Genealogy of branching populations, new descriptions of large random trees, and a mathematical framework for the evolution of senescence

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Not only do most complex organisms age, in the sense that they degenerateas time passes, but they often do so according to similar patterns. Biologistshave proposed qualitative explanations of this fact based on the concept ofmutation-selection balance: that natural selection does not oppose mutationswith deleterious effects if those effects are felt primarily after the bearer has been able to reproduce. However, it has been a challenge to provide a rigorous quantitative version of this notion because it involves large numbers of genes with complex interactions, and yet the resulting patterns of aging should be fairly insensitive to the underlying specifics. One part of the project aims to improve understanding of this important problem in evolutionary biology and biodemography using ideas from infinite-dimensional, non-linear dynamical systems.Another common question in evolutionary biology is whether evolutionaryprocesses leave some sort of signature in the shape of the phylogenetic tree ofa collection of present day species. Similarly, with the increasing availability of large amounts of data, computer scientists wonder if the current structure of a large network that has grown over time reveals something about the dynamics of that growth. Motivated by such questions, one seeks "statistics" that somehow summarize the shape of trees and more general graphs, and to determine the behavior of these quantities when the graphs are generated by specific mechanisms. Mathematically natural candidates for such descriptors are the so-calledeigenvalues of the graph. Two further goals of the project are, firstly, to userelatively simple techniques from linear algebra and probability to understandthe eigenvalues of broad classes of large random graphs, and, secondly, to useideas from the theory of Martin boundaries to delineate fully all the possibleways that certain models for growing trees can behave when the tree becomes large.The last objective of the proposal is to study how the age of the most-recent-common ancestor (MRCA) of a population changes over time. Any asexually reproducing population has a unique MRCA, from whom the entire population is descended. In sexually reproducing species, the same is true for each non-recombining piece of DNA. For instance, our "mitochondrial Eve" from whom all modern-day humans inherited their mitochondrial DNA is estimated to have lived around 180,000 years ago. As time progresses into the future, eventually the mitochondrial lineages of all but one of the daughters of the current mitochondrial Eve will die out, at which point the new mitochondrial Eve will have lived somewhat later in time. The proposed research will investigate the dynamics by which the age of the MRCA varies.