An alternative reconciliation of these contradictions is found in the hypothesis that current human genetic variation is the product of a very long history of small population size in equilibrium (Takahata 1993 ; Donnelly et al. 1996 ; Weiss and von Haeseler 1996 ; Fu and Li 1997 ; Harding et al. 1997, 1998 ; Hammer et al. 1998 ; Harpending et al. 1998 ; Zietkiewicz et al. 1998 ). In this long-necked-bottle model, either Ne remained constantly small, or it oscillated frequently to low levels due to periodic events such as glaciations. This hypothesis differs from a single, short-term bottleneck explanation in that the population size is posited to be small for a long enough period for an equilibrium to be reached in most, if not all, neutral gene systems. This could account for differences in coalescence between recombining autosomal and haploid genetic systems. The fact that Ne in haploid systems is expected to be one quarter of that in recombining autosomal systems predicts that we will calculate a fourfold difference in coalescence times if small ancestral population size, and not a single population size bottleneck, is the cause of the variation (Takahata 1993 ).
The long-necked bottle model was developed by Harpending et al. (1998) as part of their analysis of Alu variation. It addresses the implications of an Ne value on the order of 104–105 for a long period in humans. If the Ne calculated from Alu variation (17,500 according to Harpending et al. 1998 ) is a significant fraction of the number of breeding adults in the human species (as Harpending et al.  assume, following Wood  ), there must have been far too few people to occupy all of the continents inhabited during the Pleistocene, or even to inhabit a significant part of one continent. Such a population spread around the world would have a density so low that there would be only about 22 breeding couples in Germany and 35 in France (Takahata and Klein 1998 ). To account for this problem, Harpending et al. (1998, p. 1967) conclude that a population on the order of 104 could not have occupied the entire Old World, but lived for a million years or more "in an African area the size of Rhode Island or Swaziland" as a separate species. This species would presumably be the direct ancestor of modern humans, H. sapiens.
If correct, this would mean that the vast majority of known archaeological sites represent the remains of the activities of extinct human species. These sites are direct evidence of somebody’s behavior, and they show that expansions of the geographic range of humans from Africa to the rest of the Old World may have begun shortly after the appearance of significant changes in human mobility. These changes are suggested by the much larger size, particularly the longer legs, of our earliest direct ancestors some 2 MYA. An early range expansion, with the implication of increasing population size, is indicated by Late Pliocene/Early Pleistocene dates published for the first Indonesian hominids (Swisher et al. 1994 ) and by the early dates variously suggested for the Yuanmou incisors from China (Qing 1985 ) and the Dmanisi mandible from Georgia (Gabunia and Vekua 1995 ). These dates range between 1.9 and 1.6 Myr and are compatible with range expansions that might have quickly followed the 2 Myr African appearance of our lineage. The sites involved are far from Africa and are more likely to be lucky findspots within a large range of new habitations than isolated migrant populations that moved from Africa to the places where they were found. Even if these earliest dates of colonization are incorrect, significant habitation in many areas of the Old World were certainly established by 1.4 and 1.2 MYA (Liu and Ding 1983 ; DeVos 1985 ; Hyodo et al. 1993 ; Wu and Poirier 1995 ). Colonizations at this later time were range expansions that may reflect the adaptive changes in human populations marked by the Acheulean industry, which first appears abruptly in eastern Africa some 1.4 MYA (Asfaw et al. 1992 ). Important behavioral changes are reflected at these earliest sites, where bifaces and picks (rare tools in earlier industries) dominate the Acheulean, and the first case of butchering of mature adults of large mammalian species is found. After these early colonizations, archaeological evidence shows that the range of humans continued to expand into more marginal habitats, and despite significant fluctuations, occupation densities appear to have steadily increased (Klein 1989 ).
But is this archaeological picture of human paleodemography actually shown to be incorrect by the long-bottleneck interpretation? There are some specific reasons to believe that living human populations have multiple roots in widespread past populations. For example, nested cladistic analyses of mtDNA (Templeton 1993 ) and ß-globin (Templeton 1998 ) indicate a long-term occupation of different areas of the Old World over the past 200,000 years or more. The contradiction between a geographically limited ancestral population and the worldwide habitation indicated by both archaeological distributions and some genetic analyses must be resolved.
We examined the consequences of a small population size bottleneck at the beginning of the human lineage to see whether the Alu analysis really requires that human ancestors lived in a very restricted geographic area for a million years. To do so, we developed a compound growth equation to model the paleodemographic history of humanity, much as it has been modeled in the past (Coale 1974 ; Hassan 1981 ; Keyfitz 1966 ). Our exponential growth equation is of the form:
is the initial population size, i
is the rate of increase, and t
is time. We determined parameters for this equation from the small population size bottleneck that subsequently expanded quickly to an initial species size of 10,000 individuals at the time of humanity’s origins some 2 MYA, and from the onset of the Neolithic some 10,000 years ago, when paleodemographers (such as Weiss 1984
) estimate a population of 6 million. Assuming an average generation length of 23 years, as above, we calculated a generational growth rate of 7 x 10-5
. An estimate of effective size from this equation will be conservative in being an overestimate, since inbreeding effective size is smaller than variance effective size in a growing population. This calculation takes into account only the effects of growth common to both effective sizes, so using it to estimate inbreeding effective size establishes base conditions that we can examine deviations from.
The question of interest is what long-term average Ne the simple growth model describes. To calculate the long-term effective population size implied by this simple exponential model, we summed the population sizes over the period from 2 Myr to 10,000 years in 23-year generational intervals. The harmonic mean of the human population, calculated at these generational intervals, is approximately 64,000. If we assume a 1:3 ratio of effective to census population size, as observed over the short term in some human groups (Wood 1987 ) and as assumed by Harpending et al. (1998) , the long-term effective population size over the past 2 Myr can be estimated at about 21,000. This cannot be significantly different from the estimate Harpending et al. (1998) give for the effective population size for humans determined from Alu insertions: 17,500; as noted (and see below), our assumptions generally maximize the Ne estimate.