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Second-Order Predation and Pleistocene Extinctions:A System Dynamics Model

By

Elin Whitney-Smith

B.A. June 1975, Rutgers University, New Brunswick, NJ

M.S. June 1985, San Jose State University, San Jose, CA

Ph.D. February 1991, Old Dominion University, Norfolk, VA

A Dissertation submitted to

The Faculty of

Columbian School of Arts and Sciences

of the George Washington University in partial satisfaction

of the requirements for the degree of Doctor of Philosophy

May 20, 2001

Dissertation directed by

Henry Merchant, Ph.D.

Associate Professor of Biology

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For

Christoph Berendes

who has never lost faith

seldom lost patience

and continues to be there.

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Abstract

At the end of the Pleistocene, there were significant climate changes and, following the

appearance of Homo Sapiens on each major continent, significant megafaunal

extinctions.

The leading extinction theories, climate change and overkill, are inadequate. Neither

explains why: (1) browsers, mixed feeders and non-ruminant grazer species suffered

most, while ruminant grazers generally survived, (2) many surviving mammal species

were sharply diminished in size; and (3) vegetative environments shifted from plaid to

striped (Guthrie, 1980.)

Nor do climate change theories explain why mammoths and other megaherbivores

survived changes of similar magnitude.

Although flawed, the simple overkill hypothesis does link the extinctions and the

arrival of H. sapiens. Mosimann & Martin(1975) and Whittington & Dyke( 1984)

quantitatively model the impact of H. Sapiens hunting on prey. However, they omit the

reciprocal impact of prey decline on H. Sapiens; standard predator-prey models, which

include this effect, demonstrate that predators cannot hunt their prey to extinction

without themselves succumbing to starvation.

I propose the Second-Order Predation Hypothesis , a “boom/bust” scenario: upon

entering the New World, H. sapiens reduced predator populations, generating a

megaherbivore boom, then over-consumption of trees and grass, and, finally,

environmental exhaustion and the extinctions.

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The systems dynamic model developed in this work (available in the CDROM

attached or from http://quaternary.net/extinct2000/) specifies interrelationships between

high and low quality grass, small and large trees, browsers, mixed feeders, ruminant

grazers and non-ruminant grazers, carnivores, and H. sapiens driven by three inputs: H.

sapiens in-migration, H. sapiens predator kill rates, and H. sapiens food requirements It

permits comparison of the two hypotheses, through the setting of H. sapiens predator kill

rates. For low levels of the inputs, no extinctions occur. For certain reasonable values

of the inputs, model behavior consistent with Second-Order-Predation: carnivore killing

generates herbivore overpopulation, then habitat destruction, and ultimately differential

extinction of herbivores. Without predator killing, extinctions occur only at unreasonable

levels of in-migration. Thus, Second-Order-Predation appears to provide a better

explanation.

Further, the boom-bust cycles suggest we “over-interpret” the fossil record when we

infer that the populations decreased steadily, monotonically to extinction.

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Contents -- Click to Access Page

Abstract .......................................................................................................... iii

Contents ......................................................................................................... v

Tables ............................................................................................................ xi

Figures..........................................................................................................xii

Chapter I: Introduction and Literature Review............................................... 1

Introduction ..................................................................................................................... 1

Characteristics of the Pleistocene–Holocene Transition and Its Extinctions.............. 3

Hypotheses Regarding the Cause of the Pleistocene-Holocene Extinctions............. 10

Criteria for New Hypotheses of Extinction............................................................... 28

An Alternative Hypothesis of Extinction...................................................................... 32

The Argument for Second-Order Predation .............................................................. 32

A Proposed Scenario of Pleistocene Extinctions Due to Second-Order Predation... 38

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Chapter II: A Method for Testing Hypotheses of Pleistocene Extinctions in

the New World .............................................................................................. 45

Introduction ................................................................................................................... 45

The Modeling Process............................................................................................... 46

General Conventions and Definition of Terms ......................................................... 47

General Overview of the Model................................................................................ 57

Criteria for Success ................................................................................................... 58

Base Model: Dynamic Equilibrium – Step 1 ................................................................ 59

Overview ................................................................................................................... 59

Model Diagram ......................................................................................................... 59

Conventions, Definitions and Equations................................................................... 62

Graph of the Base Model – Step 1 ............................................................................ 68

Second Predator: Overkill – Step 2a ............................................................................. 72

Overview ................................................................................................................... 72

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Model Diagram for Step 2a....................................................................................... 72

Conventions, Definitions, and Equations.................................................................. 74

Results Second Predator (Overkill) – Step 2a............................................................... 76

Graph of the Model – Step 2a ................................................................................... 76

Second-Order Predation – Step 2b ................................................................................ 78

Overview ................................................................................................................... 78

Model Diagram ......................................................................................................... 78

Conventions, Definitions, and Equations.................................................................. 78

Results Second-Order Predation – Step 2b ................................................................... 80

Graph of Model – Step 2b ......................................................................................... 80

Step 3 – Three Herbivores – (Browsers, Grazers and Mixed Feeders)......................... 83

Overview ................................................................................................................... 83

Diagram of the Model ............................................................................................... 83

Conventions, Definitions, and Equations.................................................................. 85

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Plants ......................................................................................................................... 89

Herbivores ............................................................................................................... 103

Results of Step 3: Three-Herbivore Model ................................................................. 129

Graph of the Model ................................................................................................. 129

Step 4 – Four Herbivores (Browsers, Ruminant Grazers, Non-ruminant Grazers and

Mixed Feeders)............................................................................................................ 136

Overview ................................................................................................................. 136

Diagram of the Model ............................................................................................. 136

Conventions and definition of terms used in Step 4................................................ 136

Results of the– Four-Herbivore Model ....................................................................... 163

Graph of the Model ................................................................................................. 163

Chapter III: Testing and Validity................................................................ 171

Introduction ................................................................................................................. 171

Tests for Suitability of Structure ................................................................................. 172

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Dimensional Consistency........................................................................................ 172

Extreme Conditions................................................................................................. 173

Boundary Adequacy................................................................................................ 179

Tests for Suitability of Model Behavior...................................................................... 181

Parameter sensitivity ............................................................................................... 181

Structural Sensitivity............................................................................................... 183

Tests for the Consistency of the Model with the Real system .................................... 183

Face Validity ........................................................................................................... 183

Parameter Values..................................................................................................... 184

Replication of Reference Modes ............................................................................. 185

Surprise behavior..................................................................................................... 186

Additional Characteristics Contributing to Model Utility and Effectiveness ............. 186

Appropriateness of Structure:.................................................................................. 186

Counterintuitive Behavior: ...................................................................................... 187

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Generation of Insights ............................................................................................. 188

Chapter IV: Conclusions and Significance................................................. 189

Introduction ................................................................................................................. 189

Conclusions ................................................................................................................. 189

Discussion ................................................................................................................... 190

Climate and Vegetation........................................................................................... 190

Animals ................................................................................................................... 192

Archaeological Evidence......................................................................................... 194

Implications for Further Research............................................................................... 196

Significance of the Model for Science and Research.................................................. 197

A Broader Significance for the Modern World........................................................... 200

Appendix A: Equations............................................................................... 201

Appendix B - Summary Graphs.................................................................. 218

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Appendix C – Model and Runtime Software on CD Rom......................... 222

Bibliography................................................................................................ 223

Tables

Table 1: Variables and starting values of the Mosimann and Martin and the Whittington

and Dyke models....................................................................................................... 19

Table 2: Features of the Pleistocene-Holocene transition................................................. 31

Table 3: Values taken directly from Whittington and Dyke (1984) ................................. 56

Table 4: Modified values based on Whittington and Dyke (1984) ................................... 56

Table – 3 – Comparison of Second Predator (Overkill) and Second-Order Predation

Ending Values ........................................................................................................... 82

Table – 4 – Array Illustration.......................................................................................... 139

Table 5 – Validity matrix based on Richardson and Pugh (1981) ................................. 172

Table – 6. – Carnivore population reduction................................................................... 183

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Figures

Fig. 1. Prorated rates of mammalian extinction (after Webb, 1989)................................... 2

Fig. 2 Correlation of the strata of the Pleistocene – Holocene transition in North America

(adapted from Haynes, 1984) ...................................................................................... 9

Fig. 3. The march of extinction (after Martin, 1984): ....................................................... 16

Fig. 4 Oscillation of predator and prey populations.......................................................... 24

Fig. 5 The path of extinction held by various hypotheses................................................. 34

Fig. 6. Effect of the arrival of a new predator on the populations of North American prey

and predators ............................................................................................................. 40

Fig. 7. Role of trees and grass in climate change:............................................................. 44

Fig. 8. Dynamic equilibrium reference mode ................................................................... 49

Fig. 9. Second-predator (overkill) reference mode ........................................................... 50

Fig. 10. Second-order predation reference mode .............................................................. 51

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Fig. 11. Illustration of the step function ............................................................................ 53

Fig. 12. Illustration of the pulse function .......................................................................... 54

Fig. 13. Base model diagram............................................................................................. 60

Fig. 14. Hunting function .................................................................................................. 66

Fig. 15. Graph of the base model ...................................................................................... 69

Fig. 16. Pulse outflow from plants, herbivores and carnivores......................................... 70

Fig. 17. Second predator (overkill) model diagrams......................................................... 73

Fig. 18. Graph of the second predator (overkill) mode..................................................... 77

Fig. 19. Second-order predation diagram.......................................................................... 79

Fig. 20. Graph of the second order predation model......................................................... 81

Fig. 21. Continent, trees and grass, diagram. Three herbivore model. ............................. 84

Fig. 22. Herbivores. – browsers, grazers, and mixed feeders diagram. Three herbivore

model......................................................................................................................... 86

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Fig. 23. Density diagram. Three herbivore model. ........................................................... 87

Fig. 24. Carnivores, Hsapiens diagram. Three herbivore model....................................... 88

Fig. 25. WoodMix function. Three herbivore model. ....................................................... 90

Fig. 26. MixedFeeder efficiency. Three herbivore model................................................. 95

Fig. 27. GzEffGr. Actual efficiency of Grazers given the amount of grass available in the

system Three herbivore model. ............................................................................... 100

Fig. 28. Effect of Browser density, as Browser density declines Browser birth function

drops toward zero. Three herbivore model. ............................................................ 105

Fig. 29. The rate at which Carnivores kill Browsers. Three herbivore model. ............... 107

Fig. 30. The rate at which Hsapiens kill Browsers. Three herbivore model. .................. 109

Fig. 31. Actual efficiency of Grazers given the amount of grass available in the system.

Three herbivore model. ........................................................................................... 111

Fig. 32. The birth rate of Grazers. Three herbivore model ............................................. 112

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Fig. 33. Effect of Grazer density, as Grazer density declines Grazer birth function drops

toward zero. Three herbivore model ....................................................................... 114

Fig. 34. The rate at which Carnivores kill Grazers. Three herbivore model................... 115

Fig. 35. The rate at which Hsapiens kill Grazers. Three herbivore model...................... 117

Fig. 36. The birth rate of MixedFeeders. Three herbivore model. .................................. 120

Fig. 37. Effect of MixedFeeder density, as MixedFeeder density declines MixedFeeder

birth function drops toward zero. Three herbivore model. ..................................... 121

Fig. 38. The rate at which Carnivores kill MixedFeeders. Three herbivore model. ....... 123

Fig. 39. The rate at which Hsapiens kill MixedFeeders. Three herbivore model. .......... 125

Fig. 40. The death rate of MixedFeeders according to the amount of grass in the

environment relative to the amount needed. Three herbivore model...................... 127

Fig. 41. The rate Hsapiens hunts Carnivores relative to their density. Three herbivore

model....................................................................................................................... 128

Fig. 42. Equilibrium mode graph. Three herbivore model.............................................. 130

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Fig. 43. - Second predator (overkill) mode, aggregated view. Three herbivore model. . 131

Fig. 44. Second-order predation, aggregated view. Three herbivore model ................... 133

Fig. 45. Second-order predation, herbivores Three herbivore model ............................. 134

Fig. 46. Second-order predation, plants Three herbivore model..................................... 135

Fig. 47. Grass, diagram. Four-herbivore model. ............................................................. 137

Fig. 48. Grazers diagram. Four-herbivore model............................................................ 138

Fig. 49. Actual efficiency of ruminant grazers (Grazers[Ruminant]) given the amount of

grass available in the system. Four-herbivore model. ............................................. 142

Fig. 50. Actual efficiency of non–ruminant grazers(Grazers[NonRuminant]) given the

amount of grass available in the system. Four herbivore model............................. 144

Fig. 51. The birth rate of ruminants Grazers[Ruminant] Four herbivore model............. 147

Fig. 52. Effect of Grazer[Ruminant] density, as Grazers[Ruminant] density declines

Grazers[Ruminant] birth function drops toward zero. Four-herbivore model. ....... 149

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Fig. 53. The rate at which Carnivores kill ruminant grazers (Grazers[Ruminant]). Four

Herbivore Model. .................................................................................................... 151

Fig. 54. The rate at which Hsapiens kill ruminant grazers (Grazers[Ruminant]). Four

Herbivore Model. .................................................................................................... 153

Fig. 55. The birth rate of non-ruminants (Grazers[NonRuminant]). Four-herbivore model.

................................................................................................................................. 156

Fig. 56. Effect of non-ruminant density, as Grazers[NonRuminant] density declines

Grazers[NonRuminant] birth function drops toward zero. Four-herbivore model. 157

Fig. 57. The rate at which carnivores kill non-ruminant grazers (Grazers[NonRuminant]).

Four-herbivore model.............................................................................................. 159

Fig. 58. The rate at which Hsapiens kill non-ruminant grazers (Grazers[NonRuminant]).

Four-herbivore model.............................................................................................. 161

Fig. 59. Equilibrium mode graph. Four-herbivore model. .............................................. 164

Fig. 60. Second predator (Overkill), Hsapiens enters the New World. Four-herbivore

model....................................................................................................................... 165

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Fig. 61. Second-order predation, aggregated view. Four-herbivore model. ................... 167

Fig. 62. Second-order predation, herbivores. Four-herbivore model.............................. 168

Fig. 63. Second-order predation, plants. Four-herbivore model. .................................... 169

Fig. 64. A. Second-order predation comparative graphs of A. browsers and B.mixed

feeders C.Aggregate with AmtHsKillCrn=0.015. Four herbivore model ............... 170

Fig. 65. Comparative populations predicated on varying migration of H. sapiens over

time.......................................................................................................................... 174

Fig. 66. Comparative population sizes predicated on varying food needs of H. sapiens

over time.................................................................................................................. 175

Fig. 67. Comparative populations predicated on varying food needs of H. sapiens and an

absence of second-order predation over time.......................................................... 176

Fig. 68. Herbivore populations where AmtHsMIgrate is set at 100,000, FoodNeedHs is

set at 10, and AmtHsKillCrn=0.075........................................................................ 178

Fig. 69. Interface to the model: ....................................................................................... 199

Chapter I: Introduction and Literature Review

Introduction

The greatest mammalian extinction event of the last ten million years occurred in North

America at the end of the Pleistocene epoch (called the Wisconsin by geologists and the

Rancholabrean by mammalogists). During a thousand-year period, more than 40

mammalian genera disappeared, an extinction rate of 77 percent prorated throughout the

stratigraphic interval. Thirty-nine of the 40 genera were large mammals. In geologic

terms, this extinction, occurring in thousands rather than millions of years, was

extraordinarily fast. By way of contrast, the second fastest mammalian extinction took

place in the Late Hemiphillian. It extended over 1.5 million years and involved 62

genera, 35 of which were large mammals (Webb, 1984). Prorated rates of mammalian

extinction during various periods are shown in Figure 1.

Throughout time, the origin and extinction of species and genera have been part and

parcel of how evolution happens. As Raup (1992) states, “most species [that once

existed] are [now] extinct.” He discusses the five massive extinction events of the

Ordovician, Devonian, Permian, Triassic, and Cretaceous periods. An average of 65

percent of all species disappeared during these periods. Following hydrologists’ use of

extreme value statistics, Raup has developed a “kill curve,” which allows scientists to

make intelligent guesses about the likely time span of extinction events of a given

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Fig. 1. Prorated rates of mammalian extinction (after Webb, 1989)

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magnitude. Thus, it can be determined that during the Cretaceous-Tertiary (K-T) periods,

the extinction of virtually all plants and animals, on land and in the sea, from dinosaurs to

plankton, took place over a hundred million years. At the other end of the scale is a low

level of background extinction that occurs more or less all the time, although it appears to

take place over a long time frame when compared to the relatively short span of human

life. Few of these extinctions are considered complete, because replacements usually

occur. (Replacement happens when one species of animal disappears and another, similar

species takes its place, thereby utilizing the same or a similar niche.)

Characteristics of the Pleistocene–Holocene Transition and Its Extinctions

Late Pleistocene extinctions fall somewhere between the two extremes described above.

They were nowhere near the magnitude of the “big five” and did not have the same

characteristics. They involved primarily large mammals and birds, rather than all life

forms as in the K-T period. On the other hand, the Pleistocene extinctions, such as the

disappearance of the saber tooth cat, the mammoth, and the mastodon, were without

replacements. Others were geographically limited; the camel and the horse, for example,

became extinct in the New World, but survived in parts of Europe, Africa, and Asia.

Gingerich (1984) suggests that the rate of replacement may be more significant than

the rate of extinction. For example, 56 percent of the large mammals that disappeared in

the Rancholabrean were not replaced by new species. That is, more than half of the

mammalian extinctions did not result in one species evolving into another with similar

niche demands (e.g., Bison antiquus into Bison bison). Instead, there were many species

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that disappeared completely (Webb, 1984; Gingerich, 1989). As a result, the

Rancholabrean extinctions were far in excess of those low-level background extinctions

that take place continually among all life forms.

It is possible that the extinctions of the Rancholabrean were simply the result of

random events. However, the magnitude and rate of extinction, the high proportion of

non-replacement, and the bias toward the extinction of large mammals strongly suggest

that they were caused by factors other than chance.

Ecosystem changes have been identified and described as a possible cause of the

extinctions of this period. The first such change was a general global warming, which

resulted in the disappearance of the Wisconsin ice sheet. In North America during the

height of glaciation, a sheet of ice that averaged more than a mile in thickness spread out

from Hudson Bay to bury all of eastern Canada, New England, and much of the Midwest.

A second ice sheet spread out from the Canadian Rockies and other highlands in western

North America to cover parts of Alaska, all of western Canada, and portions of Idaho,

Montana, and Washington. The final extent of the ice sheets’ edges and their subsequent

retreat can be traced in moraines (sedimentary deposits that accumulate at the terminus of

glaciers). In addition to these geomorphological signposts, evidence of planetary

warming comes from changes in the ratio of oxygen isotopes found in deep–sea cores

(Broecker & Van Donk, 1970; CLIMAP, 1976; Imbrie, 1985). These independent lines of

investigation are consistent in their demonstration of an increase in temperature of

roughly 6o Celsius. It is generally held that astronomical cycles are the mechanisms

forcing the alternation of glacials and interglacials (Imbrie & Imbrie, 1986).

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Some investigators have identified a second ecological change that took place at the

end of the Pleistocene, namely a decrease in moisture and a resultant increase in the

continentality of the climate. In North America, in comparison with other glacial-

interglacial transitions, the summers became hotter and the winters colder. Conversely,

Taylor (1965), working on mollusks found in the mid-continent, deduced that during

periods in the early part of the Pleistocene, winters were milder and frost-free, while the

summers were cooler. Further evidence for this change in continentality was a change in

the pattern of species association (Guthrie, 1968, 1980, 1990; Hoffman & Jones, 1970).

Plants (Martin & Mehringer, 1965; Davis, 1976; Delcourt & Delcourt, 1987), insects

(Ashworth 1977, 1980; Coope, 1967, 1977) and fauna (Kurten & Anderson, 1980;

Russell et al, 1984) that had lived together throughout most of the Pleistocene and other

temperate periods, were not found to be living together in Holocene environments. This

suggests that some species that once were able to live in the earlier, more temperate

climate, eventually found the summers too hot, while other found the winters too cold

As a result of these climate transformations, the pattern of vegetation changed. The

earlier, more heterogeneous, patchy environment was transformed into one comprised of

more specialized vegetation zones, i.e., grasslands in the center of the continent and

forests on the continental edges. Guthrie (1980) has described this change as one from

“plaid” to “striped” environments. The geographic region that today is associated with

prairie, or grassland, was more of a mixed woodland-parkland, a cross between a

temperate savanna and an open-canopy forest. Areas that today are closed-canopy forests

were once open, with many grassy places (Hopkins et al, 1967; Bryson et al, 1970;

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Hibbard, 1970; Wendorf, 1970; Birks & West, 1973; Anderson, 1974; Morgan &

Morgan, 1979, 1981; Brumley, 1978; Graham & Lundelius, 1989). According to Guilday

(1989):

The deterioration and disintegration of the eastern and western segments of this

parkland were, in some respects, mirror images of one another. In the western

segment, the Great Plains tree cover disappeared almost completely except for

scattered firebreak ridges and along river courses where corridor woodlands persisted.

