  In this paper, we focus on the nonlocal dispersal monostable equation with
seasonal succession, which can be used to describe the dynamics of species in
an environment alternating between bad and good seasons. We first prove the
existence and uniqueness of global positive solution, and then discuss the long
time behaviors of solution. It is shown that its dynamics is completely
determined by the sign of the principal eigenvalue, i.e., the time periodic
problem has no positive solution and the solution of the initial value problem
tends to zero when principal eigenvalue is non-negative, while the time
periodic positive solution exists uniquely and is globally asymptotically
stable when principal eigenvalue is negative.
