The authors consider the birth-death-multiple immigration (BDMI) process characterized by the rate equation \[ \text{d}P_ N(t)/\text{d}t =\mu(N+1)P_ {N+1} -(\lambda+\mu)NP_ N +\lambda(N-1)P_ {N-1} -P_ N\sum_ {m=1}^ \infty\alpha_ m +\sum_ {m=1}^ N\alpha_ mP_ {N-m}, \] where \(P_ N(t)\) is the probability that the population comprises \(N\) members at time \(t\), births and deaths occur at constant rates \(\lambda\) and \(\mu\), respectively, and the population size also increases through immigration of \(m\)-tuplets arriving at rates \(\alpha_ m\geq0\) for \(m=1,2,\ldots\).
The authors calculate the immigration rates \(\alpha_ m\) at which the BDMI process has a stationary discrete stable distribution with power-law tail \(P_ N\sim1/N^ {1+\nu}\), provided that \(0<\nu<1-2\lambda/\mu\). The generating function of the stationary distribution is of the form \(Q(s)=\exp\{-as^ \nu/\nu(\mu-\lambda)\}\). The partial case of a death-multiple immigration (DMI) process (\(\lambda=0\)) is also considered.
The separate problem of monitoring and characterizing the fluctuations is studied by analyzing the statistics of individuals that leave the population.