Traveling Wave Front Solutions Research Articles

On wavefront patterns in a fractional reaction–diffusion model for predator–prey system with anti-predator behavior

AbstractIn recent studies, much attention has been paid to reaction–diffusion systems with anomalous diffusion. In this paper, we investigate the formation of wavefront patterns in a predator–prey model with anti-predator behavior under the influence of anomalous subdiffusion. We use methods of traveling wave analysis and numerical integration to establish the existence of traveling wavefront solutions. Further, obtained traveling wavefront solutions are validated through direct computer simulations of time-dependent solutions for fractional partial differential equation system. It is found that wavefronts exist in a range of system parameters, which travel faster in the subdiffusive system than in the normal diffusive one.

Semi-waves and spreading speeds of stage-structured diffusive competition model with a free boundary

A stage-structured diffusive competition model with a free boundary is considered. The main aim of this paper is to study the evolutions of mature invasive species with competitive ability and determine whether it disrupts the stable ecosystem provided by native species. According to the long-time behaviors under the case of weak-strong competition, strong-weak competition, and weak competition, we obtain respectively the exact asymptotic spreading speed by delay-induced semi-wave problems if the spreading of invasive species occurs. And we demonstrate that, as the moving coefficient goes to infinite, the corresponding Cauchy problem can be viewed as the limiting profile of the model, and the asymptotic spreading speed approaches the minimal speed of the corresponding traveling wavefront solutions. Moreover, under the strong-weak competition situation, the asymptotic spreading speed of invasive species will decrease when the maturation time is longer.

Traveling wave fronts for a diffusive Nicholson’s Blowflies equation accompanying mature delay and feedback delay

For the diffusive Nicholson’s Blowflies equation accompanying disparate mature delay and feedback delay, the traveling wave front analysis has not been touched. By using the inequality technique and constructing suitable upper and lower solutions, we successfully built up a novel criterion to ensure the existence of traveling wave front solutions of the proposed model, which is essentially new and supplement some existing literatures. In particular, we afford an explanatory example to substantiate the validness and efficacy of the theoretical results.

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 > c∗ by applying the perturbation method and implicit function theorem.

Analytic iteration procedure for solitons and traveling wavefronts with sources

A method is presented for calculating solutions to differential equations analytically for a variety of problems in physics. An iteration procedure based on the recently proposed beyond-linear-use-of-equation-superposition function method is shown to converge for nonlinear ordinary differential equations. Case studies are presented for solitary wave solutions of the Camassa–Holm equation and for traveling wavefront solutions of the Burgers equation, with source terms. The convergence of the analytical approximations towards the numerically exact solution is exponentially rapid. In practice, the zeroth-order approximation (a simple convolution) is already useful and the first-order approximation is already accurate while still easy to calculate. The type of nonlinearity can be chosen rather freely, which makes the method generally applicable.

A Leslie-Gower predator-prey model with a free boundary

In this paper, we consider a Leslie-Gower predator-prey model in one-dimensional environment. We study the asymptotic behavior of two species evolving in a domain with a free boundary. Sufficient conditions for spreading success and spreading failure are obtained. We also derive sharp criteria for spreading and vanishing of the two species. Finally, when spreading is successful, we show that the spreading speed is between the minimal speed of traveling wavefront solutions for the predator-prey model on the whole real line (without a free boundary) and an elliptic problem that follows from the original model.

Strong and weak properties of travelling waves for degenerate diffusion/aggregation–diffusion models with non-smooth reaction term

In this paper, we investigate a degenerate reaction–diffusion/reaction–aggregation–diffusion model with non-smooth reaction term. The strong and weak properties of travelling wave front solutions for this model are analyzed. Under degenerate reaction–diffusion/reaction–aggregation–diffusion term and non-smooth reaction term, we obtain how strong and weak properties change.

Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations

Physically motivated modified Fisher equations are studied in which nonlinear convection and nonlinear diffusion is allowed for besides the usual growth and spread of a population. It is pointed out that in a large variety of cases separable functions in the form of exponentially decaying sharp wavefronts solve the differential equation exactly provided a co-moving point source or sink is active at the wavefront. The velocity dispersion and front steepness may differ from those of some previously studied exact smooth traveling wave solutions. For an extension of the reaction-diffusion-convection equation, featuring a memory effect in the form of a maturity delay for growth and spread, also smooth exact wavefront solutions are obtained. The stability of the solutions is verified analytically and numerically.

Dynamical analysis of a logistic equation with spatio-temporal delay

This paper is concerned with a logistic equation with spatio-temporal delay. The local asymptotic behavior of positive constant steady-state solution of the equation is considered. In particular, by using the iterative technique, sufficient conditions are established for the existence of traveling wave front solution connecting the zero and the positive constant steady-state solution. Finally, numerical simulations supporting the theoretical analysis are also included.

On some free boundary problems of the prey–predator model

In this paper we investigate some free boundary problems for the Lotka–Volterra type prey–predator model in one space dimension. The main objective is to understand the asymptotic behavior of the two species (prey and predator) spreading via a free boundary. We prove a spreading–vanishing dichotomy, namely the two species either successfully spread to the entire space as time t goes to infinity and survive in the new environment, or they fail to establish and die out in the long run. The long time behavior of solution and criteria for spreading and vanishing are also obtained. Finally, when spreading successfully, we provide an estimate to show that the spreading speed (if exists) cannot be faster than the minimal speed of traveling wavefront solutions for the prey–predator model on the whole real line without a free boundary.

Global stability and wavefronts in a cooperation model with state-dependent time delay

This article deals with a diffusive cooperative model with state-dependent delay which is assumed to be an increasing function of the population density with lower and upper bounds. For the cooperative DDE system, the positivity and boundedness of solutions are firstly given. Using the comparison principle of the state-dependent delay equations obtained, the stability criterion of model is analyzed both from local and global points of view. When the diffusion is properly introduced, the existence of traveling waves is obtained by constructing a pair of upper–lower solutions and Schauder's fixed point theorem. Calculating the minimum wave speed shows that the wave is slowed down by the state-dependent delay. Finally, the traveling wavefront solutions for large wave speed are also discussed, and the fronts appear to be all monotone, regardless of the state dependent time delay. This is an interesting property, since many findings are frequently reported that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.

Traveling Waves in Spatial SIRS Models.

We study traveling wavefront solutions for two reaction–diffusion systems, which are derived respectively as diffusion approximations to two nonlocal spatial SIRS models. These solutions characterize the propagating progress and speed of the spatial spread of underlying epidemic waves. For the first diffusion system, we find a lower bound for wave speeds and prove that the traveling waves exist for all speeds bigger than this bound. For the second diffusion system, we find the minimal wave speed and show that the traveling waves exist for all speeds bigger than or equal to the minimal speed. We further prove the uniqueness (up to translation) of these solutions for sufficiently large wave speeds. The existence of these solutions are proved by a shooting argument combining with LaSalle’s invariance principle, and their uniqueness by a geometric singular perturbation argument.

Existence of Wave Front Solutions of an Integral Differential Equation in Nonlinear Nonlocal Neuronal Network

An integral-differential model equation arising from neuronal networks with very general kernel functions is considered in this paper. The kernel functions we study here include pure excitations, lateral inhibition, lateral excitations, and more general synaptic couplings (e.g., oscillating kernel functions). The main goal of this paper is to prove the existence and uniqueness of the traveling wave front solutions. The main idea we apply here is to reduce the nonlinear integral-differential equation into a solvable differential equation and test whether the solution we get is really a wave front solution of the model equation.

