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For the systems modeled by a generalized reaction–diffusion equation with the higher‐order quintic nonlinearity, we explore the soliton dynamics via the F‐expansion method. By the novel F‐base function ansatz, we first derived the bright soliton and kink soliton solutions for the one‐dimensional case of the reaction–diffusion equation with quintic nonlinearity. Furthermore, we employed self‐similar techniques to analyze the higher‐dimensional dynamics of bright soliton and kink soliton solutions supported by the (2+1)‐dimensional reaction–diffusion equation system with quintic nonlinearity. Additionally, we conducted stability analysis of derived soliton solutions. Our theoretical results demonstrate that under certain parametric setting, the reaction–diffusion equation model with higher‐order nonlinearity supports bright soliton and kink soliton in higher‐dimensional as well as lower dimensional setting, which provides guidance for observing and investigating soliton behavior in systems modeled by the reaction–diffusion equation with higher‐order quintic nonlinearity. [ABSTRACT FROM AUTHOR]

Constrained mixture models have successfully simulated many cases of growth and remodeling in soft biological tissues. So far, extensions of these models have been proposed to include either intracellular signaling or chemo-mechanical coupling on the organ-scale. However, no version of constrained mixture models currently exists that includes both aspects. Here, we propose such a version that resolves cellular signal processing by a set of logic-gated ordinary differential equations and captures chemo-mechanical interactions between cells by coupling a reaction-diffusion equation with the equations of nonlinear continuum mechanics. To demonstrate the potential of the model, we present 2 case studies within vascular solid mechanics: (i) the influence of angiotensin II on aortic growth and remodeling and (ii) the effect of communication between endothelial and intramural arterial cells via nitric oxide and endothelin-1. [ABSTRACT FROM AUTHOR]

In this study, to solve the singularly perturbed delay convection–diffusion–reaction problem, we proposed a hybrid numerical scheme that converges uniformly. Parabolic right boundary layer outcomes from the presence of the small perturbation parameter. To grip this layer behaviour, the problem is solved by Bakhvalov–Shishkin mesh for spatial domain discretization and uniform mesh for temporal domain discretization. A hybrid scheme consisting of a non-polynomial spline scheme for fine mesh and a midpoint upwind scheme for coarse mesh is used to discretize the spatial derivative, while an implicit Euler scheme is used to discretize the time derivative. To make computed solutions more accurate and increase rate of convergence of the scheme, we applied Richardson extrapolation technique. The stability and convergence of the scheme are established. The scheme has a second order of convergence in the discrete supreme norm and is parametric uniformly convergent. The scheme’s application is demonstrated through two test problems. [ABSTRACT FROM AUTHOR]

In this paper, we investigated a distributed optimal control problem for the viscous Cahn-Hilliard-Oono system. In this system, chemotaxis, active transport, and nonlocal interaction of Oono's type were also taken into account. The system consisted of a viscous Cahn-Hilliard equation for the difference in volume fractions of the two components $ \varphi $ and a diffusion-reaction equation for the nutrient concentration $ \sigma $. Our aim was to achieve a desired goal of the difference in volume fractions of the two components by minimizing the cost functional of standard tracking type. To begin, the strong well-posedness of the state system was addressed. Then, we establish the existence of optimal controls. Afterward, we presented the Fréchet differentiability of the control-to-state operator $ \mathcal S $. Lastly, we derived the first-order necessary optimality conditions by considering the backward adjoint system. [ABSTRACT FROM AUTHOR]

We consider a non-local tumour growth model of phase-field type, describing the evolution of tumour cells through proliferation in presence of a nutrient. The model consists of a coupled system, incorporating a non-local Cahn-Hilliard equation for the tumour phase variable and a reaction-diffusion equation for the nutrient. First, we establish novel regularity results for such a model, by applying maximal regularity theory in weighted L p spaces. This technique enables us to prove the local existence and uniqueness of a regular solution, including also chemotaxis effects. By leveraging time-regularisation properties and global boundedness estimates, we further extend the solution to a global one. These results provide the foundation for addressing an optimal control problem, aimed at identifying a suitable therapy, guiding the tumour towards a predefined target. Specifically, we prove the existence of an optimal therapy and, by studying the Fréchet-differentiability of the control-to-state operator and introducing the adjoint system, we derive first-order necessary optimality conditions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of time-varying impulsive recursive motions. We investigate the connections among minimal sets, non-wandering points, the Birkhoff center, and attractors. As an illustration, the obtained results are applied to a semilinear reaction-diffusion equation with impulses and to an integrate-and-fire neuron model. [ABSTRACT FROM AUTHOR]

