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.     

Perron Theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations

In this paper we revisit the existence of traveling waves for delayed reaction-diffusion equations by the monotone iteration method. We show that Perron Theorem on existence of bounded solution provides a rigorous and constructive framework to find traveling wave solutions of reaction-diffusion systems with time delay. The method is tried out on two classical examples with delay: the predator-prey and Belousov-Zhabotinskii models.     

Generalized fronts in reaction-diffusion equations with mono-stable nonlinearity

We consider transition fronts (generalized traveling fronts) of mono-stable reaction-diffusion equations with spatially inhomogeneous nonlinearity. By constructing a cutoff function and using an approximate method, we establish the existence of transition fronts of the equation. Furthermore, we give the uniform non-degeneracy estimates of the solutions, such as a lower bound on the time derivative on some level sets, as well as an upper bound on the spatial derivative.     

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. Supported by the NNSF of China (10571078) and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China.     

Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays

A diffusive Lotka-Volterra type model with nonlocal delays for two competitive species is considered. The existence of a traveling wavefront analogous to a bistable wavefront for a single species is proved by transforming the system with nonlocal delays to a four-dimensional system without delay. Furthermore, in order to prove the asymptotic stability (up to translation) of bistable wavefronts of the system, the existence, regularity and comparison theorem of solutions of the corresponding Cauchy problem are first established for the systems on R by appealing to the theory of abstract functional differential equations. The asymptotic stability (up to translation) of bistable wavefronts are then proved by spectral methods. In particular, we also prove that the spreading speed is unique by upper and lower solutions technique. From the point of view of ecology, our results indicate that the nonlocal delays appeared in the interaction terms are not sensitive to the invasion of species of spatial isolation.     

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.     

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.     

Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity

This paper is concerned with the existence and stability of periodic traveling curved fronts for reaction-diffusion equations with time-periodic bistable nonlinearity in two-dimensional space. By constructing supersolution and subsolution, we prove the existence of periodic traveling wave fronts. Furthermore, we show that the front is globally stable.     

Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay

This paper is concerned with the existence, asymptotic stability and uniqueness of traveling wavefronts in a nonlocal diffusion equation with delay. By constructing proper upper and lower solutions, the existence and asymptotic behavior of traveling wavefronts are established. Then the asymptotic stability with phase shift as well as the uniqueness up to translation of traveling wavefronts are proved by applying the idea of squeezing technique.     

Traveling waves connecting equilibrium and periodic orbit for reaction-diffusion equations with time delay and nonlocal response

A class of reaction-diffusion equations with time delay and nonlocal response is considered. Assuming that the corresponding reaction equations have heteroclinic orbits connecting an equilibrium point and a periodic solution, we show the existence of traveling wave solutions of large wave speed joining an equilibrium point and a periodic solution for reaction-diffusion equations. Our approach is based on a transformation of the differential equations to integral equations in a Banach space and the rigorous analysis of the property for a corresponding linear operator. Our approach eventually reduces a singular perturbation problem to a regular perturbation problem. The existence of traveling wave solution therefore is obtained by the application of Liapunov-Schmidt method and the Implicit Function Theorem.     

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.     

Traveling wavefronts for time-delayed reaction-diffusion equation: (II) Nonlocal nonlinearity

This is the second part of a series of study on the stability of traveling wavefronts of reaction-diffusion equations with time delays. In this paper we will consider a nonlocal time-delayed reaction-diffusion equation. When the initial perturbation around the traveling wave decays exponentially as x→−∞ (but the initial perturbation can be arbitrarily large in other locations), we prove the asymptotic stability of all traveling waves for the reaction-diffusion equation, including even the slower waves whose speed are close to the critical speed. This essentially improves the previous stability results by Mei and So [M. Mei, J.W.-H. So, Stability of strong traveling waves for a nonlocal time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 551-568] for the speed with a small initial perturbation. The approach we use here is the weighted energy method, but the weight function is more tricky to construct due to the property of the critical wavefront, and the difficulty arising from the nonlocal nonlinearity is also overcome. Finally, by using the Crank-Nicholson scheme, we present some numerical results which confirm our theoretical study.     

Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system

In this paper, we investigate the spatial dynamics of a nonlocal and time-delayed reaction-diffusion system, which is motivated by an age-structured population model with distributed maturation delay. The spreading speed c^*, the existence of traveling waves with the wave speed c?c^*, and the nonexistence of traveling waves with c<c^* are obtained. It turns out that the spreading speed coincides with the minimal wave speed for monotone traveling waves.     

Traveling wave solutions in a delayed diffusive competition system

In this paper we first investigate the existence of traveling wave fronts in a delayed diffusive competition system by constructing a pair of upper and lower solutions. Then we consider the asymptotic behavior of traveling wave solutions at the minus/plus infinity by means of the bilateral Laplace transform. Finally, the monotonicity and uniqueness (up to the translation) of traveling wave solutions are also obtained by the strong comparison principle and the sliding method.     

Regular traveling waves for a nonlocal diffusion equation

In this paper, we study a nonlocal diffusion equation with a general diffusion kernel and delayed nonlinearity, and obtain the existence, nonexistence and uniqueness of the regular traveling wave solutions for this nonlocal diffusion equation. As an application of the results, we reconsider some models arising from population dynamics, epidemiology and neural network. It is shown that there exist regular traveling wave solutions for these models, respectively. This generalized and improved some results in literatures.     

Entire solutions in nonlocal dispersal equations with bistable nonlinearity

We consider entire solutions of nonlocal dispersal equations with bistable nonlinearity in one-dimensional spatial domain. A two-dimensional manifold of entire solutions which behave as two traveling wave solutions coming from both directions is established by an increasing traveling wave front with nonzero wave speed. Furthermore, we show that such an entire solution is unique up to space-time translations and Liapunov stable. A key idea is to characterize the asymptotic behaviors of the solutions as t→−∞ in terms of appropriate subsolutions and supersolutions. We have to emphasize that a lack of regularizing effect occurs.     

Traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity

This paper deals with the existence of traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity. Based on two different mixed-quasimonotonicity reaction terms, we propose new definitions of upper and lower solutions. By using Schauder's fixed point theorem and a new cross-iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The general results obtained have been applied to type-K monotone and type-K competitive nonlocal diffusive Lotka-Volterra systems.     

Traveling wave fronts for generalized Fisher equations with spatio-temporal delays

We study the existence of traveling wave fronts for a reaction-diffusion equation with spatio-temporal delays and small parameters. The equation reduces to a generalized Fisher equation if small parameters are zero. We present two results. In the first one, we deal with the equation with very general kernels and show the persistence of Fisher wave fronts for all sufficiently small parameters. In the second one, we deal with some particular kernels, with which the nonlocal equation can be reduced to a system of singularly perturbed ODEs, and we are then able to apply the geometric singular perturbation theory and phase plane arguments to this system to show the existence of the minimal wave speed, the existence of a continuum of wave fronts, and the global uniqueness of the physical wave front with each wave speed.     

Persistence of wavefronts in delayed nonlocal reaction-diffusion equations

We develop a perturbation argument based on existing results on asymptotic autonomous systems and the Fredholm alternative theory that yields the persistence of traveling wavefronts for reaction-diffusion equations with nonlocal and delayed nonlinearities, when the time lag is relatively small. This persistence result holds when the nonlinearity of the corresponding ordinary reaction-diffusion system is either monostable or bistable. We then illustrate this general result using five different models from population biology, epidemiology and bio-reactors.     

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.