281886

Society for American Archaeology

Simulating Success or Failure: Another Look at Small-Population DynamicsAuthor(s): Sylvia W. Gaines and Warren M. GainesSource: American Antiquity, Vol. 62, No. 4 (Oct., 1997), pp. 683-697Published by: Society for American ArchaeologyStable URL: http://www.jstor.org/stable/281886 .Accessed: 21/12/2010 07:39

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SIMULATING SUCCESS OR FAILURE: ANOTHER LOOK AT SMALL-POPULATION DYNAMICS

Sylvia W. Gaines and Warren M. Gaines

Simulation can be an effective toolfor investigating the demography of small, prehistoric Southwest Pueblo communities. The model presented here incorporates biological and physiological, cultural, and behavioral characteristics and tracks each individual as the simulation of a small population is carried forward through 70 years of annual iterations. Sensitivity analyses are performed for a suite of critical parameter values. Many of parameters and functions are probabilistic, and Monte Carlo techniques are used to obtain statistically significant results. Simulation results are collected on numerous variables that profile the individual and group characteristics such as mortality, immigration to emigration ratio, nuclear family for- mination, and distribution of population size and mix. Initial success is dependent on the attributes of the founding population and its gender mix. The long-term survival of a small population is extremely sensitive to the mortality schedule, attributes of the founding population, and marriage-residence rules. Small shifts in the age-specific mortality statistics dramatically affect the population growth and the frequency of site collapse. The consequences of inaccuracies in mortality statistics are highlighted.

El tema de este articulo es la simulacion comprensiva de una pequena comunidad prehistorica Pueblo del suroeste norteamer- icano que incluye un modulo del ciclo de vida de la poblacion. El modulo del ciclo de vida de la poblacion incorpora caracteristicas biologicas, culturales y conductuales y sigue la trayectoria de cada individuo al llevar la simulacion de esta pequeina poblacion a lo largo de 70 anios de repeticiones anuales. Los pardmetros y lasfunciones empleadas son fdcilmente ajustables y pueden ser examinadas en una amplia escala de condiciones. Se utilizan las tecnicas Monte Carlo y se presentan los resultados para un grupo de valores de pardmetros criticos. La supervivencia de una pequenia poblacion demuestra ser extremadamente sensible al calendario de supervivencia y a las reglas de residencia marital. El exito inicial es dependiente de la esperanza de vida de la poblacion fundadora y de la mezcla de sexos de sus descendientes. Los resultados de la simulacion se tomaron de numerosas variables las cuales perfilan las caracteristicas individuales y grupales tales como record de mortalidad, tendencias en la proporcion inmigracion/emigracion, organizacion de la familia nuclear y mezcla poblacional. Cambios pequeniios en las estadisticas de edad especifica de sobrevivencia demuestran que estos influyen dramaticamente en el potencial de crecimiento de la poblacion y en la frecuencia de su colapso. Se destacan las consecuencias de las inexactitudes en las estadisticas de mortalidad.

A model can be more accurate than the data used to build it because it amplifies hidden patterns and discards unwanted noise.-Gauch (1993:468)

As archaeologists have become more sophisticated in the use of modeling systems, the scope has broadened from sim-

ple models illustrating a set of events to one of analyzing the impact of underlying causes. Numerical simulation forces the transformation of conjecture into formal models, which allow explo- ration of a multitude of approaches and ideas. Hypothesized explanations can be evaluated in the

context of alternates, and this "virtual archaeology" permits extraction of schema from data that may not be evident from more conventional approaches.

This expanded theoretical arsenal for treating phenomena not directly observable permits analysis of cultural and physiological influences addressed in this paper. The impetus of the research stemmed from questions concerning the great number of small habitation sites in the American Southwest. While many sites were seasonal or used for limited activities, a significant number have been identified as permanent habitations (Hantman 1989). It is this

Sylvia W. Gaines * Department of Anthropology, Arizona State University, Tempe, AZ 85287-2402 Warren M. Gaines * CSD&A, 3602 North 49th Street, Phoenix, AZ 85018

American Antiquity, 62(4), 1997, pp. 683-697. Copyright ? by the Society for American Archaeology

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-Annual Iteration

Cultural Behavior: -Marriage of eligible females; age randomly selected

-New husband given attributes, added to role -Departure of males of marriagable age

-Widowers remarried or placed in marriage pool -Departure of widowers not remarried after suitable period

Behavioral Rules and Survival Schedule

ConceDtion: If married female, check for

a) Steriity; b) Post partum infertility -Conception randomly determined from fertility probability schedule

Fertility Schedule

Birth Results (if preQnant): >\ 0 ~~-Probability of birth-crisis or infant-death checked

-Child's life span randomly selected from survival schedule

/r4 shdl -Other attributes randomly assigned: sex, year marries, etc. -Child added to role

Sural schedule

Summary; -Individual, family, and population statistics

-Inivdua, um__v

Figure 1. Sequence of operations in the Life Cycle Model. "RND" is input from a pseudorandom number generator.

group that is considered here. Quite frequently, these sites were abandoned after a short occupation of 20 to 30 years or less; however, some small populations managed to survive and to prosper.