This paper is concerned with the propagation phenomena for the nonlocal reaction-diffusion monostable equation with delay of the form ∂ u (t , x) ∂ t = d Δ u (t , x) + f (u (t , x) , ∫ R J (x − y) u (t − τ , y) d y) , t > 0 , x ∈ R. It is well-known that if we take J (x) = δ (x) and τ = 0 , there exists a minimal wave speed c ⁎ > 0 , such that this equation has no traveling wave front for 0 < c < c ⁎ and a traveling wave front for each c ≥ c ⁎ , which is unique up to translation and is globally asymptotically stable. Furthermore, when J is symmetry and exponentially bounded and τ > 0 , Wang et al. (2008) [27] considered the effects of delay and nonlocality on the spreading speed and proved that (i) if ∂ 2 f (0 , 0) > 0 , then the delay can slow the spreading speed of the wave fronts and the nonlocality can increase the spreading speed; and (ii) if ∂ 2 f (0 , 0) = 0 , then the delay and nonlocality do not affect the spreading speed. However, when J is asymmetry or exponentially unbounded , the question was left open. In this paper we obtain a rather complete answer to this question. More precisely, we show that for exponentially bounded kernels the minimal speed of traveling waves exists and coincides with the spreading speed. We also investigate the case of exponentially unbounded kernels where we prove the non-existence of traveling wave solutions and obtain upper and lower bounds for the position of any level set of the solutions. These bounds allow us to estimate how the solutions spread, depending on the kernels. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The reaction-diffusion equations have been studied in various aspects of nature, such as heat transfer and biological dynamic modelling. In this paper, we study a nonlocal reaction-diffusion model with time delay associated with the density and diffusion behaviour of biological populations. Based on our previous conclusions and numerical strategy of the direct problem, we continue to study the inverse problem about the diffusion coefficient as well as the parameters in the birth function, and also perform numerical simulations with error analysis. In addition, we further generalize the constant diffusion coefficient to the position-dependent diffusion rate, which is intended to improve the generalizability to multiple organisms diffusion in nature. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain. We first obtain the existence of a globally attractive absorbing set for the dynamical system generated by the equation under the assumption that the nonlinear term is bounded. Then, we construct exponential attractors of the equation directly in its natural phase space, i.e., a Banach space with explicit fractal dimension by combining squeezing properties of the system as well as a covering lemma of finite dimensional subspaces of a Banach space. Our result generalizes the methods established in Hilbert spaces and weighted spaces, and the fractal dimension of the obtained exponential attractor does not depend on the entropy number but only depends on some inner characteristic of the studied equation. [ABSTRACT FROM AUTHOR]

In this paper we provide general results on the uniform upper semicontinuity of pullback attractors with respect to the time parameter for non-autonomous delay dynamical systems. Namely, we establish a criteria in terms of the multi-index convergence of solutions for the delay system to the non-delay one, locally pointwise convergence and local controllability of pullback attractors. As an application, we prove the upper semicontinuity of pullback attractors for a non-autonomous delay reaction-diffusion equation to the corresponding nondelay one over any bounded time interval as the delay parameter tends to zero. [ABSTRACT FROM AUTHOR]

Four formulations based on the Kirchhoff transformation and time linearization for the numerical study of one-dimensional reaction–diffusion equations, whose heat capacity, thermal inertia and reaction rate are only functions of the temperature, are presented. The formulations result in linear, two-point boundary-value problems for the temperature, energy or heat potential, and may be solved by either discretizing the second-order spatial derivative or piecewise analytical integration. In both cases, linear systems of algebraic equations are obtained. The formulation for the temperature is extended to two-dimensional, nonlinear reaction–diffusion equations where the resulting linear two-dimensional operator is factorized into a sequence of one-dimensional ones that may be solved by means of any of the four formulations developed for one-dimensional problems. The multidimensional formulation is applied to a two-dimensional, two-equation system of nonlinearly coupled advection–reaction–diffusion equations, and the effects of the velocity and the parameters that characterize the nonlinear heat capacities and thermal conductivity are studied. It is shown that clockwise-rotating velocity fields result in wave stretching for small vortex radii, and wave deceleration and thickening for counter-clockwise-rotating velocity fields. It is also shown that large-core, clockwise-rotating velocity fields may result in large transient periods, followed by time intervals of apparent little activity which, in turn, are followed by the propagation of long-period waves. [ABSTRACT FROM AUTHOR]

This paper explores the impact of spatial memory on the double-Hopf bifurcation and dynamics of the memory-based diffusion system. First, employing the center manifold theory and the normal form method, the explicit formulae for the coefficients in the normal form for the double-Hopf bifurcation of the general reaction-diffusion equations with memory-based self-diffusion and cross-diffusion are derived, which are expressed in terms of the original system parameters and can be used to simplify the system to analyze the spatiotemporal dynamics revealed by the double-Hopf bifurcation. In addition, considering the effect of memory-based diffusion on the population dynamics of the predator-prey model with a Holling-Tanner-type functional response function, we improve the conditions for the occurrence of Hopf bifurcation and establish the conditions for the constant steady state to lose its stability through double-Hopf bifurcation. Further, by analyzing the normal form of the double-Hopf bifurcation, we prove that memory-based diffusion can lead to a bistable phenomenon, i.e., two stable spatially asynchronous periodic orbits with different wave numbers coexist in the model and can also lead to a spatially inhomogeneous quasi-periodic orbit, revealing that species of animals with spatial memory may survive in diverse patterns. [ABSTRACT FROM AUTHOR]