In the eastern segment, as closed-canopy deciduous forest evolved, grasslands

became restricted primarily to river valley corridors. (p. 225)

In Alaska, sediments have yielded insect remains (Matthews, 1979) and pollen and

plant remains (Davis, 1976) that are associated with well-drained soil. They suggest a

parkland of low, grassy vegetation and scattered tree cover, rather than today’s tundra,

which is typified by treeless, poorly-drained, cold soils. The remains of a large and

diverse complement of animals, dominated by mammoth, bison, horse, and their

predators, lived in this parkland. Guthrie (1982) has named it the Mammoth Steppe

biome (Pruitt, 1960; Fuller & Bayrock, 1965; Flerov, 1967; Hoffmann & Taber, 1967;

Pewe & Hopkins, 1967; ; Hopkins et al, 1967;Repenning, 1967; Sainsbury, 1967;

Frenzel, 1968 Ritchie & Hare, 1971; Kurten, 1972, 1988; Yurtsev, 1972; Sorenson, 1977;

Batzli et al, 1980; Calef, 1984; Harrington, 1984; Guthrie, 1989).

Because climate and vegetation changes are strongly supported by the evidence

described above, it is highly probable that important changes in consumer populations

also occurred. Evidence from archaeological and paleontological sites shows a change in

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the spatial distribution of animal associations in the temperate zone. Many species of

animals that today are allopatric (occurring in disjunct geographic areas) were sympatric

(occurring in overlapping areas) during most of the Pleistocene. The literature from many

different fields is full of references to “defunct species associations” (Slaughter 1967),

“communities without modern or extant counterparts” (Matthews 1979), and

“disharmonious species associations” (Graham, 1976; Graham & Lundelius, 1989).

According to Guilday (1989):

. . .the broad belt of ecologically diverse, predominantly coniferous parkland that

extended from at least Wyoming east to the Atlantic Coastal Plain. . . disintegrated as

a biological unit within a relatively short period of time. . . its component species

either becoming extinct or regrouping themselves into assemblages that continued to

polarize to the present day. . . Neither of these corridors, wooded in the Plains,

grassed in the East, was extensive enough to support more than a few large mammals

on a sustained basis. (p. 225)

In addition to changes in the distribution of species associations, the Pleistocene–

Holocene transition was notable for its uneven impact on different groups of ungulate

mammals. A survey of extinct vs. extant animals (Anderson, 1989;Guthrie,1989)

indicates that the North American ungulate fauna of the Pleistocene was generally larger,

with comparatively more monogastrics, than those of the Holocene. According to Owen-

Smith (1992), the incidence of generic extinction correlates positively with body size. All

megaherbivore genera over 1,000 kilograms disappeared, compared with 76 percent of

genera in the 100 to 1,000 kilogram range, and 41 percent of genera between 5 and 100

kilograms.

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The final major change in the ecosystems of the New World during the late

Pleistocene was the arrival of H. sapiens. Haynes (1984) has done a survey of well–dated

(14C) and well–stratified sites in the United States. From this survey, he has extrapolated

a simplified pattern of the stratigraphy of the Pleistocene-Holocene. In each of the sites

surveyed, he has identified a turning point, from degradation (loss of material due to

glacial outflow) to aggradation (deposition of material due to sedimentation). He uses this

switch as a benchmark that can be traced across the country, tying together sites in

various locations. In all of these sites, evidence of megafauna is found below the

benchmark, with no evidence of artifacts. Above the benchmark, Paleo–Indian artifacts

occur for the first time in association with remains of megafauna. In sites that give

evidence of human occupation, the sequence is 1) Paleo–Indian artifacts with remains of

megafauna; 2) Paleo–Indian artifacts in association with transitional fauna remains; 3)

archaic artifacts, and then ceramics, both in association with the remains of modern fauna

(Figure 2). Thus, the picture drawn by Haynes’s survey shows that megafauna remains

exist at the lower levels , but became scarcer up through the stratigraphic sections, and

disappear entirely during the transition to the Holocene.

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Fig. 2 Correlation of the strata of the Pleistocene – Holocene transition in North America

(adapted from Haynes, 1984)

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Hypotheses Regarding the Cause of the Pleistocene-Holocene Extinctions

Since the timing of the Pleistocene-Holocene extinctions in North America correlates

roughly with climate change and the appearance of H. sapiens, it is not surprising that

they have been linked causally. This linkage has been expressed in two major categories

of hypotheses: those related to climate change and those associated with hunting by H.

sapiens.

Climate Change Hypotheses

When scientists first realized that there had been glacial and interglacial ages, and that

they were somehow associated with the prevalence or disappearance of certain species

and genera, they surmised that the termination of the Pleistocene ice age might be a

sufficient explanation for certain mammalian extinctions.

Increased Temperature

The most obvious change associated with the termination of an Ice Age is the increase in

temperature. Between 15kya and 10kya, a 6o Celsius increase in global temperatures

occurred at the climatic optimum. This was generally thought to present appropriate

conditions for an extinction event.

According to this hypothesis, a temperature increase sufficient to melt the Wisconsin

ice sheet also could have provided sufficient thermal stress to cold-adapted mammals to

cause them to die. The heavy fur of these mammals, which functioned to conserve body

heat in the glacial cold, might have impeded the dumping of excess heat into the warmer

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environment, so that they would have died of heat exhaustion. Large mammals, with their

reduced surface-area-to-body ratio, would have fared worse than small mammals.

Shortcomings of the Temperature Hypothesis

Perhaps the strongest argument against the temperature hypothesis is that since it was

first proposed it has become evident that today’s annual mean temperature is no higher

than that of previous interglacials (Andersen, 1973; Birks, 1973; Davis, 1976; Ashworth,

1980; Birks & Birks, 1980; Bradely, 1985). Because the large mammals survived similar

temperature increases in previous interglacials, warmer temperature alone is not a

sufficient explanation for the extinction of the Pleistocene megafauna. Furthermore, cold-

adapted animals, such as polar bears, are able to survive the warm summers in zoos

located in temperate zones.

Increased Continentality of Climate

The increase in continentality (hotter summers and colder winters) is also cited as either a

direct or indirect climate-related cause of extinction (Bryson et. al, 1970; Graham &

Lundelius, 1989; King & Saunders, 1989). Axelrod (1967) and Slaughter (1967) argue

that along with colder winters and hotter summers, the late Pleistocene experienced less

rainfall, which was also less predictable.

According to the supporters of this group of hypotheses, the changes in the amount

and/or distribution of rainfall could have affected the survival of the Pleistocene

mammalian megafauna by changing the amount and kind of vegetation that served as the

basis for the energy/nutrition relationships within the ecosystem. Graham & Lundelius

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(1989), Guthrie (1989), and Guilday (1989) attribute the extinctions to changes in the

floral environment. Graham & Lundelius (1989) say that because of the change in

continentality and the ensuing change from mixed woodland-parkland to separate prairie

and woodland environments (see above), there was a change in the kinds of food

available. They suggest that herbivores were unable to find the plants with which they

had co–evolved. As a result, they fell prey to the anti–herbivory toxins of the plants they

were able to find.

In a related vein, Hoppe (1978) and Guthrie (1980, 1989) argue that the extinction of

large herbivores and the dwarfing of many others was due to changes in the length of the

growing season. Based on their observations of modern fauna, they surmise that large

ruminants, such as bison, fared better than monogastrics, such as horses and elephants.

The large ruminants may have succeeded because they were able to extract more

nutrition from limited quantities of high-fiber food, and thus were better able to deal with

anti–herbivory toxins (Hoppe, 1978; Guthrie, 1980, 1989). However, this hypothesis has

been challenged by those who observe that increased continentality effected ecosystem

changes, which, in turn, resulted in an increased prevalence of grasses. McDonald (1981,

1989) suggests that the animals that became extinct actually should have prospered

during the shift from mixed woodland-parkland to prairie, because their primary food

source, grass, was increasing rather than decreasing (Birks & West, 1973; McDonald

1981, 1989).

Another hypothesis connecting megafaunal extinction to increased continentality

suggests that changes in rainfall restricted the amount of time favorable for reproduction.

13

Axelrod (1967) and Slaughter (1967) posit that large animals, with their longer, more

inflexible mating periods, often produced young at unfavorable seasons (i.e., when

sufficient food, water, or shelter was unavailable because of shifts in the growing season).

In this vein, Kilti (1989) suggests that the better survival rate of small animals may have

been due indirectly to the unpredictability of rainfall. He observes that the relationship

between the timing of gestation and the periods of available resources may have been

more important than the actual length of gestation per se. He says that small mammals,

with their shorter life cycles, shorter reproductive cycles, and shorter gestation periods,

were better able to adjust to the increased unpredictability of the climate, both as

individuals and as species. This better adjustment came about, in part, because their

reproductive efforts often coincided opportunistically with conditions favorable for

offspring survival. Even when their efforts did not coincide with good conditions, they

still risked less and lost fewer offspring than the large mammals, because they were better

able to repeat the reproductive effort later on, when circumstances once more favored

offspring survival.

Shortcomings of the Continentality Hypotheses

Critics of the continentality hypotheses have suggested a number of shortcomings. One

difficulty is that megaherbivores seem to have prospered in other continental climates.

For example, Holocene climates of North America are more continental today than they

were in the Pleistocene, but they are not more continental than the climate of Siberia

during the Pleistocene, in which megaherbivores were abundant (Flereov, 1967; Frenzel,

1968; McDonald, 1989).

14

Another argument against the continentality hypotheses addresses the presence of horses

in Holocene environments. The critics point out that although horses became extinct in

the New World, they were successfully reintroduced by the Spanish in the sixteenth

century. Today there are feral horses still dealing with environments similar to those in

the Holocene. They find a sufficient mix of food to avoid toxins, and they extract enough

nutrition from forage to reproduce effectively.

A third argument suggests that large mammals should have been able to migrate,

permanently or seasonally, if they found the temperature too extreme, the breeding

season too short, or the rainfall too sparse or unpredictable (Pennycuick, 1979). As a

modern-day example of this adaptation technique, African elephants migrate during

periods of drought to places where there is apt to be water (Wing and Buss, 1970).

Furthermore, season length is not simply a condition of temperature and humidity; it is

also a function of latitude. By moving south during the Pleistocene-Holocene transition,

Holarctic herbivores could have found areas with growing seasons more favorable for

finding food and breeding successfully.

Additionally, in studies of modern megaherbivores, Owen-Smith (1992) found that large

animals store more fat in their bodies than do medium-sized animals. Hence, increased

body mass should have encouraged an adaptation that compensated for extreme seasonal

fluctuations in food availability.

Finally, one of the soundest pieces of evidence against a purely climatic explanation is

the survival of members of extinct genera on isolated islands. According to Burney

15

In Europe, for instance, the last members of the elephant family survived climatic

warming not on the vast Eurasian land mass, but on small islands in the

Mediterranean—an even warmer climate. Radiocarbon dates suggest, for instance,

that dwarf elephants persisted on Tilos, a tiny island in the Aegean, until perhaps

4,000 to 7,000 years ago. (Burney, 1993)

Dwarf mammoths also survived on Wrangel Island in the Siberian Arctic until , until

4,000 to 7,000 years ago. (Vartanyan, 1993)

The theory that held the combined impact of hunting and fire caused the extinctions

now appears to be overly simplistic, according to the research of Burney and others on

Madagascar. The analysis of fossil pollen and charcoal in sediment cores from

throughout the island have shown that wildfires and vegetation changes were a normal

part of the environment for 35,000 years or more before the arrival of people. (Burney,

1993)

The Overkill Hypothesis: Hunting by H. sapiens

The overkill hypothesis suggests that humans hunted New World megaherbivores to

extinction. As a result, carnivores and scavengers that depended upon those animals

became extinct from lack of prey (Martin, 1963, 1967, 1984, 1986, 1988; Reeves, 1978;

Scott, 1984). This hypothesis is based on the observation that throughout the world,

extinctions have followed the emergence of H. sapiens. As shown in Figure 3, large

segments of fauna disappeared soon after humans entered the scene. Moreover, the

severity of extinction was greatest in areas where humans arrived relatively late, such as

16

Fig. 3. The march of extinction (after Martin, 1984):

A. loss of mammalian species relative to the immigration of H. sapiens; and B. extinct

fauna and archaeological evidence

A

B

17

the New World and Madagascar, and was least where humans arrived early, such as

Africa.

The Rationale Behind the Hypothesis

The rationale behind this hypothesis is that “naive” animals (ones who have no

experience with humans), are more easily killed than animals that evolve in the presence

of H. sapiens. Following this line of reasoning, the argument goes that because H.

sapiens originated in Africa, African animals had the most experience with H. sapiens

and presumably learned and evolved in response to the new hunting techniques that

threatened them. When H. sapiens entered other areas later in time, the animals they

encountered were naive about hunting and hence more vulnerable. There was a greater

gap between the ability of the hunter to kill and the ability of the animal to avoid being

killed. Flannery (1995) and Diamond (1984, 1997) have both given modern examples of

hunters coming within easy killing range of naive animals.

The rationale described above not only explains the timing of the disappearance of large

components of the fauna; it also explains why mammalian megafauna still exists in

Africa, but is depauperate in the rest of the world (Martin, 1966; Dawkins & Krebs, 1979;

Foley, 1984; Flannery, 1995).

In the New World, the presence of projectile points found imbedded in the bones of

extinct animals , as well as the presence of bones of extinct animals found in association

with archaeological sites, suggest that hunting by H. sapiens was directly responsible for

the extinction of the Pleistocene mammalian megafauna. Applying the overkill

18

hypothesis to this evidence, one can assume that evolution did not equip these animals to

withstand the onslaught of technologically-supported predation when humans finally

arrived on the scene. (Hester, 1967; Frison, 1974, 1978; Brumley, 1978).

Computer Simulations of the Overkill Hypothesis

Mosimann and Martin (1975) and Whittington and Dyke (1984) have designed computer

simulations of their “blitzkrieg” version of the overkill hypothesis (Martin 1974). In the

model designed by Mosimann and Martin (1975), human population is assumed to have

begun with 100 individuals around Edmonton, Canada. This population, with a densely

consolidated front of hunters, subsequently fanned out across the continent in an arc that

expanded geometrically (simulations were run at various population growth rates,

between 2.4 and 3.5 percent per year). The designers of this model likened the advance to

a blitzkrieg, or military invasion of great force and speed, in which the front remained

stationary only until all the megafauna in a given area were extinct. Then it would move

on at a rate of 20 miles per year. It was assumed that the hunters in the front were so

densely positioned that megafauna were not able to cross through the line. In fact, it was

assumed that the major factor causing extinction was the density of the front rather than

the overall density of the human population behind it. It was also assumed that the reason

for the lack of archaeological evidence was the speed with which the front advanced. In

the simulation, the front reached the coast in about 300 years and megafaunal extinction

occurred within three years of that time. Biologically, the model was based on

observations of the spread of snails, fish, and large herbivores into previously

uninhabited, but now habitable regions (Mosimann & Martin, 1975; Whittington & Dyke,

19

1984). Table 1 reports the variables and starting values of the Mosimann and Martin

model.

Table 1: Variables and starting values of the Mosimann and Martin and the

Whittington and Dyke models

Mosimann & Martin (1975) Whittington & Dyke (1984)Description of Variable

Source Source

Human population size

(individuals)

100 Arbitrary 200 Budyko, 1967, 1974

Human population growth rate

(percentage per year)

0.024 Birdsell, (1957);

Caughley, (1969)

0.0443 Birdsell, (1957)

Prey carrying capacity (a.u. *

per sq. mile)

25 Martin, (1973) 25 Mosimann and Martin,

(1975)

Prey biomass replacement rate

(a.u. * per sq. mile)

0.25 0.25 Mosimann and Martin,

(1975)

Human carrying capacity

(individuals per sq. mile)

1.295 Budyko, (1967, 1974)

Initial prey biomass (a.u. * per

sq. mile.)

25 25 Mosimann and

Martin, (1975)

Prey destruction rate (a.u.* per

person, per year)

3.862 Derived

* a.u. = animal units = 1K lb. of herbivore

The model created by Whittington and Dyke (1984) is based on the Mosimann and

Martin model (1975), but does not assume a front. For the most part, it contains the same

base values with some additions, as indicated in Table 1. In this model, the human growth

rate begins to decrease, thus doubling the time needed for population density to force the

20

periphery of the inhabited area to move at 20 miles a year. The baseline values include,

human carrying capacity, which is defined as 1.295 individuals per square mile, based on

the upper limit of estimates of human density in Europe at the end of the Upper

Paleolithic (Budyko, 1967, 1974, in Whittington & Dyke, 1984).

Whittington and Dyke’s (1984) prey destruction rate is 38.6 pounds of animal per pound

of H. sapiens per year. They derived the baseline value for the prey destruction rate by

running the model with the baseline values and varying the prey destruction rate until it

resulted in megafaunal extinction. They found that it represents 3,862 pounds of animal

killed per person annually, or, if we assume 100 lbs. average weight, 38.6 pounds of

animal per pound of H. sapiens per year.

Whittington and Dyke ran the model, employing various starting values and rates.

They found that extinction of megafauna will occur once the human population reaches

its carrying capacity, defined as “critical density” by Mosimann and Martin (1975), and

as “human carrying capacity” by Whittington and Dyke (1984). They found this to be

true even though the value of the prey-destruction is only 0.0001 a.u. above what the prey

biomass replacement rate will support. They state:

The only way to avoid megafaunal extinction is to reduce the human population

growth rate to zero before the [human] density reaches the threshold [carrying

capacity] . No matter how slowly the human population continues to grow while

hunting at its old rate, once the threshold is reached the megafaunal population will

fall quickly in density and size. This will occur even though the value of the prey

destruction rate is only 0.0001 a.u. (animal unit, 1 a.u. = 1,000 pounds of living prey)

above what the prey biomass replacement rate will support, an amount imperceptible

21

to a hunter. Even if humans switch to more readily available prey when the animal

biomass begins its crash, as long as they continue to occupy the entire continent and

kill megafaunal prey when they come upon it, the extinction process will continue.

(Whittington and Dyke, p. 463)

Shortcomings of the Overkill Hypothesis

Computer simulations add credence to the overkill hypothesis. However, all computer

models are based on the modeler’s assumptions. Some of the assumptions in question

here should receive further examination.

First, the models assume that a destruction rate greater than the replacement rate will

always lead to extinction. However, because human hunting rates are not driven by prey

replacement, it does not matter that humans cannot perceive the difference

Second, the derived hunting rate seems unrealistically large. There is simply no evidence

that H. sapiens needs more meat than non-human predators. Yet Whittington and Dyke’s

rate of 38.6 pounds of prey per pound of H. sapiens per year is almost double the needs

of a large felid, which requires 20 pounds of food per pound of cat per year (The Cat

House, 1996). It may be argued that H. sapiens is more wasteful than are cats. However,

to balance that, H. sapiens is an omnivore, who eats vegetables as well as meat,

The third assumption is that the carrying capacity for humans remains constant.

“Carrying capacity” is a term that expresses the concept that the resources of an area can

satisfy the needs of only a finite number of individuals. According to this concept, each

individual requires some constant fraction of the area to meet its resource needs. Thus,

the value of the carrying capacity can be expressed as the amount of area required to

22

support one individual, or more commonly, as the maximum number of individuals that

can be supported by a given area. Definition of the carrying capacity on an areal basis

assumes that the resources available to meet the needs of the consumers are everywhere

the same all of the time, and remain so throughout the entire simulation. This assumption

is appropriate under conditions in which the environment remains homogeneous and

unchanging, and one in which the constant consumption of resources is exactly balanced

by the constant replenishment of those same resources. Both the “critical mass” of the

Mosimann and Martin model (1975) and the “human carrying capacity” of the

Whittington and Dyke model (1984) express carrying capacity in these terms. But, in

fact, the many changes in abiotic and biotic features at the end of the Pleistocene are

themselves clear evidence that this assumption is unrealistic.

The introduction of hunter-gatherer H. sapiens into the ecosystem of North America

must have disrupted the existing predator–prey relations in the same ways that adding an

exotic predator disrupts a present–day balanced predator–prey system. Specifically, the

prey population decreases as it bears the impact of an additional predator. Therefore the

carrying capacity for predators decreases. Because the Pleistocene megafauna became

extinct, it is reasonable to assume that the prey populations could not replenish

themselves, and so the carrying capacity for the new predators (H. sapiens) had to have

declined. Yet the overkill hypothesis argues for a constant carrying capacity, based on a

steady rate of resource replenishment.

In formal models, it is generally assumed that predators and prey are linked in a mutually

causal loop (Caughley, 1970; May, 1973; Roff, 1975; Gluckenheimer et al, 1976; Hanby

23

& Bygott, 1979; Smuts, 1979; Hilborn & Sinclair, 1979; Cushing, 1984; Schaffer, 1988).

Thus, as prey populations increase, predators increase. As prey populations decrease, so

do predator populations. This makes it impossible for predators to kill off all their prey as

illustrated in Figure 4. In the real world, the kind of oscillations shown in the formal

model have been observed in simple ecosystems (Elton & Nicholson, 1942; Bulmer,

1974; Batzili et al, 1980). In complex ecosystems, however, as a prey species becomes

scarce, predators hunt prey that is more plentiful. This smoothes the oscillations and

makes it less likely that the prey will become extinct.

In the argument above, it is assumed that human and non- human predators behave

similarly – that they tend to switch prey as it becomes scarce, or that predator populations

will be reduced in response to a reduction in prey populations. Even so, Whittington and

Dyke (1984) imply that human predators are less likely than non- human predators to

switch prey.

At this point, it may be appropriate to argue that human predators are quite dissimilar

to non-human predators. I would suggest that humans are unlike non-human predators in

two respects. First, they are able to think economically; and second, they are omnivores.

