In this paper we study activity fluctuations in an asymmetric death-branching process in one-dimension. The model, which is a variant of the asymmetric Glauber model, has already been studied in Torkaman and Jafarpour (2015 J. Stat. Mech. P01007). It is known that in the low-activity region i.e. below the typical activity in the steady-state, the dynamical free energy of the system can be calculated exactly. However, the behavior of the system in the high-activity region is different and more interesting. The system undergoes a series of dynamical phase transitions. In present work we justify the hierarchy of dynamical phase transitions in terms of effective interactions in the system. It turns out that the effective interactions are long-range and that they can be described in terms of interactions between repelling shock fronts.
