April 24, 2014

Program in Probability and Its Applications
Coxeter Lecture Series

Rick Durrett, Cornell University
May 11 - 12, 1999

May 11, 1999

Repeats of short patterns such as CACACA... appear in our DNA at a frequency far greater than if they were due to chance. These repeats have a high mutation rate compared to nucleotide substitutions, so they are useful for locating genes and for understanding population structure on the scale of thousands of years. In this talk we will describe a simple model of the evolution of these repeat sequences developed in joint work with Semyon Kruglyak, Malcolm Schug, and Chip Aquadro, and we will fit this model to DNA sequence data from yeast, fruit flies, mice, and humans. The model fits the observed data and gives parameter estimates that are consistent with experimental work.


May 11, 1999

In many situations in biology it is useful to consider a model that represents space as a grid of sites, each of which can be in one of a finite number of states and changes at a rate that depends on the state of finitely many neighbors. Durrett and Levin proposed in 1994 that the behavior of these systems can be inferred by looking at the associated "mean field" ODE that is obtained by pretending that all sites are always independent. We will describe the answers that result from this approach for a number of systems of interest in biology and illustrate our results by a videotape of computer simulations.


May 12, 1999

It has long been known that scaling limits of critical branching random walks lead to an interesting measure valued process, now called super-Brownian motion. Here we describe recent results which show that in two or more dimensions super-Brownian motion is the limit of rescaled contact processes and voter models. To get more interesting limits in d=2 or 3 one can (we think) take multitype contact processes like the colicin systems of Durrett and Levin (1997) and let the interaction parameters between species get large at the right rate to get convergence to models that generalize the interacting super-Brownian motions constructed by Evans and Perkins (1998). Some theorems and simulations will be shown in support of this picture.