For both of these reasons, humans are more likely to switch prey than non- human

predators. In support of the economic argument, Hawks and O’Connell (1994), in

observations of current-day hunter-gatherer groups, found that it is possible to predict

which foods will be utilized or discarded on the basis of frequency of resource encounter

and relative profitability (i.e., the rate of time spent on post–encounter pursuit, capture,

24

Fig. 4 Oscillation of predator and prey populations

25

and processing). Thus, as the frequency of encounter decreases, other resources will be

utilized. This suggests that human predators are likely to switch prey sooner than non-

human predators and are therefore less likely to overhunt their prey than non- human

predators.

Finally, animals that were not known to have been hunted by H. sapiens, such as the

Shasta ground sloth, became extinct at the same time as other Pleistocene mammals. The

overkill hypothesis does not specifically address this issue. In fact, by following the

reasoning behind the hypothesis, one would have to assume that because the ground sloth

became extinct, it must have been hunted by H. sapiens.

Combination Hypotheses

Diamond (1984) observes that extinctions in historic time have been brought about by a

variety of causes, ranging from simple overhunting, such as happened to the Great Auk

(Pinguinus impennis), through complex combinations of effects. He says of climate-

induced extinctions:

In considering the modern effects of climate, we had to content ourselves with

examining range contractions and local extinctions, because almost no modern cases

exist of total extinction due clearly to climate. (p.838)

By contrast, Diamond says of human-induced extinctions:

Modern man has proven versatile as an exterminator, with at least six major methods

long at his disposal (and a seventh, chemical pollution, recently added): overkill;

habitat destruction by logging, fire, induced browsing and grazing animals and

26

draining; introduction of predators; introduction of a competitor; introduction of

diseases; and extinctions secondary to other extinctions. All six modes were probably

effective in prehistoric extinctions as well. (p. 839)

This way of thinking about extinctions is useful. Diamond (1984) gives examples of

secondary extinctions, or “trophic cascades” (a succession of events based on nutritional

relationships) in history. One example is the extinction of ground-nesting birds on Barro

Colorado Island.

…insularization led to the loss of the largest predators (jaguar, puma, Harpy Eagle)

leading to a population explosion of smaller predators such as monkeys, pecaries,

coatimundis, and possums that served as their prey and that in turn rob bird nests.(p.

845)

In the same paper, Diamond also points out that just as there are a variety of

individual methods that have produced extinction, there are also cases in which a

combination of factors were at work.

The Heath Hen (Tympanuchus cupido cupido) was. . . shot by the thousands for food,

preyed upon by introduced cats, and afflicted with diseases of introduced poultry, all

the while its grassland habitat was being converted to farmland. By1830 it was

confined to the island of Martha’s Vineyard where its numbers rose to 20,000 by

1916. In that year its numbers were decimated by a fire in the summer, followed by a

harsh winter and the invasion of Goshawks. Cats, inbreeding and a disease introduced

with turkeys reduced its number to 13 in 1928, 2 in 1929, and in 1939 one, which

died in 1932. (p. 846)

The theory that held the combined impact of hunting and fire caused the extinctions

now appears to be overly simplistic, according to the research of Burney and others on

27

Madagascar. The analysis of fossil pollen and charcoal in sediment cores from

throughout the island have shown that wildfires and vegetation changes were a normal

part of the environment for 35,000 years or more before the arrival of people. (Burney,

1993)

Tim Flannery (1995) examines the ecology of the Australasian islands and describes a

variety of anthropogenic extinctions. In his work, he says that humans either adapted or

failed to adapt to the ecology of the area. On many of the Australasian islands, humans

hunted the fauna to extinction; then they came close to going extinct themselves. For

example, Flannery suggests that the success of Australian aborigines was due largely to

their use of “firestick farming.” He suggests that the use of fire was a cultural adaptation

necessitated by the extinction of Australia’s megaherbivores. He states:

After their extinction, fire became the main consumer of Australian vegetation.

Without human interference the fire pattern in Australia would probably have been

one of vast, periodic wildfires that ravaged huge areas of the continent. Indeed, this is

precisely the kind of fire regime that has emerged over much of the continent since

Aboriginal firestick farming ceased. . . It seems entirely possible that firestick

farming initially evolved as a response to the threat that the natural fire regime posed

to middle sized mammals. (p.240)

Australian megaherbivores, before their extinction, were what Owen-Smith (1992) calls

“keystone species.” By this he means that they served to maintain the balance of the

ecosystem. Once a keystone species is removed, the balance is broken and the ecosystem

becomes unstable until another equilibrium is found. Fire became the way aboriginal

people kept the forests from covering the environment and taking over grazing land. All

28

the early explorers of Australia record the use of fire by aboriginal people and describe

the open-woodland landscapes that the fires helped to sustain. Since the domination of

the continent by Europeans, aboriginal use of fire has been suppressed. This has resulted

in a transition from open woodlands to rainforest. In the absence of fire, vegetation has

filled in spaces that once were open (Flannery, 1995).

Burney suggests from the evidence from Madagascar that

…many of our sites show evidence for a combination of simultaneous changes,

including natural climate change, activities of the first human hunters, changes in fire

regime and vegetation structure, and the arrival of exotic species. I have come to refer

to the extinction event in Madagascar about 1,000 years ago as a “recipe for disaster.”

Instead of finding overwhelming evidence for the actions of a single cause in the

extinctions, these four factors, and perhaps others, seem to have been functioning

simultaneously …(Burney, 1993)

Criteria for New Hypotheses of Extinction

It seems eminently reasonable to think about extinctions as resulting from a variety of

anthropogenically-related factors, which caused a collapse of one ecosystem into another.

To understand these factors, a closer examination of the characteristics of an extinction

event may yield criteria to use in evaluating various hypotheses.

In one study, a condition of scarcity is suggested by the narrowness of growth rings on

mammoth tusks, which are wide when conditions are good, and narrow when conditions

are poor (Haynes, 1995, 1998). Dietary stress is also implicated as a cause of extinction

in an isotopic analysis of mammoth and mastodon remains (Koch, 1998). However,

29

neither climate change nor overkill by H. sapiens should have resulted in scarcity

conditions during the Pleistocene-Holocene transition. As herbivores were reduced in

numbers due to hunting, the competition between herbivores for food would have been

reduced. In fact, there would have been a net increase in available food. In addition, the

retreat of the ice sheet would have freed up more land for vegetation; overkill, by

eliminating some of the herbivores, would have resulted in more plants per remaining

animal.

Holocene fauna, in contrast to Pleistocene fauna, had a bias in favor of ruminants

(Graham, 1998; Guthrie, 1989; Koch, 1998). Ruminants extract energy and nutrients

more efficiently from their food than do monogastrics. Therefore, they need less food to

survive and to reproduce (Janis, 1975; Guthrie, 1989). The bias in favor of ruminants

suggests to this researcher a condition of scarcity of plant food.

These facts and their implications suggest that any new hypothesis should evaluate a

combination or factors and should address and explain 1) the extinction of horses in

North America; 2) the extinction of the ground sloth; 3) the bias in favor of ruminants;

and 4) the bias in favor of small mammal size.

The development of better simulation technologies presents an opportunity to develop an

alternative simulation model that takes into account these new observations. Such a

model may be able to eliminate the unrealistic assumptions in the existing models, which

were developed in support of the overkill hypothesis.

30

For an alternative hypothesis to be accepted, it must take into account all of the

features that are generally accepted as characteristic of the Pleistocene-Holocene

transition. These features are reported in summary form in Table 2. In addition, the

alternative hypothesis includes relevant information and understandings that come from

the ecological study of the interactions within and among present–day animal and plant

populations and communities. For example, Box A of Figure 5 presents the change in

mammalian megafauna populations as revealed by the fossil evidence. The fossil record

indicates only that extinction took place, not how that extinction occurred. Box B of

Figure 5 shows the monotonic decline of megafauna, using the assumptions of the

climate change and the overkill hypotheses. Box C of Figure 5 shows similar, but discrete

monotonic decline lines for both herbivores and predators, again using the assumptions of

the climate change and overkill hypotheses.

31

Table 2: Features of the Pleistocene-Holocene transition

Pleistocene Holocene

Climate Colder, less continental, higher

relative humidity, ice sheets

Warmer, more continental, lower relative

humidity, ice sheets melted

Vegetation Mixed woodland-parkland, more

patchiness in eastern and western

forests, more trees in the center of the

country

Unbroken tree cover in the east and west,

prairie-grassland in the center of the country

Animals More genera, many large animals.

more monogastrics

Fewer and smaller animals, a larger

percentage of ruminants than in the

Pleistocene

Archaeology Paleo artifacts, stylistic homogeneity,

evidence of hunting megafauna

Archaic artifacts, stylistic heterogeneity, less

megafauna

32

An Alternative Hypothesis of Extinction

An alternative hypothesis to those discussed previously proposes that the megafaunal

extinctions at the end of the Pleistocene were the result of complex interactions involving

changes in climate, changes in herbivore food supplies, changes in predator–prey

relationships, and changes in the activities of H. sapiens. In brief, the hypothesis is that

upon entry into the New World, H. sapiens reduced predator populations such that

herbivore populations expanded, which, in turn, resulted in environmental exhaustion and

ecosystem collapse due to overgrazing. As edible plants dwindled, the megaherbivores

lost their food supply and eventually became extinct by reason of starvation.

The Argument for Second-Order Predation

In support of this general approach, studies of present-day populations of herbivores

indicate that their numbers are often kept small by the action of their dependent predators

(Petersen, 1979; Carbyn et.al., 1993; Dale et.al., 1995; Seip, 1995;.Klein, 1995) Without

predator control, a population of herbivores expands to the point at which its food

becomes scarce, a situation leading to its rapid decline. If the “boom phase” of this

population cycle is large enough, significant changes within the ecosystem occur. As a

result, vegetative support for the herbivore population dwindles, and this deprivation

leads in extreme cases to a “bust phase,” which may be so severe that extinction of the

herbivore species occurs.

In the transition from the Pleistocene to the Holocene, more carnivores than herbivores

were lost (Graham, 1998). As a result, it is possible that the populations of Pleistocene

33

herbivores experienced a temporary release from control by predators. This may have led

to dramatic increases in herbivore population size, followed by such severe effects upon

and changes in the vegetation that the ecosystem was no longer able to support many of

its herbivore species. This pattern of decline is presented in Box D of Figure 5.

The boom- and-bust cycles of herbivore populations have been mathematically

described by Pitelka (1964). May (1973), Gluckenheimer et al (1976), Schaffer (1988),

and Swart (1990). They also have been observed in field studies involving caribou

introduced into predator–free islands in the Antarctic (Leader–Williams, 1988). The role

of predators in controlling herbivore populations is well illustrated by the expansion to

pest densities of white-tailed deer in predator-free suburbs of the eastern United States. It

is also evident in field studies of reindeer, caribou, moose, feral horses, burros, and bison

(Scheffer, 1951; Petersen, 1977; Leader–Williams, 1989, Carbyn et al, 1993).

An objection to this approach might be that H. sapiens would not have hunted the

predators in question. However, predators, both human and non-human, could have been

influenced indirectly by complex changes or disruptions in the ecosystem. This has been

demonstrated in some of the interactions of humans, predators, and prey in Africa. For

example, at the time of the European colonization of Africa, rinderpest, a pathogen

functioning as a predator, was introduced into the wildebeest and buffalo populations.

The loss of these animals to rinderpest was so severe that the population of lions was

deprived of a major source of food. Prior to this time, lions and humans had established a

relationship of mutual avoidance. But now, with the severe loss of their traditional prey,

the lions turned to attacking and eating humans. This threat triggered an anti–lion

34

Fig. 5 The path of extinction held by various hypotheses

35

response among humans, and the anthropogenic death of lions, though not killed for food,

contributed importantly to the reduction of their numbers (Schaller, 1972; Sinclair, 1979).

On the other hand, the reduced populations of wildebeest and buffalo, while not attaining

levels reached in pre–rinderpest times, stabilized and remained viably large (Schaller,

1972).

Another objection might be that predators do not hunt predators. However, predators

may change their behavior after a disruption in the usual predator–prey relationship. The

events surrounding the local extirpation and subsequent reintroduction of the gray, or

timber, wolf in parts of North America illustrate this pattern. Prior to the twentieth

century, foxes, coyotes, and timber wolves coexisted as predators upon a variety of

mammalian prey. During that time, foxes and coyotes avoided wolves. Perhaps because

of this avoidance, wolves were not known to kill foxes or coyotes. After the timber wolf

was extirpated from much of continental North America, generations of foxes and

coyotes hunted without competition or interference. Over the years, as they became

habituated to being the dominant predator, they apparently lost the behavior habits that

had helped them avoid the larger and more aggressive wolf. Today, when gray wolves are

reintroduced to their former lands, it is not unusual for them to kill foxes and coyotes

(Petersen, 1995). On the other hand, elk, which also experienced the absence of gray

wolves for the same amount of time, maintained the same avoidance and protection–of–

young behaviors that they had exhibited before the wolf was driven to local extinction

(Mlot, 1998). This contrast suggests that elk have a genetic avoidance response to

36

wolves, whereas foxes and coyotes seem to have learned and then forgotten their

avoidance response.

The sensitivity of predators to changes in the predator–prey relationship and the

subsequent killing of one group of predators (in the case cited, the foxes and coyotes), by

another group of predators (the gray wolves) is an example of second-order predation

(Peterson, 1995). In the instances of wolves killing foxes and coyotes, there is no

suggestion that they were killed for food. Rather, they seem to have been killed to

eliminate competition. H. sapiens, the newly-introduced, second-order predator we are

considering in the context of Holocene extinctions, did not eat other carnivores, although

these early humans may have been aware of competition, and probably killed carnivores,

utilizing their fur for clothing and their teeth for ornaments. Soffer (1985) documents

numerous instances of fur-bearing mammal bones especially the foot bones, in

Pleistocene archaeological sites (e.g. White, R. 1993. Technological and social

dimensions of Aurignacian-age body ornaments across Europe)

Second-order predation, as explained earlier, can lead to a very rapid decline in the

preyed-upon population of predators, especially if prey have no mechanisms to resist,

because of their previous status as dominant carnivores. Therefore, when non-human

predator populations are reduced, herbivore populations increase and thus trigger the

boom-and-bust cycles that result in ecosystem collapse.

In support of the theory of ecosystem collapse resulting from the impact of a

megaherbivore population explosion, the severity of such an impact on the ecosystem of

37

African megaherbivores (elephant, rhinoceros, and giraffe) has been well documented by

a variety of investigators (vanWyk & Fairfall, 1969; Laws, 1970; Leuthold, 1977; Short,

1981; Yoaciel, 1981; Anderson & Walker, 1984; Barnes, 1985; Owen-Smith, 1992). All

have found that under population stress, megaherbivores can destroy a woodland or

savanna. Anderson and Walker (1984) report that under food stress, African elephants

have turned mixed woodland into grassland by knocking over trees to get leaves at the

treetops. Wing and Buss (1970) report that the effects of elephant damage combined with

destruction by fire have converted forest into grassland in only 50 years.

According to Smithers (1983), the difference between savanna woodland and

unwooded savanna grassland is caused by over-utilization. To this researcher, his

conclusion could apply equally to the transition from mixed woodland to prairie that took

place during the Pleistocene-Holocene transition.

Proponents of the theory that environmental stress results in climate change have

observed that when trees are removed from an area to such an extent that a forest is

converted to a grassland, one result is a general drying out of the local and regional (and

perhaps even global) climate because of a loss in transpired water (Charney et al., 1975;

Potter et al., 1975). Martin (1993) writes:

With regard to transpiration, which is vaporization at the leaf surface of water

extracted from the soil by the plant, the physiological responses of the vegetation and

its physical characteristics determine the partitioning between the sensible and latent

heat at the surface of the earth. This, in turn, affects atmospheric motion, and the

water balance. (p.133)

38

Other studies indicate that there is a considerable reduction in humidity after a forest

is clear-cut. Longman and Jenik (1987) write of the effect of deforestation in Brazil:

…that the forests influence the rainfall. At 17 sites throughout the Amazon Basin, the

proportion of river water originating by evaporation from the South Atlantic Ocean or

from the forest has been estimated, using the natural frequency of the isotopes

oxygen-18 and deuterium as tracers (Salati et al. 1979). About 75 per cent of rainfall

evaporates directly or via the trees, and provides much of the moisture for cloud

formation and rain further inland. Significantly, deforestation near the coast seems to

break the cycle, which propagates the repeated succession of rainstorms moving

rapidly westwards thus threatening the survival of otherwise untouched tropical forest

ecosystems, far inland. (p.15)

A Proposed Scenario of Pleistocene Extinctions Due to Second-Order Predation

Modeling is like a laboratory experiment in that the researcher seeks to isolate the

relevant variables from extraneous factors. Economic models do this by stating at the

beginning “all things being equal,” even though everyone knows that things are never

economically equal. By the same token, real ecosystems are seldom, if ever, in a state of

equilibrium. But to isolate the relevant variables in a model of possible causes of

Pleistocene megafaunal extinction in North America, it must be assumed that prior to the

entry of new, or second-order, predators (H. sapiens), the plants, herbivores, and original,

or first-order, predators existed in a steady state—a state of dynamic equilibrium. Starting

from this steady state, the proposed scenario proceeds in the steps briefly described

below.

39

Step 1: H. sapiens Enters the New World

It is assumed that with the arrival of a new predator, namely H. sapiens, the populations

of the original, or first-order North American predators declined. This assumption is

represented graphically in Figure 6 and is based on three prior assumptions. First,

because the non-human predators did not evolve in the presence of H. sapiens, they were

naive and lacked both avoidance and defense behaviors. Second, because humans were

hunters, they frequently occupied the same territory as the non-human predators. This

propinquity gave humans the opportunity to recognize the non-human predators as actual

or potential competitors for prey species. Third, because of the pressure of an additional

predator, the prey populations declined, at least initially, thereby reducing the food

available for both humans and the non-human predators. This food shortage probably

encouraged the non-human predators to switch prey, perhaps even causing them to begin

to prey on H. sapiens.

Step 2: Second-Order Predation Begins

Because human hunters had the opportunity to learn the habits of the non-human

predators, and because they possibly saw themselves as competing with or preyed upon

by these animals, it is reasonable to assume that they eventually began to kill them. Also

since humans in cold climates require some kind of clothing, it is likely that they hunted

carnivores for their fur. At first, killing the competition relieved predatory pressure on the

prey, thus allowing the herbivores to regain at least some of their numbers. The increased

40

Fig. 6. Effect of the arrival of a new predator on the populations of North American prey and

predators

41

abundance of prey would have convinced H. sapiens that the non-human predators were

competitors that kept the food supply low. Therefore, H. sapiens would have continued to

kill them, especially if they had themselves begun to kill H. sapiens.

Step 3: Prey Populations Are No Longer Well Controlled by Predation

The continual killing non-human predators by H. sapiens is assumed to have reduced

their numbers to a point where these animals no longer regulated the size of the prey

populations.

Step 4: Prey Populations Trigger Boom-and-Bust Cycles

It is assumed that as prey populations suddenly expanded, they overgrazed and over-

browsed the land. Soon the environment was no longer able to support them. As a result,

many herbivores starved, and some species became extinct.

Step 5: Mixed Parkland Becomes Grassland

Under conditions of scarcity, mixed feeders, including mammoths and mastodons, ate

less grass and more browse (tender twigs and leaves of trees and shrubs). They pushed

over trees to get at the leaves on the tops of trees. This killed more trees. This shift in

vegetation put competitive pressure on animals that were exclusively browsers. Together,

the mixed feeders and browsers, by virtue of their eating habits, had a profound impact

on trees, eventually turning mixed parkland into prairie, or grassland.

The loss of tree cover favored the grazers, by selecting out those animals that needed

mixed food. In a denuded environment, herbivores that were able to survive on the

42

newest shoots of grass got the most nutrition out of poor quality forage. Also within

species smaller animals who could reproduce on the least amount of forage where

selectively favored. Others, such as horses, mammoths, mastodons, and sloths, could not

compete and eventually became extinct. The only surviving large herbivore was the

bison, which was smaller than ever before. It became part of a stable Holocene prairie.

The “bust” phase of the boom-and-bust cycle had placed a premium upon efficient

use of available energy, and the less-efficient monogastrics showed poorer staying power

than the more-efficient ruminants. If H. sapiens populations experienced food stress

during the time of scarcity, they would probably have strengthened their efforts to kill

any remaining non-human predators, whom they would have regarded at this point as

even more serious competitors for food. In this way, they would have unknowingly

exacerbated the problems associated with the next boom phase of the cycle.

The loss of tree cover that occurred as a result of these ecological boom-and-bust

cycles decreased atmospheric moisture, which, in turn, resulted in a more continental

climate. Continentality, as mentioned previously, is largely a factor of the relative aridity

of the environment (Deshmukh, 1986; Martin, 1993). Figures 7a and 7b show the role of

tree cover in this shift in climate. Figure 7a shows the vegetative regime under normal

conditions. Trees recruit slowly, and there is a time lag before a lost tree can replaces its

biomass. The death of an individual tree diminishes the amount of aggregate biomass of

the community of trees, but proportionally few trees die from herbivore browsing. The

obverse is true of grass. There is little loss of accumulated biomass per individual grass

plant, many grass plants suffer from herbivore grazing, and there is heavy replenishment

43

of biomass. Figure 7b shows that under scarcity conditions, there is an increase in

herbivore impact on both trees and grass, but the impact on trees is far greater because

large herbivores knock over trees to get at the tops. This kills trees for a small marginal

increase in nutrition for the herbivore.

A model can be created to test this scenario. The task of the model is to explore the

hypothesis that second-order predation resulted in an overpopulation of herbivores which

overgrazed their environment resulting in widespread extinction.

