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