Are there bushes in a forest?

In this paper we consider Markov processes in which the state at time t is & : Zd + (0, 1,2}. We think of 0 = grass, 1 = bushes, and 2 = trees, and formulate the evolution as follows: (i) l’s and 2’s each die (i.e. become 0) at rate 1. (ii) l’s (resp. 2’s) give birth at rate A, (resp. A,). (iii) If the birth occurs at x the offspring is sent to a site chosen at random from {y: y -x E X}, K = the set of neighbors of 0. (iv) If l,(y) 2 t,(x) then the birth is suppressed. The last rule reflects the fact that grass, bushes, trees is a successional sequence, i.e. each plant can displace its predecessor. Since 2’s can replace l’s or O’s, it should be clear that 5, = {y: t,(y) = 2) is a Markov process. In the terminology of Liggett (1985) or Durrett (1988), it is the contact process with neighborhood set N. Let A,=inf{A,: P(l:#@ for all r~0)>0}, where ly={y:[:“(y)=2} when &o(O) = 2 and [y(x) = 0 for x f 0. Here the superscript 20 on 5 suggests a 2 in a sea of O’s, and the superscript 0 on t indicates that this set valued process has Si = (0). If A, < A, then the 2’s die out and the process reduces to a one type contact process, so we will only be interested in what happens when A2 > A,. Our first result shows that the one dimensional nearest neighbor case is not interesting.