Undoubtedly, many complex factors contributed to survival or early site abandonment, including environmental conditions, availability of resources, and

competitive pressures. How human factors con-

Founding Porulation Attributes preset gender, age at death, date of marriage, fertility

; i

Uodate Individual Records: -Delete deaths and emigrants from active list

-Update age classification: infant, chdd, juvenile, adult, non-productive

RND

RND

RND

[Vol. 62, No. 4, 1997] 684

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tributed to site collapse has received less attention. Preliminary work (Gaines and Gaines 1994, 1996) found significant differences between predictions based on "average characteristics" representing cultural behavior and simulation results reflecting individual attributes. Expanding on these earlier studies, we have focused on the influence of biological characteristics on small-population demography when constrained by cultural rules of marriage and residency. The results offer promise of a better understanding of small-site abandonment.

The attributes of the individuals who comprise the initial population of small sites define the character of early occupation, while the inherent mortality defines longer term survival of the community. Within this framework, success depends on several factors: reproductive life span (opportunity to produce more children), fraction of time married (greater realization of reproductive potential), and, in a matrilocal society, a pre- dominance of female births (girls marry, adding to population growth, while boys emigrate). These observations are intuitively obvious. What is not so obvious is the relative importance of physiological and cultural factors on these trends. Our objective was to quantify these interactions through the use of simulation.

Specifically, the simulation was targeted at three areas:

1. Exploring the impact of variations in the underlying biological and physiological characteristics on the vitality of small populations. In particular, we focus on sensitivity to differences in mortality and factors influencing effective birth rate observed in the prehistoric American Southwest, as well as the biological attributes of the initial group that populates the site.

2. Assessing implications of strict adherence to cultural rules on the viability of the population; in particular, the impact of monogamous relationships and constraints on remarriage.

3. Investigating correlation between population size and nuclear family formation.

Methodology

Computer simulations representing prehistoric activities are prevalent in current literature (see e.g., Biskowski 1992; Black 1978; Dyke and MacCluer 1973; Gilbert and Doran 1994; Hegmon 1989; Hodder 1978; Horvitz et al. 1971; Kohler et

al. 1996; McArthur et al. 1973; MacCluer 1967, 1973; Mithen 1990; Roth 1981; Sabloff 1981; Skolnick and Cannings 1973; Ward et al. 1973). Any model is an abstraction of the world it emulates; depending on the researcher's interest, different models are created to represent the complex and varied phenomena with which the archaeolo- gist is concerned. Our Life Cycle Model (LCM) emphasizes the interaction of cultural and biological-physiological factors and provides a fine- grained, microdemographic perspective. Many parameters are probabilistic and the LCM is for- mulated to explore the spectrum of consequences as the stochastic nature of the variables is allowed to operate.'

Approach

The questions posed are treated in the context of a single, small matrilocal group with exogamous marriage patterns. Diachronic simulation is carried forward, year by year, from the founding to 70 years or to site abandonment.2 Quantification for the simulation is based on empirical evidence from archaeology, ethnography, and early historic reports. Where this was not possible, values are drawn from standard physiological characteristics and contemporary Native American customs.

Structure of Model

The short-term occupancy of a site demands con- sideration of the specific attributes of the initial, what we label "founding," population in Figure 1. For consistency, the studies discussed involve three families (10 individuals) in the founding group.

Each individual is tracked through his or her life-from birth to death. The trajectory of a particular individual's life is dependent on the attributes assigned and the varying constraints of the cultural environment. Because of the probabilistic nature, every individual's life is unique and will be different if the simulation is repeated.

Cultural customs and rules relating to marriage, remarriage, polygyny, birth spacing, age of last birth, and widowers' and locally born males' resi- dencies are addressed. Marriage occurs within defined age ranges. An immigrant spouse's age is based on the marriage rules, his age at death is cal- culated from the age-specific survival schedule using conditional probability, and an ID is

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assigned to him. Locally born males leave the community at marriage age (subject to certain family responsibility constraints).

Each married female's fertility status is checked annually. If no birth spacing constraints apply (either biological or cultural), whether conception occurs is determined by use of an age-specific fertility probability function.

The probability of fetal death or infant mortality is checked. If a birth crisis or infant death does not occur, age at death is assigned using an age- specific survival schedule with a pseudorandom number generator. The child is assigned the other random attributes (sex, fertility, marriage age, etc.) and is given an ID in the population pool. No social status is assigned.

The entire process is iterated annually. Regional issues are not addressed; however, the migration of males who leave and those who enter for marriage with locally born females is considered. While inclusion of regional or site interaction (fissioning or mutual intersite support) is beyond the scope of the immediate research, the results of the simulation allow identification of factors that may lead to success or failure on a broader scale.

Marriage, family association, life expectancy, and 22 other variables are monitored on each individual.

