Appendix: Details of simulation parameters, equations, and algorithms
Geographic range: For each species, presence was assumed at the start of a trial
throughout each one-by-one degree grid cell falling partially or entirely within a
minimum convex polygon drawn around occurrences of the species in the
FAUNMAP database (13). Grid cells were excluded if their midpoints fell in an
ocean or if they contained only one species, indicating an extreme lack of data.
Body mass: The 11 surviving species and their estimated masses in kg are Alces
alces (457), Antilocapra americana (68), Bison bison (422), Cervus elaphus (500),
Odocoileus hemionus (118.1), O. virginianus (106.85), Oreamnos americanus
(91), Ovibos moschatus (286), Ovis canadensis (= "O. catclawensis": 91), Pecari
("Tayassu") tajacu (30), and Rangifer tarandus (60.6). The 30 extinct species and
their masses are Bison priscus (522.8), Cervalces scotti (485.6), Bootherium
bombifrons (753.0), Camelops hesternus (995), Capromeryx minor (21), Equus
complicatus (439), E. conversidens (306), E. francisi (368), E. niobrarensis
(533.4), E. occidentalis (574), E. scotti (555), Euceratherium collinum (498.8),
Glyptotherium floridanum (665.8), Hemiauchenia macrocephala (= "H.
blancoensis": 238.1), Holmesina septentrionalis (312), Mammut americanum
(3297.7), Mammuthus columbi (= "M. imperator," "M. jeffersonii": 5826.6), M.
primigenius (3174.2), Megalonyx jeffersonii (1320), Mylohyus fossilis (= "M.
nasutus": 73.5), Navahoceros fricki (222.6), Nothrotheriops shastensis (614),
Oreamnos harringtoni (45.0), Palaeolama mirifica (244.7), Paramylodon
("Glossotherium") harlani (1990), Platygonus compressus (52.5), Stockoceros
conklingi (52.5), S. onusrosagris (53.7), Tapirus veroensis (324), and Tetrameryx
shuleri (60.6). FAUNMAP identifications (13) were modified as noted.
Initial population density: The proportion si of fossil collections including the ith
species was computed as the number including the species / number falling within
a minimum convex polygon surrounding the species' range. Collections including
just one species were excluded from these counts because they often derived from
taxonomic reviews and were unlikely to constitute complete faunal censuses.
Assuming that sampling is a Poisson process, si = 1 - exp(-di), where is a
constant and di is the true population density in individuals/km2. Thus, di can be
derived if the constant  can be estimated. This is done by treating the average
sampling process for a model species of the same body mass as sˆ = 1 - exp(- dˆ ).
Thus,  = ln(1 - sˆ )/(- dˆ ) and di = ln(1 - si)/-= ln(1 - si)/(-ln[1 - sˆ ]/[- dˆ ]). sˆ was
set using the empirically derived regression equation logit ( sˆ ) = 0.02905 ln(body
mass, g) - 2.0762. This insignificant body mass/preservability relationship
explains only 4.7% of the variance, perhaps because slower decay and weathering
of large species is counter-balanced by higher per-annum mortality of smaller
species. dˆ was set to exp(1.592 - 0.2622 ln[body mass, g]) (18).
Prey population growth: Dynamics were based on the discrete logistic growth
equation Nt+1 = Ntexp(r(1-Nt/K)), where Nt = population in a grid cell at time t; r =
the intrinsic (maximal) rate of increase, set to equal exp(1.4967 - 0.37 ln[body
mass, g]) (19); and k = carrying capacity, set to equal the initial population size.
Logistic density dependence implies that extinction is impossible without a strong
perturbation.
Secondary food resources: Productivity of plant and small game foods was
assumed to be 200 g/m2 for all geographic regions because areas with higher net
primary productivity than this probably yield relatively less edible plant resources
(9). Of this amount, 2% was assumed to be edible (9). Regrowth was modelled
with logistic dynamics, with the intrinsic rate of increase set to 22% the initial
value (9). Caloric values were set to 3 kcal/g (13), which is similar to estimates
used in other simulations (10). Because a significant amount of small game is
incorporated in the secondary resource budget, the caloric value of big game and
secondary resources is probably very similar (21), so no difference was assumed.
