Advection Diffusion Reaction Research Articles

Multivalued Random Dynamics of Fractional Reaction–Diffusion–AdvectionEquations Driven by Superlinear Colored Noise on Unbounded Domains

This article investigates the multivalued random dynamics generated by solution operators of fractional random reaction–diffusion–advection equations driven by superlinear colored noise on unbounded domains. The equations under consideration exhibit non-Lipschitz continuous nonlinear drift and diffusion terms, which lead to the non-uniqueness of solutions and hence generate multivalued random dynamical systems. Our goals are to demonstrate: (i) the existence of solutions for fractional reaction–diffusion–advection equations driven by superlinear colored noise on unbounded domains; and (ii) the existence and uniqueness of pullback random attractors for these systems. The measurability of the random attractors is demonstrated using a method that relies on the weak upper semicontinuity of the solutions. The asymptotic compactness of solution operators is obtained by employing Ball’s approach to energy equations in order to address the non-compactness of Sobolev embeddings on the entire space.

Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems

Abstract Trefftz schemes are high-order Galerkin methods whose discrete spaces are made of elementwise exact solutions of the underlying partial differential equation (PDE). Trefftz basis functions can be easily computed for many PDEs that are linear, homogeneous and have piecewise-constant coefficients. However, if the equation has variable coefficients, exact solutions are generally unavailable. Quasi-Trefftz methods overcome this limitation relying on elementwise ‘approximate solutions’ of the PDE, in the sense of Taylor polynomials. We define polynomial quasi-Trefftz spaces for general linear PDEs with smooth coefficients and source term, describe their approximation properties and, under a nondegeneracy condition, provide a simple algorithm to compute a basis. We then focus on a quasi-Trefftz DG method for variable-coefficient elliptic diffusion–advection–reaction problems, showing stability and high-order convergence of the scheme. The main advantage over standard DG schemes is the higher accuracy for comparable numbers of degrees of freedom. For nonhomogeneous problems with piecewise-smooth source term we propose to construct a local quasi-Trefftz particular solution and then solve for the difference. Numerical experiments in two and three space dimensions show the excellent properties of the method both in diffusion-dominated and advection-dominated problems.

Reconstructing Waddington Landscape from Cell Migration and Proliferation.

The Waddington landscape was initially proposed to depict cell differentiation, and has been extended to explain phenomena such as reprogramming. The landscape serves as a concrete representation of cellular differentiation potential, yet the precise representation of this potential remains an unsolved problem, posing significant challenges to reconstructing the Waddington landscape. The characterization of cellular differentiation potential relies on transcriptomic signatures of known markers typically. Numerous computational models based on various energy indicators, such as Shannon entropy, have been proposed. While these models can effectively characterize cellular differentiation potential, most of them lack corresponding dynamical interpretations, which are crucial for enhancing our understanding of cell fate transitions. Therefore, from the perspective of cell migration and proliferation, a feasible framework was developed for calculating the dynamically interpretable energy indicator to reconstruct Waddington landscape based on sparse autoencoders and the reaction diffusion advection equation. Within this framework, typical cellular developmental processes, such as hematopoiesis and reprogramming processes, were dynamically simulated and their corresponding Waddington landscapes were reconstructed. Furthermore, dynamic simulation and reconstruction were also conducted for special developmental processes, such as embryogenesis and Epithelial-Mesenchymal Transition process. Ultimately, these diverse cell fate transitions were amalgamated into a unified Waddington landscape.

Dynamics of a Reaction–Diffusion–Advection System From River Ecology With Inflow

ABSTRACTThis paper is to study the population dynamics of a single species model and a two‐species competition model from river ecology with inflow. One interesting feature in these models is that species can flow into the river through the upstream end due to advective movement while both diffusive and advective movements will cause population loss at the downstream end. We first determine necessary and sufficient conditions for persistence of a single species, in terms of the critical habitat size and the critical advection rate. For the competition model, we investigate the joint effects of advection rates, interspecific competition intensities, and boundary conditions on global dynamics of the system. It shows that the strategy of smaller advection is beneficial for species to survive. Our results partially extend the previous works.

