Wave Front Solutions Research Articles

Delay induced travelling wavefronts in a diffusive vector disease model with distributed delay

The existence of travelling wavefront solutions is established for a reaction diffusion equation with distributed delay, which models vector disease. Instead of geometric perturbation method, the approach used here is the upper---lower solution technique and monotone iteration recently developed by Wang et al. [J. Differ. Equations 222 (2006), pp. 185---232)] and the result shows that delay can indeed induce slow travelling wavefronts.

Wavefront solutions in diffusive Nicholson’s blowflies equation with nonlocal delay

This paper is concerned with the diffusive Nicholson’s blowflies equation with nonlocal delay incorporated as an integral convolution over the entire past time up to now and the whole one-dimensional spatial domain ℝ. Assume that the delay kernel is a strong generic kernel. By the linear chain techniques and the geometric singular perturbation theory, the existence of travelling front solutions is shown for small delay.

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.

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.

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.

Turing Patterns and Wavefronts for Reaction-Diffusion Systems in an Infinite Channel

This paper deals with reaction-diffusion systems on an infinitely long strip in $\boldsymbol{R}^2$. Through a pitchfork bifurcation, spatially heterogeneous patterns exist in a neighborhood of Turing instability. Motivated by the works of Kondo and Asai, we study wavefront solution heteroclinic to Turing patterns. It will be seen that the dynamics of a wavefront can be approximated by a fourth order equation of buckling type.

LS method and qualitative analysis of traveling wave solutions of Fisher equation

LS method，a new method for solving nonlinear evolution equations，is proposed. It is based on the （G′/G）–expansion method and the extended hyperbolic tangent function method，and the Poincaré’s qualitative theory is also led in. Then Fisher equation is tested as an example. The properties of integral curves for traveling wave system of Fisher equation are obtained through qualitative analysis，and then a monotonically decreasing wave-front solution of Fisher equation as a dissipative system and a monotonically increasing wave-front solution of Fisher equation as an expansion system are obtained too. Some results agree with that of Ablowitz et al. and some new results for Fisher equation are also obtained as an expansion system. The LS method is used to look for the exact solutions under the condition that the property of solution curves have been obtained through the qualitative analysis，and the target is clear. The LS method also reveals that a linear system can also be used as auxiliary equations to solve nonlinear systems.

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.

Existence of solutions to Cauchy problems for a mixed continuum-discrete model for supply chains and networks

We consider a continuum-discrete model for supply chains based on partial differential equations. The state space is formed by a graph: The load dynamics obeys to a continuous evolution on each arc, while at nodes the good density is conserved, while the processing rate is adjusted. To uniquely determine the dynamics at nodes, the through flux is maximized, with the minimal possible processing rate change. Existence of solutions to Cauchy problems is proven. The latter is achieved deriving estimates on the total variation of the density flux, density and processing rate along a wave-front tracking approximate solution. Then the extension to supply networks is showed.

Traveling wave front in diffusive and competitive Lotka–Volterra system with delays

Existence of travelling wave front solution is established for diffusive and competitive Lotka–Volterra system with delays. The approach used in this paper is the upper-lower solution technique and the monotone iteration. The same results are suitable to Belousov–Zhabotinskii model with delays and cooperative Lotka–Volterra system with delays.

Wavefront solution in the theory of boiling liquids

In this note we discuss a heat equation with a nonlinear Neumann boundary condition introduced by Professor W. Marquardt to model the phase transition from nucleate to transient boiling. We show that for certain initial configuration the solution indeed has a wave front like behavior typical for the mathematical description of phase transitions.

Heteroclinic contours and self-replicated solitary waves in a reaction–diffusion lattice with complex threshold excitation

The space–time dynamics of the network system modeling collective behavior of electrically coupled nonlinear cells is investigated. The dynamics of a local cell is described by the FitzHugh–Nagumo system with complex threshold excitation. Heteroclinic orbits defining traveling wave front solutions are investigated in a moving frame system. A heteroclinic contour formed by separatrix manifolds of two saddle-foci is found in the phase space. The existence of such structure indicates the appearance of complex wave patterns in the network. Such solutions have been confirmed and analyzed numerically. Complex homoclinic orbits found in the neighborhood of the heteroclinic contour define the propagation of composite pulse excitations that can be self-replicated in collisions leading to the appearance of complex wave patterns.

