and repulsive predator-taxis compress the spatial patterns, while repulsive prey-taxis and attractive predator-taxis help to generate spatial patterns. Our results are applied to the Holling-Tanner predator-prey model to demonstrate the pattern formation mechanism. [ABSTRACT FROM AUTHOR]

The goal of this work is to understand and quantify how a line with nonlocal diffusion given by an integral enhances a reaction-diffusion process occurring in the surrounding plane. This is part of a long term programme where we aim at modelling, in a mathematically rigorous way, the effect of transportation networks on the speed of biological invasions or propagation of epidemics. We prove the existence of a global propagation speed and characterise in terms of the parameters of the system the situations where such a speed is boosted by the presence of the line. In the course of the study we also uncover unexpected regularity properties of the model. On the quantitative side, the two main parameters are the intensity of the diffusion kernel and the characteristic size of its support. One outcome of this work is that the propagation speed will significantly be enhanced even if only one of the two is large, thus broadening the picture that we have already drawn in our previous works on the subject, with local diffusion modelled by a standard Laplacian. We further investigate the role of the other parameters, enlightening some subtle effects due to the interplay between the diffusion in the half plane and that on the line. Lastly, in the context of propagation of epidemics, we also discuss the model where, instead of a diffusion, displacement on the line comes from a pure transport term., (© The Author(s) 2024. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.)

The coupling between energy-consuming molecular processes and the macroscopic dimension plays an important role in nature and in the development of active matter. Here, we study the temporal evolution of a macroscopic system upon the local activation of a dissipative self-assembly process. Injection of surfactant molecules in a substrate-containing hydrogel results in the local substrate-templated formation of assemblies, which are catalysts for the conversion of substrate into waste. We show that the system develops into a macroscopic (pseudo-)non-equilibrium steady state (NESS) characterized by the local presence of energy-dissipating assemblies and persistent substrate and waste concentration gradients. For elevated substrate concentrations, this state can be maintained for more than 4 days. The studies reveal an interdependence between the dissipative assemblies and the concentration gradients: catalytic activity by the assemblies results in sustained concentration gradients and, vice versa, continuous diffusion of substrate to the assemblies stabilizes their size. The possibility to activate dissipative processes with spatial control and create long lasting non-equilibrium steady states enables dissipative structures to be studied in the space-time domain, which is of relevance for understanding biological systems and for the development of active matter., (© 2024 Wiley-VCH GmbH.)

Thin-films patterned with complex motifs are of fundamental interest because of their advanced optical, mechanical and electronic properties, but fabrication of these materials remains challenging. Self-organization strategies, such as immersion controlled reaction-diffusion patterning, have shown great potential for production of patterned thin-films. However, the autonomous nature of such processes limits controllable pattern customizability and complexity. Here, it is demonstrated that photography inspired manipulation processes can overcome this limitation to create highly-complex tapestries of micropatterned films (MPF's). Inspired by classical photographic processes, MPF's are developed, bleached, exposed, fixed, and contoured into user-defined shapes and photographic toning reactions are used to convert the chemical composition MPF's, while preserving the original stripe patterns. By applying principles of composite photography, highly complex tapestries composed of multiple MPF layers are designed, where each layer can be individually manipulated into a specific shape and composition. By overcoming fundamental limitations, this synergistic approach broadens the design possibilities of reaction-diffusion processes, furthering the potential of self-organization strategies for the development of complex materials., (© 2024 The Authors. Advanced Science published by Wiley‐VCH GmbH.)

Supramolecular self-assembly is ubiquitous in living system and is usually controlled to proceed in time and space through sophisticated reaction-diffusion processes, underpinning various vital cellular functions. In this contribution, we demonstrate how spatiotemporal self-assembly of supramolecular hydrogels can be realized through a simple reaction-diffusion-mediated transient transduction of pH signal. In the reaction-diffusion system, a relatively faster diffusion of acid followed by delayed enzymatic production and diffusion of base from the opposite site enables a transient transduction of pH signal in the substrate. By coupling such reaction-diffusion system with pH-sensitive gelators, dynamic supramolecular hydrogels with tunable lifetimes are formed at defined locations. The hydrogel fibers show interesting dynamic growing behaviors under the regulation of transient pH signal, reminiscent of their biological counterpart. We further demonstrate a proof-of-concept application of the developed methodology for dynamic information encoding in a soft substrate. We envision that this work may provide a potent approach to enable transient transduction of various chemical signals for the construction of new colloidal materials with the capability to evolve their structures and functionalities in time and space., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2024 Elsevier Inc. All rights reserved.)

