Dynamics of Diversification in State Space

America spread west in the mid- to late nineteenth century. In our own century, the story of mass migration has been told and retold in countless ballads and novels. And in now-classic movie westerns, it has been epitomized in scenes of long wagon trains winding across the prairie toward the setting sun. Census data from that period confirm the demographic aspect of the story: the mean location of Americans--the country’s center of gravity, so to speak--shifted westward. However, the data also show that from two broad belts occupying the middle third of the country, the west-central states and the mountain states, about the same number of people moved east as moved west, at least during the first part of the migration from 1850 to 1880 (data from Mitchell, 1983). Actually slightly more moved east during this period. The pattern is puzzling at first. The country as a whole spread west, while slightly more people migrated east. But it is consistent with a simple explanation, illustrated abstractly in Figure 1A. The horizontal axis is location in space, reduced to a single dimension, east and west, and the vertical axis is time. Suppose for simplicity that we are dealing with an asexually reproducing species instead of people. The population begins as a single individual. As time passes, the individual reproduces or splits, and its progeny do likewise, causing the population to grow. Across the country, individuals migrate east slightly more often than they migrate west, except at one point in space where migration to the east is blocked by a boundary, the Atlantic Ocean. The result is that, despite the slight eastward bias in the movement of individuals, the population as a whole diffuses slowly to the west. More generally, the behavior of the system at the large scale seems to be independent of its dynamics at the small scale. ...