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Spatial dynamics of a periodic population model with dispersal

This paper is devoted to the study of spatial dynamics of a class of periodic integro-differential equations which describe the population dispersal process via a dispersal kernel. By appealing to the theory of asymptotic speeds of spread and travelling waves for monotonic periodic semiflows, we establish the existence of the spreading speed c* and the nonexistence of continuous periodic travelling wave solutions with wave speed c < c*. We also prove the existence of left-continuous periodic travelling waves with wave speed c ⩾ c*. In the autonomous case, the continuity of monotonic wave profiles with wave speed c ⩾ c* is obtained.