Inhomogeneous birth-death and birth-death-immigration processes and the logarithmic series distribution. Part 2

A simple graphical argument described in a previous paper is used to show that the zero-modified geometric form of the population-size distribution of a time-inhomogeneous birth-and-death model is maintained when the death rates of individuals depend on their ages and times of birth. An explicit form for the population-size distribution is found. Certain models incorporating immigration, but again with general lifetime distributions, continue to lead to Fisher's logarithmic series distribution for the abundance of families of a particular size. It is shown that the zero-modified geometric form no longer holds if the model is extended to incorporate age-dependent birth rates.