Title:Matched Asymptotics for Reaction Models

Speaker:Cameron Smith and Raymond Puzio

Abstract:
When modelling biochemical and biological processes as reactionnetworks, one encounters situations in which a new species appears andgrows or goes extinct. In order to study such situations properly, thereaction network should be modeled stochasticly to describe dynamics involving small numbers. To analyze the stochastic dynamics, we have developed a technique of matched asymptotics which has three components:
Firstly, when the numbers of some of the species are small, we employ a series in which the leading term is a branching process describing non-interacting individuals and the higher order terms account for the interactions perturbatively. Secondly, when the numbers are high, we employ a series whose leading term is the deterministic dynamics and whose higher order terms describe fluctuations about it in powers of 1/N. Thirdly, we have a matching condition which unifies these two descriptions into a coherent whole. We shall explain these methods in the context of a simple Lotka-Volterra example and sketch applications of the technique to movement between chemical organizations, origin of hypercycles, and community assembly.