A disparate collection of mathematical ideas and methods for quantifying populations is presented in the literature. This paper attempts to achieve the multiple purposes of review, summary, and guidance to new areas to better characterize the system and offer illumination on the alternatives. Mathematical approaches are given for modeling the total population growth and decline, for two competing species, and for an age-dependent gender mixture model, along with the differential Lancaster equation system for conflict with a solution. A simple fractional calculus differential equation model generalizing the integer calculus models is defined and solved. Then a continuum description of both gender and age-dependent birth processes which has not been previously specified is developed. Discrete individuals suspended in an evolving continuous fluid motion is applied. Several population studies for siting nuclear faciliies are presented.