Stochastics Models in Population Genetics and Ecology
9626764 Krone ABSTRACT The projects in this proposal fall into two categories: population genetics (especially genealogical processes and sampling theory in models with selection) and interacting particle systems (especially multitype models motivated by ecology). Due in large part to the recent explosion of DNA sequence data, two significant trends have emerged in population genetics over the past 20 years. Firstly, the theory now focuses on sampling distributions and the inferences that can be drawn from them. Secondly, the retrospective view (often referred to as coalescent theory) replaced the former prospective view. By studying ancestry, much of the unnecessary structure is stripped away and simple flexible models emerge. Coalescent theory has had much success in analyzing neutral models (i.e., models with no selection); however, the corresponding selective theory has, for the most part, remained out of reach. Recent progress by one of the PI's has opened up many avenues of research on biologically relevant problems. It is shown that genealogical information is embedded in the dual processes of certain interacting particle systems. The investigators propose to extend these techniques to include a variety of models, most notably models involving DNA polymorphism and recombination. Sampling theory for models with selection will also be developed. The sampling theory for a large class of models is characterized and is analogous to the exponential family. Thus the theory of inference, including parameter estimation, information criteria, hypothesis testing, and model selection will be developed rigorously. Interacting particle systems are ideal for modeling biological systems with spatial structure. In recent years, tools have been developed for dealing with multitype particle systems. These multitype systems are particularly relevant in ecological models. A number of multitype models of competition and succession are proposed. Some of the key features in the proposed model s are the effects of different dispersal ranges and random environments. The problems in this proposal are motivated by important issues in population genetics and ecology. Mathematical models are useful for several reasons. The time scales of interest in many biological settings are often hundreds or thousands of years (e.g., forest succession, evolutionary changes, etc.), and hence it is important to be able to predict long-term effects when certain characteristics of the population are changed. Another example of a timely issue which precludes a "wait and see" approach is the extinction of endangered species. The models and analysis proposed will allow for qualitative and quantitative predictions, as well as efficient simulation strategies. Significant contributions to the mathematical underpinnings of these models are also expected. The proposed work on population genetics will address the effects of natural selection on genealogy. This is crucial for developing efficient tests for selection, determining how powerful these tests are, deciding which selective models are realistic and what their properties are. Spatial models in ecology will be approached through "interacting particle systems." Issues such as the spread and control of disease in plant communities, extinction of populations in subdivided habitats, and the effects of inhomogeneous environments will be addressed. The models proposed are much more complicated than most models in the literature because they take into account the spatial structure that is inherent (and important) in real populations.