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We study the quenching behavior of the solution of a semilinear reaction-diffusion system with nonlinear boundary conditions. We prove that the solution quenches in finite time and its quenching time goes to the one of the solution of the differential system. We also obtain lower and upper bounds for quenching time of the solution. [ABSTRACT FROM AUTHOR]

We address the uniqueness of the nonzero stationary state for a reaction–diffusion system of Fisher–KPP type that does not satisfy the comparison principle. Although the uniqueness is false in general, it turns out to be true under biologically natural assumptions on the parameters. This Liouville-type result is then used to characterize the wake of traveling waves. All results are extended to an analogous nonlocal reaction–diffusion equation that contains as a particular case the cane toads equation with bounded traits. [ABSTRACT FROM AUTHOR]

In a recent paper by Cantrell et al. [9], two-component KPP systems with competition of Lotka–Volterra type were analyzed and their long-time behaviour largely settled. In particular, the authors established that any constant positive steady state, if unique, is necessarily globally attractive. In the present paper, we give an explicit and biologically very natural example of oscillatory three-component system. Using elementary techniques or pre-established theorems, we show that it has a unique constant positive steady state with two-dimensional unstable manifold, a stable limit cycle, a predator–prey structure near the steady state, periodic wave trains and point-to-periodic rapid travelling waves. Numerically, we also show the existence of pulsating fronts and propagating terraces. [ABSTRACT FROM AUTHOR]

Models issued from ecology, chemical reactions and several other application fields lead to semi-linear parabolic equations with super-linear growth. Even if, in general, blow-up can occur, these models share the property that mass control is essential. In many circumstances, it is known that this L 1 control is enough to prove the global existence of weak solutions. The theory is based on basic estimates initiated by M. Pierre and collaborators, who have introduced methods to prove L 2 a priori estimates for the solution. Here, we establish such a key estimate with initial data in L 1 while the usual theory uses L 2. This allows us to greatly simplify the proof of some results. We also establish new existence results of semilinearity which are super-quadratic as they occur in complex chemical reactions. Our method can be extended to semi-linear porous medium equations. [ABSTRACT FROM AUTHOR]

We introduce a model designed to account for the influence of a line with fast diffusion–such as a road or another transport network–on the dynamics of a population in an ecological niche.This model consists of a system of coupled reaction-diffusion equations set on domains with different dimensions (line / plane). We first show that, in a stationary climate, the presence of the line is always deleterious and can even lead the population to extinction. Next, we consider the case where the niche is subject to a displacement, representing the effect of a climate change. We find that in such case the line with fast diffusion can help the population to persist. We also study several qualitative properties of this system. The analysis is based on a notion of generalized principal eigenvalue developed and studied by the authors (2019). [ABSTRACT FROM AUTHOR]

In this paper, we propose an explicit time-stepping scheme for the pattern formation in reaction–diffusion systems on evolving surfaces. The proposed numerical method is based on a simple discretization scheme of Laplace–Beltrami operator over triangulated surface. On the static and evolving domains, we perform various numerical experiments for effect of domain growth and pattern formations. The computational results demonstrate that our proposed method can simulate pattern formation in reaction–diffusion systems on evolving surfaces. The actual zebra skin pattern and computational results are compared. In the computational results, we can observe different pattern formations on the evolving surface with specific rotation speed. • An explicit time-stepping scheme is proposed for the pattern formation in the reaction–diffusion systems on evolving surfaces. • The proposed numerical method is based on discretization of Laplace–Beltrami operator over triangulated surface. • The proposed method can simulate the pattern formation in the reaction–diffusion systems on evolving surfaces. [ABSTRACT FROM AUTHOR]

