Class Of Reaction-diffusion Equations Research Articles

Dynamical behavior of a temporally discrete non-local reaction-diffusion equation on bounded domain

This paper focuses on the study of global dynamics of a class of temporally discrete non-local reaction-diffusion equations on bounded domains. Similar to classical reaction diffusion equations and integro-difference equations, temporally discrete reaction-diffusion equations can also be used to describe the dispersal phenomena in population dynamics. In this paper, we first derived a temporally discrete reaction diffusion equation model with time delay and nonlocal effects to model the evolution of a single species population with age-structured located in a bounded domain. By establishing a new maximum principle and applying the monotone iteration method, the global stabilities of the trivial solution and the positive steady state solution are obtained respectively under some appropriate assumptions.

Global dynamics of a diffusive SARS-CoV-2 model with antiviral treatment and fractional Laplacian operator

In this paper, we propose and investigate the global dynamics of a SARS-CoV-2 infection model with diffusion and antiviral treatment. The proposed model takes into account the two modes of transmission (virus-to-cell and cell-to-cell), the lytic and nonlytic immune responses. The diffusion into the model is formulated by the regional fractional Laplacian operator. Furthermore, the global asymptotic stability of equilibria is rigorously established by means of a new proposed method constructing Lyapunov functions for a class of partial differential equations (PDEs) with regional fractional Laplacian operator. The proposed method is applied to the classical reaction-diffusion equations with normal diffusion.

Numerical solution of a malignant invasion model using some finite difference methods

Abstract In this article, one standard and four nonstandard finite difference methods are used to solve a cross-diffusion malignant invasion model. The model consists of a system of nonlinear coupled partial differential equations (PDEs) subject to specified initial and boundary conditions, and no exact solution is known for this problem. It is difficult to obtain theoretically the stability region of the classical finite difference scheme to solve the set of nonlinear coupled PDEs, this is one of the challenges of this class of method in this work. Three nonstandard methods abbreviated as NSFD1, NSFD2, and NSFD3 are considered from the study of Chapwanya et al., and these methods have been constructed by the use of a more general function replacing the denominator of the discrete derivative and nonlocal approximations of nonlocal terms. It is shown that NSFD1, which preserves positivity when used to solve classical reaction-diffusion equations, does not inherit this property when used for the cross-diffusion system of PDEs. NSFD2 and NSFD3 are obtained by appropriate modifications of NSFD1. NSFD2 is positivity-preserving when the functional relationship [ ψ ( h ) ] 2 = 2 ϕ ( k ) {\left[\psi \left(h)]}^{2}=2\phi \left(k) holds, while NSFD3 is unconditionally dynamically consistent with respect to positivity. First, we show that NSFD2 and NSFD3 are not consistent methods. Second, we tried to modify NSFD2 in order to make it consistent but we were not successful. Third, we extend NSFD3 so that it becomes consistent and still preserves positivity. We denote the extended version of NSFD3 as NSFD5. Finally, we compute the numerical rate of convergence in time for NSFD5 and show that it is close to the theoretical value. NSFD5 is consistent under certain conditions on the step sizes and is unconditionally positivity-preserving.

Asymptotic profiles for positive solutions of diffusive logistic equations

In this paper, we study the asymptotic profiles of positive solutions for diffusive logistic equations. The aim is to study the sharp effect of linear growth and nonlinear function. Both the classical reaction-diffusion equation and nonlocal dispersal equation are investigated. Our main results reveal that the linear and nonlinear parts of reaction term play quite different roles in the study of positive solutions.

A Feynman Path Integral-like Method for Deriving Reaction-Diffusion Equations.

