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In this paper, we study the traveling wave solutions of a Lotka-Volterra diffusion competition system with nonlocal terms. We prove that there exists traveling wave solutions of the system connecting equilibrium (0, 0) to some unknown positive steady state for wave speed c > c * = max 2, 2 √ dr and there is no such traveling wave solutions for c < c * , where d and r respectively corresponds to the diffusion coefficients and intrinsic rate of an competition species. Furthermore, we also demonstrate the unknown steady state just is the positive equilibrium of the system when the nonlocal delays only appears in the interspecific competition term, which implies that the nonlocal delay appearing in the interspecific competition terms does not affect the existence of traveling wave solutions. Finally, for a specific kernel function, some numerical simulations are given to show that the traveling wave solutions may connect the zero equilibrium to a periodic steady state.Here the unknown functions u(x, t), v(x, t) denote the densities of two competing species, the positive constants d 1 , d 2 are the diffusion coefficients of species u and v, the positive constants a 1 , a 2 are the inter-specific competition coefficients of species u and v, the positive constants b 1 , b 2 are the intra-specific competition coefficients of species u and v, and the positive constants r 1 , r 2 are the growth rates of species

In this paper, we study a nonlocal diffusive single species model with Allee effect. We prove that the model admits positive traveling wave solutions connecting the equilibrium 0 to some unknown positive steady state if and only if the wave speed c ≥ 2 √ r, where r > 0 is the intrinsic rate of increase of the species. For the sufficient large wave speed c, we show that the unknown steady state is the unique positive equilibrium. For two types of convolution kernel functions, we investigate the change of the wave profile as the non-locality increases, and illustrate that the unknown steady state may be a positive periodic solution.

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