Gene drive alleles that can bias their own inheritance could engineer populations for control of disease vectors, invasive species and agricultural pests. There are successful examples of suppression drives and confined modification drives, but developing confined suppression drives has proven more difficult. However, CRISPR-based toxin-antidote dominant embryo (TADE) suppression drive may fill this niche. It works by targeting and disrupting a haplolethal target gene in the germline with its gRNAs while rescuing this target. It also disrupts a female fertility gene by driving insertion or additional gRNAs. Here, we used a reaction-diffusion model to assess drive performance in continuous space, where outcomes can be substantially different from those in panmictic populations. We measured drive wave speed and found that moderate fitness costs or target gene disruption in the early embryo from maternally deposited nuclease can eliminate the drive's ability to form a wave of advance. We assessed the required release size, and finally we investigated migration corridor scenarios. It is often possible for the drive to suppress one population and then persist in the corridor without invading the second population, a potentially desirable outcome. Thus, even imperfect variants of TADE suppression drive may be excellent candidates for confined population suppression.

Cellular self-organization is essential to physiological tissue and organ development. We previously observed that vascular mesenchymal cells, a multipotent subpopulation of aortic smooth muscle cells, self-organize into macroscopic, periodic patterns in culture. The patterns are produced by cells gathering into raised aggregates in the shape of nodules or ridges. To determine whether these patterns are accounted for by an oriented pattern of cell divisions or postmitotic relocation of cells, we acquired time-lapse, videomicrographic phase-contrast, and fluorescence images during self-organization. Cell division events were analyzed for orientation of daughter cells in mitoses during separation and their angle relative to local cell alignment, and frequency distribution of the mitotic angles was analyzed by both histographic and bin-free statistical methods. Results showed a statistically significant preferential orientation of daughter cells along the axis of local cell alignment as early as day 8, just before aggregate formation. This alignment of mitotic axes was also statistically significant at the time of aggregate development (day 11) and after aggregate formation was complete (day 15). Treatment with the nonmuscle myosin II inhibitor, blebbistatin, attenuated alignment of mitotic orientation, whereas Rho kinase inhibition eliminated local cell alignment, suggesting a role for stress fiber orientation in this self-organization. Inhibition of cell division using mitomycin C reduced the macroscopic pattern formation. Time-lapse monitoring of individual cells expressing green fluorescent protein showed postmitotic movement of cells into neighboring aggregates. These findings suggest that polarization of mitoses and postmitotic migration of cells both contribute to self-organization into periodic, macroscopic patterns in vascular stem cells.

Mechanistic capabilities found in nature have influenced a variety of successful functional designs in engineering. However, the unique combinations of mechanical properties found in natural materials have not been readily adapted into synthetic materials, due to a disconnect between biological principles and engineering applications. Current biomimetic material approaches tend to involve mimicking nature's microstructure geometries or mimicking nature's adaptive design process through brute force element-by-element composite optimization techniques. While the adaptive approach promotes the generation of application-specific microstructure geometries, the element-by-element optimization techniques encompass a large design space that is directly related to the number of elements in the system. In contrast, a novel methodology is proposed in this paper that merges biological pattern generation mechanisms observed in the Gray-Scott model, with an evolutionary-inspired genetic algorithm to create adaptive bio-inspired composite geometries optimized for stiffness and toughness. The results reveal that this methodology significantly reduces the optimization parameter space from tens of thousands of parameters to only four or five. In addition, the resultant composite geometries improved upon the overall combination of stiffness and toughness by a factor of 13, when compared to published brute force element-by-element techniques.

In this article, we consider non-negative solutions of the nonlinear cancer invasion mathematical model involving proliferation and growth functions with homogeneous Neumann and Robin type boundary conditions. We first obtain lower bounds for the finite time blow-up of solutions in $\mathbb{R}^3$ with assumed boundary conditions. Finally, we extend the blow-up results of the given system in $\mathbb{R}^2$ using first-order differential inequality techniques and under appropriate assumptions on data.

In this paper, the existence and stability of mild periodic solutions to the stochastic reaction-diffusion neural networks (SRDNNs) with S-type distributed delays are studied. First, the key issues of the Markov property of mild solutions to the SRDNNs with S-type distributed delays in Cb-space are investigated. Next, the existence of mild periodic solutions is discussed by the dissipative theory and the operator semigroup theory. Then, some sufficient conditions ensuring the stability of mild periodic solutions are derived by the Lyapunov method. To overcome the difficulties created by the special features possessed by S-type distributed delays, the truncation method is applied. Finally, a numerical example is given to illustrate the feasibility of our results.

