Traveling Wave Fronts Research Articles

Effect of stochastic perturbations for front propagation in Kolmogorov Petrovskii Piscunov equations

This article considers equations of Kolmogorov Petrovskii Piscunov type in one space dimension, with stochastic perturbation: ∂tu=κ2uxx+u(1−u)dt+ϵu∂tζu0(x)=1(−∞,−1Nlog2)(x)+12e−Nx1[−1Nlog2,+∞)(x)where the stochastic differential is taken in the sense of Itô and ζ is a Gaussian random field satisfying Eζ=0 and Eζ(s,x)ζ(t,y)=(s∧t)Γ(x−y). Two situations are considered: firstly, ζ is simply a standard Wiener process (i.e. Γ≡1): secondly, Γ∈C∞(R) with lim|z|→+∞|Γ(z)|=0.The results are as follows: in the first situation (standard Wiener process: i.e. Γ(x)≡1), there is a non-degenerate travelling wave front if and only if ϵ22<1, with asymptotic wave speed max2κ(1−ϵ22),1N(1−ϵ22)+κN21{N<2κ(1−ϵ22)}; the noise slows the wave speed. If the stochastic integral is taken instead in the sense of Stratonovich, then the asymptotic wave speed is the classical McKean wave speed and does not depend on ϵ.In the second situation (noise with spatial covariance which decays to 0 at ±∞, stochastic integral taken in the sense of Itô), a travelling front can be defined for all ϵ>0. Its average asymptotic speed does not depend on ϵ and is the classical wave speed of the unperturbed KPP equation.

Nonlinear Stability of Traveling Waves in a Monostable Epidemic Model with Delay

This paper is concerned with the nonlinear stability of traveling waves of a delayed monostable epidemic model with quasi-monotone condition. We prove that the traveling wave front is exponentially stable by means of the weighted-energy method and the comparison principle to perturbation in some exponentially weighted $L^{\infty}$ spaces, when the difference between initial data and traveling wave front decays exponentially as $x \to -\infty$, but the initial data can be suitable large in other locations. Finally, we present two examples to support our theoretical results.

Traveling wave front for partial neutral differential equations

Traveling wave front for partial neutral differential equations

Global stability of traveling wave fronts for a reaction–diffusion system with a quiescent stage on a one-dimensional spatial lattice

ABSTRACTThis paper is concerned with the stability of traveling wave fronts for a coupled system of non-local delayed lattice differential equations with a quiescent stage. It shows that under certain conditions all non-critical traveling wave fronts are globally exponentially stable, and critical ftraveling wave fronts are globally algebraically stable by applying the weighted energy method and the semi-discrete Fourier transform.

Front-like entire solutionsfor a delayed nonlocal dispersalequation with convolution typebistable nonlinearity

This paper is concerned with front-like entire solutions of a delayed nonlocal dispersal equation with convolution type bistable nonlinearity. Here, a solution defined for all $(x, t)\in \mathbb {R}^2$ is an entire solution. It is known that the equation has an increasing traveling wavefront with nonzero wave speed under some reasonable conditions. We first give the asymptotic behavior of traveling wavefronts at infinity. Then, by the comparison argument and sub-super-solutions method, we construct new types of entire solutions other than traveling wavefronts and equilibrium solutions of the equation, which behave like two increasing traveling wavefronts propagating from both sides of the $x$-axis and annihilate at a finite time. Finally, various qualitative properties of the entire solutions are also investigated.

Existence, uniqueness and asymptotic behavior of traveling wave fronts for a generalized Fisher equation with nonlocal delay

This paper is concerned with existence, uniqueness and asymptotic behavior of traveling wave fronts for a generalized Fisher equation with nonlocal delay. The existence of traveling wave fronts is established by linear chain trick and geometric singular perturbation theory. The strategy is to reformulate the problem as the existence of a heteroclinic connection in R4. The problem is then tackled by using Fenichel’s invariant manifold theory. The asymptotic behavior and uniqueness of traveling wave fronts are also obtained by using standard asymptotic theory and sliding method.

