Traveling waves for nonlocal and non-monotone delayed reaction-diffusion equations

We study the existence of traveling wave solutions for a nonlocal and non-monotone delayed reaction-diffusion equation. Based on the construction of two associated auxiliary reaction diffusion equations with monotonicity and by using the traveling wavefronts of the auxiliary equations, the existence of the positive traveling wave solutions for c 〉 c. is obtained. Also, the exponential asymptotic behavior in the negative infinity was established. Moreover, we apply our results to some reactiondiffusion equations with spatio-temporal delay to obtain the existence of traveling waves. These results cover, complement and/or improve some existing ones in the literature.     

Existence and uniqueness of traveling waves for non-monotone integral equations with application

This paper is concerned with the traveling waves in a class of non-monotone integral equations. First we establish the existence of traveling waves. The approach is based on the construction of two associated auxiliary monotone integral equations and a profile set in a suitable Banach space. Then we show that the traveling waves are unique up to translations under some reasonable assumptions. The exact asymptotic behavior of the profiles as ξ→−∞ and the existence of minimal wave speed are also obtained. Finally, we apply our results to an epidemic model with non-monotone “force of infection”.     

状态依赖时滞非局部扩散方程的波前解

本文主要研究状态依赖时滞非局部扩散方程的波前解,当出生函数单调时,可以得到单调行波解的存在性和非存在性,然后,由先验估计和Ikehara定理,进一步得到临界波前解的渐近性;当出生函数非单调时,通过引进两个辅助拟单调方程,也可以得到相应非拟单调条件下的存在性结果.     

Propagation dynamics of a time periodic diffusion equation with degenerate nonlinearity

This paper is on study of traveling wave solutions and asymptotic spreading of a class of time periodic diffusion equations with degenerate nonlinearity. The asymptotic behavior of traveling wave solutions is investigated by using auxiliary equations and a limit process. In addition, the monotonicity and uniqueness, up to translation, of traveling wave solution with critical speed are determined by sliding method. Finally, combining super and sub-solutions and the stability of steady states, some sufficient conditions on asymptotic spreading are given, which indicates that the success or failure of asymptotic spreading are dependent on the degeneracy of nonlinearity as well as the size of compact support of initial value.     

Oscillatory waves in reaction–diffusion equations with nonlocal delay and crossing-monostability

This paper is concerned with the existence of oscillatory waves in reaction–diffusion equations with nonlocal delay and crossing-monostability, which include many population models, and two main results are presented. In the first one, we establish the existence of non-monotone traveling waves from the trivial solution to the positive equilibrium. 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 traveling fronts of the auxiliary equations. In the second one, we obtain the existence of periodic waves around the positive equilibrium by using Hopf bifurcation theorem.     

Asymptotic speeds of spread and traveling wave solutions of a second order integrodifference equation without monotonicity

This paper is concerned with the propagation modes of a second order integrodifference equation without monotonicity. The equation cannot generate monotone semifolws. By constructing auxiliary functions/equations and applying some known results, the minimal wave speed of traveling wave solutions and asymptotic speeds of spread are established.     

Entire solutions in reaction-advection-diffusion equations with bistable nonlinearities in heterogeneous media

This paper is concerned with entire solutions ( t ∈ R) for bistable reaction-advection-diffusion equations in heterogeneous media. By using traveling curved fronts connecting a constant unstable stationary state and a stable stationary state, we proved that there exist entire solutions behaving as two traveling curved fronts coming from opposite directions, and approaching each other. Furthermore, we prove that such an entire solution is unique and Liapunov stable. The key technique is to characterize the asymptotic behavior of solutions at infinity in term of appropriate subsolutions and supersolutions.     

EVANS FUNCTIONS AND ASYMPTOTIC STABILITY OF TRAVELINGWAVE SOLUTIONS

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Propagation dynamics of a three-species nonlocal competitive–cooperative system

This paper is concerned with the existence, monotonicity, asymptotic behavior and uniqueness of traveling wave solutions for a three-species competitive–cooperative system with nonlocal dispersal and bistable dynamics. By considering a related truncated problem, we first establish the existence and strict monotonicity of traveling waves by means of a limiting argument and a comparative lemma. Then the asymptotic behavior of traveling waves is investigated by using Ikehara’s lemma and bilateral Laplace transform. Finally, we obtain the uniqueness of wave speed and traveling wave by sliding method.     

Traveling waves in a convolution model with infinite distributed delay and non-monotonicity

In this paper, we study a population model with nonlocal diffusion and a non-monotonic reaction term with infinite distributed delay. Some existence results of traveling wavefronts for the system with a monotonic reaction term are firstly obtained by the construction of upper-lower solutions and the application of Schauder’s fixed point theorem. Then by constructing a couple of auxiliary equations with monotonicity and using the comparison method, we prove the existence of traveling waves for the system without monotonicity. We also give some discussion on the asymptotic behavior of the traveling waves as $\xi =x+ct\to ?\infty$.     

