  We study a competitive stochastic growth model called chase-escape in which
red particles spread to adjacent uncolored sites and blue only to adjacent red
sites. Red particles are killed when blue occupies the same site. If blue has
rate-1 passage times and red rate-$\lambda$, a phase transition occurs for the
probability red escapes to infinity on $\mathbb Z^d$, $d$-ary trees, and the
ladder graph $\mathbb Z \times \{0,1\}$. The result on the tree was known, but
we provide a new, simpler calculation of the critical value, and observe that
it is a lower bound for a variety of graphs. We conclude by showing that red
can be stochastically slower than blue, but still escape with positive
probability for large enough $d$ on oriented $\mathbb Z^d$ with passage times
that resemble Bernoulli bond percolation.