We consider reaction‐diffusion equations, which are parabolic systems of partial differential equations. Assuming periodic boundary conditions in space, a multidimensional Fourier transform can be used. A spectral method yields an initial value problem of a stiff system of ordinary differential equations for time‐dependent Fourier coefficients. We investigate the application of Rosenbrock‐Wanner W‐schemes in the time integration of the ordinary differential equations including step size selection. Results of numerical computations are presented using a Turing system in two space dimensions. [ABSTRACT FROM AUTHOR]

In this study, the dynamics of a reaction–diffusion equation on a bounded interval at the first dynamic transition point is investigated. The main difference of this study from the existing ones in the dynamic transition literature is that in this work, no specific form of the nonlinear operator is assumed except that it is an analytic function of u$u$ and ux$u_x$. The linear operator is also assumed to be a general operator with sinusoidal eigenvectors. Moreover, we assume that there is a first transition as a real simple eigenvalue changes sign. With this general framework, we make a rigorous analysis of the dependence of the first transition dynamics on the coefficients of the Taylor expansion of the nonlinear operator. The main tool we use in this study is the center manifold reduction combined with the classification of the dynamic transition theory. This study is a generalization of a recent work where the nonlinear operator contains only low‐order (quadratic and cubic) nonlinearities. The current generalization requires certain technical difficulties such as the validity of the genericity conditions on the Taylor coefficients of the nonlinear operator and a bootstrapping argument using these genericity conditions. The results of this work can be generalized in various directions, which are discussed in the conclusions section. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and uniqueness of isolated periodic traveling wave solutions of a family of nonlinear reaction–convection–diffusion equations using the monotonicity of the ratio of Abelian integrals. A numerical study is also presented. [ABSTRACT FROM AUTHOR]

We obtain the one-dimensional symmetry and monotonicity of the entire positive solutions to some reaction-diffusion equations involving fractional p-Laplacian by virtue of the sliding method. More precisely, we consider the following problem { ∂ ⁡ u ∂ ⁡ t ⁢ (x , t) + (- Δ) p s ⁢ u ⁢ (x , t) = f ⁢ (t , u ⁢ (x , t)) , (x , t) ∈ Ω × ℝ , u ⁢ (x , t) > 0 , (x , t) ∈ Ω × ℝ , u ⁢ (x , t) = 0 , (x , t) ∈ Ω c × ℝ , where s ∈ (0 , 1) , p ≥ 2 , (- Δ) p s is the fractional p-Laplacian, f ⁢ (t , u) is some continuous function, the domain Ω ⊂ ℝ n is unbounded and Ω c = ℝ n ∖ Ω . Firstly, we establish a maximum principle involving the parabolic p-Laplacian operator. Then, under certain conditions of f, we prove the asymptotic behavior of solutions far away from the boundary uniformly in t ∈ ℝ . Finally, the sliding method is implemented to derive the monotonicity and uniqueness of the bounded positive entire solutions. To our best knowledge, there has not been any results on the symmetry and monotonicity properties of solutions to the parabolic fractional p-Laplacian equations before. [ABSTRACT FROM AUTHOR]

The reaction-diffusion equation is widely used in mathematical models of cancer. The calibration of model parameters based on limited clinical data is critical to using reaction-diffusion equation simulations for reliable predictions on a per-patient basis. Here, we focus on cell-level data as routinely available from tissue biopsies used for clinical cancer diagnosis. We analyze the spatial architecture in biopsy tissues stained with multiplex immunofluorescence. We derive a two-point correlation function and the corresponding spatial power spectral distribution. We show that this data-deduced power spectral distribution can fit the power spectrum of the solution of reaction-diffusion equations that can then identify patient-specific tumor growth and invasion rates. This approach allows the measurement of patient-specific critical tumor dynamical properties from routinely available biopsy material at a single snapshot in time. [ABSTRACT FROM AUTHOR]

A necessary and sufficient condition for the existence or nonexistence of global solutions to the following fractional reaction-diffusion equations u t = Δ α u + ψ (t) f (u) , in R N × (0 , ∞) , u (· , 0) = u 0 ≥ 0 , in R N , has not been known and remained as an open problem for a few decades, where N ≥ 2 , Δ α = - - Δ α / 2 denotes the fractional Laplace operator with 0 < α ≤ 2 , ψ is a nonnegative and continuous function, and f is a convex function. The purpose of this paper is to resolve this problem completely as follows: There is a global solution to the equation if and only if ∫ 1 ∞ ψ (t) t N α f ϵ t - N α d t < ∞ , for some ϵ > 0. [ABSTRACT FROM AUTHOR]

