  How do seasonal successions influence the propagation dynamics of an
age-structured invasive species? We investigate this problem by considering the
scenario that the offsprings are reproduced in spring and then reach maturation
in fall within the same year. For this purpose, a reaction-diffusion system is
proposed, with yearly periodic time delay and spatially nonlocal response
caused by the periodic developmental process. By appealing to the recently
developed dynamical system theories, we obtain the invasion speed $c^*$ and its
coincidence with the minimal speed of time periodic traveling waves. The
characterizations of $c^*$ suggest that (i) time delay decreases the speed and
its periodicity may further do so; (ii) the optimal time to slow down the
invasion is the season without juveniles; (iii) the speed increases to infinity
with the same order as the square root of the diffusion rate.