Monotonicity, uniqueness, and stability of traveling waves in a nonlocal reaction‐diffusion system with delay

The purpose of this paper is to study the traveling wave solutions of a nonlocal reaction‐diffusion system with delay arising from the spread of an epidemic by oral‐faecal transmission. Under monostable and quasimonotone it is well known that the system has a minimal wave speed c* of traveling wave fronts. In this paper, we first prove the monotonicity and uniqueness of traveling waves with speed c⩾c∗. Then we show that the traveling wave fronts with speed c&gt;c∗ are exponentially asymptotically stable.

Existence and stability of traveling wave fronts for a reaction‐diffusion system with spatio‐temporal nonlocal effect

AbstractIn this paper, traveling wave fronts for a two dimensional quasi‐monotone reaction‐diffusion system with spatio‐temporal delays are investigated. We use the upper‐lower solution method and the fixed‐point theorem to establish the existence of traveling waves for the system, and the technique of comparison principle combined with the weighted energy function to study the global exponential stability of traveling waves of the equations with large initial perturbation. The initial perturbation around the traveling waves decays exponentially as , but it can be arbitrarily large in other locations.

Spatial Dynamics of Multilayer Cellular Neural Networks

The purpose of this work is to study the spatial dynamics of one-dimensional multilayer cellular neural networks. We first establish the existence of rightward and leftward spreading speeds of the model. Then we show that the spreading speeds coincide with the minimum wave speeds of the traveling wave fronts in the right and left directions. Moreover, we obtain the asymptotic behavior of the traveling wave fronts when the wave speeds are positive and greater than the spreading speeds. According to the asymptotic behavior and using various kinds of comparison theorems, some front-like entire solutions are constructed by combining the rightward and leftward traveling wave fronts with different speeds and a spatially homogeneous solution of the model. Finally, various qualitative features of such entire solutions are investigated.

Evans functions and bifurcations of nonlinear waves of some nonlinear reaction diffusion equations

The main purposes of this paper are to accomplish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations ut=uxx+α[βH(u−θ)−u]−w, wt=ε(u−γw) and to establish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation ut=uxx+α[βH(u−θ)−u], under different conditions on the model constants.To establish the bifurcation for the system, we will study the existence and instability of a standing pulse solution if 0<2(1+αγ)θ<αβγ; the existence and stability of two standing wave fronts if 2(1+αγ)θ=αβγ and γ2ε>1; the existence and instability of two standing wave fronts if 2(1+αγ)θ=αβγ and 0<γ2ε<1; the existence and instability of an upside down standing pulse solution if 0<(1+αγ)θ<αβγ<2(1+αγ)θ. To establish the bifurcation for the scalar equation, we will study the existence and stability of a traveling wave front as well as the existence and instability of a standing pulse solution if 0<2θ<β; the existence and stability of two standing wave fronts if 2θ=β; the existence and stability of a traveling wave front as well as the existence and instability of an upside down standing pulse solution if 0<θ<β<2θ. By the way, we will also study the existence and stability of a traveling wave back of the nonlinear scalar reaction diffusion equation ut=uxx+α[βH(u−θ)−u]−w0, where w0=α(β−2θ)>0 is a positive constant, if 0<2θ<β.To achieve the main goals, we will make complete use of the special structures of the model equations and we will construct Evans functions and apply them to study the eigenvalues and eigenfunctions of several eigenvalue problems associated with several linear differential operators. It turns out that a complex number λ0 is an eigenvalue of the linear differential operator, if and only if λ0 is a zero of the Evans function. The stability, instability and bifurcations of the nonlinear waves follow from the zeros of the Evans functions.A very important motivation to study the existence, stability, instability and bifurcations of the nonlinear waves is to study the existence and stability/instability of infinitely many fast/slow multiple traveling pulse solutions of the nonlinear system of reaction diffusion equations. The existence and stability of infinitely many fast multiple traveling pulse solutions are of great interests in mathematical neuroscience.

