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This paper is concerned with the travelling waves for a class of non-local dispersal non-cooperative system, which can model the prey-predator and disease-transmission mechanism. By the Schauder's fixed-point theorem, we first establish the existence of travelling waves connecting the semi-trivial equilibrium to non-trivial leftover concentrations, whose bounds are deduced from a precise analysis. Further, we characterize the minimal wave speed of travelling waves and obtain the non-existence of travelling waves with slow speed. Finally, we apply the general results to an epidemic model with bilinear incidence for its propagation dynamics. [ABSTRACT FROM AUTHOR]

This paper is concerned with the new types of entire solutions other than traveling wave solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats. We first establish the existence and properties of spatially periodic solutions connecting two steady states. Then new types of entire solutions are constructed by combining the rightward and leftward pulsating traveling fronts with different speeds and a spatially periodic solution. Finally, for a class of special heterogeneous reaction, we further establish the uniqueness of entire solutions and the continuous dependence of such an entire solution on parameters, such as wave speeds and the shifted variables. In other words, we build a five-dimensional manifold of solutions and the traveling wave solutions are on the boundary of the manifold. [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]