We consider the spatio‐temporal dynamics of a Gray‐Scott reaction diffusion system driven by a time periodic force. The drive excites the system through a direct transition and a parametric transition. The destructive interference of two simultaneous transitions gives rise to an antiresonance dip in the response function‐frequency spectrum. This is manifested in the stabilization of the dynamics in an intermediate frequency range centered at the dip, outside which one observes spatio‐temporal patterns due to instability. A theoretical scheme based on linear analysis followed by detailed numerical simulations of the reaction‐diffusion system has been carried out to explore this antiresonance‐induced dynamical stabilization. This is the first realization of anti‐resonance, an analog of Fano Resonance in auto ionization, in a reaction‐ diffusion system. [ABSTRACT FROM AUTHOR]

Mathematical models for self-propelled motions are often utilized for understanding the mechanism of collective motions observed in biological systems. Indeed, several patterns of collective motions of camphor disks have been reported in experimental systems. In this paper, we show the existence of asymmetrically rotating solutions of a two-camphor model and give necessary conditions for their existence and non-existence. The main theorem insists that the function describing the surface tension should have a concave part so that asymmetric motions of two camphor disks appear. Our result provides a clue for the dependence between the surfactant concentration and the surface tension in the mathematical model, which is difficult to be measured in experiments. [ABSTRACT FROM AUTHOR]

Urban crime such as residential burglary is a social problem in every major urban area. As such, many mathematical models have been proposed to study the collective behavior of these crimes. In [V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi, M. B. Short, M. R. D'Orsogna and L. B. Chayes, A statistical model of crime behavior, Math. Methods Appl. Sci107 (2008) 1249–1267; M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression, SIAM J. Appl. Dyn. Syst.9 (2010) 462–483], Short et al. proposed an agent-based statistical model of residential burglary to model the crime hotspot phenomena. From the point of view of reaction–diffusion systems, the model is a chemotactic system with cross diffusion that exhibit hotspot phenomena. In this paper, we first construct a radial hotspot solution of this system, then study the linear stability of this hotspot solution by studying a nonlocal eigenvalue problem. It turns out that the stability of the hotspot is completely different depending on which spatial dimension the system is on. The main mathematical difficulty of the system involves treating the steep change of diffusion near the core of the hotspot, because of the quasilinearity induced by the cross diffusion. We believe that the techniques used in this paper can be developed to treat many other chemotactic systems. [ABSTRACT FROM AUTHOR]

A reaction–diffusion system is formulated to describe three interacting species within the Hastings–Powell (HP) food chain structure with chemotaxis produced by three chemicals. We construct a finite volume (FV) scheme for this system, and in combination with the non-negativity and a priori estimates for the discrete solution, the existence of a discrete solution of the FV scheme is proven. It is shown that the scheme converges to the corresponding weak solution of the model. The convergence proof uses two ingredients of interest for various applications, namely the discrete Sobolev embedding inequalities with general boundary conditions and a space–time L 1 compactness argument. Finally, numerical tests illustrate the model and the behavior of the FV scheme. [ABSTRACT FROM AUTHOR]

For the nonlinear predator–prey system (PPS), although a variety of numerical methods have been proposed, such as the difference method, the finite element method, and so on, but the efficient numerical method has always been the direction that scholars strive to pursue. Based on this question, a sinc function interpolation method is proposed for a class of PPS. Numerical simulations of a class of PPS with complex dynamical behaviors are performed. Time series plots and phase diagrams of a class of PPS without self-diffusion are shown. The pattern is obtained by setting up different initial conditions and the parameters in the system according to Turing bifurcation condition. The numerical simulation results have a good agreement with theoretical results. Simulation results show the effectiveness of the method. [ABSTRACT FROM AUTHOR]

Existence of travelling waves is studied for a delay reaction–diffusion system of equations describing the distribution of viruses and immune cells in the tissue. The proof uses the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces. [ABSTRACT FROM AUTHOR]

