Propagation and its failure in a lattice delayed differential equation with global interaction

We study the existence, uniqueness, global asymptotic stability and propagation failure of traveling wave fronts in a lattice delayed differential equation with global interaction for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment. In the bistable case, under realistic assumptions on the birth function, we prove that the equation admits a strictly monotone increasing traveling wave front. Moreover, if the wave speed does not vanish, then the wave front is unique (up to a translation) and globally asymptotic stable with phase shift. Of particular interest is the phenomenon of “propagation failure” or “pinning” (that is, wave speed c = 0), we also give some criteria for pinning in this paper.     

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.     

Existence of fast positive wavefronts for a non-local delayed reaction-diffusion equation

We establish the existence of a continuous family of fast positive wavefronts u(t,x)=?(x+ct), ?(−∞)=0, ?(+∞)=κ, for the non-local delayed reaction-diffusion equation . Here 0 and κ>0 are fixed points of g∈C²(R₊,R₊) and the non-negative K is such that is finite for every real λ. We also prove that the fast wavefronts are non-monotone if .     

Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay

This paper is concerned with the existence, uniqueness and globally asymptotic stability of traveling wave fronts in the quasi-monotone reaction advection diffusion equations with nonlocal delay. Under bistable assumption, we construct various pairs of super- and subsolutions and employ the comparison principle and the squeezing technique to prove that the equation has a unique nondecreasing traveling wave front (up to translation), which is monotonically increasing and globally asymptotically stable with phase shift. The influence of advection on the propagation speed is also considered. Comparing with the previous results, our results recovers and/or improves a number of existing ones. In particular, these results can be applied to a reaction advection diffusion equation with nonlocal delayed effect and a diffusion population model with distributed maturation delay, some new results are obtained.     

Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay

In this paper, we study the existence, uniqueness and asymptotic stability of travelling wavefronts of the following equation:

u_t(x,t)=D[u(x+1,t)+u(x-1,t)-2u(x,t)]-du(x,t)+b(u(x,t-r)),     

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.     

Monotone traveling wavefronts of the KPP-Fisher delayed equation

In the early 2000's, Gourley (2000), Wu et al. (2001), Ashwin et al. (2002) initiated the study of the positive wavefronts in the delayed Kolmogorov-Petrovskii-Piskunov-Fisher equation

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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.     

Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models

The theory of asymptotic speeds of spread and monotone traveling waves is generalized to a large class of scalar nonlinear integral equations and is applied to some time-delayed reaction and diffusion population models.     

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.     

Diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations

For abstract functional differential equations and reaction-diffusion equations with delay, an exponential ordering is introduced which takes into account the spatial diffusion. The induced monotonicity of the solution semiflows is established and applied to describe the threshold dynamics (extinction or persistence/convergence to positive equilibria) for a nonlocal and delayed reaction-diffusion population model.     

On the existence of traveling waves for delayed reaction-diffusion equations

We study the existence of traveling wave solutions for reaction-diffusion equations with nonlocal delay, where reaction terms are not necessarily monotone. The existence of traveling wave solutions for reaction-diffusion equations with nonlocal delays is obtained by combining upper and lower solutions for associated integral equations and the Schauder fixed point theorem. The smoothness of upper and lower solutions is not required in this paper.     

On a class of abstract neutral functional differential equations

By using the theory of semigroups of growth α, we discuss the existence of mild solutions for a class of abstract neutral functional differential equations. A concrete application to partial neutral functional differential equations is considered.     

Asymptotic speed of propagation and traveling wave solutions for a lattice integral equation

We investigate the asymptotic speed of propagation and monotone traveling wave solutions for a lattice integral equation which is an epidemic model while the population is distributed on one-dimensional lattice Z$\mathbb{Z}$. It is proved that the asymptotic speed of propagation _c∗$c_{\ast }$ coincides with the minimal wave speed.     

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.     

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.     

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 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.     

Oscillation of even order partial differential equations with distributed deviating arguments

Some Kamenev-type oscillation criteria are established for a class of boundary value problems associated with even-order partial differential equations with distributed deviating arguments. Our approach is to reduce the high-dimensional oscillation problem to a one-dimensional oscillation one, and the general means developed by Philos and Wong is used as the main tool. The results obtained here extend and improve some known results in the literature.