Criticality of a dissipative self-organizing process in a dynamic population

We derive a general formulation of the self-organized branching process by considering sandpile dynamics in an evolving population characterized by "birth" (excitation) and "death" (de-excitation) of active sites (

z=1

). New active sites are born in empty sites (

z=0

) with a probability of

\eta

, whereas active sites die, thus becoming empty, with a probability

\lambda

. Subsequently, when an active site becomes unstable (

z=2

), it topples by transferring two grains to two randomly chosen sites with probability

\alpha

or, by transferring only one grain to a randomly selected site (while retaining the other) with probability

\beta=1+\frac{\lambda}{\eta}-2\alpha

, thus remaining active after toppling. We show that when sandpile dynamics occurs in an evolving population, self-organized criticality, characterized by a power-law avalanche size distribution with exponent