Sustained oscillations and density dependent Markov chains.
Peter Baxendale, University of Southern California
ABSTRACT: A number of recent papers have considered the phenomenon of sustained oscillations in density dependent Markov models of processes in areas such as interacting populations, epidemics, and chemical kinetics. This talk will consist of three parts. One part will describe a recent stochastic averaging result which explains the existence of sustained oscillations for a class of 2-dimensional diffusion processes. A second part will survey the transition from the original density dependent Markov chain to a diffusion approximation. (For mathematicians this is work of Kurtz from the 1970s; for physicists, chemists and biologists this is the Van Kampen expansion). The last part will bring these ideas together to provide both qualitative and quantitative results on sustained oscillations for density dependent Markov chains. This is joint work with Cindy Greenwood.