In this paper, we present an approach to characterizing fast-reaction limits of systems with nonlinear diffusion, when there are either two reaction–diffusion equations, or one reaction–diffusion equation and one ordinary differential equation, on unbounded domains. Here, we replace the terms of the form u x x in usual reaction–diffusion equation, which represent linear diffusion, by terms of form ϕ (u) x x , representing nonlinear diffusion. We prove the convergence in the fast-reaction limit k → ∞ that is determined by the unique solution of a certain scalar nonlinear diffusion problem. [ABSTRACT FROM AUTHOR]

Calcium signal transduction is essential for cellular activities such as gene transcription, death, and neuronal plasticity. Dynamical changes in the concentration of calcium have a profound effect on the intracellular activity of neurons. The Caputo fractional reaction-diffusion equation is a useful tool for modeling the intricate biological process involved in calcium concentration regulation. We include the Amyloid Beta, STIM-Orai mechanism, voltage-dependent calcium entry, inositol triphosphate receptor (IPR), endoplasmic reticulum (ER) flux, SERCA pump, and plasma membrane flux in our mathematical model. We use Green's function and Hankel and Laplace integral transforms to solve the membrane flux problem. Our simulations investigate the effects of various factors on the spatiotemporal behavior of calcium levels, with a simulation on the buffers in Alzheimer's disease-affected neurons. We also look at the effects of calcium-binding substances like the S100B protein and BAPTA and EGTA. Our results demonstrate how important the S100B protein Amyloid beta and the STIM-Orai mechanism are, and how important they are to consider when simulating the calcium signaling system. As such, our research indicates that a more realistic and complete model for modeling calcium dynamics may be obtained by using a generalized reaction-diffusion technique. [ABSTRACT FROM AUTHOR]

The functioning of several cellular processes in neuron cells relies on the interplay between multiple systems, such as calcium ([Ca2+]), inositol 1, 4, 5-trisphosphate (IP3), and dopamine. But, their individual dynamics provide very little insight into the various regulatory and dysregulatory cellular processes. The interaction of two systems dynamics offers some useful information about cell functioning in neurons. But, no attempt has been noted in the literature about the cooperation of three systems dynamics of [Ca2+], IP3, and dopamine in neurons. A mathematical model was utilized to examine the dynamic interactions of [Ca2+], IP3, and dopamine in neurons, considering their spatiotemporal aspects. Numerical findings were obtained using the finite element technique in conjunction with the Crank–Nicholson scheme. The effects of different component events like IP3-receptor (IP3R), sodium–calcium exchanger (NCX), calbindin-D28K buffer, etc. on the synergetic calcium, IP3, and dopamine dynamics have been studied in neuronal cells. The present model offers novel insights into the effects of regulation and dysregulation in different mechanisms like IP3R, NCX, calbindin-D28K, etc. on the synergetic systems of [Ca2+], IP3 and dopamine in neurons and their association with multiple neurological disorders, including Alzheimer's disease and Parkinson's disease. [ABSTRACT FROM AUTHOR]

In this paper, the Fuzzy Sumudu Transform Iterative method (FSTIM) was applied to find the exact fuzzy solution of the fuzzy space-time fractional telegraph equations using the Fuzzy Caputo Fractional Derivative operator. The Telegraph partial differential equation is a hyperbolic equation representing the reaction-diffusion process in various fields. It has applications in engineering, biology, and physics. The FSTIM provides a reliable and efficient approach for obtaining approximate solutions to these complex equations improving accuracy and allowing for fine-tuning and optimization for better approximation results. The work introduces a fuzzy logic-based approach to Sumudu transform iterative methods, offering flexibility and adaptability in handling complex equations. This innovative methodology considers uncertainty and imprecision, providing comprehensive and accurate solutions, and advancing numerical methods. Solving the fuzzy space-time fractional telegraph equation used a fusion of the Fuzzy Sumudu transform and iterative approach. Solution of fuzzy fractional telegraph equation finding analytically and interpreting its results graphically. Throughout the article, whenever we draw graphs, we use Mathematica Software. We successfully employed FSTIM, which is elegant and fast to convergence. [ABSTRACT FROM AUTHOR]

This paper proposes a physics-based model based on possible chemical processes responsible for the resistive switching of sputter-deposited silicon oxide films. Diffusion–reaction differential equations are utilized to pursue physical and chemical origins of the switching phenomenon. Based on the theoretical model, the chemical reaction process is analytically and numerically solved, and an analytical model is proposed to elucidate the phenomenon. Theoretical simulation results are examined from the point of view of suitability of parameter values, and the analytical model is used to interpret the simulation results. Simulation results greatly assist in understanding the switching processes of silicon oxide films; that is, the diffusion processes of hydrogen and water molecules primarily rule the switching processes, and the displacement of oxygen atoms is assisted by those processes. The analytical model predicts that high-speed switching requires a large number of traps in the oxide, a relatively large binding energy, and a low leakage current; all of them can easily be satisfied for sputter-deposited oxide films. A combination of the theoretical simulation model and the analytical model gives a guideline of how the sputter-deposited silicon oxide films can be made suitable for high-speed resistive switching applications. [ABSTRACT FROM AUTHOR]

