Mathematical Sciences: Modelling Territorial Patterns and Stability of Wolf-Deer Interactions

研究了狼（Canis lupus）领地形成的时空动态及其对白尾鹿（Odocoileus virginians）分布的影响。他们的模型基于明尼苏达州西北部的实地研究，该研究表明，相邻的狼群形成了不同的领土，这些领土被“缓冲区”隔开，狼群很少侵入。这些缓冲区可以为主要猎物——白斑鹿提供空间避难所。野外研究表明，鹿群密度在领土间区域（保护区）较高，而在已建立的领土中心附近（鹿群可能受到强烈的捕食）较低。缓冲带还可以稳定狼鹿之间的相互作用，因为它提供了一个鹿群，可以让狼在遭受严重捕食损失的地区重新定居。研究人员建立的数学模型用非线性偏微分方程组耦合描述狼与狼、狼与鹿的行为相互作用，用非线性差分方程组耦合描述狼与鹿的长期生态相互作用。模型方程的分析将采用诸如摄动理论和有限差分数值方法等领域的数学和数值技术。这将导致对狼领地模式形成的详细定量理解，并说明领地性如何影响鹿的分布和丰度。该研究将在一般的领土格局形成领域有更广泛的应用；这种模式为许多哺乳动物和鸟类的社会组织提供了基础。

The investigators study the spatial and temporal dynamics of territory formation for wolves (Canis lupus) and the resulting impact on the distribution of white-tailed deer(Odocoileus virginians). Their model is based on field studies from northwestern Minnesota that indicate that adjacent wolf packs form distinct territories that are separated by "buffer zones" where packs rarely trespass. These buffer zones can provide a spatial refuge for the primary prey species: white-tailored deer. The field studies show that deer density is higher in the inter- territorial areas (refuges) and is lower near the center of established territories (where deer populations may be subject to intense predation). The buffer zones may also stabilize wolf- deer interactions by providing a reservoir of deer that can recolonize areas of wolf territory that have suffered heavy predation losses. The mathematical model the investigators develop describes the behavioral wolf-wolf and wolf-deer interactions using a system of coupled of nonlinear partial differential equations, and describes the long-term ecological wolf-deer interactions using a system of coupled nonlinear difference equations. The analysis of the model equations will employ mathematical and numerical techniques from areas such as perturbation theory and finite difference numerical methods. This should lead to a detailed quantitative understanding of wolf territorial pattern formation, and an illustration of how territoriality influences the distribution and abundance of deer. The research will have a wider application to the general area of territorial pattern formation; such patterns provide the basis for social organization in many mammals and birds.