Problems in Probability Motivated by Questions in Ecology

这位调查员将研究一些由生态学引起的问题。这项研究的一个重要方向将是证明支持Durrett和Levin(1994)所提出的图景的一般结果：随机空间模型的行为可以从通过假设所有站点总是独立而获得的平均场节点的行为来预测。他还将继续杂交区的工作，现在进展到这些系统的PDE极限的推导，当选择趋于0时，以及空间不均匀模型的研究，以了解镶嵌杂交区的结构，这些结构已经在板球物种Gryllus Firus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极限，研究进化军备竞赛的渐近行为，以及证明吉普赛飞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？这种比较被比作一个均匀的混合系统，在这个系统中，每个人都与所有其他人平等地相互作用。在过去的五年里，这位研究人员和他的合作者开发了一些预测这个问题的答案的规则，并将它们应用于各种生物系统。在接下来的三年里，他们将把这一理论应用于一些新的例子。他们将研究的最重要的问题之一是：多样性的极限是什么？他们将通过调查随机竞争模型中可以共存的物种数量来数学地解决这个问题。其他问题将涉及吉普赛飞蛾种群大小的波动和由于空间上不同的选择而产生的杂交区域的结构。

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.