This work is devoted to deriving a more accurate reaction-diffusion equation for an A/B binary system by summing over microscopic trajectories. By noting that an originally simple physical trajectory might be much more complicated when the reactions are incorporated, we introduce diffusion-reaction-diffusion (DRD) diagrams, similar to the Feynman diagram, to derive the equation. It is found that when there is no intermolecular interaction between A and B, the newly derived equation is reduced to the classical reaction-diffusion equation. However, when there is intermolecular interaction, the newly derived equation shows that there are coupling terms between the diffusion and the reaction, which will be manifested on the mesoscopic scale. The DRD diagram method can be also applied to derive a more accurate dynamical equation for the description of chemical reactions occurred in polymeric systems, such as polymerizations, since the diffusion and the reaction may couple more deeply than that of small molecules.

Recent developments on spatial propagation for diffusion equations in shifting environments

<p style='text-indent:20px;'>In this short review, we describe some recent developments on the spatial propagation for diffusion problems in shifting environments, including single species models, competition/cooperative models and chemotaxis models submitted to classical reaction-diffusion equations (with or without free boundaries), integro-difference equations, lattice differential equations and nonlocal dispersal equations. The considered topics may typically come from modeling the threats associated with global climate change and the worsening of the environment resulting from industrialization which lead to the shifting or translating of the habitat ranges, and also arise indirectly in studying the pathophoresis as well as some multi-stage invasion processes. Some open problems and potential research directions are also presented.</p>

Spatial movement with diffusion and memory-based self-diffusion and cross-diffusion

Spatial memory has been considered significant in animal movement modeling. In this paper, we formulate a two-species interaction model by incorporating both random walk and spatial memory-based walk in their movement. The spatial memory-based walk, described by a chemotactic-like term, is derived by a modified Fick's law involving a directed movement toward the gradient of the density distribution function at a past time. For the proposed model, local stability and bifurcations are studied at constant steady states. Unlike a classical reaction-diffusion equation, we show that the accumulation points of eigenvalues for the model will locate at a vertical line in the complex plane, which will make the model generate spatially inhomogeneous time-periodic patterns through Hopf bifurcation. As illustrations, we apply these results to competition and cooperative models with memory-based diffusion. For the competition model, it turns out that the outcomes are far more complicated than those of classic Lotka-Volterra reaction-diffusion models, due to the consideration of memory-based diffusion. In particular, the existence of periodic oscillations is proved under weak competition. Similar conclusions hold for the cooperative model.

On some theory of monostable and bistable pure birth-jump integro-differential equations

We study an integral-differential equation that models a pure birth-jump process, where birth and dispersal cannot be decoupled. A case has been made that these processes are more suitable for phenomena such as plant dynamics, fire propagation, and cancer cell dynamics. We contrast the dynamics of this equation with those of the classical reaction-diffusion equation, where the reaction term models either logistic growth or a strong Allee effect. Recent evidence of an Allee effect has been found in plant dynamics during the germination process (due to seed predation) but not in the generation of seeds. This motivates where the Allee effect is included in our model. We prove the global existence and uniqueness of solutions with bounded initial data and analyze some properties of the solutions. Additionally, we prove results related to the persistence or extinction of a species, which are analogous to those of the classical reaction-diffusion equation. A key finding is that in some cases a population which is initially below the Allee threshold in some area, even if small, will actually survive. This is in contrast to solutions of the classical reaction-diffusion with the same initial data. Another difference of note is the lack of regularization and an infinite number of discontinuous equilibrium solutions to the birth-jump model.

Pattern formation in superdiffusion predator–prey‐like problems with integer‐ and noninteger‐order derivatives

This paper focuses on the modeling and application of fractional derivative to model the interactions between two different species in which the one named predator depends on the other called prey solely for survival. The interaction between predator and prey has been one of the most intriguing and interesting subjects in applied mathematical biology and ecology. In the models, the classical reaction–diffusion equations subject to the Neumann boundary conditions are formulated on a finite but large domainx ∈ [0, L]by replacing the second‐order spatial derivatives with the fractional Laplacian operator of order1 < α ≤ 2, which is classified as superdiffusion process. We examine the resulting coupled reaction–diffusion models for linear stability analysis and derive conditions under which the spatial patterns is evolved. In a view to understand our theoretical findings, the species spatial interactions is described in one and two dimensions. Through numerical experiments, we observe that a number of patterns can arise, including Turing spots, spiral‐like structures, and seemingly complex spatiotemporal distributions.

