One of the problems that we have modeling the military time series is that there is simply not enough data to be able to produce a general model of the process that produced the series. A common way to model a time series is to decompose the series into a number of separate components:
Trend: Does the series generally increase or decrease over time? Is the trend an exponential curve as we saw with the population?
Seasonality: Does the series exhibit periodic rises and falls at a set number of intervals? For monthly data it is common to observe a period cycle of 12 months.
Cycles: Are there longer-term cycles in the dataset that span multiple seasons? For example, in financial data we might observe multi-year cycles corresponding to periods of expansion and recession.
Another way of specifying the issue with the military data is that there is not enough information to determine whether or not there is a trend, and whether the observed peak is part of a seasonal or...