In this paper, we study the inhomogeneous reaction–diffusion equation involving the infinity Laplacian: Δ ∞ u (x) = f (x , u (x)) + g (x) , x ∈ Ω , where the continuous function f satisfies 0 ≤ f (x , δ t) ≤ Λ (x) δ γ f (x , t) , a positive function Λ (x) ∈ C (Ω ¯) , γ ∈ [ 0 , 3) , t > 0 and 0 < δ ≤ 1 2 . Such a model permits existence of solutions with dead core zones, i.e. a priori unknown regions where non-negative solutions vanish identically. For γ ∈ [ 0 , 3) and the non-positive inhomogeneous term g , we establish the existence, uniqueness and stability of the viscosity solution of the corresponding continuous Dirichlet problem. Under additional structure conditions on f and g , we obtain the optimal C 4 3 − γ regularity across the free boundary ∂ { u > 0 } ∩ Ω. Moreover, we establish the porosity of the free boundary and Liouville type theorem for entire solutions. Finally, we prove that the dead core vanishes in the limit case γ = 3. [ABSTRACT FROM AUTHOR]

Intraguild predation (IGP) allows for coexistence between two consumers of a single resource, as long as the intraguild prey (IG prey) is competitively superior to the intraguild predator (IG predator) and resource population productivity is neither abundant or limiting. Here, we explore biological invasions forming IGP community modules by either introducing IG prey or IG predator species to established consumer-resource populations in homogeneous and heterogeneous landscapes, using reaction-diffusion equations as our modeling framework. Our main methods of analysis are comparing numerical solutions to linearization techniques and homogenization approximations. We find that in homogeneous landscapes, speeds are linearly determinate, i.e., depend on low invader population densities at the leading edge. We also find traveling wave solutions and dynamical stabilization regimes. On heterogeneous landscapes, our results show that depending on habitat preferences of the three species involved, coexistence regimes can occur regardless of IG-prey being least effective consumer, or be hindered even when IG-prey remains as the dominant competitor. Our work assesses how fast can organisms invade novel landscapes in presence of a established IG prey or IG predator and also demonstrates how habitat fragmentation and species habitat preference can disrupt or facilitate coexistence in IGP communities. [ABSTRACT FROM AUTHOR]

Recent experiments suggest that mesoscale catalysts are active materials that power their motion with chemical free energy from their environment and also "chemotax" with respect to substrate gradients. In the present work, we explore a thermodynamic framework for relating this chemotaxis to the evolution of a system down the gradient of its free energy. This framework builds upon recent studies that have employed the Wasserstein metric to describe diffusive processes within the Onsager formalism for irreversible thermodynamics. In this work, we modify the Onsager dissipation potential to explicitly couple the reactive flux to the diffusive flux of catalysts. The corresponding gradient flow is a modified reaction-diffusion equation with an advective term that propels the chemotaxis of catalysts with the free energy released by chemical reactions. In order to gain first insights into this framework, we numerically simulate a simplified model for spherical catalysts undergoing artificial chemotaxis in one dimension. These simulations investigate the thermodynamic forces and fluxes that drive this chemotaxis, as well as the resulting dissipation of free energy. Additionally, they demonstrate that chemotaxis can delay the relaxation to equilibrium and, equivalently, prolong the duration of nonequilibrium conditions. Although future simulations should consider a more realistic coupling between reactive and diffusive fluxes, this work may provide insight into the thermodynamics of artificial chemotaxis. More generally, we hope that this work may bring attention to the importance of the Wasserstein metric for relating nonequilibrium relaxation to the thermodynamic free energy and to large deviation principles. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem of a time-fractional nonlinear diffusion equation. According to Kaplan's first eigenvalue method, we first prove the blow-up of the solutions in finite time under some sufficient conditions. We next provide sufficient conditions for the existence of global solutions by using the results of Zhang and Sun. In conclusion, we find the second critical exponent for the existence of global and non-global solutions via the decay rates of the initial data at spatial infinity. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiple solutions of parametric quasi-linear systems of the gradient-type on the Sierpiński gasket arising in physical problems leading to nonlinear models involving reaction-diffusion equations, in problems on elastic fractal media or fluid flow through fractal regions. By using some critical point theorems, we give some new results to establish one, two and three weak solutions for our system. Finally, we give four examples to illustrate the main results. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of a class of reaction-advection-diffusion equations of the form u t − Δ u + β (y) u x = f (t , u) in cylinders, where f is a tristable or multistable nonlinearity satisfying f (⋅ , 0) = f (⋅ , 1) = 0. We establish the alternative of wave solutions, that is, either there is a unique, asymptotically stable periodic traveling wave connecting 0 to 1 directly, or there is a unique propagating terrace connecting 0 to 1. Under the assumption that all wave speeds in propagating terrace are not equal to each other, we further obtain that the propagating terrace is asymptotically stable if it exists. Moreover, the sufficient and necessary conditions for the existence of periodic traveling waves connecting 0 to 1 are given respectively. Our arguments are based on the parabolic maximum principle, sliding method and the super- and sub-solution method. [ABSTRACT FROM AUTHOR]