Abstract A 3-D continuum mixture model describing the corrosion of concrete with sulfuric acid is built. Essentially, the chemical reaction transforms slaked lime (calcium hydroxide) and sulfuric acid into gypsum releasing water. The model incorporates the evolution of chemical reaction, diffusion of species within the porous material and mechanical deformations. This model is applied to a 1-D problem of a plate-layer between concrete and sewer air. The influx of slaked lime from the concrete and sulfuric acid from the sewer air sustains a gypsum creating chemical reaction (sulfatation or sulfate attack). The combination of the influx of matter and the chemical reaction causes a net growth in the thickness of the gypsum layer on top of the concrete base. The model allows for the determination of the plate layer thickness h=h(t) as function of time, which indicates both the amount of gypsum being created due to concrete corrosion and the amount of slaked lime and sulfuric acid in the material. The existence of a parameter regime for which the model yields a non-decreasing plate layer thickness h(t) is identified numerically. The robustness of the model with respect to changes in the model parameters is also investigated.

In this paper, we investigate the dynamics of a diffusive Gompertz population model with nonlocal delay effect and Dirichlet boundary condition. The stability of the positive spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcations with the change of the time delay are discussed by analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system. Then we derive the stability and bifurcation direction of Hopf bifurcating periodic orbits by using the normal form theory and the center manifold reduction. Finally, we give some numerical simulations.

The asymptotic behavior of dissipative evolution problems, determined by complex networks of reaction-diffusion systems, is investigated with an original approach. We establish a novel estimation of the fractal dimension of exponential attractors for a wide class of continuous dynamical systems, clarifying the effect of the topology of the network on the large time dynamics of the generated semi-flow. We explore various remarkable topologies (chains, cycles, star and complete graphs) and discover that the size of the network does not necessarily enlarge the dimension of attractors. Additionally, we prove a synchronization theorem in the case of symmetric topologies. We apply our method to a complex network of competing species systems modeling an heterogeneous biological ecosystem and propose a series of numerical simulations which underpin our theoretical statements. [ABSTRACT FROM AUTHOR]

This study considers the problem of control and synchronization between fractional‐order and integer‐order, N‐components reaction‐diffusion systems with nonidentical coefficients and different nonlinear parts. The control scheme is designed using the Lyapunov direct method. The results are exemplified by two significant biochemical models, namely, the fractional‐order Lengyel‐Epstein model and the Gray‐Scott model. To illustrate the effectiveness of the proposed scheme, numerical simulations are performed in one and two space dimensions using Homotopy Analysis Method (HAM). [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of a general reaction-diffusion-advection equation with nonlocal delay effect and Dirichlet boundary condition. The existence and stability of positive spatially nonhomogeneous steady state solution are shown. By analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized equation, the existence of Hopf bifurcation is proved. We introduce the weighted space to overcome the hurdle from advection term. We also show that the effect of adding a term advection along environmental gradients to Hopf bifurcation values for a Logistic equation with nonlocal delay. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a delay reaction–diffusion–advection model with ideal free dispersal. The stability of positive steady-state solutions and the existence of the associated Hopf bifurcation are obtained by analyzing the principal eigenvalue of an elliptic operator. By the normal form theory and the center manifold reduction, the stability and bifurcation direction of Hopf bifurcating periodic solutions are obtained. Moreover, numerical simulations and a brief discussion are presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

Glioblastoma Multiforme is a malignant brain tumor with poor prognosis. There have been numerous attempts to model the invasion of tumorous glioma cells via partial differential equations in the form of advection–diffusion–reaction equations. The patient-wise parametrization of these models, and their validation via experimental data has been found to be difficult, as time sequence measurements are mostly missing. Also the clinical interest lies in the actual (invisible) tumor extent for a particular MRI/DTI scan and not in a predictive estimate. Therefore we propose a stationalized approach to estimate the extent of glioblastoma (GBM) invasion at the time of a given MRI/DTI scan. The underlying dynamics can be derived from an instationary GBM model, falling into the wide class of advection-diffusion-reaction equations. The stationalization is introduced via an analytic solution of the Fisher-KPP equation, the simplest model in the considered model class. We investigate the applicability in 1D and 2D, in the presence of inhomogeneous diffusion coefficients and on a real 3D DTI-dataset. [ABSTRACT FROM AUTHOR]