Existence and asymptotic behavior of traveling wave fronts for a food-limited population model with spatio-temporal delay

This paper is concerned with the existence and asymptotic behavior of traveling wave fronts for a food-limited population model with spatio-temporal delay. Using geometric singular perturbation theory and Fredholm alternative, we establish the existence of traveling wave fronts of this model for sufficiently small time delay. The approach is to reformulate the problem as an existence question for a heteroclinic connection in \({\mathbb {R}}^6\). Furthermore, employing standard asymptotic theory, we obtain the asymptotic behavior of traveling wave fronts of this model for the first time.

Bistable front dynamics in a contractile medium: Travelling wave fronts and cortical advection define stable zones of RhoA signaling at epithelial adherens junctions.

Mechanical coherence of cell layers is essential for epithelia to function as tissue barriers and to control active tissue dynamics during morphogenesis. RhoA signaling at adherens junctions plays a key role in this process by coupling cadherin-based cell-cell adhesion together with actomyosin contractility. Here we propose and analyze a mathematical model representing core interactions involved in the spatial localization of junctional RhoA signaling. We demonstrate how the interplay between biochemical signaling through positive feedback, combined with diffusion on the cell membrane and mechanical forces generated in the cortex, can determine the spatial distribution of RhoA signaling at cell-cell junctions. This dynamical mechanism relies on the balance between a propagating bistable signal that is opposed by an advective flow generated by an actomyosin stress gradient. Experimental observations on the behavior of the system when contractility is inhibited are in qualitative agreement with the predictions of the model.

Traveling wave fronts in a delayed lattice competitive system

This paper is concerned with the existence, asymptotic behavior, strict monotonicity, and uniqueness of traveling wave fronts connecting two half-positive equilibria in a delayed lattice competitive system. We first prove the existence of traveling wave fronts by constructing upper and lower solutions and Schauder’s fixed point theorem, and then, for sufficiently small intraspecific competitive delays, prove that these traveling wave fronts decay exponentially at both infinities. Furthermore, for system without intraspecific competitive delays, the strict monotonicity and uniqueness of traveling wave fronts are established by means of the sliding method. In addition, we give the exact decay rate of the stronger competitor under some technique conditions by appealing to uniqueness.

Traveling wavefronts for a reaction-diffusion-chemotaxis model with volume-filling effect

In this paper, we study the propagation of the pattern for a reaction-diffusionchemotaxis model. By using a weakly nonlinear analysis with multiple temporal and spatial scales, we establish the amplitude equations for the patterns, which show that a local perturbation at the constant steady state is spread over the whole domain in the form of a traveling wavefront. The simulations demonstrate that the amplitude equations capture the evolution of the exact patterns obtained by numerically solving the considered system.

Uniqueness and exponential stability of traveling wave fronts for a multi-type SIS nonlocal epidemic model

This paper is concerned with the traveling wave fronts of a multi-type SIS nonlocal epidemic model. From Weng and Zhao (2006), we know that there exists a critical wave speed c∗>0 such that a traveling wave front exists if and only if its wave speed is above c∗. In this paper, we first prove the uniqueness of certain traveling wave fronts with non-critical wave speed. Then, we show that all non-critical traveling wave fronts are asymptotically exponentially stable. The exponential convergent rate is also obtained.

Influence of resistant varieties on the speed of propagation of simple epidemics in crops

We determine the speeds of traveling wave fronts for simple epidemics in crops that contain a mixture of resistant and non-resistant hosts. For our model the wave speeds are found to be proportional to the fraction of resistant individuals. The conclusions are reached through the application of the linear conjecture, and verified by comparing the results with numerical solutions of the non-linear model.

Traveling waves and entire solutions for an epidemic model with asymmetric dispersal

This paper is concerned with traveling waves and entire solutions of one epidemic model with asymmetric dispersal kernel function arising from the spread of an epidemic by oral-faecal transmission. The asymmetry of the kernel function will have an influence on two aspects: (ⅰ) the minimal wave speed of traveling wave fronts may be nonpositive, but we give a new restrictive condition on the kernel function to guarantee it is positive; (ⅱ) the two traveling wave solutions with the same speed spreading from right and left of $x$-axis may be different in shape, which further makes that the entire solutions with five or four parameters may be asymmetric and the entire solutions with three parameters increasing in $x$ may be different from those decreasing in $x$ in shape. As for traveling wave solutions, we get the existence, asymptotic behavior and uniqueness of the two traveling wave solutions spreading from right and left of $x$-axis, respectively. We further construct three new entire solutions with five, four or three parameters. Two comparison principles also be established.

