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 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.     

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.     

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.     

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.     

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 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.     

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.     

EVANS FUNCTIONS AND ASYMPTOTIC STABILITY OF TRAVELINGWAVE SOLUTIONS

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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.     

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.     

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.     

Cnoidal and snoidal wave solutions to coupled nonlinear wave equations by the extended Jacobi’s elliptic function method

This paper studies two nonlinear coupled evolution equations. They are the Zakharov equation and the Davey–Stewartson equation. These equations are studied by the aid of Jacobi’s elliptic function expansion method and exact periodic solutions are extracted. In addition, the Zakharov equation with power law nonlinearity is solved by traveling wave hypothesis.     

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.     

Existence of traveling wave solutions for influenza model with treatment

To investigate the spreading speed of influenza and the influence of treatment on the spreading speed, a reaction–diffusion influenza model with treatment is established. The existence of traveling wave solutions is shown by introducing an auxiliary system and applying the Schauder fixed point theorem. The non-existence of traveling wave solutions is proved by a two-sided Laplace transform, which needs a new approach for the prior estimate of exponential decay of traveling wave solutions.     

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.

     

Asymptotic stability of traveling wave fronts in nonlocal reaction–diffusion equations with delay

This paper is concerned with the asymptotic stability of traveling wave fronts of a class of nonlocal reaction–diffusion equations with delay. Under monostable assumption, we prove that the traveling wave front is exponentially stable by means of the (technical) weighted energy method, when the initial perturbation around the wave is suitable small in a weighted norm. The exponential convergent rate is also obtained. Finally, we apply our results to some population models and obtain some new results, which recover, complement and/or improve a number of existing ones.     

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.     

Orbital stability of periodic traveling wave solutions to the generalized Long-Short wave equations

This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Long-Short wave equations $\left\{\begin{array}{l}i\varepsilon_{t}+\varepsilon_{xx}=n\varepsilon+\alpha|\varepsilon|^{2}\varepsilon,\\n_{t}=(|\varepsilon|^{2})_{x}, x\in R.\end{array} \right.$ Firstly, we show that there exist a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period $L$ for the generalized Long-Short wave equations. Then, combining the classical method proposed by Benjamin, Bona et al., and detailed spectral analysis given by using Lame equation and Floquet theory, we show that the dnoidal type periodic wave solution is orbitally stable by perturbations with period $L$. As the modulus of the Jacobian elliptic function $k\rightarrow 1$, we obtain the orbital stability results of solitary wave solution with zero asymptotic value for the generalized Long-Short equations. In particular, as $\alpha=0$, we can also obtain the orbital stability results of periodic wave solutions and solitary wave solutions for the long-short wave resonance equations. The results in the present paper improve and extend the previous stability results of long-shore wave equations and its extension equations.     

The blowup for 2-D quasilinear wave equations with cubic nonlinearity

For 2-D quasilinear wave equations with cubic nonlinearity and small initial data, we not only show that the solutions blow up in finite time but also give a complete asymptotic expansion of the lifespan of classical solutions. Hence we solve a problem posed by S. Alinhac and A. Hoshiga. Moreover, as an application of this result, we prove the blowup of solutions for the nonlinear vibrating membrane equations.