Traveling Wave Fronts Research Articles

Global stability of traveling wave fronts for a population dynamics model with quiescent stage and delay

This paper is concerned with the globally exponential stability of traveling wave fronts for a class of population dynamics model with quiescent stage and delay. First, we establish the comparison principle of solutions for the population dynamics model. Then, by the weighted energy method combining comparison principle, the globally exponential stability of traveling wave fronts of the population dynamics model under the quasi-monotonicity conditions is established, which extends the results of nonlocal scalar equations to the system.

Existence of traveling wave fronts of delayed Fisher-type equations with degenerate nonlinearities

In this paper, two different kinds of degenerate n-degree Fisher-type equations with delays are considered. Due to the difference of the reaction terms, the existence of traveling front are proved by different methods. More precisely, when the reaction term satisfies the weak quasimonotonicity condition, for c>2, the existence result is given by the super-sub solution method and the fixed point theorem. Then for c∗<c⩽2, where c∗ is the minimal speed of degenerate p-degree Fisher-type equations without delays, the existence result is proved by the perturbation method and the implicit function theory. For the other type reaction term, we apply the monotone iteration method and the super-sub solution method to obtain the existence conclusion.

Traveling waves for reaction-diffusion PDE coupled to difference equation with nonlocal dispersal term and time delay

We consider a class of biological models represented by a system composed of reactiondiffusion PDE coupled with difference equations (renewal equations) in n-dimensional space, with nonlocal dispersal terms and implicit time delays. The difference equation generally arises, by means of the method of characteristics, from an age-structured partial differential system. Using upper and lower solutions, we study the existence of monotonic planar traveling wave fronts connecting the extinction state to the uniform positive state. The corresponding minimum wave speed is also obtained. In addition, we investigate the effect of the parameters on this minimum wave speed and we give a detailed analysis of its asymptotic behavior.

Spatio-temporal analysis of coinfection using wavefronts of Escherichia coli O157: H7 in a dairy cattle farm

This study proposes a new methodology to analyze the impacts of host and pathogen dispersal on coinfection dynamics using the example of Escherichia coli O157:H7 in a dairy farm. A multi-strain Susceptible–Infected–Susceptible model is extended to a Reaction–Diffusion (RD) coinfection model of E. coli O157:H7 transmission, which includes intermittent shedding and pathogen growth in the environment. Analysis of the new RD coinfection model consists of existence and stability of constant equilibria, calculation of the basic reproduction number, and existence of traveling and stationary wave solutions. A stationary wave solution of the RD model defines a nonconstant endemic equilibrium. Whereas a traveling wavefront represents an epidemic wave of infection passing through the farm with a specific speed, direction, and amplitude. The numerical simulations of the RD model demonstrate stable traveling and stationary wavefronts of the RD coinfection model. The stationary wavefront connects two constant coexistence equilibria and the traveling wavefront represents the gradual establishment of an endemic coexistence equilibrium in the dairy farm. The significance of this study lies in utilizing the nonlinear wave theory to analyze dynamics of coinfection in a dairy farm both with respect to location and time. Hence, in addition to Turing theory, the nonlinear wave theory can be used to determine the likelihood of acquiring the infection based on the location of cattle at each specific time.

Traveling Waves Solutions for Delayed Temporally Discrete Non-Local Reaction-Diffusion Equation

This paper deals with the existence of traveling wave solutions to a delayed temporally discrete non-local reaction diffusion equation model, which has been derived recently for a single species with age structure. When the birth function satisfies monotonic condition, we obtained the traveling wavefront by using upper and lower solution methods together with monotonic iteration techniques. Otherwise, without the monotonicity assumption for birth function, we constructed two auxiliary equations. By means of the traveling wavefronts of the auxiliary equations, using the Schauder’ fixed point theorem, we proved the existence of a traveling wave solution to the equation under consideration with speed c&gt;c*, where c*&gt;0 is some constant. We found that the delayed temporally discrete non-local reaction diffusion equation possesses the dynamical consistency with its time continuous counterpart at least in the sense of the existence of traveling wave solutions.

Travelling wave fronts of Lotka-Volterra reaction-diffusion system in the weak competition case

This paper is concerned with spreading phenomena of the classical two-species Lotka-Volterra reaction-diffusion system in the weak competition case. More precisely, some new sufficient conditions on the linear or nonlinear speed selection of the minimal wave speed of travelling wave fronts, which connect one half-positive equilibrium and one positive equilibrium, have been given via constructing types of super-sub solutions. Moreover, these conditions for the linear or nonlinear determinacy are quite different from that of the minimal wave speeds of travelling wave fronts connecting other equilibria of Lotka-Volterra competition model. In addition, based on the weighted energy method, we give the global exponential stability of such solutions with large speed $c$. Specially, when the competition rate exerted on one species converges to zero, then for any $c&gt;c_0$, where $c_0$ is the critical speed, the travelling wave front with the speed $c$ is globally exponentially stable.

