Spreading Speeds and Traveling Waves for Non-cooperative Reaction�CDiffusion Systems

Much has been studied on the spreading speed and traveling wave solutions for cooperative reaction–diffusion systems. In this paper, we shall establish the spreading speed for a large class of non-cooperative reaction–diffusion systems and characterize the spreading speed as the slowest speed of a family of non-constant traveling wave solutions. Our results are applied to a partially cooperative system describing interactions between ungulates and grass.     

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.     

Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity

This paper is concerned with the traveling wave solutions and the spreading speeds for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity, which is motivated by an age-structured population model with time delay. We first prove the existence of traveling wave solution with critical wave speed c = c*. By introducing two auxiliary monotone birth functions and using a fluctuation method, we further show that the number c = c* is also the spreading speed of the corresponding initial value problem with compact support. Then, the nonexistence of traveling wave solutions for c < c* is established. Finally, by means of the (technical) weighted energy method, we prove that the traveling wave with large speed is exponentially stable, when the initial perturbation around the wave is relatively small in a weighted norm.     

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.     

Traveling waves for a periodic Lotka–Volterra predator–prey system

This paper is concerned with the time periodic traveling wave solutions for a periodic Lotka–Volterra predator–prey system, which formulates that both species synchronously invade a new habitat. We first establish the existence of periodic traveling wave solutions by combining the upper and lower solutions with contracting mapping principle and Schauder’s fixed point theorem. The asymptotic behavior of nontrivial solution is given precisely by the stability of the corresponding kinetic system that has been widely investigated. Then, the nonexistence of periodic traveling wave solutions is confirmed by applying the theory of asymptotic spreading. We show the conclusion for all positive wave speed and obtain the minimal wave speed.     

Propagation of second order integrodifference equations with local monotonicity

This paper is concerned with the spreading speeds and traveling wavefronts of second order integrodifference equations with local monotonicity. By introducing two auxiliary integrodifference equations, the spreading speed and traveling wave solutions are studied. In particular, we obtain the nonexistence of monotone traveling wave solutions for an example if it is local monotone. These results are applied to a model which is obtained by introducing the spatial variable to a difference equation used by the International Whaling Commission.     

Traveling wave solutions in a diffusive system with two preys and one predator

This paper is concerned with the traveling wave solutions in a diffusive system with two preys and one predator. By constructing upper and lower solutions, the existence of nontrivial traveling wave solutions is established. The asymptotic behavior of traveling wave solutions is also confirmed by combining the asymptotic spreading with the contracting rectangles. Applying the theory of asymptotic spreading, the nonexistence of traveling wave solutions is proved.     

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].     

Propagation Speed of the Bistable Traveling Wave to the Lotka–Volterra Competition System in a Periodic Habitat

We study propagation direction of the traveling wave for the diffusive Lotka–Volterra competition system with bistable nonlinearity in a periodic habitat. By directly proving the strong stability of two semitrivial equilibria, we establish a new and sharper result on the existence of traveling wave. Using the method of upper and lower solutions, we provide two comparison theorems concerning the direction of traveling wave propagation. Several explicit sufficient conditions on the determination of the speed sign are established. In addition, an interval estimation of the bistable-wave speed reveals the relations among the bistable speed and the spreading speeds of two monostable subsystems. Biologically, our idea and insight provide an effective approach to find or control the direction of wave propagation for a system in heterogeneous environments.

     

Spatiotemporal propagation of a time-periodic reaction–diffusion SI epidemic model with treatment

This work is concerned with the spatiotemporal propagation phenomena for a time-periodic reaction-diffusion susceptible-infectious (SI) epidemic model with treatment in terms of the asymptotic speed of spread and periodic traveling waves. First, the asymptotic speed of spread $c^{\ast }$ is characterized and the spreading properties of the model are analyzed by combining the periodic principal eigenvalue problem, comparison method, and the uniform persistence idea for a dynamical system. Second, by constructing suitable super- and subsolutions for truncation problems corresponding to the traveling wave system, the existence of periodic traveling waves is established via the fixed point theorem twice. It turned out that the asymptotic speed of spread coincides with the minimum wave speed of periodic traveling waves. Finally, via numerical simulation, the effects of some important parameters (such as diffusion rate, treatment rate, etc.) on the spreading speed are discussed, and the asymptotic properties of the periodic traveling waves are explored.     

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.     

Minimal wave speed and spread speed of competing pioneer and climax species

In this article, for a diffusive population model describing interaction of pioneer-climax species, we explore the issues of spreading speed, linear determinacy and traveling wave fronts. Applying the theory developed by Weinberger et al. [J. Math. Biol. 2002;45:183–218], we identify some ranges of model parameters within which, the model is shown to have a single spreading speed which is linearly determinate and coincides with the corresponding minimal speed for the traveling wave fronts connecting two relevant equilibria, one being a boundary equilibrium and the other being a coexistence equilibrium.     

Spreading speed for a nonlocal dispersal vaccination model with general incidence

This paper is concerned with the spreading speed for a nonlocal dispersal vaccination model with general incidence. We first prove the existence and uniform boundedness of solutions for this model by using the Schauder’s fixed point theorem. Then, applying comparison principle, we establish the existence of spreading speed for the infective individuals. According to our result, one can see that the spreading speed coincides with the critical speed of traveling wave solution connecting the disease-free and endemic equilibria. In addition, the diffusion rate of the infected individuals can increase the spread of infectious diseases, while the vaccination rate reduces the spread of infectious diseases.     

Spatial Dynamics of Multilayer Cellular Neural Networks

The purpose of this work is to study the spatial dynamics of one-dimensional multilayer cellular neural networks. We first establish the existence of rightward and leftward spreading speeds of the model. Then we show that the spreading speeds coincide with the minimum wave speeds of the traveling wave fronts in the right and left directions. Moreover, we obtain the asymptotic behavior of the traveling wave fronts when the wave speeds are positive and greater than the spreading speeds. According to the asymptotic behavior and using various kinds of comparison theorems, some front-like entire solutions are constructed by combining the rightward and leftward traveling wave fronts with different speeds and a spatially homogeneous solution of the model. Finally, various qualitative features of such entire solutions are investigated.     

Spreading speeds and traveling wavefronts for second order integrodifference equations

This paper is concerned with the spreading speeds and traveling wavefronts for second order integrodifference equations. By introducing an auxiliary integrodifference system, the spreading speed is established for the integrodifference equation. It is shown that the spreading speed coincides with the minimal wave speed for monotonic traveling wavefronts. Furthermore, we prove that the traveling wavefronts are stable by applying the squeezing technique. Finally, we analyze the different effects of the delay term appearing in the integrodifference equation from the viewpoint of ecology.     

Spreading speed and traveling waves for a nonmonotone reaction–diffusion model with distributed delay and nonlocal effect

In this paper, spreading speed and traveling waves for reaction–diffusion model with distributed delay and nonlocal effect without monotonicity are investigated. It is shown that there exists the spreading speed c^∗ which coincides with the minimal wave speed, and its limiting integral equation has an unique traveling wave with speed c > c^∗, and no traveling wave with c < c^∗. Moreover, the dependence of the spreading speed on the delay and the nonlocal effect is considered.     

Spreading and generalized propagating speeds of discrete KPP models in time varying environments

The current paper deals with spatial spreading and front propagating dynamics for spatially discrete KPP (Kolmogorov, Petrovsky and Paskunov) models in time recurrent environments, which include time periodic and almost periodic environments as special cases. The notions of spreading speed interval, generalized propagating speed interval, and traveling wave solutions are first introduced, which are proper modifications of those introduced for spatially continuous KPP models in time almost periodic environments. Among others, it is then shown that the spreading speed interval in a given direction is the minimal generalized propagating speed interval in that direction. Some important upper and lower bounds for the spreading and generalized propagating speed intervals are provided. When the environment is unique ergodic and the so called linear determinacy condition is satisfied, it is shown that the spreading speed interval in any direction is a singleton (called the spreading speed), which equals the classical spreading speed if the environment is actually periodic. Moreover, in such a case, a variational principle for the spreading speed is established and it is shown that there is a front of speed c in a given direction if and only if c is greater than or equal to the spreading speed in that direction.      

Bifurcation to locked fronts in two component reaction–diffusion systems

We study invasion fronts and spreading speeds in two component reaction–diffusion systems. Using a variation of Lin's method, we construct traveling front solutions and show the existence of a bifurcation to locked fronts where both components invade at the same speed. Expansions of the wave speed as a function of the diffusion constant of one species are obtained. The bifurcation can be sub or super-critical depending on whether the locked fronts exist for parameter values above or below the bifurcation value. Interestingly, in the sub-critical case numerical simulations reveal that the spreading speed of the PDE system does not depend continuously on the coefficient of diffusion.     

Existence of Generalized Traveling Waves in Time Recurrent and Space Periodic Monostable Equations

This paper is concerned with the extension of the concepts and theories of traveling wave solutions of time and space periodic monostable equations to time recurrent and space periodic ones.&nbsp; It first introduces the concept of generalized traveling wave solutions of time recurrent and space&nbsp;periodic monostable equations, which extends the concept of periodic traveling wave solutions of time and space periodic monostable equations to time recurrent and space periodic ones.&nbsp;It then proves that in the direction of any unit vector \(\xi\), there is \(c^*(\xi)\) such that for any \(c&gt;c^*(\xi)\), a generalized traveling wave solution in the direction of \(\xi\) with averaged propagation speed \(c\) exists. It also proves that if the time recurrent and space periodic&nbsp;monostable equation is indeed time periodic, then \(c^*(\xi)\) is the minimal wave speed in the direction of&nbsp;\(\xi\)&nbsp;and the generalized traveling wave solution in the direction of&nbsp;\(\xi\)&nbsp;with averaged speed \(c&gt;c^*(\xi)\) is a periodic traveling wave solution with speed \(c\), which recovers the existing results on the existence of periodic traveling wave solutions in the direction of&nbsp;\(\xi\)&nbsp;with speed greater than the minimal speed in that direction.     

Boundedness and persistence of traveling wave solutions for a non-cooperative lattice-diffusion system with time delay

In this paper we study traveling wave solutions of a non-cooperative lattice-diffusion system with time delay, which includes predator–prey models and disease-transmission models. Minimal wave speed of traveling wave solutions is given. Schauder’s fixed-point theorem is applied to show the existence of semi-traveling wave solutions. The boundness and persistence of traveling wave solutions are overcome by using rescaling method and Laplace transform, where the application of Laplace transform to persistence is very novel and creative. The traveling wave solutions for some specific models are shown to connect to a positive equilibrium by using Lyapunov function and LaSalle’s invariance principle.