# Discrete or geometric growth in populations with nonoverlapping generations

The use of an appropriate model depends first on the life history of the organism. So you first need basic information on the life cycle of the species. In this first model of density-independent growth, the population has a life history with discrete, non-overlapping generations. That is, there are no adult survivors from one generation to the next. Examples include annual plants, annual insects, salmon, periodical cicadas, century plants, and certain species of bamboo. In most of these cases the organism passes through a dormant period as a spore, a seed, or an egg, and/or a juvenile stage such as a larva or pupa. Once the adults reproduce, they perish, and the future of the population is based on the dormant or juvenile stage of the organism. As noted above, when modeling such populations we usually collapse fertility and mortality into one constant, R, the net replacement rate or net growth rate per generation - or X, the finite rate of increase, when measuring growth per specific time period. When we are discussing annual plants or insects, X, the growth rate per year, and R, the growth rate per generation, are identical, since generation time equals one year. However, in some populations, such as the periodical cicada (Magicicada septendecim), generation time equals 13 or 17 years, and in these cases it is useful to make a distinction between the growth rate per generation and a finite rate of increase. That is, R ^ X, when T, the generation time, ^ 1 year.

To find R we often count one life stage of the population in successive years. For gypsy moths (Lymantria dispar) we estimate R by counting egg masses in successive years (see Example 1.1). R is estimated from the ratio of egg masses at time t + 1 versus time t. For the periodical cicada (Example 1.2), however, we would have to wait 17 years between generations before we could estimate R. The overall model is based on finding successive estimates of the growth rate based on: