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.     

Non-monotone traveling waves and entire solutions for a delayed nonlocal dispersal equation

This paper is concerned with the traveling waves and entire solutions for a delayed nonlocal dispersal equation with convolution- type crossing-monostable nonlinearity. We first establish the existence of non-monotone traveling waves. By Ikehara’s Tauberian theorem, we further prove the asymptotic behavior of traveling waves, including monotone and non-monotone ones. Then, based on the obtained asymptotic behavior, the uniqueness of the traveling waves is proved. Finally, the entire solutions are considered. By introducing two auxiliary monostable equations and establishing some comparison arguments for the three equations, some new types of entire solutions are constructed via the traveling wavefronts and spatially independent solutions of the auxiliary equations.     

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.     

Stability of non-monotone traveling waves for a discrete diffusion equation with monostable convolution type nonlinearity

This paper is concerned with the stability of non-monotone traveling waves for a discrete diffusion equation with monostable convolution type nonlinearity. By using the anti-weighted energy method and nonlin-ear Halanay’s inequality, we prove that all noncritical traveling waves (waves with speeds c > c_*, c_* is minimal speed) are time-exponentially stable, when the initial perturbations around the waves are small. As a corollary of our stability result, we immediately obtain the uniqueness of the traveling waves.     

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.     

Traveling waves for nonlinear cellular neural networks with distributed delays

In this paper, we will establish the existence and nonexistence of traveling waves for nonlinear cellular neural networks with finite or infinite distributed delays. The dynamics of each given cell depends on itself and its nearest m left or l right neighborhood cells where delays exist in self-feedback and left or right neighborhood interactions. Our approach is to use Schauder?s fixed point theorem coupled with upper and lower solutions of the integral equation in a suitable Banach space. Further, we obtain the exponential asymptotic behavior in the negative infinity and the existence of traveling waves for the minimal wave speed by the limiting argument. Our results improve and cover some previous works.     

Separation dichotomy and wavefronts for a nonlinear convolution equation

This paper is concerned with a scalar nonlinear convolution equation, which appears naturally in the theory of traveling waves for monostable evolution models. First, we prove that, at each end of the real line, every bounded positive solution of the convolution equation should either be separated from zero or be exponentially converging to zero. This dichotomy principle is then used to establish a general theorem guaranteeing the uniform persistence and existence of semi-wavefront solutions to the convolution equation. Finally, we apply our theoretical results to several well-studied classes of evolution equations with asymmetric non-local and non-monotone response. We show that, contrary to the symmetric case, these equations can possess simultaneously stationary, expansion and extinction waves.     

Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks

The author is concerned with the existence and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks. Previous methods do not apply in solving these problems because there is no maximum principle or conservation laws available to the integral differential equations. He applies fixed point theorems to prove the existence of the traveling waves. Then, he makes use of linearization technique as well as eigenvalue functions to study the exponential stability of the waves.     

Traveling waves in discrete models of biological populations with sessile stages

A common approach to describing invasions of non-native species into previously unoccupied habitat is to consider the speed of population expansion and the existence of traveling waves. Typical existence theorems for traveling waves require some compactness properties of the next-generation operator. Many realistic modeling assumptions, however, give rise to non-compact operators; for example the occurrence of sessile stages during the life cycle of an individual. Recent results have extended the existence theory of traveling waves to a large class of weakly compact operators, but conditions can be difficult to check and not easily accessible to theoretical ecologists. In this paper, we give a new proof for the existence of traveling waves in a large class of equations where the next generation operator is not compact, but rather the sum of an integral operator and a contraction. We illustrate our proof with a model for the dispersal of a plant species with a seed-bank and a model for dispersal of stream insects with larval stages.     

