Setting r to be above 3.0 and you'll start to see periodic behaviour: the graph doesn't settle down to a single value. Inch it up further and you'll see periods of 4 (r=3.5), 8, and so on. The doubling of periods start to come faster and faster, and then, at about r=3.7 you're into chaos. Mess around with a few values. My favourite is r=3.88989. (Oh, don't set r below 1 because you'll kill off all the fish).
Ecologists at the time assumed that the numbers were fluctuating around some underlying equilibrium point. (Those folk didn't have PCs to crank the calculations out...ouch). Gleick says: "...in the back of the ecologist's minds was always the assumption that an erratic string of numbers probably meant that the calculator was acting up or, or just lacked accuracy."