Dynamics in ecological population genetics
Time and place
1 PM on Thursday, March 12th, 2009; NAC 4/113
Prof. Judy Miller (Georgetown University)
Abstract
Population genetics is the study of how the genetic makeup of a population— of any species— changes over time in response to evolutionary forces. It is a fruitful source of mathematical problems in both deterministic and stochastic dynamics. I will discuss two problems arising in this field.
One problem concerns a system of ordinary differential equations (ODEs) modeling a physically linked pair of loci (or genes), in a population consisting of two demes (or subpopulations). Using geometrical singular perturbation theory, it is possible to establish rigorously the long-time behavior of the system. The analysis provides an unusually complete picture of the dynamics of an ODE system that generalizes and extends systems with a long history of study in mathematical population genetics. It also yields a practical conclusion: it allows biologists to determine the density of genomic markers (roughly speaking, the amount of detail in a genetic map) they must achieve in order to make reliable inferences about a gene from a nearby marker.
The second problem concerns the fate of a mutation that arises in a population while it is invading new territory. The probability of survival of such a mutation, and the long-term spatial distribution of mutants when survival occurs, have attracted much recent interest. A careful statistical analysis of simulation results in this setting corrects and illuminates a number of assertions that have been published to date, and motivates rigorous analysis of both stochastic and deterministic (integrodifference equation) models of invasion genetics.