For determining the final, steady-state behavior, it is sometimes interesting to compare several initial values. However, for verification of the precise steady state, you should always use an initial of 1/2. This is because is a ``critical value'' of the map, or a point at which . An orbit arising from a critical value must lie in the ``basin of attraction''  for an attracting periodic point  (provided that it exists).
Then, a definite change in behavior occurs for the following a values:
(The very adventurous may wish to examine .) You are now looking at the famous period-doubling cascade to chaos. Do you see why the period of the population doubles for each new case? Each of the above a values is in the center of a period-multiplying region; how would you characterize each region? The boundaries of these regions are specified in Figure 1.