Variational approach of critical sharp front speeds in degenerate diffusion model with time delay

For the classical reaction diffusion equation, the a priori speed of fronts is determined exactly in the pioneering paper (Benguria and Depassier 1996 Commun. Math. Phys. 175 221–227) by variational characterization method. In this paper, we study the age-structured population dynamics using a degenerate diffusion equation with time delay. We show the existence and uniqueness of sharp critical fronts, where the sharp critical front is C1-smooth when the diffusion degeneracy is weaker with 1 < m < 2, and the sharp critical front is non-C1-smooth (piecewise smooth) when the diffusion degeneracy is stronger with m ⩾ 2, and the non-critical waves are C2-smooth. We give a new variational approach for the critical wave speed and investigate how the time delay affects the propagation mechanism of fronts. It is shown that the time delay slows down the critical wave speed.

On a Class of Reaction-Diffusion Equations with Aggregation

Abstract In this paper, global-in-time existence and blow-up results are shown for a reaction-diffusion equation appearing in the theory of aggregation phenomena (including chemotaxis). Properties of the corresponding steady-state problem are also presented. Moreover, the stability around constant equilibria and the non-existence of nonconstant solutions are studied in certain cases.

Sharp profiles for periodic logistic equation with nonlocal dispersal

In this paper, we study the nonlocal dispersal logistic equation $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=J*u-u+\lambda u-[b(x)q(t)+\delta ]u^p &{}\text {in}\,\bar{\Omega }\times (0,\infty ),\\ u(x,t)=0 &{}\text {in}\,{\mathbb {R}^N\setminus \bar{\Omega }}\times (0,\infty ),\\ u(x,t)=u(x,t+T) &{}\text {in}\,\bar{\Omega }\times [0,\infty ), \end{array}\right. } \end{aligned}$$here $$\Omega \subset \mathbb {R}^N$$ is a bounded domain, J is a nonnegative dispersal kernel, $$p>1$$, $$\lambda $$ is a fixed parameter and $$\delta >0$$. The coefficients b, q are nonnegative and continuous functions, and q is periodic in t. We are concerned with the asymptotic profiles of positive solutions as $$\delta \rightarrow 0$$. We obtain that the temporal degeneracy of q does not make a change of profiles, but the spatial degeneracy of b makes a large change. We find that the sharp profiles are different from the classical reaction–diffusion equations. The investigation in this paper shows that the periodic profile has two different blow-up speeds and the sharp profile is time periodic in domain without spatial degeneracy.

Asymptotic behavior of solutions for the classical reaction-diffusion equation with nonlinear boundary conditions and fading memory

Asymptotic behavior of solutions for the classical reaction-diffusion equation with nonlinear boundary conditions and fading memory

Scaling effects and front propagation in a class of reaction-diffusion equations: From classic to anomalous diffusion

A simulation study is proposed where a reaction-diffusion equation in a semi-infinite medium is numerically solved by a second-order approximate finite-difference method. The non-linearities present both in the diffusivity and in the kinetic term required a proper iterative method that proved to be satisfactory in robustness and efficiency. The dynamics of the moving front were investigated in the absence and presence of a chemical reaction. In the former case, a non-linear diffusivity led to correlations between the speed of the diffusion front and the strength of interactions between diffusing species, though preserving the classical time scaling. In the latter, a genuine non-classical trend of the travelling front was detected for long times, in agreement with the recent studies concerning anomalous diffusion. An analytical approach based on scaling considerations motivated the numerical results. This study may explain the onset of a subdiffusive trend in reaction-diffusion front dynamics observed in permeation experiments through layers of different substrates, such as catalytic beds and building materials.