Pinning of reaction–diffusion travelling waves in one-dimensional annular geometry

We analyse the complex dynamics arising in a one-dimensional bistable advection–reaction–diffusion equation on a bounded annular domain, along with its simplest nontrivial approximation in terms of a Fourier expansion. We investigate the pinning of travelling waves produced by spatial anisotropies and suggest possible biological implications in the context of cardiac action potential propagation failure due to acute ischemia.

Well-posedness of Keller–Segel systems on compact metric graphs

Chemotaxis phenomena govern the directed movement of microorganisms in response to chemical stimuli. In this paper, we investigate two Keller–Segel systems of reaction–advection–diffusion equations modeling chemotaxis on thin networks. The distinction between two systems is driven by the rate of diffusion of the chemo-attractant. The intermediate rate of diffusion is modeled by a coupled pair of parabolic equations, while the rapid rate is described by a parabolic equation coupled with an elliptic one. Assuming the polynomial rate of growth of the chemotaxis sensitivity coefficient, we prove local well-posedness of both systems on compact metric graphs, and, in particular, prove existence of unique classical solutions. This is achieved by constructing sufficiently regular mild solutions via analytic semigroup methods and combinatorial description of the heat kernel on metric graphs. The regularity of mild solutions is shown by applying abstract semigroup results to semi-linear parabolic equations on compact graphs. In addition, for logistic-type Keller–Segel systems we prove global well-posedness and, in some special cases, global uniform boundedness of solutions.

Opposite effects of a reaction-driven viscosity decrease on miscible viscous fingering depending on the injection flow rate

When a less-viscous solution of a reactant $A$ displaces a more-viscous solution of another reactant $B$ , a fast bimolecular $A + B \rightarrow C$ reaction decreasing locally the viscosity can influence the viscous fingering (VF) instability taking place between the two miscible solutions. We show both experimentally and numerically that, for monotonic viscosity profiles, this decrease in viscosity has opposite effects on the fingering pattern depending on the injection flow rate. For high flow rates, the reaction enhances the shielding effect, creating VF patterns with a lower surface density, i.e. thinner fingers covering a smaller area. In contrast, for lower flow rates, the reaction stabilises the VF dynamics, i.e. delays the instability and gives a less-deformed displacement, reaching in some cases an almost-stable displacement. Nonlinear simulations of reactive VF show that these opposite effects at low or high flow rates can only be reproduced if the diffusivity of $A$ is larger than that of $B$ , which favours a larger production of the product $C$ and, hence, a larger viscosity decrease. The analysis of one-dimensional viscosity profiles reconstructed on the basis of a one-dimensional reaction–diffusion–advection model confirms that the VF stabilisation at low Péclet number and in the presence of differential diffusion of reactants originates from an optimum reaction-driven decrease in the gradient of the monotonic viscosity profile.

Time periodic travelling waves for an advection–reaction–diffusion SIR epidemic model with seasonality and bilinear incidence

In order to investigate the influence of spatial convection effect and the periodic environment on the spreading behaviour of epidemic, in this paper, an SIR epidemic model with time varying coefficients and diffusion advection is proposed. The periodic travelling waves satisfying certain boundary conditions are discussed by constructing operators on bounded closed convex sets consisting of periodic upper and lower solutions, utilizing the twice fixed point theorem and some limit techniques. The results show that the existence of travelling waves depends on the reproduction number R0 and the critical wave speed c∗. Specifically, when R0>1 and c>c∗, the existence of periodic travelling waves satisfying some boundary condition is obtained, and the nonexistence of such travelling waves for two cases (i) R0>1 and 0<c<c∗, (ii) R0≤1 and c≥0 are also obtained. Finally, some brief simulations are shown to verify the theoretical results, and the effects of spread speed and advection phenomena on disease spread were further investigated.

Study of two-dimensional nonlinear coupled time-space fractional order reaction advection diffusion equations using shifted Legendre-Gauss-Lobatto collocation method

In this article, the nonlinear coupled two-dimensional space-time fractional order reaction-advection–diffusion equations (2D-STFRADEs) with initial and boundary conditions is solved by using Shifted Legendre-Gauss-Lobatto Collocation method (SLGLCM) with fractional derivative defined in Caputo sense. The SLGLC scheme is used to discretize the coupled nonlinear 2D-STFRADEs into the shifted Legendre polynomial roots to convert it to a system of algebraic equations. The efficiency and efficacy of the scheme are confirmed through error analysis while applying the scheme on two existing problems having exact solutions. The impact of advection and reaction terms on the solution profiles for various space and time fractional order derivatives are shown graphically for different particular cases. A drive has been made to study the convergence of the proposed scheme, which has been applied on the proposed mathematical model.