Existence,uniqueness and stability of pyramidal traveling fronts in reaction-diffusion systems

In the one-dimensional space, traveling wave solutions of parabolic differential equations have been widely studied and well characterized. Recently, the mathematical study on higher-dimensional traveling fronts has attracted a lot of attention and many new types of nonplanar traveling waves have been observed for scalar reaction-diffusion equations with various nonlinearities. In this paper, by using the comparison argument and constructing appropriate super- and subsolutions, we study the existence, uniqueness and stability of threedimensional traveling fronts of pyramidal shape for monotone bistable systems of reaction-diffusion equations in R³. The pyramidal traveling fronts are characterized as either a combination of planar traveling fronts on the lateral surfaces or a combination of two-dimensional V-form waves on the edges of the pyramid. In particular, our results are applicable to some important models in biology, such as Lotka-Volterra competition-diffusion systems with or without spatio-temporal delays, and reaction-diffusion systems of multiple obligate mutualists.     

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 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.     

On camel-like traveling wave solutions in cellular neural networks

This paper is concerned with the existence of camel-like traveling wave solutions of cellular neural networks distributed in the one-dimensional integer lattice . The dynamics of each given cell depends on itself and its nearest m left neighbor cells with instantaneous feedback. The profile equation of the infinite system of ordinary differential equations can be written as a functional differential equation in delayed type. Under appropriate assumptions, we can directly figure out the solution formula with many parameters. When the wave speed is negative and close to zero, we prove the existence of camel-like traveling waves for certain parameters. In addition, we also provide some numerical results for more general output functions and find out oscillating traveling waves numerically.     

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

A class of integral equations without monotonicity is investigated. It is shown that there is a spreading speed c^∗>0 for such an integral equation, and that its limiting integral equation admits a unique traveling wave (up to translation) with speed c?c^∗ and no traveling wave with c<c^∗. These results are also applied to some nonlocal reaction-diffusion population models.     

Spreading speed and traveling waves for a multi-type SIS epidemic model

The theory of asymptotic speeds of spread and monotone traveling waves for monotone semiflows is applied to a multi-type SIS epidemic model to obtain the spreading speed c^∗, and the nonexistence of traveling waves with wave speed c<c^∗. Then the method of upper and lower solutions is used to establish the existence of monotone traveling waves connecting the disease-free and endemic equilibria for c?c^∗. This shows that the spreading speed coincides with the minimum wave speed for monotone traveling waves. We also give an affirmative answer to an open problem presented by Rass and Radcliffe [L. Rass, J. Radcliffe, Spatial Deterministic Epidemics, Math. Surveys Monogr. 102, Amer. Math. Soc., Providence, RI, 2003].     

Quasi-Stable Structures in Circular Gene Networks

A new mathematical model is proposed for a circular gene network representing a system of unidirectionally coupled ordinary differential equations. The existence and stability of special periodic motions (traveling waves) for this system is studied. It is shown that, with a suitable choice of parameters and an increasing number m of equations in the system, the number of coexisting traveling waves increases indefinitely, but all of them (except for a single stable periodic solution for odd m) are quasistable. The quasi-stability of a cycle means that some of its multipliers are asymptotically close to the unit circle, while the other multipliers (except for a simple unit one) are less than unity in absolute value.     

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.     

Traveling Wave Solutions for Planar Lattice Differential Systems with Applications to Neural Networks

We obtain some existence results for traveling wave fronts and slowly oscillatory spatially periodic traveling waves of planar lattice differential systems with delay. Our approach is via Schauder's fixed-point theorem for the existence of traveling wave fronts and via S¹-degree and equivarant bifurcation theory for the existence of periodic traveling waves. As examples, the obtained abstract results will be applied to a model arising from neural networks and explicit conditions for traveling wave fronts and global continuation of periodic waves will be obtained.     

Traveling waves in an SEIR epidemic model with a general nonlinear incidence rate

ABSTRACT

We study the traveling waves of reaction-diffusion equations for a diffusive SEIR model with a general nonlinear incidence. The existence of traveling waves is determined by the basic reproduction number of the corresponding ordinary differential equations and the minimal wave speed. Its proof is showed by introducing an auxiliary system, applying Schauder fixed point theorem and then a limiting argument. The non-existence proof is obtained by two-sided Laplace transform when the speed is less than the critical velocity. Finally, we present some examples to support our theoretical results.