We study a system of particles which evolve on the lattice in discrete generations: Each particle produces a Poissonian number of offspring which independently move to a uniformly chosen site within a fixed distance from their parent's position. Whenever a site is occupied by at least two particles, all the particles at that site are annihilated.
This can be interpreted as a very strong form of local competition and implies that the system is not monotone. We prove that the system dies out when the mean offspring number is too small or too large, for a set of intermediate values and sufficiently large jump range the system survives with positive probability. In a restricted parameter range, we can strengthen this to complete convergence with a non-trivial invariant measure. A central tool in the proof is comparison with oriented percolation on a coarse-grained level, using suitably tuned density profiles which expand in time and are reminiscent of discrete travelling  wave solutions.

Joint work with Alice Callegaro (TU Munich), Jiří Černý (University of Basel), Nina Gantert (TU Munich) and Pascal Oswald (University of Basel)