Biological control of the chestnut gall wasp with T. sinensis: A mathematical model

The Asian chestnut gall wasp 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 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, for realistic parameters, the map would support the hypothesis of biological control, the full model, in the same parameter range, does not give such a clear-cut answer. In particular, according to the spatially explicit model, the introduction of T. sinensis would spark a traveling wave of the parasitoid population that destroys the pest on its passage. Then, depending on the value of the diffusion coefficients of the two species, the pest may 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.