The dynamics of a one-dimensional stochastic system of classical particles consisting of asymmetric death and branching processes is studied. The dynamical activity, defined as the number of configuration changes in a dynamical trajectory, is considered as a proper dynamical order parameter. By considering an ensemble of dynamical trajectories and applying the large deviation method, we have found that the system might undergo both continuous and discontinuous dynamical phase transitions at critical values of the counting field. Exact analytical results are obtained for an infinite system. Numerical investigations confirm our analytical calculations.
