Modeling the Expansion of an Introduced Tree Disease

Pine wilt disease is caused by the introduced pinewood nematode, Bursaphelenchusxylophilus, for which the vector is the pine sawyer beetle, Monochamus alternatus. Native Japanese pines, black pine (Pinus thunbergii) and red pine (P. densiflora), are extremely sensitive to the nematode's infection, and the parasite has been expanding nationwide in the last few decades, despite intensive control efforts. To understand the parasite's range expansion in Japan, we modeled the dynamics of the pines and the beetle that disperses the nematode, using an integro-difference equation in a one-dimensional space. Based on field data collected in Japan, we investigated the dependence of the parasite's rate of range expansion on the eradication rate of the beetle, the initial pine density, and the beetle dispersal ability. Our model predicts several results. (1) The Allee Effect operates on beetle reproduction, and consequently the parasite cannot invade a pine stand, once the beetle density decreases below a threshold. (2) The distribution of the dispersal distance of the beetles critically affects the expansion rate of the disease. As the fraction of the beetles that travel over long distance increases from zero, the range expansion accelerates sharply. (3) However, too frequent long-range dispersal results in a failure of the parasite invasion due to the Allee Effect, suggesting the importance of correctly assessing the beetle's mobility to predict the speed of range expansion of the parasite. (4) As the eradication rate is increased, the range expansion speed decreases gradually at first and suddenly drops to zero at a specific value of the eradication rate.