Continuous phase transitions in the $k$-creation process without stirring in $d=1$

This paper is a companion to one in which we prove there is a discontinuous phase transitions in the $k$-creation process with fast stirring in $d=1$ when $k \ge 2$. Dickman and Tomé (1991) introduced models on $Z$ in which $k$ consecutive occupied sites give birth at rate $λ$ and individual particl