Colloquia — Fall 2005
Wednesday, November 23, 2005
Quasi-stationarity and exit times for a size-dependent branching process
Åbo Akademi University
A size-dependent branching model of the evolution of a single species population is introduced. The short-term behavior mimics the corresponding deterministic dynamical system very closely, but the branching model hits \(0\) eventually, i.e., the population goes extinct in contrast with the deterministic model which remains positive at all times. It turns out that the so-called quasi-stationary distribution (q.s.d.) of the branching model gives the right asymptotics. The q.s.d. approximates the attracting limit cycles of the deterministic system. I will also discuss some results, mainly due to Klebaner and Liptser, on the order of magnitude of the extinction times.
Tuesday, November 1, 2005
Some Problems on the Chromatic Number of Infinite graphs
We survey some problems concerning the chromatic number of infinite graphs. A recent negative solution to Taylor's problems will be sketched.