Numerous color images are digitally reproduced today, mainly using the cyan, magenta, yellow and black ink set. The information in a printed image, however, may be distorted or lost due to the image's gradual color degradation as the result of ink fading and diffusion on the substrate. The degradation behavior of the inks can be difficult to predict when catalytic fading is introduced by a variety of ink mixtures produced in the printing process. In this article, we prove that the fading or diffusion rate has an equivalent interpretation as the hazard function in reliability modeling, and the degradation model of printed images with constant fading and diffusion rates can be reformulated as a Markov process model. Hence, the degradation of printed images with time-varying fading and diffusion rates can be modeled as a semi-Markov process. The semi-Markov process model provides a deterministic degradation prediction. We further propose a spatio-temporal Wiener process model for stochastic prediction of the ink fading and diffusion of printed images. Numerical examples and a real-life case study are provided to illustrate the proposed approach. [ABSTRACT FROM AUTHOR]

This article deals with a mass conservation reaction-diffusion system. As a model for studying cell polarity, we are interested in the existence of spike solutions and some properties related to its dynamics. Variational arguments will be employed to investigate the existence questions. The profile of a spike solution looks like a standing pulse. In addition, the motion of such spikes in heterogeneous media will be derived. [ABSTRACT FROM AUTHOR]

In this tutorial, recent development of pattern generation algorithms based on DNA computing will be overviewed. Natural pattern generation, especially in biological organisms, are often driven by spatio-temporal chemical reactions. Various reaction–diffusion systems have been proposed to generate artificial patterns out of DNA sequences. To program DNA reaction–diffusion systems, in addition to the design of the reaction, diffusion of each DNA species must be considered. This is realized by immobilizing or suppressing diffusion of DNA molecules in the reaction field. Here, several typical methods to build patterns by 1-D or 2-D reaction–diffusion systems are introduced and how to implement the system by DNA molecules is explained. The direction of future research and possible applications of this technology will be also discussed. [ABSTRACT FROM AUTHOR]

A blow-up analysis for a nonlocal reaction-diffusion system with time-dependent coefficients is investigated under null Dirichlet boundary conditions. Based on the Kaplan's method, comparison principle and modified differential inequality technique, we establish a blow-up criteria and derive the bounds for the blow-up time under the appropriate measures in whole-dimensional space. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that a time fractional reaction diffusion system with reaction terms of the Brusselator type admits a global solution by using the feedback method of F. Rothe [20]. Furthermore, some results on the large time behavior of the solutions are obtained. We give a positive answer to Problem 6 of the valuable paper of Gal and Warma [6]. [ABSTRACT FROM AUTHOR]

The CO oxidation on platinum-group metals under ultra-high-vacuum conditions is one of the most studied surface reactions. However, the presence of disturbing species and competing reactions are often neglected. One of the most interesting additional gases to be treated is hydrogen, due to its importance in technical applications and its inevitability under vacuum conditions. Adding hydrogen to the reaction of CO and O2 leads to more adsorbed species and competing reaction steps towards water formation. In this study, a model for approaching the competing surface reactions CO+O2 + H2 is presented and discussed. Using the framework of bifurcation theory, we show how the steady states of the extended system correspond to a swallowtail catastrophe set with a tristable regime within the swallowtail. We explore numerically the possibility of reaching all stable states and illustrate the experimental challenges such a system could pose. Lastly, an approximative first-principle approach to diffusion illustrates how up to three stable states balance each other while forming heterogeneous patterns. [ABSTRACT FROM AUTHOR]

For distributed evolutionary dynamical systems of the "reaction-diffusion" and "reaction-diffusion-advection" types, we analyze the behavior of invariant numerical characteristics of the attractor as the diffusion coefficients decrease. We consider the phenomenon of multimode diffusion chaos, one of whose signatures is an increase in the Lyapunov dimensions of the attractor. For several examples, we perform broad numerical experiments illustrating this phenomenon. [ABSTRACT FROM AUTHOR]

