Frequency distributions from birth, death, and creation processes

The time-dependent frequency distribution of groups of individuals versus group size was investigated within a continuum approximation, assuming a simplified individual growth, death and creation model. The analogy of the system to a physical fluid exhibiting both convection and diffusion was exploited in obtaining various solutions to the distribution equation. A general solution was approximated through the application of a Greenʹs function. More specific exact solutions were also found to be useful. The solutions were continually checked against the continuum approximation through extensive simulation of the discrete system. Over limited ranges of group size, the frequency distributions were shown to closely exhibit a power-law dependence on group size, as found in many realizations of this type of system, ranging from colonies of mutated bacteria to the distribution of surnames in a given population. As an example, the modeled distributions were successfully fit to the distribution of surnames in several countries by adjusting the parameters specifying growth, death and creation rates.