Model Qualifications

Most assumptions related to quantification are noted in the "Implementation" section. Those with broader implication for the conceptual framework include:

* The autonomous nature of the LCM restricts regional relationships and aggregation.

* Cognitive-based corrective actions (decision processes) are not operative; rules relating relationships are set at initialization and remain fixed for each case.3

* Subsistence deficiency and other physiological stresses are innate to the mortality data and are not explicitly defined.

* The model is structured with assumptions relevant to small populations, and some can be chal- lenged as group size increases; e.g., no intrasite marriage between those born locally.

* Matrilocality and exogamous marriage patterns with the associated implications on male migration are central to the model structure.

Implications of these restrictions are examined further in "Interpretation and Discussion."

Sensitivity Analysis

Sensitivity analysis is used to generate the results. A set of default values and functions defines a "reference case." Extensive testing of the model for validity has provided insight to the more critical parameters. Our approach was to define a set of relevant incremental changes from the reference case for each of these parameters. This combina- tion of parameter changes composes the suite of variations explored in detail during the sensitivity studies. The magnitude of the perturbations in population vitality resulting from a change suggests the sensitivity to that particular parameter. These parameters are detailed in "Implementation."

Primarily, first order sensitivity analyses were performed, i.e., only one parameter is varied at a time.

Implementation

In quantifying the LCM, we drew heavily from data reflecting the Colorado Plateau of the American Southwest. Therefore, the model most realistically emulates a small Puebloan community in the A.D. 1050-1350 time frame. The model is not representative of any particular site. Since a wide range of parameter values is investigated, ascribing these to a particular site could be misin- terpreted.

Characterizing Biological/Physiological Aspects

Fifty-one parameters and age-specific survival and fertility functions are employed in establish-

ing the character of a particular individual. Only key demographic parameters of mortality (survival schedules), infant mortality, and fertility will be discussed.

Survival Schedules. To assure a range of survival characteristics for the sensitivity study, several different mortality data sets are used to develop the five survival schedules employed in the analyses. Many reports derive life tables (or abridged life tables) from skeletal records in the Southwest (see e.g., Bennett 1973, Point of Pines; Berry 1985, Grasshopper; Hayes 1981, Gran Quivira; Martin et al. 1991, Black Mesa; Mobley 1980, Pecos; Palkovich 1980, 1983, Arroyo

686 [Vol. 62, No. 4, 1997]

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j 0.6

?0.4-- \ ^. L - REFERENCE

0.2

20 25 30 35 40 45 50 55 60 AGE AT DEATH

Figure 2. Comparison of alternate survival schedules. The "REV-P," "WEST-5," and "REFERENCE" survival schedules are identical at age 20 and younger. The derivation of the Long-Life survival schedule results in slightly higher survival at age 20. All schedules shown are based on survivors at age two.

Hondo; Stodder 1990, Hawikki). Unfortunately, most Southwest prehistoric skeletal series are faulted as not representing the mortality statistics of the population. Methods of constructing life tables under these circumstances, as well as the skepticism of using skeletal data at all, have been discussed extensively in the literature (Boquet- Appel and Masset 1982, 1985; Buikstra and Konigsberg 1985; Buikstra and Mielke 1985; Holland 1989; Howell 1976; Martin et al. 1991; Moore et al. 1975; Swedlund 1975; Weiss 1973, 1975). Our interest is not to debate these issues but to use representative skeletal records and generic life tables to explore the impact of variations in mortality.

Alternate Survival Schedules

The survival schedules that follow were all derived from published data; however, each schedule deviates from the original data for appli- cation in our model. In particular, all are based on survivors at age two; all assume equal mortality rates for male and female; all terminate at age 60 or before; and the data are smoothed. The rationale for these conditions are discussed later.

* The Point of Pines skeletal record (Bennett 1973) was selected for constructing the "reference case" survival schedule. While not representing an extreme (e.g., see the Hawikki data in Stodder

[1990]), the Point of Pines data set reflects relatively short adult life expectancies. As such, it formed a lower bound for the sensitivity studies.

. The composite Pecos data, as analyzed by Mobley (1980), were used to construct an alternate site-based survival schedule. Although the Pecos data set suffers from problems of sampling and preservation bias (also, recent analysis has questioned the age classifications [Ruff 1981]), this data set typifies several skeletal series at ages above 20 years that have higher survivorship than Point of Pines data. Since it was modified extensively for adaptation to our model, the term "Pecos" would be inappropriate; so the revised schedule is referred to as the "Rev-P" schedule.

* Consistent error in age classification is a con- tinuing concern with empirically based mortality data. To assess the importance, a schedule was synthesized from the reference schedule assuming a 20 percent older classification for each age, e.g., the fraction surviving at age 20 was then at age 24, 30 was 36, etc. As the simulations progressed, finer resolution was needed and a second synthesized schedule using a 10 percent upgrade was created. These are called the "Long-Life" and "Long-Life 10" schedules, respectively.

