Mathematical Study of Spatial Spread of Biological Invasion

1.几十年来，松树枯萎病在日本的广大地区蔓延。该病害是由松材线虫引起的，以松墨天牛（Monochamus alternatus）为媒介，以松索耶木蠹蛾（Monochamus alternatus）为病原。我们提出了一个数学模型来描述松树和松锯木虫携带线虫的种群动态和松树枯萎病的发病率在位于日本西北海岸的一个研究地点的详细数据的基础上，宿主-寄生虫之间的相互作用。通过分析模型，我们模拟了发病率的时间变化，并预测了如何通过根除松树索耶来控制流行病。主要结果如下：1）松褐腐病的成功入侵存在一个最小的起始密度阈值。然而，即使松树密度超过最小阈值，由于松树索耶的Allee效应，当其密度很低时，病害也不能入侵。2)最小阈值密度随着根除率的增加而不成比例地增加。3)当根除率较低时，随着初始密度的增加，健康树逃避感染直至流行病消失的概率急剧下降.本文分析了入侵物种的范围扩展，并推导出两种破碎化环境下物种持续生存的条件：（1）分带的环境，有利斑块和不利斑块交替排列; (2)在水平轴和垂直轴上分割的环境，正方形的有利斑块规则地分布在不利环境中。

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.