The study of pattern-forming instabilities in reaction–diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental chemistry. Analyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction–diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace–Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as a sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing–Hopf instabilities. Extensions to higher-order spatial systems are also included as a way of demonstrating the generality of the approach. [ABSTRACT FROM AUTHOR]

A nonlinear reaction-diffusion partial differential equation occurring in models of transient controlled-potential experiments in electroanalytical chemistry is investigated analytically and numerically, with a view to determining a relationship between the concentration of a chemical species and its flux at a reacting electrode. It is shown that a previously known relation that holds for the steady-state case can be used as the first term in a singular perturbation expansion for the time-dependent case. However, in trying to determine the second term, so as to extend the range of validity of the solution, it is found that a phenomenon akin to switchbacking occurs, with the asymptotic details being strongly dependent on the reaction order; this appears to be a consequence of the spatial algebraic decay of the leading-order solution far from the electrode. Comparison of asymptotic results with numerical solutions obtained using finite element methods indicates a relation involving the homogeoneous reaction order for which the two-term asymptotic approximation would work best for all time. Links to problems that involve algebraically decaying boundary layers are briefly discussed. [ABSTRACT FROM AUTHOR]

We consider in this paper the maximization problem for the quantity ∫ Ωu(t, x)dx with respect to u0 =: u(0, ⋅), where u is the solution of a given reaction diffusion equation. This problem is motivated by biological conservation questions. We show the existence of a maximizer and derive optimality conditions through an adjoint problem. We have to face regularity issues since non-smooth initial data could give a better result than smooth ones. We then derive an algorithm enabling to approximate the maximizer and discuss some open problems. [ABSTRACT FROM AUTHOR]

We consider two-point, reaction–diffusion type, singularly perturbed boundary value problems of order 2 ν ∈ Z + , and the approximation of their solution using isogeometric analysis. In particular, we use a Galerkin formulation with B-splines as basis functions, defined using appropriately chosen knot vectors. We prove robust exponential convergence in the energy norm, independently of the singular perturbation parameter. Numerical examples are also presented, which illustrate (and extend) the theory. [ABSTRACT FROM AUTHOR]

This article is concerned with the existence of a weak solution to the initial boundary problem for a cross-diffusion system which arises in the study of two cell population growth. The mathematical challenge is due to the fact that the coefficient matrix is non-symmetric and degenerate in the sense that its determinant is. The existence assertion is established by exploring the fact that the total population density satisfies a porous media equation. [ABSTRACT FROM AUTHOR]

This paper focuses on the quasi-fixed time synchronization of memristive Cohen-Grossberg neural networks with reaction diffusion. First, a new definition of finite/fixed-time attraction set and the concept of quasi finite/fixed-time synchronization are proposed. Then, a quasi fixed-time synchronized lemma is given and an example is used to illustrate the correctness of the lemma. At the same time, the new results given in this article are compared with the existing ones under certain conditions, and the conclusions of this paper are less conservative in remark. In addition, quasi fixed-time synchronization theorem and full synchronization theorem are derived by designing different controllers, finally two examples are given to illustrate the validity of these theorems. [ABSTRACT FROM AUTHOR]

Dynamic phenomena in social and biological sciences can often be modeled by reaction-diffusion equations. When addressing the control from a mathematical viewpoint, one of the main challenges is that, because of the intrinsic nature of the models under consideration, the solution, typically a proportion or a density function, needs to preserve given lower and upper bounds (taking values in [ 0 , 1 ])). Controlling the system to the desired final configuration then becomes complex, and sometimes even impossible. In the present work, we analyze the controllability to constant steady states of spatially homogeneous monostable and bistable semilinear heat equations, with constraints in the state, and using boundary controls. We prove that controlling the system to a constant steady state may become impossible when the diffusivity is too small due to the existence of barrier functions. We build sophisticated control strategies combining the dissipativity of the system, the existence of traveling waves, and some connectivity of the set of steady states to ensure controllability whenever it is possible. This connectivity allows building paths that the controlled trajectories can follow, in a long time, with small oscillations, preserving the natural constraints of the system. This kind of strategy was successfully implemented in one-space dimension, where phase plane analysis techniques allowed to decode the nature of the set of steady states. These techniques fail in the present multi-dimensional setting. We employ a fictitious domain technique, extending the system to a larger ball, and building paths of radially symmetric solution that can then be restricted to the original domain. [ABSTRACT FROM AUTHOR]