We consider a general impulsive reaction-diffusion equation with shifting environment and birth pulse, both induced by climate changes, to capture essential features of the dynamics of species exhibiting distinct stages of reproduction and dispersal. We convert this model into a discrete-time semiflow, and hence to a discrete-time recursion system. We establish the existence of forced waves and asymptotic spreading properties of solutions, and obtain sufficient conditions for the global asymptotic stability of the forced waves. [ABSTRACT FROM AUTHOR]

In this paper, we provide several techniques to explicitly calculate fractional powers of 2×2$2\times 2$ operator matrices Λ=Λ11Λ12Λ21Λ22,$$\begin{equation*}\hspace*{10pc} \Lambda = \def\eqcellsep{&}\begin{bmatrix} \Lambda _{11} & \Lambda _{12} \\ \Lambda _{21} & \Lambda _{22} \end{bmatrix}, \end{equation*}$$focusing on creating a theory that can be applied to distinct situations. To illustrate the abstract results developed, we consider its application in systems of coupled reaction–diffusion equations and in (strongly damped) wave equations. We also discuss how these techniques can be applied to higher order matrices and we specifically calculate the fractional powers of a 4×4$4\times 4$ operator matrix associated to a weakly coupled system of wave equation. In addition, we deal with the applicability of this analysis with respect to solvability, stabilization, regularity, smooth dynamics, and connection with evolutionary classic equation and its fractional counterpart. [ABSTRACT FROM AUTHOR]

Mass transfer limitations may considerably affect the rate of a heterogeneous catalytic process. The catalyst effectiveness factor is a quantitative measure of the impact of the diffusion process inside a catalyst particle. The effectiveness factor is derived from the solution of the steady-state reaction-diffusion problem. Herein, we simulate the steady-state reaction-diffusion equation with space-fractional derivative and linear reaction kinetics. The solution to the problem is obtained numerically using the Haar wavelet collocation method. The effect of the anomalous diffusion exponent on the catalyst effectiveness factor and process parameters, e.g. reactor volume and catalyst mass, is demonstrated. We anticipate that the process efficiency will be notably improved by changing the diffusion regime from standard to superdiffusive. [ABSTRACT FROM AUTHOR]

Viral infection develops in the organism due to virus replication inside infected cells and its transmission from infected to uninfected cells through the extracellular matrix or cell junctions. In this work, we model infection spreading in tissue with a delay reaction-diffusion system of equations for the concentrations of uninfected cells, infected cells and virus. We prove the wave existence, determine its speed of propagation and introduce a simplified one-equation model obtained from the complete model using a quasi-stationary approximation. [ABSTRACT FROM AUTHOR]

This paper aims to demonstrate a numerical strategy via finite difference formulations for time fractional reaction–diffusion models which are ubiquitous in chemical and biological phenomena. The time-fractional derivative is considered in the Caputo sense for both linear and nonlinear problems. First, the Caputo derivative is replaced with a quadrature formula, then an implicit method is used for the remaining part. In the linear case, the proposed strategy reduces the time fractional models into linear simultaneous equations. In nonlinear cases, Quasilinearization is utilized to tackle the nonlinear parts. With this strategy, solutions of the fractional system transform into linear algebraic systems which are easy to solve. Next, the Von Neumann method is implemented to examine the stability of the scheme which discloses that the scheme is unconditionally stable. Further, the applicability of the presented scheme is tested with different linear and nonlinear models which include the one dimensional Schnakenberg and Gray–Scott models, and one and two dimensional Brusselator models. To analyze the accuracy of the present technique two norms namely, L ∞ and L 2 , and relative error are addressed. Moreover, the obtained outcomes are shown tabulated and graphically which identifies that the scheme properly works for the time fractional reaction–diffusion systems. [ABSTRACT FROM AUTHOR]

We propose a malaria model involving the sensitive and resistant strains, which is described by reaction-diffusion equations. The model reflects the scenario that the vector and host populations disperse with distinct diffusion rates, susceptible individuals or vectors cannot be infected by both strains simultaneously, and the vector population satisfies the logistic growth. Our main purpose is to get a threshold type result on the model, especially the interaction effect of the two strains in the presence of spatial structure. To solve this issue, the basic reproduction number (BRN) Ri0 and invasion reproduction number (IRN) Ri0 of each strain (i = 1 and 2 are for the sensitive and resistant strains, respectively) are defined. Furthermore, we investigate the influence of the diffusion rates of populations and vectors on BRNs and IRNs. [ABSTRACT FROM AUTHOR]

