A SURPRISING POISSON PROCESS ARISING FROM A SPECIES COMPETITION MODELÅ

Rick Durrett and Vlada Limic

Motivated by work of Tilman (1994) and May and Nowak (1994) we consider a process in which points are inserted randomly into the unit interval and a new point kills each point to its left independently and with probability $\theta$. Intuitively this dynamic will create a negative dependence between the number of points in adjacent intervals. However, we show that the ensemble of points converges to a Poisson process with intensity $1/\theta(1-x)$, and the number of points at time $t$ grows like $(\log t)/\theta$.