The concentration of glucose in sweat recently has been measured with high sensitivity, selectivity, and reproducibility by nanostructured NiO electrodes manufactured by the glancing angle deposition (GLAD) technique. The GLAD technique allows electrode morphological properties such as porosity and film thickness to be tightly controlled, providing ample opportunity to enhance the sensor performance. Currently, the selection of optimal GLAD parameters is determined experimentally by trial and error, at high costs in time and resources. Numerical simulation allows the effects of various parameters on sensor performance to be investigated at a much lower cost compared to experimental studies. In this work, a 2D reaction–diffusion model for the surface-catalyzed reactions in the nanostructured GLAD electrodes is developed, which are then solved using the finite element method. Parametric studies on the nanocolumn thickness and nanocolumn separation of the GLAD structures are then conducted to optimize GLAD-based electrode structures with different adsorption and catalytic rates. This research offers new guidance for rapidly designing highly effective sensors with higher sensitivity and lower limits of detection. [Display omitted] • GLAD-fabricated sensor geometry affects sensor performance. • Sensitivity saturates with increasing nanocolumn height. • Optimal spacing between nanocolumns depends on nanocolumns' thickness. [ABSTRACT FROM AUTHOR]

Time-delay effect and bifurcation phenomenon are important topics in the study of reaction–diffusion equations. In this paper, we consider a three-species predator–prey system with diffusion and incubation delay for predator. The stability and Hopf bifurcation are mainly discussed. We conclude that there exists a critical value of delay, such that the internal equilibrium is stable or unstable as the delay crosses the critical value. Especially, the system emerges Hopf bifurcation phenomenon at this critical value. For bifurcation solution, the conclusions of stability, period and bifurcation direction are also presented. Additionally, numerical simulations are proceeded to support the main results. In biology, the existence of bifurcation solution means that when the delay of predator reaches to a certain extent, the predator and prey will coexist within a period of time. It turns out that the related computations and analyses are much more complicated than that of two-species time-delay systems. • A predation model with time-delay and diffusion is considered. • Results on stability and Hopf-bifurcations are obtained. • Hopf-bifurcation graphs are performed to illustrate the theoretical analysis. • It is found that time-delay has significant impacts on the pattern formations. [ABSTRACT FROM AUTHOR]

Exposing a solution to a temperature gradient can lead to the accumulation of particles on either the cold or warm side. This phenomenon is known as thermophoresis, and its microscopic origin is still debated. Here, we show that thermophoresis can be observed in any system having internal states with different transport properties, and temperature-modulated rates of transitions between the states. These internal degrees of freedom might be configurational, chemical or velocity states. We also derive an expression for the Soret coefficient, which decides whether particles accumulate on the cold or warm side. Our framework can be applied to any chemical reaction system diffusing in a temperature gradient. It also captures the possibility to observe a sign inversion of the Soret coefficient as the competition between chemical and velocity states. We establish thermophoresis as a genuine non-equilibrium effect, originating from internal microscopic currents consistent with the necessity of transporting heat from warm to cold regions.

Abstract In this paper, a reaction–diffusion system of a predator–prey model with Beddington–DeAngelis functional response is considered. This model describes a prey-taxis mechanism that is an immediate movement of the predator u in response to a change of the prey v (which leads to the collection of u). We use some approaches to prove the global existence-boundedness of classical solutions and overcome the substantial difficulty of the existence of a nonlinear prey-taxis term.

The work is aimed to study front solutions of a nonlinear system of parabolic equations in a two-dimensional region. The system can be considered as a mathematical model describing an abrupt change in physical characteristics of spatially heterogeneous media. We consider a system with small parameters raised to the different powers at a differential operator, that represents the difference of typical processes speeds for the system components. The study of the system is conducted by using the contrast structures theory methods, which allowed us to obtain conditions for the existence of front solutions contained in the neighborhood of a closed curve, to determine the front velocity depending on time and coordinate along the front curve, and to obtain the zero-order and the first-order terms of the asymptotic approximation to the solution. The scope of the system includes the description of autowave solutions in the field of ecology, biophysics, combustion physics and chemical kinetics. The approximate solution allows us to choose the model parameters so that the result corresponds to the processes observed, to explain and describe the characteristics of the solutions with sharp gradients, to create models with stable solutions and thereby to simplify the numerical analysis. Note that the numerical experiment for the two-dimensional spatial models requires a considerable amount of processing power and the use of parallel computing techniques and does not allow to effectively analyze and modify the model. In this paper, we obtain the asymptotic approximation that is to be justified, which can be done by the method of differential inequalities.