Traveling waves in a nonlocal dispersal population model with age-structure

This paper is concerned with the traveling waves in a single species population model which is derived by considering the nonlocal dispersal and age-structure. If the birth function is monotone, then the existence of traveling wavefront is reduced to the existence of a pair of super and subsolutions without the requirement of smoothness. It is proved that the traveling wavefront is strictly increasing and unique up to a translation. The asymptotic behavior of traveling wavefronts is also obtained. If the birth function is not monotone, the existence of traveling wave solution is affirmed by introducing two auxiliary nonlocal dispersal equations with quasi-monotonicity.     

Wave propagation for a two-component lattice dynamical system arising in strong competition models

We study a Lotka-Volterra type competition system with bistable nonlinearity in which the habitat is divided into discrete niches. We show that there exist non-monotone stationary solutions when the migration coefficients are sufficiently small. Also, we prove that the propagation failure phenomenon occurs. Finally, we focus on the traveling wave with nonzero wave speed. By investigating the asymptotic behavior of tails of wave profiles, we show that nonzero speed wave profiles are monotone. Moreover, the nonzero wave speed is unique in the sense that the wave cannot propagate with two different nonzero wave speeds.     

Pulsating type entire solutions of monostable reaction-advection-diffusion equations in periodic excitable media

We establish the existence of pulsating type entire solutions of reaction-advection-diffusion equations with monostable nonlinearities in a periodic framework. Here the nonlinearities include the classic KPP case. The pulsating type entire solutions are defined in the whole space and for all time t∈R. By studying a pulsating traveling front connecting a constant unstable stationary state to a stable stationary state which is allowed to be a positive function, we proved that there exist pulsating type entire solutions behaving as two pulsating traveling fronts coming from both directions, and approaching each other. The key techniques are to characterize the asymptotic behavior of the solutions as t→−∞ in terms of appropriate subsolutions and supersolutions.     

Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation

This Note investigates the properties of the traveling waves solutions of the nonlocal Fisher equation. The existence of such solutions has been proved recently in Berestycki et al. (2009) [3] but their asymptotic behavior was still unclear. We use here a new numerical approximation of these traveling waves which shows that some traveling waves connect the two homogeneous steady states 0 and 1, which is a striking fact since 0 is dynamically unstable and 1 is unstable in the sense of Turing.     

Spreading speeds and traveling waves for a time periodic DS-I-A epidemic model

This paper is devoted to studying the speed of asymptotic spreading and minimal wave speed of traveling wave solutions for a time periodic and diffusive DS-I-A epidemic model, which describes the propagation threshold of disease spreading. The main feature of this model is the possible deficiency of the classical comparison principle such that many known results do not directly work. The speed of asymptotic spreading is estimated by constructing auxiliary equations and applying the classical theory of asymptotic spreading for Fisher type equation. The minimal wave speed is established by proving the existence and nonexistence of the nonconstant traveling wave solutions. Moreover, some numerical examples are presented to model the propagation dynamics of this system.     

Bifurcation and the exact smooth,cusp solitary and periodic wave solutions of the generalized Kudryashov–Sinelshchikov equation

The main aim of this paper is to study the exact traveling wave solutions of the generalized Kudryashov–Sinelshchikov equation by using the auxiliary equation method based on the conclusion of qualitative analysis. The advantage of this method is to choose the effective and proper auxiliary equation on the base of the behaviors and traits of solutions revealed by analysis of phase portraits to study the solution of differential equations. By applying the proposed approach to the generalized Kudryashov–Sinelshchikov equation, the number, behavior and existence of smooth and non-smooth traveling wave solutions are gained, at the same time, the new exact smooth solitary, periodic wave solutions and cusp solitary, periodic wave solutions are obtained. From the dynamic point of view, the behavior of traveling wave solutions is analyzed. The profile,type and the form of exact expression of traveling wave solutions are influenced by the order of nonlinear term and nonlinear terms.

     

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.     

Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity

Entire solutions for monostable reaction-diffusion equations with nonlocal delay in one-dimensional spatial domain are considered. A comparison argument is employed to prove the existence of entire solutions which behave as two traveling wave solutions coming from both directions. Some new entire solutions are also constructed by mixing traveling wave solutions with heteroclinic orbits of the spatially averaged ordinary differential equations, and the existence of such a heteroclinic orbit is established using the monotone dynamical systems theory. Key techniques include the characterization of the asymptotic behaviors of solutions as t→−∞ in term of appropriate subsolutions and supersolutions. Two models of reaction-diffusion equations with nonlocal delay arising from mathematical biology are given to illustrate main results.