  Chase-Escape is a simple stochastic model that describes a predator-prey
interaction. In this model, there are two types of particles, red and blue. Red
particles colonize adjacent empty sites at an exponential rate $\lambda_{R}$,
whereas blue particles take over adjacent red sites at exponential rate
$\lambda_{B}$, but can never colonize empty sites directly. Numerical
simulations suggest that there is a critical value $p_{c}$ for the relative
growth rate $p:=\lambda_{R}/\lambda_{B}$. When $p<p_{c}$, mutual survival of
both types of particles has zero probability, and when $p>p_{c}$ mutual
survival occurs with positive probability. In particular, $p_{c} \approx 0.50$
for the square lattice case ($\mathbb Z^{2}$). Our simulations provide a
plausible explanation for the critical value. Near the critical value, the set
of occupied sites exhibits a fractal nature, and the hole sizes approximately
follow a power-law distribution.