The hypothesis to be tested is: Second-order predation and its subsequent boom-and-

bust cycles explains the Pleistocene-Holocene transition better than overkill alone. The

alternative hypothesis is that overkill alone explains the data better.

44

Fig. 7. Role of trees and grass in climate change:

A. trees and grass under normal conditions; and B. trees and grass under scarcity

conditions

45

Chapter II: A Method for Testing Hypotheses

of Pleistocene Extinctions in the New World

Introduction

In the previous chapter I proposed an alternative hypothesis that might explain the

widespread megafaunal extinctions of the late Pleistocene. I suggested that extinctions

might have been brought about through second-order predation rather than simple

overkill of herbivores occasioned by the addition of late-arriving predators (i.e., H.

sapiens) to the ecosystem. Designing models that simulate both these possibilities allows

me to test the consistency of each hypothesis. One requirement for such models is that

each one use the same assumptions and starting values. Another requirement is that the

simulated environments show the impact of each possibility on plants, herbivores, and

non-human predators. An approach that fulfills these several criteria is one known as

systems dynamics.

The systems-dynamics approach addresses problems instead of solutions. Both the

Mosimann and Martin (1975) and the Whittington and Dyke (1984) models simulate a

solution to the extinction problem based solely on the overkill hypothesis. The task of

this effort is to create models of the problem that will test both the overkill and the

second-order predation hypotheses.

46

The Modeling Process

The modeling process will involve four steps and two models. The first step will be the

creation of a model that represents a simple ecosystem consisting of plants, herbivores,

and, carnivores. Because these three elements will appear together in dynamic

equilibrium, I call this stage of the model “the dynamic-equilibrium mode.” The second

step will be to elaborate the model by adding a second predator, namely H. sapiens, to the

picture. The activity at this point will be to simulate an overkill situation, in which the

second predator, in addition to the traditional predator, hunts herbivores. I refer to this

stage of the model as “the second-predator overkill mode.” Next, I will simulate second-

order predation by having H. sapiens reduce carnivore populations, a stage I call “the

second-order predation mode.” The third step will be to create a new, more complex

model, by disaggregating plants into trees and grass, and herbivores into browsers,

grazers, and mixed feeders. I will run this model as well in dynamic-equilibrium, second-

predator overkill, and second-order predation modes. The fourth step will be to elaborate

on this second model by dividing grazers into ruminants and non-ruminants. I will run

this elaborated model in the three modes described above.

I will introduce these models according to the following procedure. First, I will

present general conventions and definitions that apply to the entire modeling effort. Next

will come an overview of the process. Following the overview, I will give a step-by-step

explanation of the simulation. At the beginning of each step, I will provide the relevant

procedures and conventions and then present the relevant equations. At the end of each

step, I will describe results.

47

General Conventions and Definition of Terms

The following conventions and definitions apply throughout the modeling effort:

The term model, as used here, signifies a simplified representation of some aspect of

reality.

The term feedback is defined as the transmission and return of information to the

model. Specifically, the model generates information, which it can use at a later time. For

example, consider a hypothetical model that shows a specific number of organisms in

existence in year one. During the course of that year, some of the organisms reproduce

and some die. Thus, in year two, there will be the original number of organisms, plus the

number who were born, less the number who have died. This new number, from which

we can now determine the population growth rate, is the result of feedback. The

population events that took place during the first year have been “fed back” into the

model for determining the size of the population in the second year.

Reference mode is the way in which a graph reflects either observed or inferred

behavior of certain key variables. If the behaviors are observable, then the graph is said to

have an observed reference mode. If the behaviors cannot be observed, then the graph is

said to have an inferred reference mode. In other words, it reflects a hypothesis about the

behaviors of certain variables over time. The reference modes for this modeling effort are

presented below.

The inferred reference mode for the model before the introduction of H. sapiens is

one of dynamic stability. Graphically, this suggests that the lines representing the size of

48

all populations over time should be flat. However, each population should respond to a

disturbance in one of the populations, as shown in Figure 8.

The inferred reference mode for second-predator overkill is shown in Figure 9.

The inferred reference mode for the second-order predation is shown in Figure 10.

Elaboration is a term meaning an addition or amendment to a previous model. It

implies a contrast with the original design. Step 1 in this modeling process is an original

design. Step 2 is an elaboration of Step 1, by virtue of adding H. sapiens to the model.

Step 3, on the other hand, is a new model, which was designed using the values and

thinking from Steps 1 and 2. Step 4 is an elaboration of Step 3.

The term stock refers to the net production of the unit in question. Thus, “herbivore

stock” signifies the number of herbivores at any given time.

Flow is defined as a change in the size of a stock. For example an increase to the

stock of herbivores is called an inflow; a decrease is called an outflow.

Time (t) is the term used to mean the time currently under consideration. In this model,

time is measured in years before present.

Delta time (dt) is the increment of time used. In this model, the increment of time

used is one half of one year.

Sector is the term used for all the equations and values that pertain to a certain stock

or stocks. For example, the sector for a stock of small and big trees would include their

associated recruitment rates and death rates, along with the equations apportioned to their

49

Fig. 8. Dynamic equilibrium reference mode

50

Fig. 9. Second-predator (overkill) reference mode

51

Fig. 10. Second-order predation reference mode

Key as in figure 9

52

respective carrying capacities. In diagrams, I use broken lines as a convenient way to

indicate sectors.

Graphical function is used in this modeling language to specify a relationship

between two variables on an x–y coordinate system. It is used for relationships that may

be non-linear. For example, if one believes that hunting by carnivores decreases when

there are more herbivores readily available, then one will describe the hunting effort as

tapering off when the density of herbivores is near or at its maximum.

Step addition or step reduction is defined as a one-time increase or decrease of a

specific amount that occurs at a specific time. Both possibilities are shown graphically in

Figure 11.

Pulse addition or pulse reduction is defined as a temporary increase or decrease of

some specific amount for a time period that lasts for a specific interval. Both situations

are shown graphically in Figure 12, along with alternative paths of displacement and

recovery.

Delay or lag describes a delay of a specified unit of time or delta time. For example,

the hunting rate at any given time is based on the density of herbivores at two delta-time

periods in the past (t–2dt). Thus, if the number of herbivores has decreased in the last unit

of time (t–dt), the hunting pressure is delayed until the next time period.

A switch refers to conditions under which an equation may change. This switch takes

place when an equation contains an “if” statement. “If” statements activate one condition

or another, depending on the response to the “if.” For example, for variable Y, if X equals

53

Fig. 11. Illustration of the step function

54

Fig. 12. Illustration of the pulse function

55

0, do A; if X does not equal 0, do B. The switch indicates the conditions under which one

does one or the other, A or B. In this work, a switch is indicated in plain (Roman)

typeface. The ordinary operation of the equation is indicated in italics. This format for the

equation of the example given above is Y = If X = 0 then A else B

The abbreviation a.u. stands for “animal unit.” An animal unit is equal to 1,000

pounds (approximately 450 kilograms) of herbivore, based on calculations formulated by

Whittington and Dyke (1984), as cited earlier.

Initial values is the term used for the starting values of stocks. The values of all

stocks are initialized at equilibrium using the following procedure. Plants are initialized

at 25 a.u. of plants for each unit of area, as per Whittington and Dyke (1984). Then the

model is run starting with 10 a.u. of herbivore and 1 a.u. of carnivore until all the stocks

reach equilibrium. These values are used as initial values in all subsequent runs of the

model.

The term area refers to the amount of land arbitrarily assigned to the model’s

continent, namely 30,000 square miles.

Values used in these simulations are values used in the simulations that are based

directly on those used by Whittington and Dyke (1984). They are shown in Table 3

below.

56

Table 3: Values taken directly from Whittington and Dyke (1984)

DESCRIPTION OF

VARIABLE

BASELINE SOURCE

Human population size 200 Budyko 1967, 1974

Human population growth rate 0.0443 Birdsell 1957

Table 4 shows starting values that were modified to fit the modeling paradigm.

Table 4: Modified values based on Whittington and Dyke (1984)

DESCRIPTION OF

VARIABLE

BASELINE VARIATION

Prey carrying capacity (a.u.*

per square mile)

25 Changed to the amount of (plants) per square

mile.

Prey biomass replacement

(a.u.* per square mile)

0.25 Used as base recruitment rate: birth rate minus

average death through hunting by predators

plus natural death rate = 0.25

* animal units = 1K lb. of herbivore

K : Whittington and Dyke (1989) assign 25 herbivore units to 1 unit of area (see

above). This suggests that there are 25 units of plants per unit of area. This is the

maximum carrying capacity of the continent at equilibrium.

Carnivore recruitment is the net growth of carnivore population (births minus

deaths) in any given year. From informal estimates based on extant predators (Carbyn,

1994; The Cat House, 1997; International Wolf Center, 1997), carnivore recruitment is

10 percent a year.

57

The hunting rate used in the simulation is based on food needed per pound of

predator per year. Data from extant predators suggests 20 pounds of food per year per

pound of predator (Schaller, 1972; Petersen, 1977; International Wolf Center, 1996; The

Cat House, 1996). The maximum number of herbivores per unit of area is set at 25

a.u.(MaxHrbP = 25), following Whittington and Dyke (1984)

General Overview of the Model

The task of the model is to explore the hypothesis that second-order predation resulted in

an overpopulation of herbivores, who overgrazed their environment. Overgrazing

eventually resulted in widespread extinction. Stated another way:

Second-order predation and its subsequent boom-and-bust ecological dynamic

explain the data better than overkill alone? The alternative hypothesis is that overkill

alone explains the data better.

To test the second-order predation hypothesis the models created must be able to

simulate the conditions of both overkill and second-order predation, using the same

starting assumptions and values.

I produced two models in a four–step process. The first two steps involved

constructing a model that is an aggregated simulation of a simple ecosystem consisting of

five components: a continent, plants, herbivores, predators, and H. sapiens. The second

model is a more complex, disaggregated simulation, in which I performed the third step

by disaggregating the plant component of the first model into trees and grass, and

disaggregating the herbivore component into browsers, grazers, and mixed feeders. The

58

fourth step was the same as the third, except that I disaggregated the grazers into

ruminants and monogastrics.

The Simple Model

The purpose of the first step of the modeling activity is to establish the basic model. The

purpose of the second step is to show the dynamics of the introduction of a second

predator and to contrast that with the dynamics of second-order predation.

The Disaggregated Model

The third step examines how predatory/prey dynamics would affect the vegetation and

herbivores in a more complex system. The fourth step examines how changes proposed in

the previous steps affect ruminant and monogastric grazers.

Criteria for Success

I will consider the modeling and simulation project a success if it indicates the conditions

under which the second-order predation hypothesis would have operated. If these

conditions are found to be less likely to produce extinction than those required for the

overkill hypothesis then the second-order predation hypothesis will be rejected.

59

Base Model: Dynamic Equilibrium – Step 1

Overview

I will start with an initial model based on the Lotka–Volterra predator-prey equations

(May, 1976; Schaffer, 1988; Smuts, 1979; Hilborn & Sinclair, 1979; Richardson and

Pugh, 1981).

Model Diagram

The model diagram below (Figure 13) illustrates the relationship between the various

sectors. Each stock is limited by the stock above it, which serves as its carrying capacity.

Thus Continent is the carrying capacity for Plants, Plants are the carrying capacity for

Herbivores and Herbivores are the carrying capacity for Carnivores. The box labeled

density shown at the bottom of the diagram contains copies of the stocks, and auxiliary

values and equations, which control herbivore hunting by carnivores. It is separate for

ease in understanding the diagram and does not constitute a separate trophic level.

Each limit is used up by the stock, which it limits. Thus Plants fill up the available

area, Herbivores eat plants, and Carnivores kill Herbivores.

This is the base model upon which other elaborations and adjustments will be made.

It should behave such that change in either the carnivore or the herbivore populations

results in a parallel change in the other.

60

Fig. 13. Base model diagram.

61

Fig. 13. continued

62

Conventions, Definitions and Equations

Continent

Area = Size of continent = 30,000mi2

HerbPerArea = Density of herbivores that can be supported by plants = 25 animal units

per square mile (where 1 animal unit equals the amount of plants required to

support 1,000 pounds of herbivore for one year)

AreaMultipleHrb = Carrying capacity of the continent for herbivores = Area *

HerbPerArea = 750,000 animal units of herbivore.

Plants

Plants is the stock of plants which is equivalent to the amount of plants at the previous

time plus the input to plants minus the outflow of plants times the unit time;

Plants(t) = Plants (t–dt) + (InPlants – OutPlants) * dt

AreaMultipleHrb is equivalent to the density of plants, which is equivalent to the number

of plants per unit (expressed as a number of animal units of herbivore that can be

supported by 1 square mile) (by assignment based upon Whittington and Dyke

(1984))

HerbPerArea =25 Area * HerbPerArea is equivalent to the maximum stock of plants

AreaMultipleHrb = 30,000 * 25=750,000

63

PlantRegen is equivalent to the replacement rate of plants PlantRegen = 1 (by

assignment)

(1–PlantRegen/AreaMultipleHrb) is equivalent to the maximum fraction of plants that

can be added to the continental stock of plants (expressed in number of animal

units). It is equivalent to the limit to plant growth which is equivalent to the

remaining fraction of continental carrying capacity that can be filled (i.e. that is

empty)

InPlants is equivalent to the number of plants added to plant stock in time interval

(=1year)

Plants * PlantRegen * (1–Plants/AreaMultipleHrb)

PlantEating is equivalent to the amount of plant material lost from plant stock (Plants).

PlantEating is in support of the population of herbivores (1 unit of plant eating =

the amount of plants required to support 1 animal unit of Herbivore when

measured in animal units = number of herbivores (which is equivalent to

Herbivore) in animal units.

OutPlants is equivalent to the amount of plant material (in animal units) removed from

Plants. Since plants replace themselves (PlantRegen=1), the only loss of plant

material is PlantEating and so OutPlants = PlantEating

64

Herbivores

Herbivores – the stock of herbivores which is equivalent to the number of herbivores

present at any given time, expressed in animal units;

Herbivores (t)=Herbivores (t–dt) + (InHrb–OutHrb) * dt

BrateH – the birth rate of Herbivores; reflects the disaggregation of recruitment used by

Whittington and Dyke (1984) into birth rate, death rate, and death by hunting. Set

arbitrarily at 0.9 individuals per individual in the population of herbivores.

(1–(Herbivores/Plants)) is equivalent to the limit to herbivore population growth imposed

by the carrying capacity of the continent for plants; which is equivalent to the

maximum fraction of herbivores that can be added to the continental stock of

herbivores; expressed in animal units.

InHrb is equivalent to the number of herbivores added to herbivore stock per unit time;

=Herbivores * BrateH * (1–(Herbivores/Plants))

DrateHrb is equivalent to the non- hunting death rate of Herbivores = 0.12 (arbitrarily

assigned)

MaxHerbA is equivalent to the maximum number of animal units a unit of area can

support = 25 (arbitrarily assigned by Whittington and Dyke (1984)).

HerbDensity is equivalent to the actual density of herbivores per unit area:

Herbivores/Area

65

DeltaHerbPerArea is equivalent to the difference between the actual density of

Herbivores (HerbDensity) and the maximum herbivore density (MaxHerbA):

HerbDensity/MaxHerbA

HuntingGrf is the rate at which predators kill herbivores. The input to the graph

(DeltaHerbPerArea) (shown on the X-axis) determines what value of HuntingGrf

(shown on the Y-axis) will be returned as output to the model. As herbivore

density approaches the maximum herbivore density carnivores are able to kill

more herbivores. Because it is assumed at maximum herbivore densities carnivore

hunting will be very successful and by the same token at low densities carnivores

are required to hunt hard. The graph of this function is shown in Figure 14

FoodNeedCrn is equivalent to the amount of food (animal units) needed to support a

pound of carnivore per year = 20 lbs. (Carbyn, 1995; The Cat House, 1997;

International Wolf Center, 1997).

HuntingC is equivalent to the loss of herbivore due to consumption by a single carnivore;

FoodNeedCrn * HuntingGrf * Herbivores

HerbKilledC is equivalent to the loss of herbivores due to hunting;

HuntingC * Carnivores

OutHrb is equivalent to the loss of herbivores from the stock of herbivores per unit time;

DrateH + HerbKilledC

66

Fig. 14. Hunting function

HuntingGrfCrn = GRAPH(DeltaHCrn)

(0, 0.000); (0.1, 0.150); (0.2, 0.280); (0.3, 0.390); (0.4, 0.490); (0.5, 0.590); (0.6, 0.675); (0.7,

0.760); (0.8, 0.846); (0.9, 0.925); (1, 1.000)

67

Carnivores

Carnivores is equivalent to the stock of carnivores;

Carnivores (t) = Carnivores (t – dt) + (InCrn – OutCrn) * dt

BrateCrn is equivalent to the natural birth rate of carnivores arbitrarily assigned at 0.4

individuals per individual in the population of carnivores = 0.4 (arbitrarily

assigned)

HerbDelay is equivalent to the amount of time when the population of Carnivores

responds to a change in the Herbivore population. If HerbDelay is 2 the impact of

a change in the population level of the Herbivore stock does not impact

Carnivores until 2 time periods after it occurs; time (t) minus two delta-time(2dt).

HerbDelay = 1

CarnivoreK is equivalent to Herbivores at the time period specified (t–2dt) by HerbDelay

the amount of time lag specified in HerbDelay.

(1–(Carnivores/CarnivoreK)) is equivalent to the limit to carnivore population growth

imposed by the size of the Herbivore population equivalent to the maximum

fraction of Carnivores that can be added to the stock of carnivores: expressed in

animal units.

InCarn is equivalent to the number of carnivores added to Carnivore stock per unit time:

Carnivores * BrateCrn * (1–(Carnivores/CarnivoreK))

68

DrateCrn is equivalent to the natural death rate of carnivores = 0.3 (arbitrarily assigned)

OutCrn is equivalent to the loss of carnivores from the stock of carnivores.

The model will be considered to be in dynamic equilibrium if a disturbance to any of

the stocks returns to the model to equilibrium. For the model to be judged sufficiently

linked, a pulse disturbance of any of the stocks should have an impact on the stocks with

which it is linked. This is necessary in order to establish a baseline of steady state so that

we know that the disturbances which follow are due to our experiments not to some

aberration in the model it self.

Graph of the Base Model – Step 1

Figure 15 – Graph of the Base Model – shows that when the model is run it is in

equilibrium (Sensitivity tests of step 1 are found in Appendix A).

In the graphs presented below, the model is perturbed by creating a pulse outflow of

5% of the population of the stock starting at –11,500 and again 250 years later, it rapidly

returns to stability. Figure 16a –5% Pulse outflow of Plants, 16b –5% Pulse outflow of

Herbivores, and 16c – 5% Pulse outflow of Carnivores, show the effects of perturbing

each of the stocks in turn.

Note when the model is disturbed higher on the food chain it causes more instability

and it takes longer for the model to return to stability than when it is disturbed lower on

the food chain. Disturbing the Carnivore stock by causing a pulse outflow of 2.5% creates

69

Fig. 15. Graph of the base model

Key as in figure 9

70

Fig. 16. Pulse outflow from plants, herbivores and carnivores

Key as in figure 9

A.

B.

71

Fig. 16. continued

C

72

more instability for a longer period of time than disturbing the Herbivore and more again

than disturbing Plants.

Second Predator: Overkill – Step 2a

Overview

To test the consistency of either of the hypotheses (overkill and second-order overkill)

with extinction the base model was elaborated.

This step, the first elaboration, introduces a second predator, H. sapiens. This is the

position of the overkill hypothesis suggested by Martin (1967, 1984), Mosimann and

Martin (1975), and Whittington and Dyke (1984).

Model Diagram for Step 2a

The illustration shown below (Figure 17 – Model Diagram) is the Base model from Step

1 with the addition of a sector for H. sapiens. The stock of H. sapiens (Hsapiens) inflow,

outflow and hunting equations have the same form as the corresponding equations for

Carnivores from the Base model explained in Step 1. The major difference between the

sectors is that Hsapiens is initially an empty stock. Therefore, it is necessary to create

equations, which simulate the migration of H. sapiens into an ecosystem in equilibrium.

The density box that modifies the impact of hunting on Herbivores has been changed

to add the appropriate equations and values to account for hunting by H. sapiens.

73

Fig. 17. Second predator (overkill) model diagrams

Key as in figure 13

74

Conventions, Definitions, and Equations

Modifications to Density and Herbivore Sectors

FoodNeedHs is equivalent to the amount of food (animal units) needed to support a

pound of Hsapiens per year = 10 lbs. It is arbitrarily assigned as half the food

needed by a Carnivore.

HuntingHs is equivalent to the loss of herbivore due to consumption by a single

Hsapiens;

HuntingHs = FoodNeedHs * HuntingGrf * Herbivores

HerbKilledHs is equivalent to the loss of herbivores due to hunting by Hsapiens;

HerbKilledHs = HuntingHs * Hsapiens

HerbsKilled is equivalent to the sum of the loss of herbivores due to hunting by Hsapiens

and by Carnivores;

HerbsKilled = HerbKilledC + HerbKilledHs

OutHrb from step 1 has been modified by using HerbKilled instead of HerbKilledC it is

equivalent to the loss of herbivores from the stock of herbivores per unit time;

OutHrb = DrateH + HerbKilled

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Hsapiens

Hsapiens is equivalent to the stock of H. sapiens. Since the hypothesis we are testing is

the inmigration of H. sapiens it is initialized at zero (0)

Hsapiens (t) = Hsapiens (t – dt) + (HsMigrate + InHs – OutHs) * dt

AmtHsMigrate is equivalent to the number of H. sapiens entering the continent.

