Controlling stochastic growth processes on lattices: Wildfire management with robotic fire extinguishers

Forest fires continue to cause considerable social and economic damage. Fortunately, the emergence of new robotics technologies, including capable autonomous unmanned aerial vehicles, may help improve wildfire management in the near future. In this paper, we characterize the number of vehicles required to combat wildfires, using a percolation-theoretic analysis that originated in the mathematical physics community. We model the wildfire as a stochastic growth process on a square lattice, where the local growth probabilities depend on the presence of robotic fire-extinguishing vehicles. We develop two control policies: First treats only a fraction of burning nodes at a given time, and the second treats burning nodes only at finite time intervals. We characterize the conditions under which these policies can stabilize a wildfire, i.e., ensure the fire stops eventually almost surely. We also provide computational results which demonstrate our theoretical analysis.