Coexistence in competing first passage percolation with conversion

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 $\rho>0$. Sites occupied by type 2 then spread at rate $\lambda>0$ through vacant sites \emph{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 non-empty at all times, we say type 1 \emph{survives}. In the case of a regular $d$-ary tree for $d\geq 3$, we show type 1 can survive when it is slower than type 2, provided $\rho$ is small enough. This is in contrast to when the underlying graph is $\mathbb{Z}^d$, where for any $\rho>0$, type 1 dies out almost surely if $\lambda>1$.