Whittington and Dyke (1984) arbitrarily assign it at 200 individuals.

BiomassHs is the conversion of number of H. sapiens into biomass. It is assigned a value

of one hundred pounds (100 lbs.)

TimeMigrateHs is equivalent to the time at which H. sapiens enters the continent. It is

arbitrarily assigned to –11500 BP

HsMigrate is equivalent to the amount the stock is filled using with a one–time inflow

(pulse) of H. sapiens in biomass (BiomassHs) at –11500 years ago

(TimeMigrate). Because there is no repeat of the pulse there is a zero (0)

following the TimeMigrateHs:

HsMigrate = PULSE (BiomassHs, TimeMigrateHs, 0)

BrateHs is equivalent to the birth rate of H. sapiens; it reflects the disaggregation of

recruitment used by Whittington and Dyke (1984) into birth rate, death rate, and

death by hunting. Set arbitrarily at 0.047 individuals per individual in the

population of Hsapiens.

76

(1–(Hsapiens/Herbivores)) is equivalent to limit to H. sapiens population growth

imposed by the size of the Herbivore population equivalent to the maximum

fraction of Hsapiens that can be added to the stock of H. sapiens: expressed in

biomass.

InHs is equivalent to the number of H. sapiens added to H. sapiens stock per unit time.

Because the Hsapiens stock is initialized at zero (0) it is necessary to put a switch

in the equation which allows the model to function with a zero stock. It is shown

in plain text:

If Hsapiens = 0 then 0 else Hsapiens * (BrateHs * (1–Hsapiens/Herbivores))

DrateHs is equivalent to the natural death rate of H. sapiens = 0.03 (arbitrarily assigned)

OutHs is equivalent to the loss of H. sapiens from the stock of H. sapiens.

Results Second Predator (Overkill) – Step 2a

This elaboration will be a success if we see the impact of the introduction of a second

predator This is the position of the overkill hypothesis suggested by Martin (1967, 1984),

and Mosimann and Martin (1975), and Whittington and Dyke (1984).

Graph of the Model – Step 2a

In the graph of the model (Figure 18 – Graph of the Second Predator (Overkill) Model)

below we see the impact of the introduction of a second predator into the ecological

system. Herbivores and carnivores decreased and plant stock increased.

77

Fig. 18. Graph of the second predator (overkill) mode

Key as in figure 9.

78

Second-Order Predation – Step 2b

Overview

This elaboration of the initial model is to simulate second-order predation. The values

and the components of the second predator model are the same the difference between the

two models is the addition of predator/predator hunting values.

Model Diagram

The diagram, (Figure 19) shows the new outflow from Carnivores.

Conventions, Definitions, and Equations

All conventions and definitions from Step 1, 2a and the general conventions and

definitions section remain the same.

AmtKill is the percent of the reduction of Carnivore populations by Hsapiens. The default

value is 1.5%. It was arrived through trial and error as a small percentage

reduction, which still had a significant impact.

KillTime is the time after migration that H. sapiens (Hsapiens) begins to reduce

Carnivore populations. The default is arbitrarily assigned to 300 years.

OutStepCrn is equivalent to the amount Carnivore populations are reduced by Hsapiens,

at an appointed time after the inmigration of Hsapiens. OutStepCrn has a switch

(shown in plain text) which allows the model to function when there are no

79

Fig. 19. Second-order predation diagram

Key as in figure 13

80

Hsapiens present. The equation function is shown in italic text:

OutStepCrn = If Hsapiens=0 then 0 else STEP (AmtKill, (TimeMigrationHs,

KillTime))

Carnivores is the stock of carnivores. It has been modified to include the additional

outflow so that the equation now reads:

Carnivores (t) = Carnivores (t – dt) + (InCrn – OutCrn – OutStepCrn) * dt

Results Second-Order Predation – Step 2b

Graph of Model – Step 2b

The graph below (Figure 20 – Second-Order Predation) shows the impact of a small

(1.5%) reduction in Carnivore populations on the ecosystem as a whole. There is an

immediate population explosion of herbivores and an immediate reduction in the amount

of plants. The slope of the reduction of Carnivores is affected very little. The

disequilibrium in the plant and in herbivore populations is the most salient feature of the

reduction of Carnivore populations.

Table 3 shows there are slightly fewer herbivores in the second predator model (Step

2a) than in the second-order predation model (Step 2b). H. sapiens does better in the

second-order model with an ending value of 15,194.47 as opposed to 13,922.97 in the

second predator model or approximately 1 person per every 2 miles.

81

Fig. 20. Graph of the second order predation model

Key as in figure 9

82

Table – 3 – Comparison of Second Predator (Overkill) and Second-OrderPredation Ending Values

Starting Value Second Predator Second-Order Predation

Plants 638,439.63 703,337.67 698,741.58

% starting value 110% 109%

Herbivores 94,966.08 43,759.15 47,755.17

% starting value 46% 50%

Carnivores 16,619.06 7,657.91 6,566.40

% starting value 46% 40%

Hsapiens 13,922.97 15,194.47

83

Step 3 – Three Herbivores – (Browsers, Grazers and Mixed Feeders)

Overview

Step three models a more complex ecosystem where the two hypotheses (overkill and

second-order overkill) can be tested. Since the step two model does not address the

pattern of extinction it was necessary to make a more complex model. Trees and Grass

have replaced plants. This made changes in the Continent sector necessary. Browsers,

grazers and mixed feeders, have replaced herbivores. Hunting equations have been

changed to reflect the division of herbivores and to introduce prey switching. The hunting

rate graphs have also been made more realistic. It is assumed that because carnivores are

obligate predators the hunting rate of carnivores will be more closely tied to prey density

whereas H. sapiens is an omnivore so it is assumed that at low prey density H. sapiens

hunting will drop off. Therefore the graph of Hsapiens hunting rates is more sigmoid

whereas the equivalent graph for Carnivores is linear. In the step 2 model carnivore

populations were reduced by a percentage each year. In this model the reduction in

carnivore populations is a factor of carnivore density and a kills per Homo sapiens. All

conventions and definitions, not discussed, remain as previously specified.

Diagram of the Model

An illustration of the elaboration of the continent, tree and grass equations is shown in

Figure 21

84

Fig. 21. Continent, trees and grass, diagram. Three herbivore model.

Key as in Base Model diagram.

85

In the base model hunting was a factor of the difference between the actual density of

herbivores and the maximum number of herbivores. The maximum number of herbivores

is the amount necessary for the predator (human or non-human) to achieve maximum

reproduction. In this model hunting is based on actual density of herbivores. An

illustration of the relationships specified by the density equations is shown in Figure 22.

An illustration of the redesign of Herbivores for step three is shown in Figure 23.

In the base model the reduction in carnivore populations was a percentage of the

population per year. It was designed to illustrate how little carnivore populations needed

to be reduced in order to change the balance in the ecosystem. In this model the reduction

in carnivore populations is due to the density of carnivores and H. sapiens hunting

carnivores at a set rate. An illustration of the redesign of Carnivores and Hsapiens is

shown in Figure 24.

Conventions, Definitions, and Equations

Continent

AllTrees is the sum of small and big trees;

AllTrees = BigTrees + SmallTrees

Plants is the sum of all the Plants; small and big trees, high and low quality grass;

Plants = BigTrees + SmallTrees + GrassHigh + GrassLow

86

Fig. 22. Herbivores. – browsers, grazers, and mixed feeders diagram. Three herbivore model.

Key as in Base Model diagram.

87

Fig. 23. Density diagram. Three herbivore model.

Key as in Base Model diagram.

88

Fig. 24. Carnivores, Hsapiens diagram. Three herbivore model.

Key as in Base Model diagram

89

TreePerC is the ratio of all trees to all Plants subtracted from one (1); as the number of

trees increase TreePerC decreases such that if the continent is all trees the

equations returns zero (0) and if it is all grass the equations returns one (1);

TreePrC = 1–AllTrees/Plants

WoodMix is equivalent to how much of the continent (K from the previous models) is

allocated to trees based on how much it was the previous time period. The shift in

environments is relatively slow so there is a large flat spot in the center of the

graph that preserves stability. When the extremes are reached then the system

should shift fairly quickly. Therefore, the slope of the curve at either end is more

extreme. The input to the graph is TreePerC. A graph of the function is shown

below (Figure 25);

WoodMix. = GRAPH (TreePrC)

TreeK is the portion of the carrying capacity that is allocated to trees by the graphical

function above:

TreeK = K * WoodMix

GrassK is the portion of the continent that is allocated to grass by subtracting the amount

of the carrying capacity allocated to trees from 1:

GrassK=1–TreeK

Plants

There are two sectors to Plants Trees and Grass.

90

Fig. 25. WoodMix function. Three herbivore model.

WoodMix=GRAPH(TreePrC)

(0.4, 0.697); (0.46, 0.438); (0.52, 0.354); (0.58, 0.319); (0.64, 0.301); (0.7, 0.270); (0.76, 0.249);

(0.82, 0.231); (0.88, 0.196); (0.94, 0.144); (1, 0.000)

91

Trees

Note: Small trees are the preferred food of browsers and mixed feeders. It is assumed that

small trees provide more energy per unit of biomass than big trees. If the demand for

trees exceeds the stock of small trees, big trees are eaten (see BtEating below). Eating big

trees reduces the biomass of big trees more than eating small trees reduces the biomass of

small trees. Each unit of nutritional demand for big trees has a 1.5 unit impact (see

below). This is based on the observation by Wing and Buss, (1970) of elephants under

population stress killing mature trees by knocking them over to eat the leaves at the top.

Trees. The tree sector has two stocks, small trees (SmallTrees) and big trees (BigTrees).

SmallTrees recruit quickly. The SmallTrees stock includes new shoots to existing

trees, some of them die through being eaten, natural death, and some mature into

BigTrees and ultimately die. All tree reproduction comes through small trees.

Both small and big trees contribute to reproduction. Trees are eaten by herbivores,

small trees are preferred to big trees.

SmallTrees is equivalent to the stock of SmallTrees; the amount of SmallTrees present at

any given time that can be supported at that time;

SmallTrees(t) = SmallTrees (t–dt) + (InTrees – Maturity – OutSt) * dt

Reseed is used in a switch to replant trees if all tree stocks are depleted. It is only active

under extreme conditions. It introduces 1 unit of trees for use in reproduction.

92

StRepo is the proportion of reproduction attributed to small trees. It is arbitrarily assigned

a rate of 0.6.

BtRepo is the proportion of reproduction attributed to big trees. It is arbitrarily assigned a

rate of 0.4.

TreeRepo is the sum of reproduction attributed to small trees and big trees:

TreeRepo= StRepo + BtRepo

TreeRepRate is the reproduction rate of trees set to 0.25. This is based on loosely on the

following thinking. Whittington and Dyke (1984) used a constant or fixed

carrying capacity. Their herbivore recruitment was 0.25 therefore the carrying

capacity of the environment must be sufficient to support that amount of

recruitment.

(1–AllTrees/TreeK) is equivalent to the maximum fraction of Trees that can be added to

the continental stock of Trees. It is expressed in the number of animal units that is

equivalent to the limit to Tree growth. This is equivalent to the fraction of

continental carrying capacity allotted to trees remaining, (TreeK) that can be filled

(i.e. that is empty).

InTrees is equivalent to the number of Trees added to Trees stock in time interval (=1

year); It has a switch (presented in plain text) which is activated when there are no

trees left on the continent otherwise the equation functions as it appears in italic

text.

93

InTrees = If AllTrees < 0.001 then Reseed else TreeRepo * (TreeRepRate * (1–

AllTrees/TreeK))

StDeath is the non- herbivore related death rate of small trees. It is arbitrarily assigned a

rate of 0.05.

BzEfficiency is equivalent to Browser efficiency measured in animal units

Efficiency is a measure of how much food it takes to produce one animal unit

(a.u.). Thus an “average” animal unit will have an efficiency of 1; it will require 1

unit of food. A more efficient animal will be able to produce more animal per unit

of food. Its efficiency will be less than 1. A less efficient animal will require more

than one unit of food so it’s efficiency will be greater than 1. Browser efficiency

is arbitrarily assigned a value of one; BzEfficiency=1.

TreesNeedBz is equivalent to the number of browsers multiplied by the efficiency of

browsers which is 1 so the trees needed is equal to the number of Browsers

TreesNeedBz = Browsers * BzEfficiency

MxTreePerC is equivalent to the percentage of trees in the diet of MixedFeeders = 0.5

(arbitrarily assigned).

TreeRatio has a switch in it to set the ratio to zero when there are less than 10 trees. This

becomes active only under extreme conditions TreeRatio is equivalent to the ratio

of trees to AllTrees plus AllGrass. = AllTrees/(AllTrees+AllGrass).

94

MxEfficiency is equivalent to mixed feeder efficiency. The optimum efficiency for

MixedFeeders is when the number of trees in the environment is the same, as the

number needed by MixedFeeders. Deviation from this point decreases

MixedFeeders efficiency. Thus a graph of this function is a ‘U’ shaped graph with

the optimum efficiency at the base of the ‘U’. Deviation from the optimum on

either side of the ‘U’ increases the amount of food necessary to obtain the same

nutritional benefit. The graph takes TreeRatio as its input and puts out the

efficiency of MixedFeeders. The graph of the function is shown in Figure 26

MxEfficiency = GRAPH (TreeRatio)

TreesNeedMx is equivalent to the amount of trees eaten by MixedFeeders at any given

time

TreesNeedMx = MixedFeeders * (MxEfficiency * MxTreePerC)

BzMxTreeNeed is the sum of trees needed by Browsers and MixedFeeders;

BzMxTreeNeed = TreesNeedBz + TreesNeedMx

StEating has a switch (shown in plain text) that becomes active when the amount of small

trees needed (BzMxTreeNeed) is greater then the stock of SmallTrees then

SmallTrees is used up otherwise StEating is equivalent to the amount of trees

needed (BzMxTreeNeed).

StEating = If BzMxTreeNeed>SmallTrees then SmallTrees else BzMxTreeNeed

95

Fig. 26. MixedFeeder efficiency. Three herbivore model.

MxEfficiency = GRAPH(TreeRatio)

(0, 4.00); (0.1, 2.75); (0.2, 1.60); (0.3, 1.25); (0.4, 1.05); (0.5, 1.00); (0.6, 1.05); (0.7,

1.25); (0.8, 1.60); (0.9, 2.75); (1, 4.00)

96

OutSt is equivalent to the outflow from small trees which is lost to the tree sector (i.e. it

does not pass into big trees);

OutSt = (SmallTrees * StDeath) + StEating

MatRate is equivalent to the rate at which small trees mature into big trees; it is arbitrarily

assigned a rate of 0.15.

Maturity is the number of small trees become large trees;

SmallTrees * MatRate

BigTrees is equivalent to the stock of BigTrees; the amount of BigTrees present at any

given time;

BigTrees (t) = BigTrees (t – dt) + (Maturity – OutBt) * dt

BtRatio is equivalent to the amount of increased impact on big trees when Browsers and

MixedFeeders are forced to eat a less preferred food, BigTrees. This reflects the

need for herbivores to kill more biomass of big trees to obtain the same

nourishment. It is equal to 1.5 (arbitrarily assigned)

AmtBt multiplies the amount of tree demand not filled by small trees by the amount of

increased impact (BtRatio). It is determined by subtracting the amount of

SmallTrees from the amount of tree demand (BzMxTreeNeed) and multiplying the

remainder by the amount of increased impact from BtRatio

AmtEatBt = (BzMxTreeNeed –SmallTrees) * BtRatio

97

BtNeed evaluates if there are more trees needed by all herbivores (BzMxTreeNeed) than

the stock of SmallTrees. If there are it calls AmEattBt since this is not the case

under equilibrium conditions the default is zero (0).

BtNeed = If BzMxTreeNeed > SmallTrees then AmtEatBt else 0

BTEating evaluates if the amount of BigTrees needed (BtNeed) is greater then BigTrees

if so it uses the entire stock of BigTrees if not it calls BtNeed.

BtEating = If BtNeed > BigTrees then BigTrees else BtNeed

BtDeath is the non- herbivore related death rate of big trees. It is arbitrarily assigned a

rate of 0.03.

Grass

Grass has two levels: high quality grass (GrassHigh) and low quality grass (GrassLow).

Recruitment is into the GrassHigh level but both kinds of grass participate in

reproduction. Some GrassHigh passes into GrassLow, some is eaten and some dies a

natural death. Grazers and mixed feeders prefer high quality grass to low quality grass. If

the demand for high quality grass exceeds the stock of high quality grass, low quality

grass is eaten. More low quality grass is necessary to provide the same level of

nourishment as high quality grass.

AllGrass is equivalent to the sum of high and low quality grass

AllGrass = GrassHigh + GrassLow

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High Quality Grass

GrassHigh is equivalent to the stock of high quality grass which is equivalent to the

amount of high quality grass that is present at any given time;

GrassHigh (t) = GrassHigh (t – dt) + (InHi – OutHi – HiToLo) * dt

HiRepo is equivalent to the amount that GrassHigh contributes to the total reproduction

of grass. It is considered to be half the contribution = GrassHigh * .5 (arbitrarily

assigned)

LoRepo is the proportion of reproduction attributed to GrassLow. It is considered to be

half the contribution = GrassLow * .5 (arbitrarily assigned)

GrRepo is the sum of high and low quality grass to reproduction;

GrRepo= HiRepo + LoRepo

RateGr is the reproduction rate of grass it is arbitrarily assigned a rate of 0.33.

(1–AllGrass/GrassK) is equivalent to the maximum fraction of grass that can be added to

the continental stock of grass. It is expressed in number of animal units equivalent

to the limit to grass growth, which is equivalent to the fraction of continental

carrying capacity allotted to grass left, (GrassK) that can be filled (i.e. that is

empty).

InHi is equivalent to the amount of high quality grass that is added to high quality grass

stock in time interval (=1year)

GrRepo * (RateGr * (1–AllGrass/GrassK))

99

SetEffGz is equivalent to the setting for optimal Grazer efficiency it is arbitrarily assigned

a value of 0.9

FoodNeedGz is equivalent to the amount of food necessary to support the population of

grazers;

= Grazers * SetEffGz

GzGrass is the ratio of how much food is needed by grazers to the amount of grass

present in the system subtracted from one;

1–FoodNeedGz/AllGrass

GzEffGrf is equivalent to the actual efficiency of grazers given the amount of grass

available in the system. It is a graphical function which takes as its input the ratio

of food needed compared to the grass available (GzGrass) and gives as an output

to the model the actual efficiency of grazers. As the amount of grass in the system

increases the efficiency of Grazers increases. It is assumed that at the highest

level of grass efficiency can no longer increase having reached its maximum and

by the same token that the greatest decrease will be in the center with efficiency

loss tailing off at lowest levels of grass. Therefore the graph is sigmoid in shape it

is shown in Figure 27:

= GRAPH (GzGrass)

GzEfficiency is equivalent to the realized efficiency of grazers it is the setting for optimal

efficiency multiplied by the actual efficiency of grazers;

= SetEffGz * GzEffGrf

100

Fig. 27. GzEffGr. Actual efficiency of Grazers given the amount of grass available in the system

Three herbivore model.

GzEffGrf=GRAPH(GzGrass)

(0.990, 2.000); (0.991, 1.858); (0.992, 1.662); (0.993, 1.415); (0.994, (1.190); (0.995,

1.010); (0.996, 0.853); (0.995, 0.725); (0.998, 0.620); (0.999, 0.538); (1.000, 0.515)

101

GrasNeedGz is equivalent to amount of grass taken out of the system by grazers it is the

realized efficiency of grazers times the efficiency of grazers;

= Grazers * GzEfficiency

(1–MxTreePerC) is equivalent to the amount of grass needed by MixedFeeders It is the

remainder of the diet once the fraction for trees has been set. It is therefore the

percentage of trees in the mixed feeder diet subtracted from one (1)

GrassNeedMx is equivalent to the grass needed by mixed feeders. It is the number of

mixed feeders times the efficiency of MixedFeeders times the amount of grass

needed (1–MxTreePerC);

= MixedFeeders * (MxEfficiency * (1–MxTreePerC)

GrassNeed is equivalent to the amount of grass taken out of the system by herbivores. It

is the sum of grass needed by grazers and mixed feeders;

= GrassNeedGz + GrassNeedMx)

DRateHi is the non- herbivore related death rate. It is arbitrarily assigned a value of 0.01

OutHi is equivalent to the outflow from the stock of high quality grass. The equation has

a switch that evaluates whether there is grass left over after herbivores have eaten.

If there is only sufficient grass to feed herbivores then the non herbivore death

rate does not apply if there is grass then it applies the non- herbivore death rate to

the remaining grass;

102

= If GrassHigh < GrassNeed then GrassHigh else GrassNeed + ((GrassHigh –

GrassNeed) * DRateHi)

HiLoRate is equivalent to the rate at which high quality grass passes into low quality

grass = 0.66 (arbitrarily assigned)

HiToLo is the amount of grass that passes from high quality grass to low quality grass. It

is the amount of high quality grass multiplied by the rate at which it passed into

low quality grass;

= GrassHigh * HiLoRate

Low Quality Grass

GrassLow is equivalent to the stock of low quality grass which is equivalent to the

amount of low quality grass present at any given time that can be supported by the

low quality grass present at that time;

GrassLow (t) = GrassLow (t – dt) + (HiToLo – OutLo) * dt

DrateLo is the non-herbivore death rate of GrassLow. It is arbitrarily assigned a rate of

0.1.