Traveling Wavefronts of a Diffusive Hematopoiesis Model with Time Delay

In this paper, a reaction-diffusion equation with discrete time delay that describes the dynamics of the blood cell production is analyzed. The existence of the traveling wave front solutions is demonstrated using the technique of upper and lower solutions and the associated monotone iteration.

Existence of travelling wave front solutions of a two-dimensional anisotropic model

This paper considers a two-dimensional anisotropic modelψt=rψ-1k04Δ+k022ψ-c̃k04∂y4ψ+2ηk04∂x2∂y2ψ-ψ3,introduced by Pesch and Kramer (1986) [28]. Assume that ψ travels with a speed c in the propagation direction x and is periodic in the transverse direction y. This model is formulated as a spatial dynamic system in which the variable x is a time-like variable. A center-manifold reduction technique and a normal form analysis are applied to show that this dynamic system can be reduced to a system of ordinary differential equations. A bifurcation analysis yields the persistence of the heteroclinic orbit for the reduced system when higher order terms are added and the speed c is small enough, which establishes the existence of travelling wave front solutions. In order to overcome the difficulty caused by the irreversibility, some appropriate constants are adjusted.

Existence and uniqueness of wave fronts in neuronal network with nonlocal post-synaptic axonal and delayed nonlocal feedback connections

An integral-differential model equation, arising from neuronal networks with both axonal and delayed nonlocal feedback connections, is considered in this paper. The kernel functions in the feedback channel we study here include not only pure excitations but also lateral inhibition. For the kernel functions in the synaptic coupling, pure excitations, lateral inhibition, the lateral excitations and more general synaptic couplings (e.g., oscillating kernel functions) are considered. The main goal of this paper is the study of the existence and uniqueness of the traveling wave front solutions. The main method we applied is the speed index functions and principle of linear superposition.

Hopf bifurcation of traveling wave solutions of delayed Fisher-KPP equation

The dynamical behavior of the traveling wave front solutions of the delayed Fisher-KPP equation is studied by employing the theory of functional differential equation. It is found that, when the delay term τ comes through a threshold, one equilibrium point is always unstable and there exists a periodic solution near the other equilibrium point in the traveling wave equation. By using the theory of center manifold and normal form, the stability of the bifurcating periodic solution, which corresponds to the periodic traveling wave solution of the original equation, is obtained. Finally, some numerical simulations are also given to illustrate our results of theoretical analysis.

Accurate computation of traveling wavefronts in a biological hydrodynamic model

This paper concerns a hyperbolic–elliptic system of partial differential equations for the biological cell density and cell velocity. This system appears as a mathematical model for describing the dynamics of cell motion. Traveling wavefront solutions for the system of equations are computed by using two different numerical methods. One of the methods involves the traveling wave equations and solving an initial-value problem, which leads to accurate computations of the wave profile of cell density and the distributions of the cell velocity. The second method is to solve an initial-moving boundary-value problem for the PDE system, where the traveling wave emerges as the asymptotic long time solution.

On a Free Boundary Problem for a Two-Species Weak Competition System

We study a Lotka–Volterra type weak competition model with a free boundary in a one-dimensional habitat. The main objective is to understand the asymptotic behavior of two competing species spreading via a free boundary. We also provide some sufficient conditions for spreading success and spreading failure, respectively. Finally, when spreading successfully, we provide an estimate to show that the spreading speed (if exists) cannot be faster than the minimal speed of traveling wavefront solutions for the competition model on the whole real line without a free boundary.

Analysis of propagating fronts in a nonlinear diffusion model with chemotaxis

This work is devoted to the study of traveling wave front solutions of a phenomenological model for pattering in bacterial colonies which incorporates cell movements in a nonlinear diffusion and chemotaxis. An analysis of propagating fronts in the model is performed to describe the occurrence of a sharp front with a minimum speed, as well as smooth fronts. This analysis also leads us to construct the sharp front profiles and accurately determine their minimum speed. Moreover, numerical time-dependent solutions of the model are constructed to confirm the results of the analysis.