A lattice-model representation of continuous-time random walks

We report some ideas for constructing lattice models (LMs) as a discrete approach to the reaction-dispersal (RD) or reaction-random walks (RRW) models. The analysis of a rather general class of Markovian and non-Markovian processes, from the point of view of their wavefront solutions, let us show that in some regimes their macroscopic dynamics (front speed) turns out to be different from that by classical reaction–diffusion equations, which are often used as a mean-field approximation to the problem. So, the convenience of a more general framework as that given by the continuous-time random walks (CTRW) is claimed. Here we use LMs as a numerical approach in order to support that idea, while in previous works our discussion was restricted to analytical models. For the two specific cases studied here, we derive and analyze the mean-field expressions for our LMs. As a result, we are able to provide some links between the numerical and analytical approaches studied.

A mathematical approach to biological invasive processes

AbstractIn this note we deal with a mathematical model arising in biology. The result presented here concerns the asymptotic analysis of one‐dimensional travelling wavefront solutions for this model. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the Diffusive Nicholson’s Blowflies Equation with Nonlocal Delay

This paper is concerned with the diffusive Nicholson’s blowflies model with nonlocal (or spatiotemporal) delay. When the spatial variable is one-dimensional, we establish the existence of travelling wave-front solutions by using the approach developed by Wang, Li, and Ruan (J. Differ. Equ. 222, 185–232, 2006) on the existence of travelling front solutions of reaction–diffusion systems with nonlocal delay. Moreover, we consider the dependence of the minimal wave speed on the delay and the mobility of the population. Our main finding here is that delay can induce slow travelling wave-fronts and the mobility of the population can increase fast travelling wave-fronts. In particular, if we choose some special kernel forms, then our results include and improve some known results.

Random fluctuations of the firing rate function in a continuum neural field model

We incorporate a source of noise into a continuum neural field model by allowing the firing threshold to fluctuate noisily about a mean value, and examine traveling wave front solutions. Under certain conditions we are able to calculate the first and second moments of the distributions of the resulting time varying front speed and shape. This is then compared with more complete numerical solutions. Fluctuations in the wave front speed and in the shape (i.e., fluctuations in activity at particular coordinate positions across the wave front) were found to increase as the magnitude of the fluctuations in firing threshold increased. The mean speed was found to increase as the magnitude of the fluctuations increases. The role of the correlation time for the threshold variation is also investigated. We also study the role of threshold fluctuations in the failure of front propagation, both in the fast and slow varying noise limits.

Traveling waves for non-local delayed diffusion equations via auxiliary equations

In this paper, we study the existence of traveling wave solutions for a class of delayed non-local reaction–diffusion equations without quasi-monotonicity. The approach is based on the construction of two associated auxiliary reaction–diffusion equations with quasi-monotonicity and a profile set in a suitable Banach space by using the traveling wavefronts of the auxiliary equations. Under monostable assumption, by using the Schauder's fixed point theorem, we then show that there exists a constant c ∗ > 0 such that for each c > c ∗ , the equation under consideration admits a traveling wavefront solution with speed c, which is not necessary to be monotonic.

Wave-front solution in extended quantum circuits with charge discreteness

We study the capacitively coupled, quantum transmission line with charge discreteness, discussed in an earlier paper [Flores, Phys. Rev. B 64, 235309 (2001)]. Due to the difficulties of dealing with a highly nonlinear system, only a low-lying propagating wave solution was obtained then, the so-called cirquiton. In this work, we obtain a wave-front solution, valid for the long-wavelengh approximation. The propagation velocity $v$ of the wave front depends on the (pseudo) flux parameter $f$; the physical requirement that $v$ should be real implies the existence of allowed and forbidden regions (gaps) in the space of the parameter $f$. A study of the stability of the solutions is presented. We remark that it is possible to make a connection between our system and the (quantum) Toda lattice.

Periodic solution and wave front solution for delay equation

First, a general theorem on the existence of periodic solutions for equations with small discrete delay is obtained by employing a technique which is based on a result of the existence of inertial manifold for small discrete delay equation, meanwhile, this general theorem is applied to show the existence of periodic solution for a predator–prey system with small discrete delay. Second, this technique is also used to obtain the existence of travelling wave solution for a host–vector disease model with small discrete delay.

Traveling fronts in diffusive and cooperative Lotka–Volterra system with nonlocal delays

This paper is concerned with a diffusive and cooperative Lotka–Volterra model with distributed delays and nonlocal spatial effect. By using an iterative technique recently developed by Wang, Li and Ruan (Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations 222 (2006), 185–232), sufficient conditions are established for the existence of traveling wave front solutions connecting the zero and the positive equilibria by choosing different kernels. The result is an extension of an existing result for Fisher-KPP equation with nonlocal delay and is somewhat parallel to the existing result for diffusive and cooperative Lotka–Volterra system with discrete delays.