* Inclusion of a theoretically based schedule was necessary to assure a full spectrum of mortality variation at older ages. A survival schedule

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L 0.8 -J

< 0.6 m

t 0.4

0.2

0 10 20 30 40 50 AGE, YEARS

Figure 3. Age-specific survival and fertility probability schedules used in the reference case. The survival curve relates to survivors after age two. Infant death before age two is determined by the setting of the infant mortality parameter. The fertility curve is normalized to 1.0 at the highest value.

based on the Coale and Demeny (1983:Table XIV) West-Male-5 table, referred to in the sequel as the "West-5" schedule, was added.

While not embodying the extremes, these schedules span the range of most of the published Southwest skeletal series. The differences in these schedules in the critical 20-to-60-age range are shown in Figure 2. Identical values are used for ages younger than 20 for Rev-P, West-5, and the reference survival schedules.

Infant Mortality. Considerable variation exists in projections for infant mortality (Skolnick and Canning 1973). It is believed to be high and estimates range from 20 percent to as high as 50 percent by age five (Aberle 1932; Berry 1985; Ford 1992). Rather than accept the reconciliation or reconstruction employed by other researchers to compensate for the persistent problems of under- enumeration (Palkovich 1980; Weiss 1973), we introduced a paradigm for representing infant mortality; "birth crisis" (intrauterine death or stillborn) and "infant mortality" (mortality from birth to two years) are defined as separate parameters in the model. Each of the survival schedules was modified to allow use of the birth-crisis and infant-mortality probabilities. Values of infant mortality from 20 to 40 percent were investigated. (Infanticide, if it occurs, is included in infant mortality.)

Fertility. An age-specific "fertility function" based on Weiss (1973:Figure 3) is used. The general form of this curve agrees well with data gath- ered by Kunitz (1973) on Navajo and Hopi women. Cultural influences, subsistence stress

Table 1. Founding Population Attributes.

Age at Name Sex Mate Age Death

lAXIa M 2A1 20 32 2A1 F lAXI 18 35 3BX 1 M 4B1 22 35 4B 1 F 3BX1 20 40 5B2 M 1 39 6CX1 M 7C1 34 45 7C1 F 6CX1 32 50 8C2 M 2 22 9C2 F 12 50 10C2 F 9 34

aElements of name IAX1 are interpreted as 1 = ID number; A = family by birth or first marriage; X = exogamous male; 1 = first generation.

(Frisch 1978, 1990), and pregnancy complications are treated separately from "inherent" congenital fecundity to allow for independent treatment of these factors. This change invalidates the use of "gross birth rate" or similar mechanisms to scale the fertility curve (Weiss 1975). To overcome this problem, it is assumed that a married woman in her mid-twenties, if not inhibited by cultural restrictions or physiological stress, would con- ceive within a year. Older and younger females experience a diminishing probability of fertility. The age-specific fertility and survival schedules used in the reference case are shown in Figure 3.

Characterizing the Founding Population

A configuration of three families (10 individuals: six adults, four children) is used in all cases. Individual attributes of the founding population are listed in Table 1. A larger founding group of five families was simulated but did not show suf- ficient differences in vitality from the three-family unit to warrant further discussion here.

Initial gender mix and age were at our discre- tion, and younger married couples were selected. Our rationale was that small groups budding off from ancestral populations would contain a mix of younger, more vigorous adults. Children of varying ages were included and sex of the children was established randomly. The children's ages require couple 6CX1/7C1, Table 1, to be older for consistency. Age at death of the founding population was determined using the reference case survival schedule and a random number table. The random selection of life span for the founding population resulted in a group with above-average robustness.

688 [Vol. 62, No. 4, 1997]

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Average life span of adults in the founding population is 39.7 and, for the children, 27 years. Life expectancy derived from the reference case survival schedule is 21 years at birth (30 percent infant mortality) and is 14 years at age 20 (age 34).

For assessment of sensitivity, a less robust, "Short-Life," alterative is provided by reducing the life spans of all the founding members by 20 percent. This yields an average age of death for the founding children of 21.6 years and for the founding adults 31.6 years. Founding attributes are fixed for all runs within a case.

Characterizing Cultural Rules

We relied heavily on ethnographic data to establish the cultural rules. Quantification is often obscure in the literature (for a few exceptions, see Gutierrez 1991; Levy 1992; Martin 1994; Parsons 1925). Although appropriateness of any choice can be questioned, a set of rational, consistent rules was selected from available information.

Marriage Rules. A monogamous, matrilocal residence pattern (common in some areas of the Southwest) was selected for the reference case cultural model. The exact age for a particular individual to marry is chosen randomly from within the acceptable age range.4 It is assumed that exogamous males are always available for first mar- riages but are not necessarily available for the remarriage of widows. The remarriage of widows is constrained to ensure a reasonable, long-term average balance of males entering and leaving the group. If there are no eligible widowers, remarriage may be postponed.

Two alternatives are considered: allowing 100 percent immediate remarriage and polygyny (Gutierrez 1991). Polygyny is restricted: widows must be at least 25 years old; there must be multiple surviving married couples in the group; only two wives are permitted.