In this article we study the traveling wave solutions of a monostable nonlocal reaction-diffusion system with delay arising from the spread of an epidemic by oral-faecal transmission. From [23], there exists a minimal wave speed $c_*>0$ such that a traveling wave solution exists if and only if the wave speed is above $c_*$. In this article, we first establish the exact asymptotic behavior of the traveling waves at $\pm\infty$. Then, we construct some annihilating-front entire solutions which behave like a traveling wave front propagating from the left side (or the right side) on the x-axis or two traveling wave fronts propagating from both sides on the x-axis as $t\to-\infty$ and converge to the unique positive equilibrium as $t\to+\infty$. From the viewpoint of epidemiology, these results provide some new spread ways of the epidemic.

This article studies the existence and uniqueness of a weak solution of the time-fractional cancer invasion system with nonlocal diffusion operator. Existence and uniqueness results are ensured by adapting the Faedo-Galerkin method and some a priori estimates. Further, finite element numerical scheme is implemented for the considered system. Finally, various numerical computations are performed along with the convergence analysis of the scheme.

In this paper, we investigate the stability for reaction systems with stochastic switching. Two types of switched models are considered: (i) Markov switching and (ii) independent and identically distributed switching. By means of the ergodic property of Markov chain, Dynkin formula and Fubini theorem, together with the Lyapunov direct method, some sufficient conditions are obtained to ensure that the zero solution of reaction–diffusion systems with Markov switching is almost surely exponential stable or exponentially stable in the mean square. By using Theorem 7.3 in [R. Durrett, Probability: Theory and Examples, Duxbury Press, Belmont, CA, 2005], we also investigate the stability of reaction–diffusion systems with independent and identically distributed switching. Meanwhile, an example with simulations is provided to certify that the stochastic switching plays an essential role in the stability of systems.

This paper has dealt with a tracking control problem for a class of unstable reaction–diffusion system with time delay. Iterative learning algorithms are introduced to make the infinite-dimensional repetitive motion system track the desired trajectory. A new Lyapunov–Krasovskii functional is constructed to deal with the time-delay system. Picewise distribution functions are applied in this paper to perform piecewise control operations. By using Poincaré–Wirtinger inequality, Cauchy–Schwartz inequality for integrals and Young’s inequality, the convergence of the system with time delay using iterative learning schemes is proved. Numerical simulation results have verified the effectiveness of the proposed method.

We consider the Gierer-Meinhardt system with small inhibitor diffusivity and very small activator diffusivity on a compact two-dimensional Riemannian manifold without boundary. We study steady state solutions which are far from spatial homogeneity. We construct two different spike clusters, each consisting of two spikes, which both approach the same nondegenerate local maximum point of the Gaussian curvature. We show that one of these spike clusters is stable, the other one is unstable. [ABSTRACT FROM AUTHOR]

This paper investigates the lowest‐order weak Galerkin finite element (WGFE) method for solving reaction–diffusion equations with singular perturbations in two and three space dimensions. The system of linear equations for the new scheme is positive definite, and one might readily get the well‐posedness of the system. Our numerical experiments confirmed our error analysis that our WGFE method of the lowest order could deliver numerical approximations of the order O(h1/2) and O(h) in H1 and L2 norms, respectively. [ABSTRACT FROM AUTHOR]

