Problems in Probability Motivated by Questions in Ecology

[877066]这位调查员将研究生态学中出现的一些问题。研究的一个重要方向将是证明一般结果，以支持Durrett和Levin（1994）所提倡的图景：可以通过假设所有站点总是独立的平均场ODE来预测随机空间模型的行为。他还将继续对杂交带的研究，目前正在研究这些系统在选择趋向于0时的PDE极限的推导，以及空间非均匀模型的研究，以了解马赛克杂交带的结构，这种结构已经在蟋蟀种Gryllus firmus和G. pennsylvanicus的野外工作中看到。其他项目包括通过分析有限网格上的Tilman竞争模型来理解多样性的限制，研究进化军备竞赛的渐近行为，以及证明舞舞蛾种群的简单离散时间模型中混沌振荡的存在。推动这项研究的基本问题是：个体的空间分布何时会改变生物系统的行为？将其与均匀混合系统进行比较，其中每个个体与所有其他个体平等地相互作用。在过去的五年中，研究者和他的合作者已经开发了一些规则来预测这个问题的答案，并将它们应用于各种生物系统。在接下来的三年里，他们将把这个理论应用到一些新的例子中。他们将研究的最重要的问题之一是：多样性的限制是什么？他们将通过研究在随机竞争模型中可以共存的物种数量，从数学上解决这个问题。其他问题将涉及舞毒蛾种群规模的波动以及空间选择变化导致的杂交带结构。

9877066This investigator will study a number of questions that arise from ecology. An important direction of the research will be to prove general results in support of the picture advocated by Durrett and Levin (1994): the behavior of stochastic spatial models can be predicted from that of the mean field ODE obtained by pretending all sites are always independent. He will also continue work on hybrid zones, progressing now to the derivation of PDE limits for these systems as selection tends to 0, and the study of spatially inhomogeneous models to understand the structure of mosaic hybrid zones that have been seen in field work on the cricket species Gryllus firmus and G. pennsylvanicus. Other projects include understanding limits to diversity by analyzing Tilman's competition model on a finite grid, studying the asymptotic behavior of an evolutionary arms race, and demonstrating the existence of chaotic oscillations in a simple discrete time model of gypsy moth populations.The basic question that drives this research is: when does the spatial distribution of individuals change the behavior of a biological system? The comparison is made to a homogeneously mixing system in which each individual interacts equally with all the others. In the past five years the investigator and his collaborators have developed some rules for predicting the answer to this question and have applied them to a variety of biological systems. In the next three years they will apply this theory to some new examples. One of the most important questions they will study is: What are the limits to diversity? They will address this question mathematically by investigating the number of species that can coexist in a stochastic competition model. Other questions will concern the fluctuations of the size of gypsy moth populations and the structure of hybrid zones that result from spatially varying selection.