Phase Transition for the Chase-Escape Model on 2D Lattices

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.