Bounds for the critical line of the theta -contact processes with 1<or= theta

We study a family of the one-dimensional contact processes introduced by Durrett and Griffeath (1983), which is parametrized by theta . For each theta >or=1, there is a unique critical value lambda c ( theta ) so that any process becomes extinct with probability 1 for lambda < lambda c ( theta ), but all processes starting from non-empty initial states have positive probabilities of survival for lambda > lambda c ( theta ). In this paper we give rigorous upper and lower bounds for the critical line lambda = lambda c ( theta ) for 1