Global well-posedness and uniform boundedness of 2D urban crime models with nonlinear advection enhancement

Abstract We study the global well-posedness and uniform boundedness of a two-dimensional reaction–advection–diffusion system with nonlinear advection. This strongly coupled system of nonlinear partial differential equations represents the continuum of a 2D lattice model designed to describe residential burglary, where each location is characterised by a tractability value that varies in both space and time. We show that the model with sublinear advection enhancement is globally well-posed, with a unique solution that is classical and uniformly bounded in time. Our results provide valuable insights into the development of urban crime models with nonlinear advection enhancements, making them suitable for broader applications, including nonlocal or heterogeneous near-repeat victimisation effects.

Efficient second-order accurate exponential time differencing for time-fractional advection–diffusion–reaction equations with variable coefficients

Time-fractional advection–diffusion–reaction type equations are useful for characterizing anomalous transport processes. In this paper, linearly implicit as well as explicit generalized exponential time differencing (GETD) schemes are proposed for solving a class of such equations having time–space dependent coefficients. The implicit scheme, being unconditionally stable, is robust in handling the numerical instabilities in problems where the advection term is dominant. Regarding the error analysis, uniformly optimal second-order convergence rates are derived using time-graded meshes to counter the effect of the inherent singularity of the continuous solution. Implementation of generalized exponential integrators requires computing the action of Mittag-Leffler function of matrices on a vector, or on a matrix in the case of the implicit scheme. For cost-effective implementation, using global Padé approximants these computation tasks get reduced to solving linear systems. A new approach based on Sylvester equation formulation of the resulting linear systems is developed in this paper. This technique leads to significantly faster algorithms for implementing the GETD schemes. Numerical experiments are provided to illustrate the theoretical findings and to assert the efficiency of the Sylvester equation based approach. Application of this approach to an existing GETD scheme for solving a nonlinear subdiffusion problem is also discussed.

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

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

A theoretical model for focal adhesion and cytoskeleton formation in non-motile cells

To function and survive cells need to be able to sense and respond to their local environment through mechanotransduction. Crucially, mechanical and biochemical perturbations initiate cell signalling cascades, which can induce responses such as growth, apoptosis, proliferation and differentiation. At the heart of this process are actomyosin stress fibres (SFs), which form part of the cell cytoskeleton, and focal adhesions (FAs), which bind this cytoskeleton to the extra-cellular matrix (ECM). The formation and maturation of these structures (connected by a positive feedback loop) is pivotal in non-motile cells, where SFs are generally of ventral type, interconnecting FAs and producing isometric tension. In this study we formulate a one-dimensional bio-chemo-mechanical continuum model to describe the coupled formation and maturation of ventral SFs and FAs. We use a set of reaction–diffusion–advection equations to describe three sets of biochemical events: the polymerisation of actin and subsequent bundling into activated SFs; the formation and maturation of cell–substrate adhesions; and the activation of signalling proteins in response to FA and SF formation. The evolution of these key proteins is coupled to a Kelvin–Voigt viscoelastic description of the cell cytoplasm and the ECM. We employ this model to understand how cells respond to external and intracellular cues in vitro and are able to reproduce experimentally observed phenomena including non-uniform cell striation and cells forming weaker SFs and FAs on softer substrates.