Birth Spacing. Birth spacing includes both biological and cultural factors. For convenience, conception delays, pregnancy term, and postpartum infertility are grouped and the total period is termed "birth spacing." The duration of postpartum infertility can vary with the condition of the mother's health and lactation pattern (Aberle 1931; Ellison 1990; Howell 1976; Population Information Program 1981). The default birth spacing is set at one year following a birth crisis,

40

z 30 0

i 20

CL

0

0 0 10 20 30 40

YEAR 50 60 70

Figure 4. Population profiles after 1,000 runs with the reference case conditions. Population profiles below 53 percent terminate before 70 years. The data are plotted at five-year intervals; the connecting lines are for ease of association and do not represent percentages between intervals.

two years if the infant dies before the second year, and an average of 30 months if the child lives longer than two years (Ward and Weiss 1976). A 24-month spacing is also investigated.

Residency. For modeling purposes, we assume a widower may leave the community in two years if there is no eligible female to marry. A locally born male reaching marriage age leaves. Departure in either situation will be delayed if there are no other adult males.

Results

A computer run can be envisioned as the demographic record of a single "virtual site." However, runs vary greatly because of the stochastic nature of the model, and a single run is not useful in com- paring the impact of changing parameter values. Monte Carlo techniques are used to provide rea- sonably stable results,5 and the outcome of each case is a set of 1,000 statistically independent computer runs. The process is nonstationary, and classical statistics are not an effective representa- tion. We use two graphic forms to portray the results: "population profiles" and "consolidated values."

Biological/Physiological Factors

Since population size derives from a number of random effects, there is not a single value for population. Rather, there is a probability associate with a range of population sizes occurring, as will be seen in the figures that follow.

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Table 2. Comparison of Average Consolidated Values.

Ref. Rev-P Long-Life Long-Life West-5 Variable Case Case Case 10 Case Case

Initial pop. 10.0 10.0 10.0 10.0 10.0 Livinga 8.8 23.4 23.6 15.9 30.9 Totalb 67.6 94.1 94.4 80.7 103.4 Births 46.2 69.5 69.0 57.2 77.8 Deaths 45.0 56.1 55.3 50.3 57.2 Immigrants 11.4 14.6 15.5 13.5 15.4 Emigrants 10.7 14.3 14.7 12.9 15.6 Life Expc 13.8 20.7 19.7 16.8 27.7

Note: Row values are consolidated values divided by 1,000 (except for Life Exp). aLiving is the number of individuals resident at the end of 70 Tears (average consolidated population). Total is the average number of individuals tracked/run, i.e.,

all individuals that had involvement with the community. CLife Exp is the life expectancy (years) at 20 years for that survival schedule.

Population Distribution. The distribution of the population size generated by 1,000 repetitions of the reference case is plotted in Figure 4. These curves, or population profiles,6 can be interpreted as the fraction of the sites reaching or exceeding a particular population size. For example, the 1 percent curve shows that 10 of the 1,000 runs had 31 or more individuals after 55 years. These profiles provide a picture of the probability of site growth, under these particular initial conditions and parameter values.

The hump in the profiles at 10 years is a result of using the same robust founding population attributes for every run.

70 -- Long Life

60 -

z 50 0 H 40

30

?20 -

10

0

Alternate Survival Schedules. The variation between the five survival schedules is depicted in Table 2 where average consolidated values are com- pared with the reference case. Consolidated value is the sum of a particular variable from all 1,000 runs a particular year. For example, consolidated population can be envisioned as placing 1,000 autonomous sites on a "virtual" landscape (10,000 individuals) and returning later to count noses.

There is a net decrease in regional consolidated population for the reference case of 12 percent,7 while all the other cases show a net growth.

Figure 5 is typical of the shifts in the population profiles for the Long-Life and Rev-P cases. The West-5 case shows a similar trend with larger population levels.

Other Trends and Impacts on the Reference Case

The impact of the founding population with different survival schedules can be observed in Table 3 where consolidated population growth trends are isolated for the last 40 years. Figure 6 graphically presents similar information for other parameter change cases, which employ the reference survival schedule. These results lead to the following observations.

Founding Population Life Expectancy. Reducing life expectancy of the founding population (Short-Life case) reduces initial growth and further exacerbates vitality.

0 10 20 30 40 50 60 70 YEARS

Figure 5. Comparison of the 1 percent and 50 percent population profiles of the Long-Life survival schedule with the reference case that shows the impact of a 20 percent improvement in survival.

[Vol. 62, No. 4, 1997] 690

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Table 3. Net Consolidated-Population Changes.

1-70 years 31-70 years Case Increase Change/yr Increase Change/yr

Reference case -12% -.17% -57% -1.42% Long-Life 10 59% .84% -3% -.07% Rev-P 134% 1.91% +56% +1.40% Long-Life 136% 1.94% +55% +1.38% West-5 209% 2.99% 119% +2.98%

Note: Increase = percent change during the period, relative to the founding population; Change/yr = Increase divided by the number of years in the period.

