The aim of this talk is to study the asymptotic behavior of solutions for some reaction-diffusion systems. By an entire solution, we mean a solution of the reaction-diffusion system which is guaranteed to exist for all real-time in the whole Euclidean space. We establish a Liouville-type theorem for the entire solutions of reaction-diffusion systems. Several examples of spreading phenomena from ecology and epidemiology are given to illustrate the applications of this theory.

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