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This paper is concerned with a model for the dynamics of a single species ina one-dimensional heterogeneous environment. The environment consists of twokinds of patches, which are periodically alternately arranged along the spatialaxis. We first establish the well-posedness for the Cauchy problem. Next, wegive existence and uniqueness results for the positive steady state and weanalyze the long-time behavior of the solutions to the evolution problem.Afterwards, based on dynamical systems methods, we investigate the spreadingproperties and the existence of pulsating traveling waves in the positive andnegative directions. It is shown that the asymptotic spreading speed, c * ,exists and coincides with the minimal wave speed of pulsating traveling wavesin positive and negative directions. In particular, we give a variationalformula for c * by using the principal eigenvalues of certain linear periodiceigenvalue problems.

Propagation phenomena in periodic patchy landscapes with interface conditions