Traveling Wave Fronts Research Articles

Traveling wave solutions for reaction–diffusion systems

This paper is concerned with traveling waves of reaction–diffusion systems. The definition of coupled quasi-upper and quasi-lower solutions is introduced for systems with mixed quasimonotone functions, and the definition of ordered quasi-upper and quasi-lower solutions is also given for systems with quasimonotone nondecreasing functions. By the monotone iteration method, it is shown that if the system has a pair of coupled quasi-upper and quasi-lower solutions, then there exists at least a traveling wave solution. Moreover, if the system has a pair of ordered quasi-upper and quasi-lower solutions, then there exists at least a traveling wavefront. As an application we consider the delayed system of a mutualistic model.

Nonlinear stability of travelling wave fronts for delayed reaction diffusion equations

This paper deals with the nonlinear stability of travelling wave fronts for delayed reaction diffusion equations. We prove that the travelling wave fronts are exponentially stable to perturbations in some exponentially weighted L∞ spaces, and obtain the time decay rates of by the weighted energy estimate.

Existence of travelling waves in a diffusive vector disease model with distributed delay

This paper is concerned with a diffusive vector disease model with distributed delay. The existence of travelling wave front solutions connecting the zero equilibrium and the positive equilibrium is established by using an iterative technique and a nonstandard ordering for the set of profiles of the corresponding wave system recently developed by Wang, Li and Ruan (Travelling wave fronts in reactiondiffusion systems with spatio-temporal delays. J. Differential Equations 222 (2006), 185---232). The existence of travelling front solutions shows that there is a moving zone of transition from the disease free state to the infective state.

Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications

This paper is concerned with monotone traveling wave solutions of reaction-diffusion systems with spatio-temporal delay. Our approach is to use a new monotone iteration scheme based on a lower solution in the set of the profiles. The smoothness of upper and lower solutions is not required in this paper. We will apply our results to Nicholson's blowflies systems with non-monotone birth functions and show that the systems admit traveling wave solutions connecting two spatially homogeneous equilibria and the wave shape is monotone. Due to the biological realism, the positivity of the monotone traveling wave solutions can be directly obtained by the construction of suitable upper-lower solutions.

Entire solutions for a two-component competition system in a lattice

We study entire solutions of a two-component competition system with Lotka-Volterra type nonlinearity in a lattice. It is known that this system has traveling wave front solutions and enjoys comparison principle. Based on these solutions, we construct some new entire solutions which behave as two traveling wave fronts moving towards each other from both sides of $x$-axis.

On the Form of Smooth-Front Travelling Waves in a Reaction-Diffusion Equation with Degenerate Nonlinear Diffusion

Reaction-diffusion equations with degenerate nonlinear diffusion are in widespread use as models of biological phenomena. This paper begins with a survey of applications to ecology, cell biology and bacterial colony patterns. The author then reviews mathematical results on the existence of travelling wave front solutions of these equations, and their generation from given initial data. A detailed study is then presented of the form of smooth-front waves with speeds close to that of the (unique) sharp-front solution, for the particular equation ut = (uux)x + u(1 i u). Using singular perturbation theory, the author derives an asymptotic approximation to the wave, which gives valuable information about the structure of smooth-front solutions. The approximation compares well with numerical results.

Wave fronts in neuronal fields with nonlocal post-synaptic axonal connections and delayed nonlocal feedback connections

We consider a neuronal network model with both axonal connections (in the form of synaptic coupling) and delayed non-local feedback connections. The kernel in the feedback channel is assumed to be a standard non-local one, while for the kernel in the synaptic coupling, four types are considered. The main concern is the existence of travelling wave front. By employing the speed index function, we are able to obtain the existence of a travelling wave front for each of these four types within certain range of model parameters. We are also able to describe how the feedback coupling strength and the magnitude of the delay affect the wave speed. Some particular kernel functions for these four cases are chosen to numerically demonstrate the theoretical results.