The paper presents an approach for overcoming modeling problems of typical life science applications with partly unknown mechanisms and lacking quantitative data: A model family of reaction–diffusion equations is built up on a mesoscopic scale and uses classes of feasible functions for reaction and taxis terms. The classes are found by translating biological knowledge into mathematical conditions and the analysis of the models further constrains the classes. Numerical simulations allow comparing single models out of the model family with available qualitative information on the solutions from observations. The method provides insight into a hierarchical order of the mechanisms. The method is applied to the clinics for liver inflammation such as metabolic dysfunction-associated steatohepatitis or viral hepatitis where reasons for the chronification of disease are still unclear and time- and space-dependent data is unavailable. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of large-scale stochastic reaction-diffusion systems. To prove the exponential stability of the system, we introduce the corresponding isolated subsystems. We show that the exponential stability of the isolated systems implies the exponential stability of the large-scale stochastic reaction-diffusion system under some conditions. Furthermore, we discuss a special case where the large-scale stochastic reaction-diffusion system is described in a hierarchical form. In this case, we prove that the original system is exponentially stable if and only if the corresponding subsystems are exponentially stable. [ABSTRACT FROM AUTHOR]

This paper investigates a free boundary tumor model with time-dependent in the presence of inhibitors. The model consists of two diffusion equations representing nutrients and inhibitors respectively, and an ordinary differential equation describing the radius of the tumor $ R(t) $. We know that angiogenesis is not a steady-state process, in general, it changes over time, so it is reasonable to assume that tumors stimulate angiogenesis at a rate proportional to $ \alpha (t) $. We find the properties of the tumor radius $ R(t) $ is greatly tied to the properties of $ \alpha (t) $. When $ \alpha \left(t \right) $ is time-dependent, we prove that for any sufficiently small $ c_1 $: If $ \alpha (t) $ remains uniformly bounded, then $ R(t) $ also remains uniformly bounded; If $ \alpha (t) $ tends to zero as $ t \to \infty $, so does the tumor radius $ R(t) $; If $ \lim_{t \to \infty}\inf \alpha \left (t \right)>0, $ then $ \lim_{t \to \infty} \inf R\left (t \right)>0 $. Moreover, the global asymptotic stability of the steady-state solution is proved, and it is surprising to find that when $ \tilde u + \bar v $ is sufficiently small and $ \frac{\lambda }{{\mu \bar u}} < {c_1} \le{c_2} $, the solution will blow up. [ABSTRACT FROM AUTHOR]

We study the reaction diffusion equation $ u_t-du_{xx} = f(u) $ with a monostable nonlinear function $ f(u) $ over a changing interval $ [g(t), h(t)] $, viewed as a model for the spreading of a species with population range $ [g (t), h(t)] $ and density $ u(t,x) $. The free boundaries $ x = g(t) $ and $ x = h(t) $ are not governed by the same Stefan condition as in Du and Lin [20] and other previous works; instead, they satisfy a related but different set of equations obtained from a 'preferred population density' assumption at the range boundary, which allows the population range to shrink. We obtain a rather complete understanding of the longtime dynamics of the model, which exhibits persistent propagation with a finite asymptotic propagation speed determined by a certain semi-wave solution, and the density function converges to the semi-wave profile as time goes to infinity. The asymptotic propagation speed is always smaller than that of the corresponding classical Cauchy problem where the reaction-diffusion equation is satisfied for $ x $ over the entire real line with no free boundary. Moreover, when the preferred population density used in the free boundary condition converges to 0, the solution $ u $ of our free boundary problem converges to the solution of the corresponding classical Cauchy problem, and the propagation speed also converges to that of the Cauchy problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider solutions for discrete systems arising from multi-term time-fractional diffusion equations. Using discrete sine transform techniques, we find that all-at-once systems of such equations have a structure similar to that of diagonal-plus-Toeplitz matrices. We establish a generalized circulant approximate inverse preconditioner for the all-at-once systems. Through a detailed analysis of the preconditioned matrices, we show that the spectrum of the obtained preconditioned matrices is clustered around one. We give some numerical examples to demonstrate the effectiveness of the proposed preconditioner. [ABSTRACT FROM AUTHOR]

C a 2 + signals have a crucial role in the immune system for cell functions such as proliferation, differentiation, apoptosis and gene transcription as well as the activation of Nuclear factor of activated T Cell (NFAT) which modulates the immune response of the cells. In this study, a fractional order spatiotemporal calcium dynamics model is formulated using the reaction-diffusion equation incorporating parameters like buffers, source influx, J SScyt and J PMCA . Grünwald estimation is employed to obtain the solution and stability analysis has been performed. The analysis of numerical results provides novel insights about the role of Brownian motion, super diffusion, source influx, buffers, etc., in the regulation of calcium and NFAT concentration levels in T cell and the conditions which might lead to disordered immune responses causing diseases such as HINI, HIV and HBV. [ABSTRACT FROM AUTHOR]