LoGRatio is equivalent to the amount of increased impact, which occurs when herbivores

are forced to eat low quality grass. This reflects the need for herbivores to kill

more biomass of low quality grass to get the same nourishment. =1.5 (arbitrarily

assigned).

103

AmtLo is the amount of grass needed over the amount supplied by GrassHigh multiplied

by the amount of increased impact (LoGRatio);

= GrassNeed–GrassHigh * LoGRatio

EatLo evaluates the need for low quality grass to be removed from the system. If there is

enough high quality grass then it returns zero to the model, if not it calls AmtLo;

= If GrassNeed >GrassHigh THEN AmtLo ELSE 0

OutLo is equivalent to the outflow from the stock of low quality grass. It removes the

grass eaten from the system and applies the non- herbivore death rate to the

remaining grass;

= EatLo + ((GrassLow–EatLo) * DRateLo)

Herbivores

Browsers, Grazers and Mixed Feeders

Part of the redesign of this step is to replace herbivores of the Step 1 & 2 models with

browsers, grazers and mixed feeders.

Herbivores is the sum of all herbivore stocks;

= Browsers + Grazers + MixedFeeders

Browsers

Browsers are similar in structure to Herbivores from the base model.

104

Browsers the stock of browsers which is equivalent to the number of browsers present at

any given time, expressed in animal units;

= Browsers (t – dt) + (InBz – OutBz) * dt

BzBirthRate is the birth rate of browsers it is equal to 0.6.

BzDensity is equivalent to the density of Browsers;

= Browsers/Area

BzDensityEffectGrf is a graphical function that reflects the inability of Browsers to find a

mate and continue reproducing when the density of browsers (BzDensity ) drops

too low. The graph of the function is shown in Figure 28

BzBirth is the birth rate of browsers (BzBirthRate) multiplied by the density effect graph

(BzDensityEffectGrf)

(1–Browsers/AllTrees) is equivalent to the limit to herbivore population growth imposed

by the carrying capacity of the continent for trees; which is equivalent to the

maximum fraction of browsers that can be added to the continental stock of

browsers; expressed in animal units.

InBz is equivalent to the number of Browsers added to Browsers stock per unit time;

= Browsers*(BzBirth*(1-Browsers/AllTrees))

BzHuntGrfCrn is equivalent to the rate at which carnivores kill Browsers. The input to

the graph (BzDensity) (shown on the X-axis) determines what value of

BzHuntGrfCrn (shown on the Y-axis) will be returned as output to the model. As

105

Fig. 28. Effect of Browser density, as Browser density declines Browser birth function drops

toward zero. Three herbivore model.

BzDensityEffectGrf = GRAPH(BzDensity)

(0.00, 0.00), (0.004, 0.00), (0.008, 0.03), (0.012, 0.2), (0.016, 0.75), (0.02, 0.94), (0.024,

1.00), (0.028, 1.00), (0.032, 1.00), (0.036, 1.00), (0.04, 1.00)

106

Browser density increases Carnivores are able to kill more Browsers. The graph

of this function is shown in Figure 29

= GRAPH (BzDensity)

FoodNeedCrn is the amount of food needed per Carnivore it is set at 20 lbs. per pound of

Carnivore per year (the Cat House, 1995)

BzHuntingCrn is equivalent to the hunting pressure per unit of carnivores. It is a factor of

how much food is required by carnivores and the density of Browsers

= BzHuntGrfCrn * FoodNeedCrn

PrCBz is equivalent to the percentage of Browsers in the herbivore population. It is the

number of Browsers divided by the number of herbivores

= Browsers/Herbivores

PrCCrnBz is equivalent to the percentage of carnivores hunting devoted to Browsers. It is

a function of the percentage of Browsers in the herbivore population multiplied by

the number of carnivores.

= Carnivores * PrCBz

BzHCrn is equivalent to the number of units of Browsers killed by carnivores

= BzHuntingCrn * PrCCrnBz

BzHuntGrfHs is equivalent to the rate at which Hsapiens kill Browsers. The input to the

graph (BzDensity) (shown on the X-axis) determines what value of BzHuntGrfHs

(shown on the Y-axis) will be returned as output to the model. As browser density

107

Fig. 29. The rate at which Carnivores kill Browsers. Three herbivore model.

BzHuntGrfCrn = GRAPH(GzDensity)

(0.00, -0.001), (0.02, 0.000), (0.04, 0.030), (0.06, 0.061), (0.08, 0.082), (0.10, 0.105),

(0.12, 0.122), (0.14, 0.139), (0.16, 0.160), (0.18, 0.178), (0.20, 0.196)

108

increases Hsapiens are able to kill more Browsers. The graph of this function is

shown in Figure 30

= GRAPH (BzDensity)

FoodNeedHs is the amount of food needed per pound of Hsapiens per year. It is set at 10

which is half what an obligate carnivore needs.

BzHuntingHs is equivalent to the hunting pressure per unit of Hsapiens. It is a factor of

how much food is required by Hsapiens and the density of Browsers

= BzHuntGrfHs * FoodNeedHs

PrCHsBz is equivalent to the percentage of Hsapiens hunting devoted to Browsers. It is a

function of the percentage of Browsers in the herbivore population multiplied by

the number of Hsapiens.

= Hsapiens * PrCBz

BzHHs is equivalent to the number of units of Browsers killed by Hsapiens

= BzHuntingHs * PrCHsBz

(BzHHs + BzHCrn) is the sum of hunting demand on Browsers from Carnivores and

Hsapiens

HuntDelay is the amount of time it takes for Carnivores and H. sapiens to switch prey. It

is an arbitrarily assigned value of 1.5 units of time.

HuntBz is hunting demand delayed by the amount specified in HuntDelay.

=delay((BzHHs + BzHCrn),HuntDelay)

109

Fig. 30. The rate at which Hsapiens kill Browsers. Three herbivore model.

BzHuntGrfHs = GRAPH(BzDensity)

(0, 0), (0.03, 0.008), (0.06, 0.021), (0.09, 0.039), (0.12, 0.062), (0.15, 0.095), (0.18,

0.127), (0.21, 0.159), (0.24, 0.183), (0.27, 0.193), (0.3, 0.199)

110

DeathRateBz is the non-hunting death rate of browsers and is arbitrarily assigned a rate of

0.4.

OutBz is equivalent to the loss of Browsers from the stock of Browsers per unit time;

= (Browsers * DeathRateBz) + HuntBz

Grazers

Grazers the stock of Grazers which is equivalent to the number of Grazers present at any

given time, expressed in animal units

= Grazers(t – dt) + (InGz – OutGz) * dt

GzEffGrf is equivalent to the actual efficiency of Grazers given the amount of grass

available in the system. It is a graphical function which takes as its input the ratio

of food needed compared to the grass available (GzGrass) and gives as an output

to the model the actual efficiency of Grazers. The graph is shown in Figure 31

GzEffGrf = GzGrass

GzBirthGrf is equivalent to the birth rate of Grazers. The input to the graph

(GzEfficiency) (shown on the X-axis) determines what value of GzBirthGrf

(shown on the Y-axis) will be returned as output to the model. As the efficiency

of Grazers increases, the birth rate of Grazers (GzBirthGrf) increases linearly.

The graph of this function is shown in Figure 32

= GRAPH (GzEfficiency)

111

Fig. 31. Actual efficiency of Grazers given the amount of grass available in the system. Three

herbivore model.

GzEffGrf = GRAPH(GzGrass)

(0.99, 2.00), (0.991, 1.86), (0.992, 1.66), (0.993, 1.42), (0.994, 1.19), (0.995, 1.01),

(0.996, 0.853), (0.997, 0.725), (0.998, 0.62), (0.999, 0.538), (1.00, 0.515)

112

Fig. 32. The birth rate of Grazers. Three herbivore model

GzBirthGrf = GRAPH(GzEfficiency)

(0.6, 1.000); (0.69, 0.906); (0.78, 0.816); (0.87, 0.730); (0.96, 0.640); (1.05, 0.550);

(1.24, 0.460); (1.23, 0.370); (1.32, 0.276); (1.41, 0.190); (1.5, 0.100)

113

GzDensity is equivalent to the density of Grazers;

= Grazers/Area

GzDensityEffectGrf is a graphical function that reflects the inability of Grazers to find a

mate and continue reproducing when the density of Grazers drops too low. The

graph of the function is shown in Figure 33

GzBirth is the birth rate of Grazers (GzBirthGrf ) multiplied by the GzDensityEffectGrf

(1–Grazers/(AllGrass) is equivalent to the limit to grazer population growth imposed by

the carrying capacity of the continent for plants; which is equivalent to the

maximum fraction of Grazers that can be added to the continental stock of

Grazers; expressed in animal units.

InGz is equivalent to the number of Grazers added to Grazers stock per unit time;

= (Grazers * GzBirth) * (1–Grazers/(AllGrass))

GzHuntGrfCrn is equivalent to the rate at which carnivores kill Grazers. The input to the

graph (GzDensity) (shown on the X-axis) determines what value of

GzHuntGrfCrn (shown on the Y-axis) will be returned as output to the model. As

Grazer density increases Carnivores are able to kill more Grazers. The graph of

this function is shown in Figure 34

= GRAPH (GzDensity)

114

Fig. 33. Effect of Grazer density, as Grazer density declines Grazer birth function drops toward

zero. Three herbivore model

GzDensityEffectGrf = GRAPH(GzDensity)

(0.00, 0.00), (0.004, 0.00), (0.008, 0.03), (0.012, 0.2), (0.016, 0.75), (0.02, 0.94), (0.024,

1.00), (0.028, 1.00), (0.032, 1.00), (0.036, 1.00), (0.04, 1.00)

115

Fig. 34. The rate at which Carnivores kill Grazers. Three herbivore model.

GzHuntGrfCrn = GRAPH(GzDensity)

(0.00, -0.001), (0.02, 0.000), (0.04, 0.030), (0.06, 0.061), (0.08, 0.082), (0.10, 0.105),

(0.12, 0.122), (0.14, 0.139), (0.16, 0.160), (0.18, 0.178), (0.20, 0.196)

116

GzHuntingCrn is equivalent to the hunting pressure per unit of carnivores. It is a factor of

how much food is required by carnivores and the density of Grazers

= GzHuntGrfCrn * FoodNeedCrn

PrCGz is equivalent to the percentage of Grazers in the herbivore population. It is the

number of Grazers divided by the number of herbivores

= Grazers/Herbivores

PrCCrnGz is equivalent to the percentage of carnivores hunting devoted to Grazers. It is

a function of the percentage of Grazers in the herbivore population multiplied by

the number of carnivores.

= Carnivores * PrCGz

GzHCrn is equivalent to the number of units of Grazers killed by carnivores

= GzHuntingCrn * PrCCrnGz

GzHuntGrfHs is equivalent to the rate at which Hsapiens kill Grazers. The input to the

graph (GzDensity) (shown on the X-axis) determines what value of GzHuntGrfHs

(shown on the Y-axis) will be returned as output to the model. As Grazer density

increases Hsapiens are able to kill more Grazers. The graph of this function is

shown in Figure 35

= GRAPH (GzDensity)

117

Fig. 35. The rate at which Hsapiens kill Grazers. Three herbivore model.

GzHuntGrfHs = GRAPH(GzDensity)

(0, 0), (0.03, 0.008), (0.06, 0.021), (0.09, 0.039), (0.12, 0.062), (0.15, 0.095), (0.18,

0.127), (0.21, 0.159), (0.24, 0.183), (0.27, 0.193), (0.3, 0.199)

118

GzHuntingHs is equivalent to the hunting pressure per unit of Hsapiens. It is a factor of

how much food is required by Hsapiens and the density of Grazers

= GzHuntGrfHs * FoodNeedHs

PrCHsGz is equivalent to the percentage of Hsapiens hunting devoted to Grazers. It is a

function of the percentage of Grazers in the herbivore population multiplied by

the number of Hsapiens.

= Hsapiens * PrCGz

GzHHs is equivalent to the number of units of Grazers killed by Hsapiens

= GzHuntingHs * PrCHsGz

(GzHHs + GzHCrn) is the sum of hunting demand on Grazers from Carnivores and

Hsapiens

HuntGz is hunting demand delayed by the amount specified in HuntDelay.

=delay((GzHHs + GzHCrn),HuntDelay)

GzDeathRate is the non-hunting death rate of grazers and is arbitrarily assigned a rate of

0.25.

OutGz is equivalent to the loss of Grazers from the stock of Grazers per unit time;

= (Grazers * GzDeathRate) + HuntGz

119

Mixed Feeders

Mixed feeders are dependent on both Plants sectors for different parts of their nutritional

mix. According to Wing & Buss (1970) and Anderson & Walker (1984), elephants do not

reproduce when they are deprived of browse and they need grass to give them sufficient

calories to sustain life.

MixedFeeders the stock of mixed feeders which is equivalent to the number of mixed

feeders present at any given time, expressed in animal units

= MixedFeeders(t – dt) + (InMx – OutMx) * dt

MxBirthGrf is equivalent to the birth rate of MixedFeeders. The input to the graph

(TreeFactorMx) (shown on the X-axis) determines what value of MxBirthGrf

(shown on the Y-axis) will be returned as output to the model. As TreeFactorMx

increases the birth rate of MixedFeeders (MxBirthGrf) increases linearly.. The

graph of this function is shown in Figure 36

MxDensity is equivalent to the density of MixedFeeders

= MixedFeeders/Area

MxDensityEffectGrf is a graphical function that reflects the inability of MixedFeeders to

find a mate and continue reproducing when the density of MixedFeeders drops

too low. The graph of the function is shown in Figure 37

MxBirth is the birth rate of MixedFeeders (MxBirthGrf ) multiplied by the

MxDensityEffectGrf

120

Fig. 36. The birth rate of MixedFeeders. Three herbivore model.

MxBirthGrf=GRAPH(TreeFactorMx)

(0.950, 0.000); (0.955, 0.070); (0.960, 0.137); (0.965, 0.207); (0.970, 0.277); (0.975,

0.347); (0.980, 0.417); (0.985, 0.483); (0.990, 0.557); (0.995, 0.626); (1.000, 0.697)

121

Fig. 37. Effect of MixedFeeder density, as MixedFeeder density declines MixedFeeder birth

function drops toward zero. Three herbivore model.

MxDensityEffectGrf = GRAPH(MxDensity)

(0.00, 0.00), (0.004, 0.00), (0.008, 0.03), (0.012, 0.2), (0.016, 0.75), (0.02, 0.94), (0.024,

1.00), (0.028, 1.00), (0.032, 1.00), (0.036, 1.00), (0.04, 1.00)

122

InMx is equivalent to the number of MixedFeeders added to MixedFeeder stock per unit

time. It is the number of MixedFeeders multiplied by MxBirth.

= MixedFeeders*MxBirth

MxHuntGrfCrn is equivalent to the rate at which carnivores kill MixedFeeders. The input

to the graph (MxDensity) (shown on the X-axis) determines what value of

MxHuntGrfCrn (shown on the Y-axis) will be returned as output to the model. As

MixedFeeder density increases Carnivores are able to kill more MixedFeeders.

The graph is shown in Figure 38

= GRAPH (MxDensity)

MxHuntingCrn is equivalent to the hunting pressure per unit of carnivores. It is a factor

of how much food is required by carnivores and the density of MixedFeeders

= MxHuntGrfCrn * FoodNeedCrn

PrCMx is equivalent to the percentage of MixedFeeders in the herbivore population. It is

the number of MixedFeeders divided by the number of herbivores

= MixedFeeders/Herbivores

PrCCrnMx is equivalent to the percentage of carnivores hunting devoted to

MixedFeeders. It is a function of the percentage of MixedFeeders in the herbivore

population multiplied by the number of Carnivores.

= Carnivores * PrCMx

123

Fig. 38. The rate at which Carnivores kill MixedFeeders. Three herbivore model.

MxHuntGrfCrn = GRAPH(MxDensity)

(0.00, -0.001), (0.02, 0.000), (0.04, 0.030), (0.06, 0.061), (0.08, 0.082), (0.10, 0.105),

(0.12, 0.122), (0.14, 0.139), (0.16, 0.160), (0.18, 0.178), (0.20, 0.196)

124

MxHCrn is equivalent to the number of units of MixedFeeders killed by Carnivores

= MxHuntingCrn * PrCCrnMx

MxHuntGrfHs is equivalent to the rate at which Hsapiens kill MixedFeeders. The input to

the graph (MxDensity) (shown on the X-axis) determines what value of

MxHuntGrfHs (shown on the Y-axis) will be returned as output to the model. As

MixedFeeder density increases Hsapiens are able to kill more MixedFeeders. The

graph of this function is shown in Figure 39

= GRAPH (MxDensity)

MxHuntingHs is equivalent to the hunting pressure per unit of Hsapiens. It is a factor of

how much food is required by Hsapiens and the density of MixedFeeders

= MxHuntGrfHs * FoodNeedHs

PrCHsMx is equivalent to the percentage of Hsapiens hunting devoted to MixedFeeders.

It is a function of the percentage of MixedFeeders in the herbivore population

multiplied by the number of Hsapiens.

= Hsapiens * PrCMx

MxHHs is equivalent to the number of units of MixedFeeders killed by Hsapiens

= MxHuntingHs * PrCHsMx

(MxHHs + MxHCrn) is the sum of hunting demand on MixedFeeders from Carnivores

and Hsapiens

125

Fig. 39. The rate at which Hsapiens kill MixedFeeders. Three herbivore model.

MxHuntGrfHs = GRAPH(MxDensity)

(0, 0), (0.03, 0.008), (0.06, 0.021), (0.09, 0.039), (0.12, 0.062), (0.15, 0.095), (0.18,

0.127), (0.21, 0.159), (0.24, 0.183), (0.27, 0.193), (0.3, 0.199)

126

HuntMx is hunting demand delayed by the amount specified in HuntDelay.

=delay((MxHHs + MxHCrn),HuntDelay)

GrassFactorMx is the ratio of grass needed by MixedFeeders to all grass subtracted from

one (1)

= 1 – GrassNeedMx/AllGrass

DMx is equivalent to the death rate of mixed feeders according to the amount of grass in

the environment relative to the amount needed by mixed

feeders(GrassFactorMx). As GrassFactorMx declines the death rate increases

linearly as shown in Figure 40

= GRAPH (GrassFactorMx)

OutMx is the out flow from the stock of mixed feeders. It is MixedFeeders multiplied by

the non-hunting death (DMx) plus the death from hunting (HuntMx).

Carnivores

CarnivoreDenisty is Carnivores per unit of area

= Carnivores/Area

CrnDensityGrf is a function that modifies the rate that Hsapiens can hunt Carnivores

relative to their density. As Carnivore density declines the modifier declines. The

graph of the function is shown in Figure 41

= GRAPH(CarnivoreDensity)

127

Fig. 40. The death rate of MixedFeeders according to the amount of grass in the environment

relative to the amount needed. Three herbivore model.

MxDeathGrf = GRAPH(GrassFactorMx)

(0.950, 1.000); (0.955, 0.915); (0.960, 0.825); (0.965, 0.730); (0.970, 0.620); (0.975,

0.525); (0.980, 0.430); (0.985, 0.320); (0.990, 0.225); (0.995, 0.110); (1.000, 0.015)

128

Fig. 41. The rate Hsapiens hunts Carnivores relative to their density. Three herbivore model.

CrnDensityGrf = GRAPH(CrnDensity)

(0.0158, 0.435), (0.0211, 0.775), (0.0263, 0.94), (0.0316, 1), (0.0421, 1), (0.0526, 1), (0.0632, 1),

(0.075, 1), (0.08, 1), (0.0895, 1), (0.1, 1)

129

AmtHsKillsCrn is the rate that Hsapiens kills Carnivores the default setting is 0.025.

OutStepCrn is the amount that Hsapiens kills Carnivores multiplied by CrnDensityGrf

= Hsapiens*(CrnDensityGrf*AmtHsKillCrn)

Results of Step 3: Three-Herbivore Model

Graph of the Model

The first test of the model is to see that it reproduces its reference modes. The first

reference mode to be reproduced is the equilibrium situation of Step 1 of the base model.

The Graph shows that the model returns to equilibrium when perturbed by a one time

pulse reduction of Carnivore populations of 5%. This is an increase over the perturbation

of the Base model because the other sectors did not respond visibly to the smaller

perturbation. Figure 42shows that the Three-Herbivore Model does reproduce the

reference mode.

The second reference mode to be reproduced is the introduction of the second

predator (Hsapiens) – Overkill Mode. The graph is shown in Figure 43

In the Three-Herbivore model the introduction of Hsapiens causes very little

disturbance to any of the sectors. It produces less of an impact than it does in the base

model. Populations are 90% of starting values. Carnivore populations are reduced

relatively more than Herbivores but only a fraction of a percent, food for herbivores, like

in the base model, increases slightly to 101% of its starting value

130

Fig. 42. Equilibrium mode graph. Three herbivore model

131

Fig. 43. - Second predator (overkill) mode, aggregated view. Three herbivore model.

132

The third reference mode is the second-order overkill mode, where Hsapiens hunts

carnivores. This results in a major crash of Herbivore and Carnivore populations.

Hsapiens population levels off sooner than in the Second Predator (Overkill) mode

shown in Figure 44.

To see why these results were obtained it is useful to look at herbivore populations

shown in Figure 45.

Browsers and mixed feeders expand and crash. Grazers, initially have a population

slump, as they bear all the pressure of predation by both Carnivores and Hsapiens. As

Hsapiens and Carnivore populations stabilize, grazers rebound, and stabilize. To

understand the dynamics of the herbivore populations it is necessary to look at Plants

shown in Figure 46.