An elementary proof of the existence of monotone traveling waves solutions in a generalized Klein–Gordon equation

We analyze the existence and uniqueness of monotone traveling wavefront for a generalized nonlinear Klein–Gordon model using classical arguments of ordinary differential equations, with V(x) a potentials family containing the ϕ‐four potential and the sine‐Gordon‐type potential . Also for these specific potentials, we give estimations of their monotone kink and anti‐kink solutions.

Propagation dynamics of an anisotropic nonlocal dispersal equation with delayed nonlocal response

This paper focuses on the critical wave speed and the traveling wave fronts for a general anisotropic nonlocal dispersal equation with delayed nonlocal response. Unlike most of the previous study, both of the dispersal kernel function and the nonlocal response function are asymmetric in such an equation. By analyzing the properties of eigenfunction, we first discuss the sign of the critical wave speed and the influence of the asymmetry of the two kernel functions on the critical wave speed. Then under the monostable condition, by using super- and subsolution and monotone iteration method, we obtain the existence of nondecreasing traveling wave solution for c>c∗ and a limiting argument for c=c∗. Moreover, we depict the asymptotic behavior of the traveling wave and its derivative at minus infinity.

Wave Propagation for Reaction-Diffusion Equations on Infinite Random Trees

The asymptotic wave speed for FKPP type reaction-diffusion equations on a class of infinite random metric trees are considered. We show that a travelling wavefront emerges, provided that the reaction rate is large enough. The wavefront travels at a speed that can be quantified via a variational formula involving the random branching degrees $\vec{d}$ and the random branch lengths $\vec{\ell}$ of the tree. This speed is slower than that of the same equation on the real line $\mathbb{R}$, and we estimate this slow down in terms of $\vec{d}$ and $\vec{\ell}$. Our key idea is to project the Brownian motion on the tree onto a one-dimensional axis along the direction of the wave propagation. The projected process is a multi-skewed Brownian motion, introduced by Ramirez [Multi-skewed Brownian motion and diffusion in layered media, Proc. Am. Math. Soc., Vol. 139, No. 10, pp.3739-3752, 2011], with skewness and interface sets that encode the metric structure $(\vec{d}, \vec{\ell})$ of the tree. Combined with analytic arguments based on the Feynman-Kac formula, this idea connects our analysis of the wavefront propagation to the large deviations principle (LDP) of the multi-skewed Brownian motion with random skewness and random interface set. Our LDP analysis involves delicate estimates for an infinite product of $2\times 2$ random matrices parametrized by $\vec{d}$ and $\vec{\ell}$ and for hitting times of a random walk in random environment. By exhausting all possible shapes of the LDP rate function (action functional), the analytic arguments that bridge the LDP and the wave propagation overcome the random drift effect due to multi-skewness.

Data-driven modeling of nonlinear traveling waves.

Presented is a data-driven machine learning framework for modeling traveling wave spatiotemporal dynamics. The presented framework is based on the steadily propagating traveling wave ansatz, u(x,t)=U(ξ=x-ct+a). For known evolution equations, this coordinate transformation reduces governing partial differential equations to a set of coupled ordinary differential equations (ODEs) in the traveling wave coordinate ξ. Although traveling waves are readily observed in many physical systems, the underlying governing equations may be unknown. For these instances, the traveling wave dynamical system can be modeled empirically with neural ODEs. Presented are these ideas applied to several physical systems that admit traveling waves. Examples include traveling wave fronts, pulses, and wavetrains restricted to one-way wave propagation in a single spatial dimension. Last, applicability to real-world physical systems is presented with an exploration of data-driven modeling of rotating detonation waves.

Traveling wavefronts for density‐dependent diffusion reaction convection equation with time delay

We are concerned with the existence of traveling waves for density‐dependent diffusion reaction nonlinear convection equations with small time delay. We first study the existence and uniqueness of smooth and sharp‐type traveling wavefront solution of the wave speed c ≥ c∗ for equation without time delay, where c∗ is the minimal wave speed. Meanwhile, we construct a variational principle for c∗ from which the upper bound of c∗ is determined, while the lower bound of c∗ is attained by using the phase plane analysis. Then, we obtain the existence of smooth traveling wavefront solution for equation with small time delay for c &gt; c∗ by applying the perturbation method and implicit function theorem.

Stability of traveling wave fronts for nonlocal diffusive systems

This paper is concerned with stability of traveling wave fronts for nonlocal diffusive system. We adopt [Formula: see text]-weighted, [Formula: see text]- and [Formula: see text]-energy estimates for the perturbation systems, and show that all solutions of the Cauchy problem for the considered systems converge exponentially to traveling wave fronts provided that the initial perturbations around the traveling wave fronts belong to a suitable weighted Sobolev spaces.

Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction

&lt;p style='text-indent:20px;'&gt;This paper is concerned with the traveling wave fronts for a lattice dynamical system with global interaction, which arises in a single species in a 2D patchy environment with infinite number of patches connected locally by diffusion and global interaction by delay. We prove that all non-critical traveling wave fronts are globally exponentially stable in time, and the critical traveling wave fronts are globally algebraically stable by the weighted energy method combined with the comparison principle and the discrete Fourier transform.&lt;/p&gt;

Traveling wave fronts in a diffusive and competitive Lotka-Volterra system

&lt;p style='text-indent:20px;'&gt;In this paper, we consider a two-species competitive and diffusive system with nonlocal delays. We investigate the existence of traveling wave fronts of the system by employing linear chain techniques and geometric singular perturbation theory. The existence of the traveling wave fronts analogous to a bistable wavefront for a single species is proved by transforming the system with nonlocal delays to a six-dimensional system without delay.&lt;/p&gt;

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF TRAVELING WAVES IN A HOST-VECTOR EPIDEMIC MODEL

In this paper, we are concerned with a diffusive host-vector epidemic model with a nonlocal spatiotemporal interaction. When the delay kernel takes some special form, by employing linear chain techniques and geometric singular perturbation theory, we establish the existence of travelling front solutions connecting the two spatially uniform steady states for sufficiently small delays. Furthermore, by employing standard asymptotic theory, we also obtain the asymptotic behavior of traveling wave fronts of this model.

Modeling thermodynamic trends of rotating detonation engines

The formation of a number of co- and counter-rotating coherent combustion wave fronts is the hallmark feature of the Rotating Detonation Engine (RDE). The engineering implications of wave topology are not well understood nor quantified, especially with respect to parametric changes in combustor geometry, propellant chemistry, and injection and mixing schemes. In this article, a modeling framework that relates the time and spatial scales of the RDE to engineering performance metrics is developed and presented. The model is built under assumptions of backpressure-insensitivity and nominally choked gaseous propellant injection. The Euler equations of inviscid, compressible fluid flow in one dimension are adapted to model the combustion wave dynamics along the circumference of an annular-type RDE. These adaptations provide the necessary mass and energy input and output channels to shape the traveling wave fronts and decaying tails. The associated unit processes of injection, mixing, combustion, and exhaust are all assigned representative time scales necessary for successful wave propagation. We find that the separation, or lack, of these time scales is responsible for the behavior of the system, including wave co- and counter-propagation and bifurcations between these regimes and wave counts. Furthermore, as there is no imposition of wave topology, the model output is used to estimate the net available mechanical work output and thermodynamic efficiency from the closed trajectories through pressure–volume and temperature–entropy spaces. These metrics are investigated with respect to variation in the characteristic scales for the RDE unit physical processes.

Wave propagation and its stability for a class of discrete diffusion systems

This paper is devoted to study the wave propagation and its stability for a class of two-component discrete diffusive systems. We first establish the existence of positive monotone monostable traveling wave fronts. Then, applying the techniques of weighted energy method and the comparison principle, we show that all solutions of the Cauchy problem for the discrete diffusive systems converge exponentially to the traveling wave fronts when the initial perturbations around the wave fronts lie in a suitable weighted Sobolev space. Our main results can be extended to more general discrete diffusive systems. We also apply them to the discrete epidemic model with the Holling-II type and Richer type effects.

FDTD analysis of transient fault induced travelling-wave propagation for multi-branch distribution networks

Many methods are available to analyze the process of the travelling-wave propagation. Among these methods, the Finite Difference Time Domain (FDTD) method has a distinct advantage in calculating dynamic process of the travelling wave propagation in the time domain and is thus applied to the field of power system protection for researching transient fault induced travelling-wave propagation. The novelty of this paper is that the attenuation law of the traveling wave signal affected by the fork junction in the multi-branch distribution network is summarized and the cause of failure in the fault location based on the incipient travelling wave front method in distribution networks is found.

Some Qualitative Properties of Traveling Wave Fronts of Nonlocal Diffusive Competition-Cooperation Systems of Three Species with Delays

This paper is concerned with nonlocal diffusion systems of three species with delays. By modified version of Ikehara’s theorem, we prove that the traveling wave fronts of such system decay exponentially at negative infinity, and one component of such solutions also decays exponentially at positive infinity. In order to obtain more information of the asymptotic behavior of such solutions at positive infinity, for the special kernels, we discuss the asymptotic behavior of such solutions of such system without delays, via the stable manifold theorem. In addition, by using the sliding method, the strict monotonicity and uniqueness of traveling wave fronts are also obtained.

Properties of Traveling Wave Fronts for Three Species Lotka–Volterra System

The purpose of this paper is to investigate properties of traveling wave fronts for three species Lotka–Volterra system: the asymptotic behavior and uniqueness. Applying the Ikehara’s theorem, we determine the exponential rates of traveling wave fronts at the negative infinity. We further investigate the uniqueness of traveling wave fronts with the help of the sliding method.