Founding Population Gender Mix. Only female children in the founding population provide more vigorous growth initially but, again, do not reverse the downward trend in the later years.

Infant Mortality. Reducing infant mortality from 30 to 20 percent in the reference case provides better consolidated population growth but does not reverse the long-term decline.

Marriage Rules. Permitting polygyny improved initial growth but did not reverse the long-term trend. One hundred percent remarriage by widows of marriageable age reverses the long- term decline with the reference survival schedule. While it is unrealistic to assume that every woman is married 100 percent of her reproductive life, the case provides a useful upper limit.

The population at different profile levels for the above parameters changes is summarized in Table 4.

Table 4. Comparison of Variations from Reference Case.

Case 1% Profile 10% Profile 50% Profile % Terma

Referenceb 39 22 6 47% All girlsc 41 27 12 31% All boysd 23 12 0 81% Polygynye 47 32 13 35% Remarriedf 57 41 23 8% 20%SLg 26 15 0 76% 20%IMh 49 32 13 31% Long-Life' 63 43 23 13%

Note: All figures are population size at 70 years except the last column. Reference case (default) conditions apply except as noted. a Percent of runs terminating before 70 years. bReference case with default values. c Default with only female children in founding population. dDefault with only male children in founding population. e Default with constrained polygyny. f Default with 100 percent remarriage. g Default with 20 percent shorter life span in founding population. hDefault with 20 percent infant mortality. i Long-Life survival schedule.

* 70 Year Population Change 0

1Q 30-70 Year Population Change 0-

a. -. 0

IJ 0 0

u)

z

o-100 REF GIRLS POLYG

20%SL 20%IM REMAR

Figure 6. Short- and long-term differences in consolidated population (see text for definition), with "% Net Consolidated Population" (y axis) representing the per- centage change in the consolidated population during the period, relative to the founding population. The solid bars are based on the period from year 1 to 70 and the gray bars are based on the period from year 31 to 70. "REF" = reference case conditions; "20%SL" = case with 20 percent shorter life span of founding members; "GIRLS" = case with all female children in founding population; "20% IM" = 20 percent infant mortality case; "POLYG" = polygyny case; "REMAR" = 100 percent remarriage case.

Interpretation and Discussion

In the introduction, we targeted three areas of research, and this section focuses on interpretation of the simulation in light of those objectives.

Biological/Physiological Impacts

As anticipated, survival schedules dominate the long-term results, ranging from robust, vital populations to those that are marginal or perhaps mori- bund. With its shrinking consolidated population, the reference case is in the latter category. While the interpretation of this trend is problematic, our view is that the decline indicates eventual site abandonment or extinction of the group. The reference case is at or below a "survival boundary." The boundary is a function of several parameters, particularly the marriage rules. Keeping other conditions fixed, the boundary is abrupt; a 10 percent improvement in the survival schedule (Long-Life 10 case) converts a 57 percent decay in the last 40 years to essentially a flat consolidated population trend, as seen in Table 3. (A 10 percent difference in skeletal classification is within the accuracy of osteological methods.)

Mortality. Differences in the survival schedules may be more physiological than biological-

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z o -~> REFERENCE - LONG-LIFE - WEST-5 -a- LONG-LIFE10

<0 4.0 --

O ..-X" 3.0

.0 Z 10 0 0

z o.o ! i I I , I r I ; ;I

D 0 10 20 30 40 50 60 70 YEAR

Figure 7. Consolidated-population trajectories for different survival schedules. Cases with 100 percent remarriage permitted are solid lines; default marriage rules are dashed lines. "Consolidated-Population Ratio" (y axis) is consolidated population divided by the founding population.

dependent on environmental conditions and nutri- tional stress (see Harrison and Waterlow 1990; Pearson and Greenwell 1980; Waterlow 1988; Wing and Brown 1979). One can posit a small shift in mortality patterns (e.g., caused by environmental conditions), which, together with marginal mortality, could contribute to the precipitous collapse of sites. Relating these implications to a broader regional perspective, the statistical nature of human physiological/biological characteristics will produce tremendous variations across a region, even assuming constant environmental and cultural parameters. Superimposing a reasonable mix of environmental conditions or micro-eco zones, with the attendant effects on mortality, widens the variation further and creates the potential for significant differences in growth between sites. Some sites would decline, while other more advantageously located sites would prosper.

Founding Population. While the founding population may be drawn from the general population, attributes can vary greatly from the average. This was true in the reference case where a robust founding group was created. The first 30 years of site occupancy are heavily influenced by the attributes of this initial population (Figure 6, Table 3) and, indirectly, the marriage rules. Because the founding population establishes the early population levels, its effects endure and influence much of the 70 years of the simulation. For example, the

fragile, Short-Life case, which is more typical of a region with the reference case survival characteristics, exhibits a population of little more than half the size of the reference case at the "10 percent profile" (Table 4). This is due principally to the early population surge from the robust founding group of the reference case.