Browsers and mixed feeders eat trees faster than trees can recruit, resulting in a

complete crash of small trees, followed by near extinction of large trees as well. The dip

in grazer populations after the extinction of browsers and mixed feeders allows the two

grass sectors to boom until grazer populations equilibrate.

133

Fig. 44. Second-order predation, aggregated view. Three herbivore model

134

Fig. 45. Second-order predation, herbivores Three herbivore model

135

Fig. 46. Second-order predation, plants Three herbivore model

136

Step 4 – Four Herbivores (Browsers, Ruminant Grazers, Non-ruminant

Grazers and Mixed Feeders)

Overview

Step four will establish, whether under step three conditions, ruminants are competitively

favored over monogastric herbivores. For this it will be necessary to partition grazers into

ruminants and non-ruminants and modify the equations which specify the amount and

kind of vegetation necessary for ruminants and monogastrics to extract sufficient energy

to sustain life and reproduce. This should result in a reduction of monogastrics relative to

ruminants. All conventions and definitions, not discussed, remain as previously specified.

Diagram of the Model

The only difference between the Three-Herbivore model and the– Four-Herbivore model

is the addition of equations describing the difference in grass consumption between

ruminants and non-ruminants. Ruminants are able to eat low quality grass once their

portion of high quality grass is depleted. shown in Figure 47

An illustration of the change in Grazers is shown in Figure 48.

Conventions and definition of terms used in Step 4

Array an array is a set of variables which are conceptually arranged in rows and columns.

The partitioning of grazers into ruminants and non-ruminants is done through an

array. Thus all the variables, which pertained to grazers in the Three-Herbivore

137

Fig. 47. Grass, diagram. Four-herbivore model.

Key as in Base Model diagram.

138

Fig. 48. Grazers diagram. Four-herbivore model.

Key as in Base Model diagram.

139

Model now have values apportioned by digestion type and levels accumulate by digestion

type. These are shown in the documentation as Variable [Digestion] to signify that it applies

to both ruminants and non- ruminants, Variable [Ruminant] to signify that it applies to

ruminants and Variable [NonRuminant] to signify that it applies to non- ruminants. The

concept is illustrated in Table 4 below

Table – 4 – Array Illustration

Digestion Type Ruminant Non- Ruminant

Population initialized at equilibrium equilibrium

SetEffGz 0.9 0.8

DGz 0.4 0.4

Grazers 1043.13K 747.75K

ARRAYSUM (Variable [ * ]) is the sum of the values for an arrayed variable for using the

example above the ARRAYSUM (Grazers [ * ]) is 1790.88K.

Continent

Herbivore Grass Eating Preferences

Ruminant grazers prefer high quality grass to low quality grass. If the demand for high

quality grass exceeds the stock of high quality grass, low quality grass is eaten. More low

quality grass is necessary to provide the same level of nourishment as high quality grass.

Thus each unit of nutritional demand on GrassLow has a 1.5 unit impact.

140

For the purposes of the model it is assumed that non-ruminant grazers eat only high

quality grass.

With the partitioning of grazers into ruminants and non-ruminants it is necessary to

apportion grass amongst them. It is assumed that grass is eaten in proportion to the

number of animals in the population.

SetEffGz [Ruminant] is equivalent to the setting for optimal ruminant Grazer efficiency;

it is arbitrarily assigned a value of 0.8

SetEffGz [NonRuminant] is equivalent to the setting for optimal non-ruminant grazer

efficiency; it is arbitrarily assigned a value of 0.9

(Grazers [Ruminant] * SetEffGz [Ruminant]) is equivalent to the amount of food needed

to support ruminant grazers. It is the population of ruminant grazers multiplied by

setting for optimal ruminant grazer efficiency.

(Grazers[NonRuminant] * SetEffGz [NonRuminant]) is equivalent to the amount of food

needed to support non-ruminant grazers. It is the population of non-ruminant

grazers multiplied by setting for optimal non-ruminant grazer efficiency.

FoodNeed is equivalent to the amount of food necessary to support the population of

grazers; it is the sum of the food needed by each of the grazer population;

= (Grazers [Ruminant] * SetEffGz [Ruminant]) + (Grazers[NonRuminant] *

SetEffGz [NonRuminant])

141

GzEffGrf [Ruminant] is equivalent to the actual efficiency of ruminant grazers given the

amount of grass available in the system. It is a graphical function which takes as

its input the ratio of food needed compared to the grass available (GzGrass) and

gives as an output to the model the actual efficiency of ruminant grazers. It is

assumed that efficiency will only increase to a certain extent so efficiency will

increase at a slower rate as there becomes an excess of grass by the same token

there will be a slowing in the rate of decline in the decrease of efficiency as grass

decreases. If the relationship between grass and grazer efficiency is graphed the

curve will be slightly sigmoid. The center of the graph will have the steepest slope

because it is the area of greatest change whereas the ends of the graph will be

tapered since the rate of change is less. Therefore the graph is sigmoid in shape it

is shown in Figure 49

GzEffGrf [Ruminant] = GzGrass

GzEffGrf [NonRuminant] is equivalent to the actual efficiency of non-ruminant grazers

given the amount of grass available in the system. It is a graphical function which

takes as its input the ratio of food needed compared to the grass available

(GzGrass) and gives as an output to the model the actual efficiency of non-

ruminant grazers. As the amount of grass in the system increases the efficiency of

non-ruminant grazers increases. As with ruminants the center of the graph has the

steepest slope and is the area of greatest change. The following are assumed.

Firstly that non-ruminants are less efficient, through the middle range, than are

ruminants. Secondly that non-ruminants will benefit proportionally more than

142

Fig. 49. Actual efficiency of ruminant grazers (Grazers[Ruminant]) given the amount of grass

available in the system. Four-herbivore model.

GzEffGrf[Ruminant]=GRAPH(GzGrass)

(0.990, 2.000); (0.991, 1.858); (0.992, 1.662); (0.993, 1.415); (0.994, 1.190); (0.995, 1.010);

(0.996, 0.853); (0.997, 0.725); (0.998, 0.620); (0.999, 0.538); (1.000, 0.515)

143

ruminants from an increase in grass, so the top of the graph does not level off.

And finally that as grass becomes more scarce efficiency will taper off more

slowly than it does for ruminants. Thus the curve for non-ruminant grazer

efficiency is slightly concave it is shown in Figure 50

GzEffGrf [NonRuminant] = GzGrass

GzEfficiency [Digestion] is equivalent to the realized efficiency of each of the grazers it

is the setting for optimal efficiency of each of the grazers multiplied by the actual

efficiency of each of the grazers;

= SetEffGz [Digestion] * GzEffGrf [Digestion]

GrasNeedGz [Digestion] is equivalent to amount of grass taken out of the system by each

of the grazers it is the realized efficiency of each of the grazers times the

efficiency of each of the grazers;

= Grazers [Digestion] * GzEfficiency [Digestion]

ARRAYSUM (GrasNeedGz [ * ]) is equivalent to the sum of the grass taken out of the

system by both grazers.

GrassNeed is equivalent to the grass taken out of the system by grazers and mixed

feeders.

= ARRAYSUM (GrasNeedGz [ * ]) + GrassNeedMx

144

Fig. 50. Actual efficiency of non–ruminant grazers(Grazers[NonRuminant]) given the amount of

grass available in the system. Four herbivore model

GzEffGrf[NonRuminant]=GRAPH(GzGrass)

(0.990, 2.000); (0.991, 1.603); (0.992, 1.355); (0.993, 1.168); (0.994, 1.025); (0.995, 0.890);

(0.996, 0.785); (0.997, 0.688); (0.998, 0.613); (0.999, 0.560); (1.000, 0.523)

145

SumGz is equivalent to the entire population of grazers. It is the sum of ruminant and

non-ruminant grazers

= ARRAYSUM (Grazers [ * ])

RatiosGz [Digestion] is the ratio of each of the grazers and is found by dividing the

number of grazers (SumGz) by the number of each of the grazers (Grazers

[Digestion])

= Grazers [Digestion]/SumGz

AvailHi [Digestion] is equivalent to the amount of high quality grass allotted to each of

the grazers in the population of grazers. It is the amount of high quality grass

(GrassHigh) times the ratio of each of the grazers to the entire population of

grazers RatiosGz [Digestion]

= GrassHigh * RatiosGz [Digestion]

LoGRatio is equivalent to the amount of increased impact on big trees when ruminant

grazers are forced to eat a less preferred food, low quality grass (GrassLow). This

reflects the need for herbivores to kill more biomass of low quality grass to obtain

the same nourishment. It is equal to 1.5 (arbitrarily assigned).

AmtLo is equivalent to the amount of grass taken out of the low quality grass stock by

ruminant grazers. It multiplies the amount of grass demand not filled by high

quality grass by the amount of increased impact (LoGRatio). It is determined by

subtracting the amount of high quality grass available to ruminant grazers from

the amount of grass demand by ruminant grazers (GrasNeedGz [Ruminant]) and

146

multiplying the remainder by the amount of increased impact (LoGRatio).

= (GrasNeedGz [Ruminant] –AvailHi [Ruminant]) * LoGRatio

Herbivores

Herbivores is equivalent to the entire population of herbivores. It is the sum of browsers,

mixed feeders and the sum of the array of grazers, ruminant and non- ruminant.

Herbivores = Browsers + MixedFeeders + ARRAYSUM (Grazers [ * ])

Grazers

Grazers [Ruminant] is the stock of Grazers which is equivalent to the number of

ruminant grazers present at any given time, expressed in animal units.

= Grazers [Ruminant] (t – dt) + (InGz [Ruminant] – OutGz [Ruminant]) * dt

GzBirthGrf [Ruminant] is equivalent to the birth rate of ruminants. It is a graphical

function that takes ruminant grazer efficiency GzEfficiency [Ruminant] as its

input and returns the birth rate of ruminant grazers GzBirthGrf [Ruminant] as the

output. The slope of the graph of the relationship between ruminant grazer

efficiency and ruminant grazer birth is very flat as ruminants are assumed to be

able to reproduce under most conditions. The shape of the graph is slightly

sigmoid – As efficiency decreases the birth rate decreases. The graph is shown in

Figure 51

= GzEfficiency [Ruminant]

147

Fig. 51. The birth rate of ruminants Grazers[Ruminant] Four herbivore model.

GzBirthGrf[Ruminant]=GRAPH(GzEfficiency[Ruminant])

(0.600, 0.591); (0.690, 0.569); (0.780, 0.559); (0.870, 0.551); (0.960, 0.548); (1.050, 0.545);

(1.140, 0.539); (1.230, 0.535); (1.320, 0.528); (1.410, 0.521); (1.500, 0.508)

148

GzDensity[Ruminant] is the density of Grazers[Ruminant]

= Grazers[Ruminant]/Area

GzDensityEffectGrf[Ruminant] is a graphical function that reflects the inability of

Grazers[Ruminant] to find a mate and continue reproducing when the density of

grazers (GzDensity[Ruminant]) drops too low. The graph of the function is shown

in Figure 52

GzBirth[Ruminant] is the birth rate of ruminant as found in the birth rate graph

(GzBirthGrf[Ruminant]) modified by the effect of density of ruminants

(GzDensityEffectGrf[Ruminant])

= GzDensityEffectGrf[Ruminant]*GzBirthGrf[Ruminant]

(1–Grazers [Ruminant]/(AllGrass)) is equivalent to the limit to ruminant grazer

population growth imposed by the carrying capacity of the continent for plants;

which is equivalent to the maximum fraction of ruminant grazers that can be

added to the continental stock of ruminant grazers Grazers [Ruminant] ;

expressed in animal units.

InGz [Ruminant] is equivalent to the number of ruminant grazers Grazers [Ruminant]

added to ruminant grazers Grazers [Ruminant] stock per unit time;

=Grazers [Ruminant] * (GzBirth [Ruminant]) * (1–Grazers

[Ruminant]/(AllGrass))

149

Fig. 52. Effect of Grazer[Ruminant] density, as Grazers[Ruminant] density declines

Grazers[Ruminant] birth function drops toward zero. Four-herbivore model.

GzDensityEffectGrf[Ruminant] = GRAPH(GzDensity[Ruminant])

(0.00, 0.00), (0.00273, 0.00), (0.00545, 0.03), (0.00818, 0.105), (0.0109, 0.215), (0.0136, 0.825),

(0.0164, 1.00), (0.0191, 1.00), (0.0218, 1.00), (0.0245, 1.00), (0.0273, 1.00), (0.03, 1.00)

150

GzHuntGrfCrn [Ruminant] is equivalent to the rate at which carnivores kill ruminant

grazers Grazers [Ruminant] . The input to the graph (GzDensity [Ruminant])

(shown on the X-axis) determines what value of GzHuntGrfCrn [Ruminant]

(shown on the Y-axis) will be returned as output to the model. As ruminant grazer

(Grazers [Ruminant] ) density increases Carnivores are able to kill more ruminant

grazers (Grazers [Ruminant]). The graph of this function is shown in Figure 53=

GRAPH (GzDensity [Ruminant])

GzHuntingCrn [Ruminant] is equivalent to the hunting pressure per unit of carnivores. It

is a factor of how much food is required by carnivores and the density of ruminant

grazers

= GzHuntGrfCrn [Ruminant] * FoodNeedCrn

PrCGz [Ruminant] is equivalent to the percentage of ruminant grazers (Grazers

[Ruminant]) in the herbivore population. It is the number of ruminant grazers

(Grazers [Ruminant]) divided by the number of herbivores

= Grazers [Ruminant]/Herbivores

PrCCrnGz [Ruminant] is equivalent to the percentage of carnivores hunting devoted to

ruminant grazers (Grazers [Ruminant]). It is a function of the percentage of

ruminant grazers (Grazers [Ruminant]) in the herbivore population multiplied by

the number of carnivores.

= Carnivores * PrCGz [Ruminant]

151

Fig. 53. The rate at which Carnivores kill ruminant grazers (Grazers[Ruminant]). Four

Herbivore Model.

GzHuntGrfCrn[Ruminant] = GRAPH(GzDensity[Ruminant])

(0.00, 0.000); (0.02, 0.020); (0.04, 0.040); (0.06, 0.061); (0.08, 0.081); (0.10, 0.099); (0.12,

0.120); (1.40, 0.139); (0.16, 0.160); (0.18, 0.178); (0.20, 0.199)

152

GzHCrn [Ruminant] is equivalent to the number of units of ruminant grazers (Grazers

[Ruminant]) killed by carnivores

= GzHuntingCrn [Ruminant] * PrCCrnGz [Ruminant]

GzHuntGrfHs [Ruminant] is equivalent to the rate at which Hsapiens kill ruminant

grazers (Grazers [Ruminant]). The input to the graph (GzDensity [Ruminant])

(shown on the X-axis) determines what value of GzHuntGrfHs [Ruminant]

(shown on the Y-axis) will be returned as output to the model. As ruminant grazer

(Grazers [Ruminant] ) density increases Hsapiens are able to kill more ruminant

grazers (Grazers [Ruminant]). The graph of this function is shown in Figure 54

= GRAPH (GzDensity [Ruminant])

GzHuntingHs [Ruminant] is equivalent to the hunting pressure per unit of Hsapiens. It is

a factor of how much food is required by Hsapiens and the density of ruminant

grazers (Grazers [Ruminant])

= GzHuntGrfHs [Ruminant] * FoodNeedHs

PrCHsGz [Ruminant] is equivalent to the percentage of Hsapiens hunting devoted to

ruminant grazers (Grazers [Ruminant]). It is a function of the percentage of

ruminant grazers (Grazers [Ruminant]) in the herbivore population multiplied by

the number of Hsapiens.

= Hsapiens * PrCGz [Ruminant]

153

Fig. 54. The rate at which Hsapiens kill ruminant grazers (Grazers[Ruminant]). Four Herbivore

Model.

GzHuntGrfHs[Ruminant] =GRAPH(GzDensity[Ruminant])

(0, 0); (0.03, 0.008); (0.06, 0.021); (0.09, 0.039); (0.12, 0.062); (0.15, 0.095); (0.18, 0.127);

(0.21, 0.159); (0.24, 0.183); (0.27, 0.193); (0.3, 0.199)

154

GzHHs [Ruminant] is equivalent to the number of units of ruminant grazers (Grazers

[Ruminant]) killed by Hsapiens

= GzHuntingHs [Ruminant] * PrCHsGz [Ruminant]

(GzHHs [Ruminant] + GzHCrn [Ruminant]) is the sum of hunting demand on ruminant

grazers (Grazers [Ruminant]) from Carnivores and Hsapiens

HuntGz [Ruminant] is hunting demand delayed by the amount specified in HuntDelay.

=delay((GzHHs [Ruminant] + GzHCrn [Ruminant]),HuntDelay)

DGz [Ruminant] is the non- hunting death rate of ruminant grazers (Grazers [Ruminant])

and is arbitrarily assigned a rate of 0.25.

OutGz [Ruminant] is equivalent to the loss of ruminant grazers (Grazers [Ruminant])

from the stock of ruminant grazers (Grazers [Ruminant]) per unit time;

= Grazers [Ruminant] * (DGz [Ruminant]) + HuntGz [Ruminant]

Non-ruminant grazers

BGz [NonRuminant] is equivalent to the birth rate of non- ruminants. It is a graphical

function that takes non-ruminant grazer efficiency GzEfficiency [NonRuminant]

as its input and returns the birth rate of non-ruminant grazers BGz [NonRuminant]

as the output. The slope of the graph of the relationship between non-ruminant

grazer efficiency and non-ruminant grazer birth is steep as non-ruminants are

assumed to be very sensitive to environmental conditions. The shape of the curve

155

is linear – As efficiency decreases the birth rate decreases, as shown in Figure 55

= GzEfficiency [NonRuminant]

GzDensity[NonRuminant] is the density of Grazers[NonRuminant]

= Grazers[NonRuminant]/Area

GzDensityEffectGrf[NonRuminant] is a graphical function that reflects the inability of

Grazers[NonRuminant] to find a mate and continue reproducing when the density

of grazers (GzDensity[NonRuminant]) drops too low. The graph of the function is

shown in Figure 56

GzBirth[NonRuminant] is the birth rate of NonRuminant as found in the birth rate graph

(GzBirthGrf[NonRuminant]) modified by the effect of density of non-ruminants

(GzDensityEffectGrf[NonRuminant])

= GzDensityEffectGrf[NonRuminant]*GzBirthGrf[NonRuminant]

(1–Grazers[NonRuminant]/(AllGrass)) is equivalent to the limit to non-ruminant grazer

population growth imposed by the carrying capacity of the continent for plants;

which is equivalent to the maximum fraction of non-ruminant grazers that can be

added to the stock of non-ruminant grazers Grazers[NonRuminant] ; expressed in

animal units.

InGz [NonRuminant] is equivalent to the number of non-ruminant grazers

Grazers[NonRuminant] added to non-ruminant grazers Grazers[NonRuminant]

stock per unit time;

156

Fig. 55. The birth rate of non-ruminants (Grazers[NonRuminant]). Four-herbivore model.

GzBirthGrf[NonRuminant]=GRAPH(GzEfficiency[NonRuminant])

(0.825, 0.906), (0.900, 0.816), (0.975, 0.730), (1.050, 0.640), (1.250, 0.550), (1.200, 0.460),

(1.275, 0.370), (1.350, 0.276), (1.425, 0.190), (1.500, 0.100 )

157

Fig. 56. Effect of non-ruminant density, as Grazers[NonRuminant] density declines

Grazers[NonRuminant] birth function drops toward zero. Four-herbivore model.

GzDensityEffectGrf[NonRuminant] =GRAPH(GzDensity[NonRuminant])

(0, 0); (0.03, 0.008); (0.06, 0.021); (0.09, 0.039); (0.12, 0.062); (0.15, 0.095); (0.18, 0.127);

(0.21, 0.159); (0.24, 0.183); (0.27, 0.193); (0.3, 0.199)

158

=Grazers[NonRuminant] * (BirthGz [NonRuminant]) * (1–

Grazers[NonRuminant]/(AllGrass))

GzHuntGrfCrn [NonRuminant] is equivalent to the rate at which carnivores kill non-

ruminant grazers Grazers[NonRuminant] . The input to the graph (GzDensity

[NonRuminant]) (shown on the X-axis) determines what value of GzHuntGrfCrn

[NonRuminant] (shown on the Y-axis) will be returned as output to the model. As

non-ruminant grazer (Grazer [NonRuminant]) density increases Carnivores are

able to kill more non-ruminant grazers (Grazers[NonRuminant]). The graph of

this function is shown in Figure 57

= GRAPH (GzDensity [NonRuminant])

GzHuntingCrn [NonRuminant] is equivalent to the hunting pressure per unit of

carnivores. It is a factor of how much food is required by carnivores and the

density of non-ruminant grazers (Grazers[NonRuminant])

= GzHuntGrfCrn [NonRuminant] * FoodNeedCrn

PrCGz [NonRuminant] is equivalent to the percentage of non-ruminant grazers

(Grazers[NonRuminant]) in the herbivore population. It is the number of non-

ruminant grazers (Grazers[NonRuminant]) divided by the number of herbivores

= Grazers[NonRuminant]/Herbivores

PrCCrnGz [NonRuminant] is equivalent to the percentage of carnivores hunting devoted

to non-ruminant grazers (Grazers[NonRuminant]). It is a function of the

percentage of non-ruminant grazers (Grazers[NonRuminant]) in the herbivore

159

Fig. 57. The rate at which carnivores kill non-ruminant grazers (Grazers[NonRuminant]). Four-

herbivore model.