This work deals with the theoretical study of the optimality conditions for the optimal control of the monodomain model arising in cardiac electrophysiology with mixed control-state constraints. The monodomain model is a coupled system of reaction–diffusion equation with cubic nonlinearity in the reaction term and an ordinary differential equation. The existence of optimal control is established, and the twice Fréchet differentiability of the control-to-state operator is discussed. The first-order necessary optimality condition is derived. Most significantly, a detailed proof of the sufficient second-order optimality condition is demonstrated with mixed control-state constraints. [ABSTRACT FROM AUTHOR]

A statistical linearization method is proposed for determining the response of the chain-like multi-degree-of-freedom (MDOF) hysteretic system endowed with fractional derivative elements and subjected to combined periodic and stochastic excitation. The method is developed based on representing the system response into a combination as a deterministic and a zero-mean stochastic component. Specifically, first, the equation of motion can be transformed into two sets of coupled nonlinear differential equations for the deterministic and the stochastic response components, respectively. Next, the deterministic and the stochastic differential equations are treated by the harmonic balance method and the statistical linearization method, respectively, yielding two sets of coupled nonlinear algebraic equations, which can be solved simultaneously via standard numerical methods for the deterministic response time history and the stochastic response second moment. Pertinent numerical examples demonstrate the applicability of the proposed method. [ABSTRACT FROM AUTHOR]

This paper mainly deals with the (non)existence, asymptotic behaviors and uniqueness of traveling waves to a nonlocal diffusion system with asymmetric kernels and delays for quasi-monotone case. The difference from some previous works is the asymmetry reflected in both diffusion and reaction terms, and this not only has an impact on the positivity of minimal wave speed and the wave profiles of traveling waves with the same speed spreading from the left and right of the x-axis, but also leads to some difficulties for the nonexistence and asymptotic behaviors of traveling waves, which are overcome by using new techniques. Thereby, the results for traveling waves of nonlocal diffusion equations with symmetric kernels and with (or without) delays are improved to equations with asymmetric kernels, and those conclusions for scalar equations and systems with Laplace diffusion and local nonlinearities are also generalized to the nonlocal case. Finally, some concrete applications and numerical simulations are shown to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study multi-bump solutions of the following Kirchhoff type equation: - M ∫ R 2 | ∇ u | 2 d x Δ u + μ V (x) + h (x) u = λ f (u) in R 2 , where M is continuous with inf R + M > 0 , V ≥ 0 and its zero set has several disjoint bounded components, μ , λ are positive parameters, f has exponential critical growth. When V decays to zero at infinity, we use variational methods to obtain the existence and concentration behavior of multi-bump solutions. [ABSTRACT FROM AUTHOR]

A nonlocal diffusion single population model with advection and free boundaries is considered. Our aim is to discuss how the advection rate affects dynamic behaviors of species under the case of small advection. Firstly, the well-posed global solution is obtained. Secondly, we apply the eigenvalue problem of integro-differential operator to obtain the dichotomy and sharp criteria for spreading and vanishing, which is determined by initial habitat and initial density. Further, the asymptotic spreading speed of species is estimated when spreading happens. Namely, we get the exact asymptotic spreading speed and find that if kernel function satisfies the certain condition, then the leftward asymptotic spreading speed is less than the rightward one due to the impact of advection rate. Meanwhile, we also observe that accelerated spreading happens if the certain condition does not be satisfied. [ABSTRACT FROM AUTHOR]

Mathematical solutions of the concentration of dissolved component in surrounding fluid were obtained for analyses of the dissolution phenomena of spherical, cylindrical, and slab-type particles. Calculation of the surface flux enabled material balance equations to be derived for the prediction of transient particle size. For the spherical particle, the resulting differential equation was solved numerically to compare with the analytical solution. For the cylindrical particle, the Weber transform was applied to predict the radius of the infinitely long cylinder as a function of time. After obtaining an analytical solution for the thickness of the dissolving slab, the result was compared with those of the sphere and cylinder. The spherical particle exhibited the shortest complete dissolution time (CDT), due to having the greatest surface-to-volume ratio. When external film resistance was important for the mass transfer, the Biot number was included in the governing equation, and to predict the particle size during dissolution, the concentration in the fluid phase could be derived by the combination of variables, or the Weber-type transform. When first-order or zero-order reaction occurs in the surrounding fluid, the concentration could also be predicted by the Laplace transform, and the particle size could be calculated for the solid particles. Similar to the simple diffusion process, CDT was predicted to increase in the order sphere < cylinder < slab. The experimental data of the dissolution of polymeric fiber was compared with the modeling result to estimate the diffusivity and solubility factor, which varied with the pH of the surrounding fluid. [ABSTRACT FROM AUTHOR]