Survival schedules have minor effects initially. The five different survival schedules explored in this paper exhibit only a 6 percent difference in consolidated population at 15 years. The impact increases with time, and the spread becomes significant after 30 years (Table 3).

Infant Mortality and Other Influences. All the

parameter changes in cases using the reference survival schedule, except the 100 percent remarriage case, show a declining consolidated population over the last 40 years (Figure 6). While the influences of these different variations are impor- tant, the significance is less than that of the biological attributes of the founding group, the survival schedules, and the marriage rules.

Impact of Cultural Rules

Any change that permits earlier remarriage of widows increases the potential number of second generation children, e.g., polygyny or 100 percent remarriage would benefit women 2A1 and 4B 1 (Table 1).

Although the life expectancy for West-5 is 40

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percent higher than for Long-Life (Table 2), a corresponding difference in the number of births is not observed for two reasons: not enough time has elapsed for inherent differences in growth rates to be reflected in the size of the reproductively active population, and life expectancies for both schedules are beyond the peak reproductive years (at that point, cultural influences become more significant).

Combined variation of survival characteristics and marriage rules provides interesting insight into these small-population dynamics. Consider the following extreme examples:

(a) Combined impact of 100 percent remarriage and Long-Life survival schedule. The compound effect is a dramatic 75 percent greater population growth than with the survival schedule change alone. The increase stems from several mutually interacting factors. The ratio of years married to total years available for reproduction ("marriage index") ranges from 65 to 75 percent with the default marriage rules. Reproduction potential remains. As the marriage index rises, the births increase by 50 percent, and the number of males entering the community for marriage doubles.

(b) Combined impact of 100 percent remarriage and West-5 survival schedule. The population growth is more modest in this case. Since the marriage index is already in the mid-eighties with the West-5 conditions, the potential is less when the marriage constraint is removed.

In both examples, as the growth potential satu- rates, the consolidated populations converge (Figure 7). In general, each set of customs (rules) defines a different spectrum of consolidated-population trajectories. For some sets (as with 100 percent remarriage), the effects of significantly different survival schedules are indistinguishable as in Figure 7. The trajectories become asymptotic to a "fecundity boundary," and little difference in short-term consolidated population can be observed (up to about 50 years for Long-Life and West-5 trajectories).

The fecundity boundary is the antithesis of the cases where the population size is sensitive to small changes in survival schedules, the survival boundary. These divergent responses emphasize the wide variations that, with relatively small changes in parameters, can occur during the transient conditions characterizing a small start-up community.

12

) 10 uJ

U

- 4

Z 2

0

--50

---40 z 0

--30

--20 L

-10

-0

YEAR Families Population

Run #630 f Run #33 --- Run #630 -- Run #33

Figure 8. Bar graph of the number of nuclear families corresponding to the reference case population trajectories for run #630 (largest population) and run #33 (site with population on the 50 percent profile at 70 years). The actual populations for the two sites are graphed as solid lines. The two sites have the identical number of families for the first 25 years. The six individuals remaining in run #33 after 60 years are unmarried adults and children.

Immigration. The fragility of small populations is ameliorated by immigration. Since males entering to marry local women are offset by the exodus of locally born males to marry outside the group, the ratio of immigrants to emigrants is critical. Immediate, 100 percent remarriage of widows in the reference case resulted in doubling the male immigrants without a corresponding increase in emigration. The attendant increase in population is a large factor in the beneficial response shown in Figure 6 and Table 4 for 100 percent remarriage.

Marriage. The matrilocal marriage stipulation is a very severe restriction, and adherence to a monogamous, exogamous marriage pattern is doubtful in small groups under stress.

A deviation from monogamous relationships could take many forms, and polygyny is one option. With the reference case conditions, polygyny has a positive impact on short-term survival but the long-term trend continues to show a nega- tive consolidated-population growth (Table 3). Other variations of the remarriage rules that would increase fertility are possible. Intrasite marriage, particularly as the site grows, is a possibility. Relaxing the restriction on local males leaving and allowing females to enter the community for marriage would minimize the dissolution of families with only surviving male offspring. Members of outside kin or ceremonial affiliated units could

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bolster the group in times of population decline. Many restrictions on marriage are subjective,

and the myriad of possibilities leads to the conclu- sion that, faced with declining prospects, a group would logically opt for alternatives.

Population-Nuclear Family Correlation

The number of habitation rooms is sometimes used as a population marker. To assess the efficacy of such a measurement, the number of nuclear families (married couples and children) was monitored during the simulations. The results of two specific runs from the reference case are shown in Figure 8. The number of nuclear families fluctu- ates and often appears out-of-phase with changes in population.