Existence, uniqueness and stability of traveling wave fronts of discrete quasi-linear equations with delay

This paper is concerned with the existence, uniqueness and asymptotically stability of traveling wave fronts ofdiscrete quasi-linear equations with delay. We first establish the existence of traveling wave fronts by usingthe super-sub solution and monotone iteration technique. Then we show that the traveling wave front is unique upto a translation. At last, we employ the comparison principle and the squeezing technique to prove that thetraveling wave front is globally asymptotic stable with phase shift.

A method for direct measurement of ion mobilities using a travelling wave ion guide

The relatively recent introduction of a commercial ion mobility-mass spectrometry instrument has provided access to ion mobility for many investigators for the first time. The mechanism of mobility separation in this instrument differs from classical drift tubes in that travelling voltage waves are used, which has necessitated calibration of the device when collision cross-section (CCS) data are required. Here we present a study into a mode of operation of the travelling wave mobility separator which enables direct determination of mobility and hence CCS values. It is shown that for fixed travelling wave pulse amplitudes and velocities, a minimum mobility is required for an ion species to fully surf on a single wave through the mobility device. For a species with the minimum mobility, the trajectory on the travelling wave front is unique and reproducible. SIMION modelling shows the electric potential profile of the travelling wave to be approximately Gaussian and this is used to establish the field experienced by an ion species on the wave front, ultimately allowing determination of mobility values. With the travelling wave mobility device operating with Helium at 2.5 mbar and equivalent wave velocities of 200–300 m/s, wave amplitudes between 7.9 and 16.2 V were required to establish the mobilities of various singly and multiply charged species ranging in CCS from 147 to 3815 Å 2. We show that using this method, CCS values can be obtained which are within 5% of the results obtained using drift tubes.

Numerical simulation of linear and nonlinear waves in hypoelastic solids by the CESE method

In this paper, we present a comprehensive model for the numerical simulation of linear and nonlinear elastic waves in slender rods. The mathematical model, based on conservation of mass, conservation of linear momentum, and a hypoelastic constitutive equation, consists of a set of three first-order, fully-coupled, nonlinear, hyperbolic partial differential equations. Three forms of the model equations are presented: the conservative form, the non-conservative form, and the diagonal form. The conservative form is solved numerically using the Conservation Element and Solution Element (CESE) method, a highly-accurate explicit space–time finite-volume scheme for solving nonlinear hyperbolic systems. To demonstrate this numerical approach, two elastic wave problems are directly calculated: (i) resonant standing waves arising from a time-harmonic external axial load and (ii) propagating compression waves arising from a bi-material collinear impact. Simulations of the resonance problem illustrate a linear-to-nonlinear evolution of the resonant vibrations, the emergence of super-harmonics of the forcing frequency, and the distribution of wave energy among multiple modes. For the bi-material impact problem, the CESE method successfully captures the sharp traveling wavefronts, wave reflection and transmission, and the different wave propagation speeds in each material.

Influence of sodium currents on speeds of traveling wave fronts in synaptically coupled neuronal networks

The main purpose of this paper is to apply mathematical analysis to investigate the influence of sodium currents on the speeds of traveling wave fronts. The authors use speed index functions to investigate the speeds of traveling wave fronts of some scalar integral differential equations arising from synaptically coupled neuronal networks. The mathematical model equation is u t + f ( u ) = α ∫ R K ( x − y ) H ( u ( y , t − 1 c | x − y | ) − θ ) d y , where 0 < c ≤ ∞ , α > 0 and θ > 0 are constants, satisfying the condition 0 < 2 f ( θ ) < α . The function f ( u ) represents sodium currents, the function K denotes synaptic coupling in a neuronal network, and the Heaviside step function H is defined by H ( u ) = 0 for all u < 0 , H ( 0 ) = 1 2 and H ( u ) = 1 for all u > 0 .

Front propagation for a two-dimensional periodic monostable lattice dynamical system

We study the traveling wave front solutions for a two-dimensional periodic lattice dynamical system with monostable nonlinearity. We first show that there is a minimal speed such that a traveling wave solution exists if and only if its speed is above this minimal speed. Then we prove that any wave profile is strictly monotone. Finally, we derive the convergence of discretized minimal speed to the continuous minimal speed.

