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Chase-escape percolation is a variation of the standard epidemic spreadmodels. In this model, each site can be in one of three states: unoccupied,occupied by a single prey, or occupied by a single predator. Prey particlesspread to neighboring empty sites at rate $p$, and predator particles spreadonly to neighboring sites occupied by prey particles at rate $1$, killing theprey particle that existed at that site. It was found that the prey can survivewith non-zero probability, if $p>p_c$ with $p_c<1$. Using Monte Carlosimulations on the square lattice, we estimate the value of $p_c = 0.49451 \pm0.00001$, and the critical exponents are consistent with the undirectedpercolation universality class. We define a discrete-time parallel-updateversion of the model, which brings out the relation between chase-escape andundirected bond percolation. For all $p < p_c$ in $D$-dimensions, the number ofpredators in the absorbing configuration has a stretched-exponentialdistribution in contrast to the exponential distribution in the standardpercolation theory. We also study the problem starting from the line initialcondition with predator particles on all lattice points of the line $y=0$ andprey particles on the line $y=1$. In this case, for $p_c<p < 1$, the center ofmass of the fluctuating prey and predator fronts travel at the same speed. Thisspeed is strictly smaller than the speed of an Eden front with the same valueof $p$, but with no predators. At $p=1$, the fronts undergo a depinningtransition. The fluctuations of the front follow Kardar-Parisi-Zhang scalingboth above and below this depinning transition.

Chase-Escape Percolation on the 2D Square Lattice