Coexistence in competing first passage percolation with conversion

Abstract

We introduce a two-type first passage percolation competition model on infinite connected graphs as follows. Type 1 spreads through the edges of the graph at rate 1 from a single distinguished site, while all other sites are initially vacant. Once a site is occupied by type 1, it converts to type 2 at rate $\mathit{\rho }>0$. Sites occupied by type 2 then spread at rate $\mathit{\lambda }>0$ through vacant sites and sites occupied by type 1, whereas type 1 can only spread through vacant sites. If the set of sites occupied by type 1 is nonempty at all times, we say type 1 survives. In the case of a regular d-ary tree for $\mathit{d}\ge 3$, we show type 1 can survive when it is slower than type 2, provided ρ is small enough. This is in contrast to when the underlying graph is ${\mathbb{Z}}^{\mathit{d}}$, where for any $\mathit{\rho }>0$, type 1 dies out almost surely if $\mathit{\lambda }>{\mathit{\lambda }}^{\prime }$ for some ${\mathit{\lambda }}^{\prime }<1$.