Traveling waves in a coupled reaction–diffusion and difference model of hematopoiesis

The formation and development of blood cells is a very complex process, called hematopoiesis. This process involves a small population of cells called hematopoietic stem cells (HSCs). The HSCs are undifferentiated cells, located in the bone marrow before they become mature blood cells and enter the blood stream. They have a unique ability to produce either similar cells (self-renewal), or cells engaged in one of different lineages of blood cells: red blood cells, white cells and platelets (differentiation). The HSCs can be either in a proliferating or in a quiescent phase. In this paper, we distinguish between dividing cells that enter directly to the quiescent phase and dividing cells that return to the proliferating phase to divide again. We propose a mathematical model describing the dynamics of HSC population, taking into account their spatial distribution. The resulting model is a coupled reaction–diffusion equation and difference equation with delay. We study the existence of monotone traveling wave fronts and the asymptotic speed of spread.

On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ&lt;sup&gt;3&lt;/sup&gt;

This paper is concerned with the following two-species Lotka-Volterra competition-diffusion system in the three-dimensional spatial space {partial derivative/partial derivative tu(1) (x, t) = Delta u(1) (x, t) + u(1) (x, t) [1 - u(1) (x, t) - k(1)u(2) (x, t)] , {partial derivative/partial derivative tu(2) (x, t) = d Delta u(2) (x, t) + ru(2) (x, t) [1 - u(2) (x, t) - k(2)u(1) (x, t)] , where x is an element of R-3 and t > 0. For the bistable case, namely k(1) , k(2) > 1, it is well known that the system admits a one-dimensional monotone traveling front Phi(x + ct) = (Phi(1) (x + ct), Phi(2) (x + ct)) connecting two stable equilibria E-u = (1, 0) and E-v = (0, 1), where c is an element of R is the unique wave speed. Recently, two-dimensional V-shaped fronts and high-dimensional pyramidal traveling fronts have been studied under the assumption c > 0. In this paper it is shown that for any s > c > 0, the system admits axisymmetric traveling fronts psi(x', x(3) + st) = Phi(1)(x ', x(3) + st), Phi(2)(x ', x(3) + st) in R-3 connecting E-u = (1, 0) and E-v = (0, 1), where x' is an element of R-2. Here an axisymmetric traveling front means a traveling front which is axially symmetric with respect to the x 3-axis. Moreover, some important qualitative properties of the axisymmetric traveling fronts are given. When s tends to c, it is proven that the axisymmetric traveling fronts converge locally uniformly to planar traveling wave fronts in R-3. The existence of axisymmetric traveling fronts is obtained by constructing a sequence of pyramidal traveling fronts and taking its limit. The qualitative properties are established by using the comparison principle and appealing to the asymptotic speed of propagation for the resulting system. Finally, the nonexistence of axisymmetric traveling fronts with concave/convex level set is discussed.

Wave propagation in an infectious disease model

This paper is devoted to the study of the wave propagation in a reaction-convection infectious disease model with a spatio-temporal delay. Previous numerical studies have demonstrated the existence of traveling wave fronts for the system and obtained a critical value c⁎, which is the minimal wave speed of the traveling waves. In the present paper, we provide a complete and rigorous proof. To overcome the difficulty due to the lack of monotonicity for the system, we construct a pair of upper and lower solutions, and then apply the Schauder fixed point theorem to establish the existence of a nonnegative solution for the wave equation on a bounded interval. Moreover, we use a limiting argument and in turn generate the solution on the unbounded interval R. In particular, by constructing a suitable Lyapunov functional, we further show that the traveling wave solution converges to the epidemic equilibrium point as t=+∞.