This article investigates the global stabilization problem of Takagi–Sugeno fuzzy memristor-based neural networks with reaction–diffusion terms and distributed time-varying delays. By using the Green formula and proposing fuzzy feedback controllers, several algebraic criteria dependent on the diffusion coefficients are established to guarantee the global exponential stability of the addressed networks. Moreover, a simpler stability criterion is obtained by designing an adaptive fuzzy controller. The results derived in this article are generalized and include some existing ones as special cases. Finally, the validity of the theoretical results is verified by two examples. [ABSTRACT FROM AUTHOR]

This paper considers the mean square exponential input-to-state stability (EISS) for stochastic delay reaction–diffusion neural networks (SDRDNNS). SDRDNNS with distributed input and boundary input are investigated. In addition, constant delay and time-varying delay are considered. With the help of Lyapunov–Krasovskii functional method, Itô formula and Wirtinger-type inequality, delay-dependent sufficient conditions on mean square EISS of SDRDNNS are presented. These sufficient conditions show the effects of time-delay and diffusion term on mean-square EISS. Moreover, by means of numerical simulation, the effectiveness of our theoretical results is illustrated. [ABSTRACT FROM AUTHOR]

Periodic patterning is widespread in development and can be modelled by reaction-diffusion (RD) processes.However, minimal two-component RD descriptions are vastly simpler than the multi-molecular events that actually occur and are often hard to relate to real interactions measured experimentally. Addressing these issues, we investigated the periodic striped patterning of the rugae (transverse ridges) in the mammalian oral palate, focusing on multiple previously implicated pathways: FGF, Hh, Wnt and BMP. For each, we experimentally identified spatial patterns of activity and distinct responses of the system to inhibition. Through numerical and analytical approaches, we were able to constrain substantially the number of network structures consistent with the data. Determination of the dynamics of pattern appearance further revealed its initiation by 'activators' FGF and Wnt, and 'inhibitor' Hh, whereas BMP and mesenchyme-specific-FGF signalling were incorporated once stripes were formed. This further limited the number of possible networks. Experimental constraint thus limited the number of possibleminimal networks to 154, just 0.004% of the number of possible diffusion-driven instability networks. Together, these studies articulate the principles of multi-morphogen RD patterning and demonstrate the utility of perturbation analysis for constraining RD systems. [ABSTRACT FROM AUTHOR]

Microchannel reactors are critical in biological plus energy-related applications and require meticulous design of hundreds-to-thousands of fluid flow channels. Such systems commonly comprise intricate space-filling microstructures to control the fluid flow distribution for the reaction process. Traditional flow channel design schemes are intuition-based or utilize analytical rule-based optimization strategies that are oversimplified for large-scale domains of arbitrary geometry. Here, a gradient-based optimization method is proposed, where effective porous media and fluid velocity vector design information is exploited and linked to explicit microchannel parameterizations. Reaction-diffusion equations are then utilized to generate space-filling Turing pattern microchannel flow structures from the porous media field. With this computationally efficient and broadly applicable technique, precise control of fluid flow distribution is demonstrated across large numbers (on the order of hundreds) of microchannels. [ABSTRACT FROM AUTHOR]

During a range expansion, deleterious mutations can "surf" on the colonization front. The resultant decrease in fitness is known as expansion load. An Allee effect is known to reduce the loss of genetic diversity of expanding populations, by changing the nature of the expansion from "pulled" to "pushed". We study the impact of an Allee effect on the formation of an expansion load with a new model, in which individuals have the genetic structure of a Muller's ratchet. A key feature of Muller's ratchet is that the population fatally accumulates deleterious mutations due to the stochastic loss of the fittest individuals, an event called a click of the ratchet. We observe fast clicks of the ratchet at the colonization front owing to small population size, followed by a slow fitness recovery due to migration of fit individuals from the bulk of the population, leading to a transient expansion load. For large population size, we are able to derive quantitative features of the expansion wave, such as the wave speed and the frequency of individuals carrying a given number of mutations. Using simulations, we show that the presence of an Allee effect reduces the rate at which clicks occur at the front, and thus reduces the expansion load. [ABSTRACT FROM AUTHOR]