Atherosclerosis, a continual inflammatory disease occurring due to plaque cumulation in the arterial intima, is one of the main reasons behind deaths from diverse cardiovascular diseases. The basic interactions between oxidized low density lipoprotein (LDL) and macrophages in the formation of atherosclerotic plaque are modeled here in terms of a reaction–diffusion system in one-dimensional (1D) space under Neumann boundary conditions. Two simple mathematical models are considered which differ by the influx term only in the case of the interaction of oxidized LDL. Both the spatial and nonspatial systems are simply analyzed theoretically and numerically. Numerical bifurcation analysis confirms the existence of Hopf bifurcation concerning four significant model parameters. Examining the gravity of the model offered in this investigation, an obvious insight into this inflammatory response can be achieved both qualitatively and quantitatively. [ABSTRACT FROM AUTHOR]

Many types of organisms disperse through heterogeneous environments as part of their life histories. For various models of dispersal, including reaction–advection–diffusion models in continuously varying environments, it has been shown by pairwise invasibility analysis that dispersal strategies which generate an ideal free distribution are evolutionarily steady strategies (ESS, also known as evolutionarily stable strategies) and are neighborhood invader strategies (NIS). That is, populations using such strategies can both invade and resist invasion by populations using strategies that do not produce an ideal free distribution. (The ideal free distribution arises from the assumption that organisms inhabiting heterogeneous environments should move to maximize their fitness, which allows a mathematical characterization in terms of fitness equalization.) Classical reaction diffusion models assume that landscapes vary continuously. Landscape ecologists consider landscapes as mosaics of patches where individuals can make movement decisions at sharp interfaces between patches of different quality. We use a recent formulation of reaction–diffusion systems in patchy landscapes to study dispersal strategies by using methods inspired by evolutionary game theory and adaptive dynamics. Specifically, we use a version of pairwise invasibility analysis to show that in patchy environments, the behavioral strategy for movement at boundaries between different patch types that generates an ideal free distribution is both globally evolutionarily steady (ESS) and is a global neighborhood invader strategy (NIS). [ABSTRACT FROM AUTHOR]

Many biological populations reside in increasingly fragmented landscapes, where habitat quality may change abruptly in space. Individuals adjust their movement behavior to local habitat quality and show preferences for some habitat types over others. Several recent publications explore how such individual movement behavior affects population-level dynamics in a framework of reaction–diffusion systems that are coupled through discontinuous boundary conditions. While most of those works are based on linear analysis, we study positive steady states of the nonlinear equations. We prove existence, uniqueness and global stability, and we classify their qualitative shape depending on movement behavior. We apply our results to study the question why and under which conditions the total population abundance at steady state may exceed the total carrying capacity of the landscape. [ABSTRACT FROM AUTHOR]

We consider a n × n nonlinear reaction-diffusion system posed on a smooth bounded domain Ω of ℝN. This system models reversible chemical reactions. We act on the system through m controls (1 ≤ m < n), localized in some arbitrary nonempty open subset ω of the domain Ω. We prove the local exact controllability to nonnegative (constant) stationary states in any time T > 0. A specificity of this control system is the existence of some invariant quantities in the nonlinear dynamics that prevents controllability from happening in the whole space L∞(Ω)n. The proof relies on several ingredients. First, an adequate affine change of variables transforms the system into a cascade system with second order coupling terms. Secondly, we establish a new null-controllability result for the linearized system thanks to a spectral inequality for finite sums of eigenfunctions of the Neumann Laplacian operator, due to David Jerison, Gilles Lebeau and Luc Robbiano and precise observability inequalities for a family of finite dimensional systems. Thirdly, the source term method, introduced by Yuning Liu, Takéo Takahashi and Marius Tucsnak, is revisited in a L∞-context. Finally, an appropriate inverse mapping theorem in suitable spaces enables to go back to the nonlinear reaction-diffusion system. [ABSTRACT FROM AUTHOR]