Bounds for the critical line of the theta -contact processes with 1<or= theta <or=2

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Abstract
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 1c( theta ) is given as the largest root of a cubic equation, theta lambda 3-(3 theta -2) lambda 2-3(2- theta ) lambda +(2- theta )=0. Recently Liggett (1991) reported an upper bound of the critical value for the case theta =1 (the threshold contact process) by a modified version of the Holley-Liggett argument. Our result includes these previous results as special cases.