Mathematical Study of Spatial Spread of Biological Invasion

1.几十年来，松材线虫病一直在日本大片地区蔓延。该病由松材线虫松材线虫引起，松材线虫以松墨天牛为媒介。根据日本西北海岸松材线虫种群动态和松材线虫发病率的详细数据，建立了描述松材线虫与松材线虫寄主-寄生相互作用的数学模型。通过对模型的分析，我们模拟了该病发病率的时间变化，并预测了如何通过根除松材线虫来控制疫情。主要研究结果如下：1)松材线虫成功入侵的初始密度存在一个最小阈值。然而，即使松树密度超过最低阈值，当松材线虫密度很低时，由于松材线虫的Allee效应，该病也不会入侵。2)随着根治率的提高，最低阈值密度不成比例地增加。3)当根根率较低时，随着初始松树密度的增加，一棵健康的树木逃脱感染直到疫病消亡的概率急剧下降。分析了入侵物种的范围扩展，得出了两种碎片化环境下物种持续生存的条件：(1)分割成带的环境，有利斑块和不利斑块交替排列；(2)水平和垂直轴上分割的环境，不利环境内规则分布的方形有利斑块。

1. An epidemic of pine wilt disease has been spreading in wide areas of Japan for several decades. The disease is caused by the pinewood nematode, Bursaphelenchus xylophilus, which the pine sawyer, Monochamus alternatus, as vector. We present a mathematical model to describe the host-parasite interaction between pines and pine sawyers carrying nematodes based on detailed data on the population dynamics of pine sawyers and the incidence of pine wilt disease at a study site located on the northwest coast of Japan. By analyzing the model we simulate the temporal change in the incidence of the disease and predict how the epidemic could be controlled by eradication of the pine sawyer. The main results are : 1) There is a minimum threshold of the initial pine density for successful invasion of the disease. However, even if the pine density exceeds the minimum threshold, the disease fails in invasion due to the Allee effect of the pine sawyer when its density is very low. 2) The minimum threshold density increases disproportionately with increase in the eradication rate. 3) The probability for a healthy tree to escape from infection until the epidemic dies out decreases sharply with increase in the initial pine density, when the eradication rate is low.2. We analyzed the range expansion of an invading species and derived the conditions for persistence of a species two kinds of fragmented environments :(1) an environment segmented into belts, with favorable and unfavorable patches arranged alterately.(2) an environment segmented in both horizontal and vertical axs, with square-shaped favorable patches regularly distributed within the unfavorable environment.