In Figure 8, at 20 years, a decrease in population is accompanied by an increase in the number of nuclear families. Even run #630, which has a consistent population growth over much of the 70 years, has a period from 30 to 50 years in which the net number of families did not increase while the population grew from 20 to 35 individuals. Older families are being dissolved during this period, while new families in their prime reproductive years are being formed. The flat period is followed by a rapid increase in the number of families as the female offspring mature and marry. While family size averages four to five individuals over the occupation period, from the perspective of small-scale episodes, the formation of family groups is not proportional to population size. The substantial swings in the number of nuclear families suggest significant changes in the archaeological record not related to population size.

Concluding Remarks

This research has accomplished two things: First, it has underscored the significant role that simulation can play in examining issues that are beyond the capabilities of simple analysis. Second, it has demonstrated, quantitatively, the interaction of mortality, marriage rules, and founding populations on small-site dynamics.

The initial occupational stage in small sites (perhaps large sites as well) is characterized by the attributes of the founding population, while the longer term trajectories are influenced by the underlying biological factors. The transitional, initial occupancy period is difficult to interpret and,

in small sites, this may be the duration of occupation. Small populations are very sensitive to parameter shifts, and there can be significant dissimilarities in growth related to small increases in survival schedules and in variations of cultural rules (e.g., marriage rules). Trends can be obscured by idiosyncratic behavior and, as demonstrated, by the random and stochastic ele- ments in population dynamics. The problem is to separate the orderly processes from the back- ground "noise" in cultural systems.

It is difficult for any excavation and analysis to extract the actual characteristics of a founding population. However, an appropriately structured, microdemographic simulation provides a way to isolate interdependent causes and to explore the potential impact of the variation of parameters.

Simulation does not negate the need for excavation. The quantification of the model parameters depends on empirical information. However, use of simulation can avoid the problem of limited experimental visibility. While it may not be possible to predict the response to a specific situation, simulation allows extraction of statistically significant effects of a range of conditions that could only be accomplished by excavation of many tem- porally related sites.

We have presented only a small subset of the possibilities, but our study has provided a basis for interpretations that have been previously only qualitatively explored. While ecosystems isolated from each other (as in the model) do not exist in the real world, these "virtual communities" help one to understand underlying causes of demographic and cultural change. We are a long way from creating a theoretical structure that speaks to the complexity of the relationships affecting small-population survival in a regional context, but clearly, biological/physiological factors and cultural constraints are part of the pattern.

Acknowledgments. We gratefully acknowledge our col- leagues for their patience and mentoring. They generously gave of their time and expertise as we explored the intrica- cies of this research focus. We would express particular appreciation to George Cowgill, John Martin, and Katherine Spielmann who suffered through earlier drafts of the paper and provided invaluable comments and direction. Lynne Goldstein and the reviewers offered helpful suggestions, and we thank them for their assistance. However, we remain solely responsible for any errors or fuzzy thinking. Geraldina Tercero kindly provided the Spanish translation.

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Notes 1. The LCM was designed to operate on a PC and to be eas- ily adapted as revisions or extensions became desirable. No standard simulation package could be found that met func- tionality and flexibility needs, and the LCM was written in Professional Basic. The design is modular consisting of 40- plus procedures in four modules. The program requires approximately 35 minutes/case on a 486/33 (8 MG RAM, Win3.1), and four minutes on a Pentium/166 (32 MG RAM, Win95). 2. The following conditions terminate a run before 70 years: * no surviving adults; or * no surviving males; or * no surviving females; or * no married couples and the number of adults is less than three. 3. Terminology: "run" implies a single simulation; "case" applies to a unique assembly of initial conditions, parameter values, and functions. In "Results," each case consists of 1,000 runs.

The terms "site," "region," and "population" are used throughout to relate the concepts to physical reality. These are, of course, abstractions of the model and while it would be more precise to join these with the modifier "virtual," this has been omitted for brevity. In the context of this paper, "site" and "region" are synonymous with "run" and "case" (i.e., the virtual landscape), respectively. 4. Marriage rules: * Age (female) at first marriage: 16 to 22 (for discussion of menarche, see Buikstra et al. [1986]; Ralethford [1990]).

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* Mate's age if from outside the group: 18 to 24. * Age (female) at last remarriage: 38 (Hassan 1981). * Age (locally born male) of emigration: 18-24. * Age at last remarriage of widower: 40 (Parsons 1925, 1936). * The male is assumed to be older (a minimum of two years) at first marriage (Gutierrez 1991). 5. The profiles are sample estimates; variations in magnitude occur when a case is repeated. A dispersion on the order of 5 percent or less has been observed between repeated 1,000 run cases. 6. The percentages given on the population profile charts relate to the number of sites, not population size. The average site of the initial 1,000 sites (50 percent profile) has a population of only six or less (470 have zero population) at

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70 years; of the 530 sites still existing, the average population is 16.6 individuals. 7. The high failure under the conditions of the reference case is in contrast with the site from which the survival schedule was derived; i.e., Point of Pines. The causes are probably related to differences in size of base population, to regional relationships, to diverse cultural behavior, and, quite possi- bly, to the skeletal series not being representative of the true mortality at Point of Pines.

Received July 19, 1995; accepted February 28, 1997; revised March 20, 1997.

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