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This paper is concerned with the traveling wave solutions for a class of predator-prey model with nonlocal dispersal. By adopting the truncation method, we use Schauder's fixed-point theorem to obtain the existence of traveling waves connecting the semi-trivial equilibrium to non-trivial leftover concentrations for c > c*, in which c* is the minimal wave speed. Meanwhile, through the limiting approach and the delicate analysis, we establish the existence of traveling wave solutions with the critical speed. Finally, we show the nonexistence of traveling waves for 0 < c < c* by the characteristic equation corresponding to the linearization of original system at the semi-trivial equilibrium. Throughout the whole paper, we need to overcome the difficulties brought by the nonlocal dispersal and the non-preserving of system itself. [ABSTRACT FROM AUTHOR]

This paper is concerned with traveling wave solutions of a nonlocal dispersal Susceptible–Infective–Removal–Healing (for short SIRH ) model with relapse. It is found that the existence and nonexistence of traveling waves of the system are not only determined by the critical wave speed c ∗ , but also by the basic reproduction number R 0 of the corresponding system of ordinary differential equations. More precisely, we use Schauder’s fixed-point theorem to obtain the existence of traveling waves for R 0 > 1 and c > c ∗ , and the nonexistence of traveling waves for R 0 > 1 and 0 < c < c ∗ . Some numerical simulations and discussions are also provided to illustrate our analytical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with space periodic traveling wave solutions of the following Lotka–Volterra competition system with nonlocal dispersal and space periodic dependence, { ∂ u 1 ∂ t = ∫ R N κ ( y − x ) u 1 ( t , y ) d y − u 1 ( t , x ) + u 1 ( a 1 ( x ) − b 1 ( x ) u 1 − c 1 ( x ) u 1 ) , x ∈ R N ∂ u 2 ∂ t = ∫ R N κ ( y − x ) u 2 ( t , y ) d y − u 2 ( t , x ) + u 2 ( a 2 ( x ) − b 2 ( x ) u 1 − c 2 ( x ) u 2 ) , x ∈ R N . Under suitable assumptions, the system admits two semitrivial space periodic equilibria ( u 1 ⁎ ( x ) , 0 ) and ( 0 , u 2 ⁎ ( x ) ) , where ( u 1 ⁎ ( x ) , 0 ) is linearly and globally stable and ( 0 , u 2 ⁎ ( x ) ) is linearly unstable with respect to space periodic perturbations. By sub- and supersolution techniques and comparison principals, we show that, for any given ξ ∈ S N − 1 , there exists a continuous periodic traveling wave solution of the form ( u 1 ( t , x ) , u 2 ( t , x ) ) = ( Φ 1 ( x − c t ξ , c t ξ ) , Φ 2 ( x − c t ξ , c t ξ ) ) connecting ( u 1 ⁎ ( ⋅ ) , 0 ) and ( 0 , u 2 ⁎ ( ⋅ ) ) and propagating in the direction of ξ with speed c > c ⁎ ( ξ ) , where c ⁎ ( ξ ) is the spreading speed of the system in the direction of ξ . Moreover, for c < c ⁎ ( ξ ) there is no such solution. When the wave speed c > c ⁎ ( ξ ) , we also prove the asymptotic stability and uniqueness of traveling wave solution using squeezing techniques. [ABSTRACT FROM AUTHOR]

Abstract: 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 in terms of appropriate subsolutions and supersolutions. We have to emphasize that a lack of regularizing effect occurs. [Copyright &y& Elsevier]

Abstract: This paper is concerned with entire solutions of the Fisher-KPP equation with nonlocal dispersal, i.e., , which is a one-dimensional nonlocal version of the Fisher-KPP equation describing the spatial spread of a mutant in a given population and the dispersion of the genetic characters is assumed to follow a nonlocal diffusion law modeled by a convolution operator. Here the entire solutions are defined in the whole space and for all time . A comparison principle is employed to establish the existence of entire solutions by combining two traveling wave solutions with different speeds and coming from both ends of the real axis and some spatially independent solutions. The main difficulty is that a lack of regularizing effect occurs. This is probably the first time the existence of entire solutions of reaction equations with nonlocal dispersal has been studied. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence of traveling wave solutions of a delayed predator–prey system with stage structure and nonlocal diffusion. By introducing the partial quasi-monotone condition and cross-iteration scheme, we first consider a class of delayed systems with nonlocal diffusion and deduce the existence of traveling wave solutions to the existence of a pair of upper–lower solutions. When the result is applied to the predator–prey system, we establish the existence of traveling wave solutions, as well as its precisely asymptotic behavior. Our result implies that there is a transition zone moving from the steady state with no species to the steady state with the coexistence of both species. [Copyright &y& Elsevier]

This paper is devoted to investigating the traveling wave solutions of a nonlocal dispersal SEIR epidemic model with standard incidence. We find that the existence and nonexistence of traveling wave solutions are determined by the basic reproduction number and the critical wave speed. Through considering a truncate problem, combining with Schauder's fixed-point theorem and applying a limiting argument, we prove the existence of traveling wave solutions. Meanwhile, the nonexistence of traveling wave solutions is showed by the Laplace transform method. Furthermore, the existence of traveling wave solutions with critical wave speed is also established by a delicate analysis. We also point out that both the nonlocal dispersal and coupling of system in the model bring some difficulties in the study of traveling wave solutions. [ABSTRACT FROM AUTHOR]