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

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

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We present a method for computing actions of the exponential-like φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions for a Kronecker sum K of d arbitrary matrices Aμ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_\\mu $$\\end{document}. It is based on the approximation of the integral representation of the φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions by Gaussian quadrature formulas combined with a scaling and squaring technique. The resulting algorithm, which we call phiks, evaluates the required actions by means of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-mode products involving exponentials of the small sized matrices Aμ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_\\mu $$\\end{document}, without forming the large sized matrix K itself. phiks, which profits from the highly efficient level 3 BLAS, is designed to compute different φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions applied on the same vector or a linear combination of actions of φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions applied on different vectors. In addition, thanks to the underlying scaling and squaring techniques, the desired quantities are available simultaneously at suitable time scales. All these features allow the effective usage of phiks in the exponential integration context. In fact, our newly designed method has been tested in popular exponential Runge–Kutta integrators of stiff order from one to four, in comparison with state-of-the-art algorithms for computing actions of φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions. The numerical experiments with discretized semilinear evolutionary 2D or 3D advection–diffusion–reaction, Allen–Cahn, and Brusselator equations show the superiority of the proposed μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-mode approach.

Correction: Integral representation bound of the true solution to the BVP of double-sided fractional diffusion advection reaction equation

Correction: Integral representation bound of the true solution to the BVP of double-sided fractional diffusion advection reaction equation

Predator–prey systems with a variable habitat for predators in advective environments

AbstractCommunity composition in aquatic environments can be shaped by a broad array of factors, encompassing habitat conditions in addition to abiotic conditions and biotic interactions. This paper pertains to reaction–diffusion–advection predator–prey model featuring a variable predator habitat in advective environments, governed by a unidirectional flow. First, we establish the near‐complete global dynamics of the system. In instances where the functional response to predation conforms to Holling‐type I or II, we explore the uniqueness and stability of positive steady‐state solutions via the application of particular auxiliary techniques, the comparison principle for parabolic equations, and perturbation analysis. Furthermore, we obtain the critical positions at the upper and lower boundaries of the predator's habitat, which determine the survival of the prey irrespective of the predator's growth rate. Finally, we show how the location and length of the predator's habitat affect the persistence and extinction of predators and prey in the event of a small population loss rate at the downstream end. From the biological point of view, these results contribute to our deeper understanding of the effects of habitat on aquatic populations and may have applications in aquaculture and the establishment of protection zones for aquatic species.

Multiple Time Scales and Normal Form of Hopf Bifurcation in a Delayed Reaction–Diffusion–Advection System

In the reaction–diffusion systems, the Laplacian is usually a self-adjoint operator. Incorporating advection terms generally destroys this and yields a quite complicated decomposition of phase space in the Center Manifold Reduction (CMR) method. Considering that the Multiple Time Scales (MTS) method can be directly applied to bifurcation analysis and normal form derivation based on algebraic operations, and the computational process is relatively universal, we apply the MTS method to analyze the Hopf bifurcation problem of reaction–diffusion–advection systems with time delay. First, we introduce the MTS method for the system with a specific boundary condition, which is used to derive the normal form of Hopf bifurcation. Calculating the normal form using the MTS method can be summarized as determining the bifurcation parameters, conducting Taylor expansion based on the MTS idea, and eliminating secular terms. Then, the key coefficients in the normal form are explicitly given, which are also compared with the results obtained by the CMR method, and both methods lead to the same bifurcation results. Finally, we use the MTS method to calculate the normal form of Hopf bifurcation for a predator–prey model with mixed boundary conditions. We find that the spatially nonhomogeneous periodic solutions appear near the equilibrium in the system when the time delay exceeds the critical value, and the theoretical results are illustrated by numerical simulations.

Regularity and wave study of an advection–diffusion–reaction equation

In this paper, we investigate the optimal conditions to the boundaries where the unique existence of the solutions to an advection-diffusion-reaction equation is secured by applying the contraction mapping theorem from the study of fixed points. Also, we extract, traveling wave solutions of the underlying equation. To this purpose, a new extended direct algebraic method with traveling wave transformation has been used. Achieved soliton solutions are different functions which are hyperbolic, trigonometric, exponential, and some mixed trigonometric functions. These functions show the nature of solitons. Two and three-dimensional plots are drawn using different values of parameters and coefficients for the comparison and behavior of solitons as combined bright-dark, dark, and bright solitons.

Existence and Asymptotic Stability of Solutions for Periodic Parabolic Problems in Tikhonov-Type Reaction–Diffusion–Advection Systems with KPZ Nonlinearities

Existence and Asymptotic Stability of Solutions for Periodic Parabolic Problems in Tikhonov-Type Reaction–Diffusion–Advection Systems with KPZ Nonlinearities