In this work we study global well-posedness and large time behavior for a typical reaction-diffusion system, which include degenerate diffusion, and whose nonlinearities arise from chemical reactions. We show that there is an indirect diffusion effect, i.e., an effective diffusion for the nondiffusive species which is incurred by a combination of diffusion from diffusive species and reversible reactions between the species. Deriving new estimates for such degenerate reaction-diffusion systems, we show, by applying the entropy method, that the solution converges exponentially to equilibrium, and provide explicit convergence rates and associated constants. [ABSTRACT FROM AUTHOR]

This paper conducts a dissipative analysis for comprehensive genetic regulatory network models with fractional Brownian motion (fBm), diffusion-reaction processes, Markovian jump and time-varying delay. By constructing an appropriate Lyapunov–Krasovskii functional, utilizing linear matrix inequality (LMI) technique and stochastic theory, several sufficient conditions of global dissipativity and strictly (Q , S , R) - γ -dissipativity of the solution are established. Moreover, the global attractive sets which are positive invariant are obtained. Finally, two numerical examples are given to illustrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

We consider a problem with free boundary problem for systems of quasi-linear parabolic equation reaction-diffusion type. The a priori estimates of the Ho¨lder norms are established. These estimates are used to prove the existence and uniqueness of the solution. [ABSTRACT FROM AUTHOR]

We develop a dynamic model of an evolving epidemic in a spatially inhomogeneous environment. We analyze the model, as a system of reaction–diffusion partial differential equations, to predict the outbreak and spatio-temporal spread of the disease. The model features both an asymptomatic infectious stage and symptomatic infectious stage, with both the asymptomatic and the symptomatic infected populations dispersing through the susceptible population. We prove the existence and uniform boundedness of solutions, and investigate their long-time behavior. We apply the spatially homogeneous version of the model to the current COVID-19 epidemic in Brazil. [ABSTRACT FROM AUTHOR]

This work focuses on dynamics arising from reaction-diffusion equations, where the profile of propagation is no longer characterized by a single front, but by a layer of several fronts which we call a propagating terrace. This means, intuitively, that transition from one equilibrium to another may occur in several steps, that is, successive phases between some intermediate stationary states. We establish a number of properties on such propagating terraces in a one-dimensional periodic environment under very wide and generic conditions. We are especially concerned with their existence, uniqueness, and their spatial structure. Our goal is to provide insight into the intricate dynamics arising from multistable nonlinearities. [ABSTRACT FROM AUTHOR]

To increase the Abaqus software capabilities, we propose a strategy to force the software to activate hidden degrees of freedom and to include extra coupled phenomena. As an illustration, we apply this approach to the simulation of a reaction diffusion process, the Gray-Scott model, which exhibits very complex patterns. Several setups have been considered and compared with available results to analyze the abilities of our strategy and to allow the inclusion of complex phenomena in Abaqus. [ABSTRACT FROM AUTHOR]

Large-time asymptotic solutions for the reaction-diffusion front between one static reactant and one diffusing reactant are known. These states apply to single-step reactions with a mean-field reaction rate proportional to \rho m\alpha n (with m, n \geq 1), where \rho, \alpha are concentrations of the diffusing and static reactants, respectively. Such reaction kinetics commonly arise in, for example, simple corrosion models of a porous solid, subject to a diffusing reactant. Here we address a more complex two-step corrosion reaction for oxidation of uranium in a water-vapor environment. In this case, additional complexity arises through a pair of coupled reaction fronts (one with m = 2, n = 1 and the other with m = 3, n = 1). Furthermore, we allow for material expansion owing to the corrosion process and a strong dependence of diffusion coefficients on the static reactant distribution. In the large-time limit there are four main asymptotic regions, comprising two diffusion layers and two reaction fronts. Asymptotic matching of these regions allows us to construct a large-time solution that gives analytical predictions for the positions of the two propagating fronts, thickness of the diffusion layers, and concentration of diffusing species outside of the fronts. This is the first mechanistic model of uranium oxidation in water vapor and predicts a thin propagating subsurface (hydride) layer, as recently observed in atom-probe tomography experiments. [ABSTRACT FROM AUTHOR]

In this paper, steady state bifurcations arising from the reaction–diffusion equations are investigated. Using the Lyapunov–Schmidt reduction on a square domain, a simple, and a double steady state bifurcation caused by the symmetry of spatial region is obtained. By examining the reduced bifurcation equations, complete bifurcation scenario and patterns at simple and double steady state bifurcation points are obtained. Numerical simulations support the theoretical results. [ABSTRACT FROM AUTHOR]