The archaeological data for each site provides a sequence of occupational durations interlaced with periods of no occupation

In Figure 1 For each occupation, the degree of dependence on terrestrial resources and marine resources will be quantified based on archaeological assemblages and other cultural contexts. One of the theoretical goal of the project is to derive distributions of site occupation times based on characteristics of the island ecosystems (land area, relative isolation from other islands, faunal resources at the beginning of each occupation), stochastic events (earthquakes and volcanic events), and characteristics of the occupant cultures. Additionally, we will to derive a distribution of the duration of site abandonment based on principles from island biogeography and the technology available for human colonization. These distributions, in turn, will be used with observations derived from the archaeological, geological, and botanical field observations to estimate a limited set of ecological parameters. In what follows, we briefly outline the characteristics of our modeling efforts to date.

*Duration of site
abandonment. *Call* f _{A}*(t) the
probability density function for times between abandonments, corresponding to
the alpha

In Figure 1 This distribution can be decomposed into a convolution of two component. The first component is a period of ecosystem re-establishment following the previous occupation. This duration will depend, in turn, on the type of subsistence. For the Jomon period, we expect that the previous abandonment arose by overexploitation of the dominant terrestrial mammal. By contrast, recovery of marine resources during the Okhotsk period should occur much more rapidly, and this distribution is irrelevant for the military period.

If a primary subsistence species is driven to extinction,
the distribution of times to *de novo*
recolonization can be taken from standard island biogeography theory, with the
critical parameters being island size and isolation. When the population is reestablished from low levels, the
distribution for this first component follows directly from the density-dependent
population growth model.

The second component, following ecosystem re-establishment, is the "rediscovery" period. This is treated as a constant hazard for a given transportation technology, island size, and degree of isolation. Hence, the distribution of times to resettlement is a convolution of the distribution of times to ecosystem re-establishment and a negative exponential distribution of times to rediscovery.

*Duration of site
occupation: *The probability density function, *f _{O}*(

In Figure 2
, but
this model will be refined and developed as a part of the project. A closed form for the distribution of *f _{O}*(

*Statistical methods.* Maximum likelihood will be used to estimate
parameters. For a series of *I* islands, each with *M _{i} *sites, each with a
sequence of

In Figure 1
, we can construct a
likelihood out of the distributions of occupation and abandonment times, *f _{o}*() and

where **q _{o}**
and

In Figure 1

A number of implicit assumption about independence are
made for this likelihood. First, we
assume all occupational duration's are independent of each other except for
characteristics shared through the array of measured covariates **x _{ijk}**. Additionally, the duration of each
abandonment is assumed to be independent of the preceding occupational duration
(except through

Maximum likelihood estimates for **q _{o}**
and

A number of models of different parameterizations and complexity will be estimated during the estimation phase of this project. We will use Akaike's information criterion (AIC, Akaike 1992, Burnham and Anderson 1998) to select among models with different parameterizations and structures that most parsimoniously approximates the true underlying model. For all parameter estimates, standard methods (asymptotic variance-covariace matrix, profile likelihood, or bootstrap confidence intervals) will be used to estimate parameter uncertainty. The particular method will depend on model complexity.