Non-ergodicity in a 1-D particle process with variable length

We present a 1-D random particle process with uniform local interaction, which displays some form of non-ergodicity, similar to contact processes, but more unexpected. Particles, enumerated by integer numbers, interact at every step of the discrete time only with their nearest neighbors. Every particle has two possible states, called minus and plus. At every time step two transformations occur. The first one turns every minus into plus with probability beta independently from what happens at other places and thereby favors pluses against minuses. The second one is "impartial.'' Under its action, whenever a plus is a left neighbor of a minus, both disappear with probability alpha independently from presence and fate of other pairs of this sort. If beta is small enough by comparison with alpha(2) and we start with "all minuses,'' the minuses never die out.