High-order solvers for space-fractional differential equations with Riesz derivative

This paper proposes the computational approach for fractional-in-space reaction-diffusion equation, which is obtained by replacing the space second-order derivative in classical reaction-diffusion equation with the Riesz fractional derivative of order \begin{document}$ α $\end{document} in \begin{document}$ (0, 2] $\end{document} . The proposed numerical scheme for space fractional reaction-diffusion equations is based on the finite difference and Fourier spectral approximation methods. The paper utilizes a range of higher-order time stepping solvers which exhibit third-order accuracy in the time domain and spectral accuracy in the spatial domain to solve some fractional-in-space reaction-diffusion equations. The numerical experiment shows that the third-order ETD3RK scheme outshines its third-order counterparts, taking into account the computational time and accuracy. Applicability of the proposed methods is further tested with a higher dimensional system. Numerical simulation results show that pattern formation process in the classical sense is the same as in fractional scenarios.

Numerical analysis and pattern formation process for space-fractional superdiffusive systems

In this paper, we consider the numerical solution of fractional-in-space reaction-diffusion equation, which is obtained from the classical reaction-diffusion equation by replacing the second-order spatial derivative with a fractional derivative of order \begin{document}$ α∈(1, 2] $\end{document} . We adopt a class of second-order approximations, based on the weighted and shifted Grunwald difference operators in Riemann-Liouville sense to numerically simulate two multicomponent systems with fractional-order in higher dimensions. The efficiency and accuracy of the numerical schemes are justified by reporting the norm infinity and norm relative errors as well as their convergence. The complexity of the dynamics in the equation is theoretically discussed by conducting its local and global stability analysis and Numerical experiments are performed to back-up the theoretical claims.

A multiple scale pattern formation cascade in reaction-diffusion systems of activator-inhibitor type

A family of singular limits of reaction-diffusion systems of activator-inhibitor type in which stable stationary sharp-interface patterns may form is investigated. For concreteness, the analysis is performed for the FitzHugh-Nagumo model on a suitably rescaled bounded domain in \mathbb R^N , with N \geq 2 . It is shown that when the system is sufficiently close to the limit the dynamics starting from the appropriate smooth initial data breaks down into five distinct stages on well-separated time scales, each of which can be approximated by a suitable reduced problem. The analysis allows to follow fully the progressive refinement of spatio-temporal patterns forming in the systems under consideration and provides a framework for understanding the pattern formation scenarios in a large class of physical, chemical, and biological systems modeled by the considered class of reaction-diffusion equations.

Conditional Lie-Bäcklund Symmetry Reductions and Exact Solutions of a Class of Reaction-Diffusion Equations

The method of conditional Lie-Bäcklund symmetry is applied to solve a class of reaction-diffusion equations ut+uxx+Qxux2+Pxu+Rx=0, which have wide range of applications in physics, engineering, chemistry, biology, and financial mathematics theory. The resulting equations are either solved exactly or reduced to some finite-dimensional dynamical systems. The exact solutions obtained in concrete examples possess the extended forms of the separation of variables.

Stability of a Planar Front in a Class of Reaction-Diffusion Systems

We study the planar front solution for a class of reaction diffusion equations in multidimensional space in the case when the essential spectrum of the linearization in the direction of the front t...

Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense

In this paper, we consider a numerical approach for fourth-order time fractional partial differential equation. This equation is obtained from the classical reaction-diffusion equation by replacing the first-order time derivative with the Atangana-Baleanu fractional derivative in Riemann-Liouville sense with the Mittag-Leffler law kernel, and the first, second, and fourth order space derivatives with the fourth-order central difference schemes. We also suggest the Fourier spectral method as an alternate approach to finite difference. We employ Plais Fourier method to study the question of finite-time singularity formation in the one-dimensional problem on a periodic domain. Our bifurcation analysis result shows the relationship between the blow-up and stability of the steady periodic solutions. Numerical experiments are given to validate the effectiveness of the proposed methods.