Human nutrition: The standard caloric intake i0 value of 2200 kcal/person/day
compares to those used in other simulations: 1862 (9), 1975 (21), 2200 (11), and
2505 (10) kcal/person/day. An independent source gives a range of 1700 - 2700
kcal/person/day for contemporary human populations (22). The materials
gathered:materials ratio is conservative because much higher big game wastage
values were used in some early models (5-7, 9); the value of 1.4 is in the range
assumed by later, more conservative researchers (1.33: 10; 1.54: 11).
Human hunting and gathering ability: Baseline hunting success rates h' in grams
were set by summing miNi(1-aeNh) across all prey species, where mi = body mass
and Ni = population size of the ith species in a grid cell, a = hunting ability
constant (see Table 1), and Nh = human population size. Realized rates were
modulated by a type 2 functional response (23) of the form h =
(3.5h'/hk)/(1+2.5h'/hk), where hk = hunting rate at equilibrium. These equations
imply that hunting is a random Poisson process: success is proportional to the
density of both food items and humans, and humans interfere with each other's
capture rates at high human population densities. Hunting success increases by no
more than a factor of 1.4 as projected intake approaches zero due to lack of
resources; likewise, actual intake is never greater than 1.4i0 at high prey densities.
The figure of 1.4 is justified assuming that maximum normal digestive capacity is
1050 ml/day (21) and that food has a nutritional value of 3 kcal/g and a density of
about 1.0 g/ml, which yields (3150/2200) kcal/day = 1.39 times the standard
requirement.
Human population growth: Growth rt at time t was assumed to be zero if actual
caloric intake it equalled the standard rate i0, and maximal if intake equalled the
maximum of 1.4i0. Maximal rates of growth of 4%/yr for hunter-gatherer
populations are known (4, 11, contra 24, 25), but earlier studies have used lower
values (2.0%: 11; 2.9%: 9; 3.0%: 4; < 3.5%: 7). A compromise value of 3% is
used here. Assuming this, then the relationship rt = (it /i0)0.0878 yields rt = 3.00%/yr
when intake is maximal and rt = -5.90%/yr when intake is half of the standard rate.
Realized population growth rates were never > 2% (Table 1), because input
hunting parameters never allowed maximal kill rates to be attained.
Timing: Starting simulations with an initial invading population of 100 humans at
14,000 B.P., allowed enough time before the 13,400 B.P. appearance of Clovis
artifacts (3) to make sure that detectable human populations would spread to the
Southwest by that date. Some extinct megafauna may have survived to 12,260
B.P. (or 10,370 B.P. uncalibrated: 26), which is during the Younger Dryas global
cooling event. Therefore, simulation trials were terminated at 11,500 B.P., just
after the end the Younger Dryas (11,530 B. P. calibrated: 3). The total run time of
2500 years should be long enough to encompass any realistic extinction pulse.
Human dispersal: Rates of population change in each grid cell were determined by
a standard difference equation that computes net growth as a summation across
larger populations in the eight nearest cells. A diffusion coefficient of 900 was
assumed (4), and the initial population was placed at 49˚ N 114˚ W in most
simulations, following Refs. 4, 6, and 7.
Interspecific competition: The competion model assumed that each species'
population growth was a Ricker logistic function of its own density plus the
population densities of all other species times a competition coefficient c. When c
= 0, there is no competition and all species have independent logistic population
dynamics; when c = 1, all species behave as an undifferentiated ecological unit.
Population sizes were weighted by mass0.75 so they would reflect energy intake per
unit time instead of standing population size (i.e., Kleiber's law that metabolic rates
scale to the 0.75 power in mammals: 27).
Prey species dispersal: The first set of simulations forbid dispersal between grid
cells, making population dynamics completely independent in each cell. The
second set evenly redistributed the total population of each species across all cells
falling in the species' original geographic range. Redistribution was performed
once per year.