Modeling Spatial Spread of Infectious Diseases with a Fixed Latent Period in a Spatially Continuous Domain

In this paper, with the assumptions that an infectious disease in a population has a fixed latent period and the latent individuals of the population may diffuse, we formulate an SIR model with a simple demographic structure for the population living in a spatially continuous environment. The model is given by a system of reaction-diffusion equations with a discrete delay accounting for the latency and a spatially non-local term caused by the mobility of the individuals during the latent period. We address the existence, uniqueness, and positivity of solution to the initial-value problem for this type of system. Moreover, we investigate the traveling wave fronts of the system and obtain a critical value c * which is a lower bound for the wave speed of the traveling wave fronts. Although we can not prove that this value is exactly the minimal wave speed, numeric simulations seem to suggest that it is. Furthermore, the simulations on the PDE model also suggest that the spread speed of the disease indeed coincides with c *. We also discuss how the model parameters affect c *.

Existence of traveling wavefronts of discrete reaction-diffusion equations with delay

In this paper, we investigate the temporally discrete reaction-diffusion with delay. By using Schauder’s fixed point theorem, we establish the existence of traveling wave fronts. The main result is applied to a delayed and discretely diffusive model for the population of Daphnia magna.

Spreading speeds and traveling wavefronts for second order integrodifference equations

This paper is concerned with the spreading speeds and traveling wavefronts for second order integrodifference equations. By introducing an auxiliary integrodifference system, the spreading speed is established for the integrodifference equation. It is shown that the spreading speed coincides with the minimal wave speed for monotonic traveling wavefronts. Furthermore, we prove that the traveling wavefronts are stable by applying the squeezing technique. Finally, we analyze the different effects of the delay term appearing in the integrodifference equation from the viewpoint of ecology.

Traveling wave fronts in a delayed population model of Daphnia magna

This paper deals with the traveling wave fronts of a delayed population model with nonlocal dispersal. By constructing proper upper and lower solutions, the existence of the traveling wave fronts is proved. In particular, we show such a traveling wave front is strictly monotone.

Asymptotic stability of traveling wave fronts in nonlocal reaction–diffusion equations with delay

This paper is concerned with the asymptotic stability of traveling wave fronts of a class of nonlocal reaction–diffusion equations with delay. Under monostable assumption, we prove that the traveling wave front is exponentially stable by means of the (technical) weighted energy method, when the initial perturbation around the wave is suitable small in a weighted norm. The exponential convergent rate is also obtained. Finally, we apply our results to some population models and obtain some new results, which recover, complement and/or improve a number of existing ones.

Traveling waves for delayed non-local diffusion equations with crossing-monostability

This paper is concerned with the traveling waves for a class of delayed non-local diffusion equations with crossing-monostability. Based on constructing two associated auxiliary delayed non-local diffusion equations with quasi-monotonicity and a profile set in a suitable Banach space using the traveling wave fronts of the auxiliary equations, the existence of traveling waves is proved by Schauder’s fixed point theorem. The result implies that the traveling waves of the delayed non-local diffusion equations with crossing-monostability are persistent for all values of the delay τ ⩾ 0 .

Existence, uniqueness and asymptotic behavior of traveling wave fronts for a vector disease model

Abstract This paper is concerned with the existence, uniqueness and asymptotic behavior of traveling wave fronts for a vector disease model. We first establish the existence of traveling wave fronts by using geometric singular perturbation theory. Then the asymptotic behavior and uniqueness of traveling wave fronts are obtained by using the standard asymptotic theory and sliding method. In addition, our method is also suitable to establish the uniqueness and asymptotic behavior of traveling wave fronts for a cooperative system.

Modeling the skin pattern of fishes

Complicated patterns showing various spatial scales have been obtained in the past by coupling Turing systems in such a way that the scales of the independent systems resonate. This produces superimposed patterns with different length scales. Here we propose a model consisting of two identical reaction-diffusion systems coupled together in such a way that one of them produces a simple Turing pattern of spots or stripes, and the other traveling wave fronts that eventually become stationary. The basic idea is to assume that one of the systems becomes fixed after some time and serves as a source of morphogens for the other system. This mechanism produces patterns very similar to the pigmentation patterns observed in different species of stingrays and other fishes. The biological mechanisms that support the realization of this model are discussed.