  The Asian chestnut gall wasp \emph{Dryocosmus kuriphilus}, native of China,
has become a pest when it appeared in Japan, Korea, and the United States. In
Europe it was first found in Italy, in 2002. In 1982 the host-specific
parasitoid \emph{Torymus sinensis} was introduced in Japan, in an attempt to
achieve a biological control of the pest. After an apparent initial success,
the two species seem to have locked in predator-prey cycles of decadal length.
We have developed a spatially explicit mathematical model that describes the
seasonal time evolution of the adult insect populations, and the competition
for finding egg deposition sites. In a spatially homogeneous situation the
model reduces to an iterated map for the egg density of the two species. While
the map would suggest, for realistic parameters, that both species should
become locally extinct (somewhat corroborating the hypothesis of biological
control), the full model, for the same parameters, shows that the introduction
of \emph{T. sinensis} sparks a traveling wave of the parasitoid population that
destroys the pest on its passage. Depending on the value of the diffusion
coefficients of the two species, the pest can later be able to re-colonize the
empty area left behind the wave. When this occurs the two populations do not
seem to attain a state of spatial homogeneity, but produce an ever-changing
pattern of traveling waves.
