This paper deals with traveling wave solutions for time periodic reaction-diffusion systems. The existence of traveling wave solutions is established by combining the fixed point theorem with super-and sub-solutions, which reduces the existence of traveling wave solutions to the existence of superand sub-solutions. The asymptotic behavior is determined by the stability of periodic solutions of the corresponding initial value problems. To illustrate the abstract results, we investigate a time periodic Lotka-Volterra system with two species by presenting the existence and nonexistence of traveling wave solutions, which connect the trivial steady state to the unique positive periodic solution of the corresponding kinetic system.2010 Mathematics Subject Classification. Primary: 45C05, 45M05; Secondary: 92D40.

This paper deals with the asymptotic spreading of time periodic Lotka-Volterra competition diffusion systems, which formulates the coinvasioncoexistence process. By combining auxiliary systems with comparison principle, some results on asymptotic spreading are established. Our conclusions indicate that the coinvasions of two competitors may be successful, and the interspecific competitions slow the invasion speed of one species.

This paper is concerned with the global dynamics of a Lotka–Volterra competition diffusion system having nonlocal intraspecies terms. Based on the reconstructed comparison principle and monotone iteration, the existence and uniqueness of the solution for the corresponding Cauchy problem are established. In addition, the spreading speed of the system with compactly supported initial data is considered, which admits uniform upper and lower bounds. Finally, some sufficient conditions for guaranteeing the existence and nonexistence of Turing bifurcation are given, which depend on the intensity of nonlocality. Comparing with the classical Lotka–Volterra competition diffusion system, our results indicate that a nonconstant periodic solution may exist if the nonlocality is strong enough, which are also illustrated numerically.

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