GzHuntGrfCrn[NonRuminant] = GRAPH(GzDensity[NonRuminant])

(0.00, 0.000); (0.02, 0.020); (0.04, 0.040); (0.06, 0.061); (0.08, 0.081); (0.10, 0.099); (0.12,

0.120); (1.40, 0.139); (0.16, 0.160); (0.18, 0.178); (0.20, 0.199)

160

population multiplied by the number of carnivores.

= Carnivores * PrCGz [NonRuminant]

GzHCrn [NonRuminant] is equivalent to the number of units of non-ruminant grazers

(Grazers[NonRuminant]) killed by carnivores.

= GzHuntingCrn [NonRuminant] * PrCCrnGz [NonRuminant]

GzHuntGrfHs [NonRuminant] is equivalent to the rate at which Hsapiens kill non-

ruminant grazers (Grazers[NonRuminant]). The input to the graph (GzDensity

[NonRuminant]) (shown on the X-axis) determines what value of GzHuntGrfHs

[NonRuminant] (shown on the Y-axis) will be returned as output to the model. As

non-ruminant grazer (Grazer [NonRuminant]) density increases Hsapiens are able

to kill more non-ruminant grazers (Grazers[NonRuminant]). The graph of this

function is shown in Figure 58

= GRAPH (GzDensity [NonRuminant])

GzHuntingHs [NonRuminant] is equivalent to the hunting pressure per unit of Hsapiens.

It is a factor of how much food is required by Hsapiens and the density of non-

ruminant grazers (Grazers[NonRuminant])

= GzHuntGrfHs [NonRuminant] * FoodNeedHs

PrCHsGz [NonRuminant] is equivalent to the percentage of Hsapiens hunting devoted to

non-ruminant grazers (Grazers[NonRuminant]). It is a function of the percentage

of non-ruminant grazers (Grazers[NonRuminant]) in the herbivore population

161

Fig. 58. The rate at which Hsapiens kill non-ruminant grazers (Grazers[NonRuminant]). Four-

herbivore model.

GzHuntGrfHs[NonRuminant] = GRAPH(GzDensity[NonRuminant])

(0, 0), (0.03, 0.008), (0.06, 0.021), (0.09, 0.039), (0.12, 0.062), (0.15, 0.095), (0.18, 0.127), (0.21,

0.159), (0.24, 0.183), (0.27, 0.193), (0.3, 0.199)

162

multiplied by the number of H. sapiens.

= Hsapiens * PrCGz [NonRuminant]

GzHHs [NonRuminant] is equivalent to the number of units of non-ruminant grazers

(Grazers[NonRuminant]) killed by Hsapiens

= GzHuntingHs [NonRuminant] * PrCHsGz [NonRuminant]

(GzHHs [NonRuminant] + GzHCrn [NonRuminant]) is the sum of hunting demand on

non-ruminant grazers (Grazers[NonRuminant]) from Carnivores and Hsapiens

HuntGz [NonRuminant] is hunting demand delayed by the amount specified in

HuntDelay.

=delay ((GzHHs [NonRuminant] + GzHCrn [NonRuminant]),HuntDelay)

DGz [NonRuminant] is the non-hunting death rate of non-ruminant grazers

(Grazers[NonRuminant]) and is arbitrarily assigned a rate of 0.25.

OutGz [NonRuminant] is equivalent to the loss of non-ruminant grazers

(Grazers[NonRuminant]) from the stock of non-ruminant grazers

(Grazers[NonRuminant]) per unit time;

= Grazers[NonRuminant] * (DGz [NonRuminant]) + HuntGz [NonRuminant]

163

Results of the– Four-Herbivore Model

Graph of the Model

The first test of the model is to see that it reproduces its reference modes. The first

reference mode to be reproduced is the equilibrium situation of Step 1 of the Base model.

The graph shows that the model returns to equilibrium when perturbed by a single pulse

reduction of carnivore populations of 3%. This is an increase over the perturbation of the

Base model because the other sectors did not respond visibly to the smaller perturbation.

Figure 59 shows that the– Four-Herbivore Model does reproduce the reference mode.

The second reference mode to be reproduced is the introduction of the second

predator (Hsapiens) – Overkill Mode. The graph is shown in Figure 60

In the– Four-Herbivore model the introduction of Hsapiens causes very little

disturbance to any of the sectors – less than in the base model. Populations are 90% of

starting values. Carnivore populations are reduced relatively more than herbivores but

only a fraction of a percent. Plants, like in the base model, increases slightly to 101% of

its starting value

The third reference mode is second-order predation, where Hsapiens begins to reduce

carnivore populations. Like the Three-Herbivore Model this results in a major crash of

herbivore, carnivore and Hsapiens populations. It occurs later than it does in the Three-

Herbivore model. Hsapiens initially establish a larger population than in the Three-

Herbivore Model that then diminishes when the first crash occurs. In the– Four-

164

Fig. 59. Equilibrium mode graph. Four-herbivore model.

165

Fig. 60. Second predator (Overkill), Hsapiens enters the New World. Four-herbivore model.

166

Herbivore Model there is a second dip in herbivore, carnivore and Hsapiens populations

and a second increase in plants after the first crash. Results are shown in Figure 61.

To see why these results were obtained it is useful to look at herbivore populations

shown in Figure 62.

As in the Three-Herbivore Model browsers and mixed feeders expand and crash.

Ruminant grazers boom and non-ruminant grazers slump. Oscillations in the two grazer

populations vary inversely. When browsers and mixed feeders crash ruminant grazers dip

severely as they bear all the pressure of predation from both carnivores and Hsapiens

allowing non-ruminants to boom. As Hsapiens and carnivore populations diminish,

grazers rebound, out competing non- ruminants. To understand the dynamics of the

herbivore populations it is necessary to look at the vegetation shown in Figure 63.

Browsers and mixed feeders eat trees faster than trees can recruit, resulting in a

complete crash of small trees, followed by near extinction of large trees as well. The

competition between the two grazer populations diminishes grass allowing trees to

colonize new territory. The take over by trees drives the non-ruminants to final

extinction.

At values less than 0.02 for the amount H. sapiens kills carnivores, browsers escape

extinction. Mixed feeders go extinct at anything over 0.014. At 0.014 and below there is

no extinction. This is shown in Figure 64

167

Fig. 61. Second-order predation, aggregated view. Four-herbivore model.

168

Fig. 62. Second-order predation, herbivores. Four-herbivore model.

169

Fig. 63. Second-order predation, plants. Four-herbivore model.

170

Fig. 64. A. Second-order predation comparative graphs of A. browsers and B.mixed feeders

C.Aggregate with AmtHsKillCrn=0.015. Four herbivore model

171

Chapter III: Testing and Validity

Introduction

In the previous chapter, I presented the models and the results of each step. In this

chapter, I will discuss the validity of the model. Validity consists of establishing methods

and a rationale for testing a model, as well as judging its legitimacy in relation to its

purpose. According to Richardson and Pugh, “[i]t is meaningless to try to judge validity

in the absence of a clear view of model purpose” (1981, p. 312).

There are two purposes to this modeling effort: first, that the dynamics of the second-

order predation hypothesis be explained; and second, that both the overkill hypothesis

and the second-order predation hypothesis be simulated using the same assumptions and

values. The first purpose is a necessary precondition of the second. Before simulating the

second-order predation hypothesis in the same modeling environment as the second-

predator overkill hypothesis, it is important to know how the former works and what

impacts it would have had if, in fact, it were the cause of megafaunal extinctions. The

second purpose — simulating both hypotheses using the same assumptions and values —

is necessary in order to evaluate the new theory in the same context as the old.

Once the model’s purpose has been determined, it becomes necessary to look at its

suitability and consistency. These validation characteristics occur on two levels: one

focuses on the model’s structure; the other, on its behavior. These characteristics can be

organized into a matrix, as shown in Table 5.

172

Table 5 – Validity matrix based on Richardson and Pugh (1981)

Focusing on Structure Focusing on Behavior

Testing for Suitability

of purpose (tests

focusing inward on the

model)

Dimensional consistency

Extreme conditions in equations

Boundary adequacy of variables

Parameter (in)sensitivity

behavior characteristics

Structural (in)sensitivitybehavior characteristics

Testing for

Consistency of purpose

(tests comparing the

model with the real

system)

Face validity - rates and levels

- information feedback

- delays

Parameter values - conceptual fit

- numerical fit

Replication of reference modes(boundary adequacy for behavior)

- problem behavior

- anticipated behavior

Surprise behavior

Extreme condition behavior

Contributing to Utility

& Effectiveness of a

suitable consistent

model

Appropriateness of model

characteristics for audience

- size

- simplicity/complexity

- aggregation/detail

Counter intuitive behavior

- exhibited by model

- made intuitive by model analysis

Tests for Suitability of Structure

The questions to be asked regarding the suitability of structure involve issues of

dimensional consistency, extreme conditions, and boundary adequacy.

Dimensional Consistency

Do the dimensions of the variables in every equation of the model agree with the

computation? (This question tests for internal, or mathematical, validity.)

In the previous chapter, I presented the equations for all four steps. These equations

are internally consistent, and therefore the models have mathematical validity.

173

Extreme Conditions

Do the equations in the model continue to hold when tested under all possible extremes

for all variables?

For the most part, the equations perform as expected under extreme conditions. When

H. sapiens migrates into the New World in very high numbers, the second-order

predation simulation changes to show that there is less extinction rather than more. The

increase in killing of carnivores is irrelevant to the functioning of the ecosystem, because

the impact of H. sapiens replaces the impact of the non-human carnivores being killed.

The comparison graphs of carnivores, herbivores, plants and H. sapiens are shown in

Figure 65.

The same is true if the model is tested with extreme food values. If H. sapiens requires

the same or more food then a non-human carnivore, then the impact of killing carnivores

is made irrelevant. H. sapiens replaces non-human carnivores from the perspective of

herbivores and plants. The comparison graphs of carnivores, herbivores, plants and H.

sapiens are shown in Figure 66.

If there is no killing of carnivores and if food is varied, then there is no extinction.

These situations are shown in Figure 67.

174

Fig. 65. Comparative populations predicated on varying migration of H. sapiens over time.

175

Fig. 66. Comparative population sizes predicated on varying food needs of H. sapiens over time.

176

Fig. 67. Comparative populations predicated on varying food needs of H. sapiens and an absence

of second-order predation over time.

177

Does the model behave reasonably under extreme conditions?

The model scales and behaves as expected under extreme conditions. When run so

that the immigration of H. sapiens is set at 100,000 and all other things are equal, it

shows no boom in herbivore populations and no extinction. At this high immigration

value, the pressure of hunting by H. sapiens replaces the pressure of hunting by

carnivores, so that there is no boom-and-bust pattern and, hence, no extinction. This is

consistent with the thinking behind this model.

When run so that the immigration of H. sapiens is set at 100,000 and H. sapiens food

need set at 10, and the amount H. sapiens kills carnivores set to 0.075 it shows a different

pattern of extinction. Non-ruminant grazers go extinct first followed by mixed feeders.

We said above that non-ruminant grazers respond more to competition with ruminant

grazers than to the reduction in carnivore population. At these values the destruction of

carnivores is so great and so sudden that ruminant grazers experience an immediate boom

that drives the non-ruminant grazers to extinction. The extinction of non-ruminants leaves

more food for mixed feeders. This in turn relieves the pressure on browsers until their

population equilibrates. Mixed feeders eventually are unable to compete with browsers

and ruminant grazers and go extinct. This situation is shown in Figure 68.

Running the model with the immigration of H. sapiens is set at 100,000 and

FoodNeedHs set at 30, results in instant extinction of browsers, ruminant grazers, mixed

feeders and a boom in non-ruminant grazers followed by extinction. That is instantly

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Fig. 68. Herbivore populations where AmtHsMIgrate is set at 100,000, FoodNeedHs is set at 10,

and AmtHsKillCrn=0.075.

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followed by a extinction of H. sapiens and carnivores. Since both the immigration values

and the FoodNeedHs values are extremely unlikely, it does not falsify the hypothesis.

Appendix B shows the varying patterns of extinction and non-extinction when the

model is run at various values.

Boundary Adequacy

Are all the relevant variables and feedback effects necessary? Are all the relevant

variables and feedback effects included?

A basic specification for the model is that it have the fewest possible variables, yet

still be able to achieve its purpose. The base model of Steps 1 and 2 is the simplest

possible ecosystem to have a full complement of players. However, it is too simple to

explain the scenario presented in Chapter I. It does not address the bias toward grazers

and against browsers and mixed feeders, nor does it address the success of ruminant

grazers. By the same token, it cannot address issues of tree and grass usage and survival.

Thus, the three-herbivore and four-herbivore models of Steps 3 and 4 were created out of

insights gleaned from the shortcomings of the base model.

The dynamics of the second-order predation hypothesis can be simulated with the

four-herbivore model, in which all relevant players are accounted for. This scenario holds

that changes in continentality and vegetative patchiness are endogenous, or part of the

system under consideration. During the boom in browsers and mixed feeders, there is a

loss of trees. Trees subsequently repopulate, starting from refugia on the coasts. Because

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they suffer no predation by herbivores, they expand to fill in all the space allotted to

them, stopping only when they reach the boundary of the plains. In Chapter I, above, I

suggested (following (McDonald, 1989 and Reher, 1978) that in the early Holocene, the

boundary between the plains and the eastern woodland was maintained by bison. Thus, in

the second-order predation scenario, the loss of patchiness is due to the boom-and-bust

dynamic.

Continentality is partially a function of vegetation. The loss of tree cover from the

plains would result in a loss of transpired moisture, which would, in turn, bring about an

increase in continentality. This climate change would in itself tend to discourage tree

growth. The model suggests that the initial loss of tree cover may have been due to the

boom-and-bust dynamic.

In the model, trees are constrained by the WoodMix function. After the extinction of

browsers, mixed feeders, and non-ruminant grazers, trees fill in until stopped by the

WoodMix function. It arbitrarily assumes the role of the boundary of the plains. The

model does not create the necessary conditions for that boundary. Thus, only if one

accepts that trees would be limited in the manner specified by the WoodMix function can

one say that the model is valid with respect to the in-filling of trees. In addition, the

model does not address the role of fire in maintaining grasslands. Anthropogenic and

natural outbreak of fire is a feature of Holocene grasslands. Since the model does not

include humidity or fire, it is not valid with respect to the increase in continentality.

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Current hypotheses of extinction do not include all relevant variables. Both the

overkill hypothesis and the various climate-change hypotheses presented in previous

chapters hold continentality and patchiness to be completely exogenous to the model. If

one holds the notion that the world is consistent, then theories other than those related to

extinction are necessary to explain changes in continentality and patchiness. This makes

other theories of extinction needlessly complicated and hence less parsimonious. That

said, the model presented here suggests reasons for a loss of patchiness and a possible

contribution to increased continentality, but it does not create those conditions. One can

see the in-filling of trees up to the boundary specified by the WoodMix function, but

since the model is not geographically based, there is no way of either specifying or

knowing the distribution of trees.

Tests for Suitability of Model Behavior

The questions to be addressed with regard to tests for suitability of behavior deal with

parameter sensitivity and structural sensitivity.

Parameter sensitivity

Is the behavior of the model sensitive to reasonable variations in values?

If the desired behavior is exhibited only within a narrow range of values, then the

model will be held to be too sensitive to be useful. This is especially important when

there are few variables that have values based on observation. Thus, it is important to test

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model behavior over a range of values for each variable and for each combination of

variables.

The specification for this test of validity raises interesting questions for this particular

work. In the general work of modeling, it is important that modelers exhibit desired

behavior over a wide range of values, because they want to know if the dynamics are

generally applicable to the issue under examination. However, if one assumes that

ecosystems, for the most part, tend toward equilibrium and that extinction is a fairly rare

event, then it should be relatively difficult to produce. What one expects to see is a

threshold beyond which extinction occurs Therefore, the specification for this test of

validity should, to some extent, be inverted without being a special case, meaning that

extinction will occur only under fairly narrow conditions.

As mentioned above, at extreme values the model indicates no extinction. From the

first chapter, one learned that animals survived similar climatic change in previous

interglacials. Thus, one knows that extinction is relatively difficult to produce.

Table 6, below, shows that there is a threshold value of 0.02 pounds of carnivores

killed per pound of H. sapiens per year, after which herbivores will become extinct. This

also suggests a motive for H. sapiens to begin a policy of killing predators. Killing

predators at less than the threshold value results in larger populations of H. sapiens and

herbivores. It is only when the killing exceeds 0.02 pounds of carnivores per pound of H.

sapiens per year that extinction occurs.

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Table – 6. – Carnivore population reduction

Carnivore Reduction Final Herbivore

Population

Final H. sapiens

Population

Overkill ---- 0.000 4,606.66 833.79

0.005 4,702.97 850.66

0.010 4,812.18 869.79

0.015 4,882.19 883.50

Extinction --- -0.02 882.87 180.53

Structural Sensitivity

Is the behavior of the model sensitive to reasonable alternative formulations?

The model has been built up incrementally, and each step has been separately tested.

The sensitivity of the model to alternative formulations has been demonstrated. The three

reference modes are alternative formulations of the ecosystem under consideration, and

the models are consistent with their reference modes.

Tests for the Consistency of the Model with the Real system

Tests for the consistency of the model’s structure with the real system address issues of

face validity, parameter values, and the replication of reference modes.

Face Validity

Does the model’s structure look like the real system?

The purpose of the model, which informs its various criteria, is to explain what

happened during the Pleistocene-Holocene transition. This goal suggests that the least

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complex model that still can explain the phenomena should be used. While the base

model partially meets this standard by simulating a very simple ecosystem, it fails in that

it does not produce extinction. It is also unrealistic in assuming a flat percentage

reduction in carnivore populations and not taking into account the survival of ruminants

over non-ruminants and of grazers over browsers. It is also not sufficiently complex to

differentiate between vegetation types. For all these reasons, I elaborated on the base

model in the Step 3 and Step 4 models, although disaggregation was minimal. Therefore,

in its final guise and within the explanatory mandate, the model is valid as regards

consistency with the real system.

Parameter Values

Are the parameters recognizable in terms of the real system? Are the values selected

consistent with the best information we have about the real system?

The parameters I used were based on previous models produced by Mosimann and

Martin (1975) and Whittington and Dyke (1984). Some of the values I selected were

based loosely on the literature and also on conversations with professionals working with

extant animals, as explained in the definitions provided in the previous chapters. I derived

the remaining values from running the model and making assumptions based on the role

of dynamic equilibrium.

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Replication of Reference Modes

Do the models adequately reproduce the reference modes?

The reference mode for the model before the introduction of H. sapiens is one of

dynamic equilibrium. This suggests that in graphing the levels for all stocks over time,

the lines should be flat, but each stock should respond to a disturbance in any of the other

stocks.

In all cases, the model reproduces the dynamic-equilibrium reference mode.

However, because equilibrium was one of the assumptions of the modeling effort, this

replication was to be expected.

The second-predator overkill reference mode is also replicated in all cases.

The second-order predation reference mode is not replicated in the aggregated model.

It produces a boom-and-bust pattern, but as the model is not sufficiently complex to

differentiate among herbivores, no extinction is shown. It merely serves to suggest that a

boom-and-bust pattern would occur if predator populations were reduced. This made it

necessary to create the disaggregated models that do reproduce the reference modes and

patterns that were anticipated by the extinction scenario. The three-herbivore model

produces extinction in browsers and mixed feeders, and the four-herbivore model shows

that ruminant grazers were more competitive than non-ruminant grazers.

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Surprise behavior

Does the model produce behavior that is surprising under some conditions?

Here it is possible that the mechanisms producing the anomalous behavior are real

and the behavior meaningful.

The basic model shows a boom-and-bust pattern, but no extinction; the three-

herbivore model has one extinction event; and the four-herbivore model has two

extinction events.

That there is less extinction when H. sapiens kills carnivores at lower rates may seem

surprising but it suggests reasons for the differences in extinction patterns.

These variations suggests two things: first, that the extinction event is likely to have

been more complex than is generally assumed; and second, that the more complex the

ecosystem, the more complex the extinction event is likely to have been.

Additional Characteristics Contributing to Model Utility and Effectiveness

Several other factors contribute to the utility and effectiveness of a suitably consistent

model, including appropriateness of structure, an explanation of counterintuitive

behavior, and the generation of new insights.

Appropriateness of Structure:

Since the model is meant to explain rather than to prove, its structure should serve this

purpose. Thus, an appropriate structure is one that is simple enough to be understood by

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informed members of the profession, but also one that is rich enough to produce the

desired behavior. The base model is simple enough to be understood, but it does not

produce sufficiently complex behavior. For this reason, I elaborated on the base model in

order to introduce more complexity. Nevertheless, it was necessary to begin with the

simple model, for without it, the more complex models might have been too hard to

understand. Thus, both the basic and disaggregated models are necessary for

understanding the dynamics of the scenario presented in the first chapter.

Counterintuitive Behavior:

The production of counterintuitive behavior is expected, but the model itself should make

the reasons for such behavior clear. For example, the path of extinction proposed by the

second-order predation hypothesis is initially counterintuitive; yet the model plays a role

in clarifying this behavior.

Intuitively, one would expect that the more meat H. sapiens consumed, the sooner

extinction would have occurred. In fact, the model shows that the exact opposite obtains.

The amount of food required by H. sapiens (FoodNeedHs) has a default value of 10

pounds per pound of H. sapiens per year. This is half of what is required by a top

carnivore (lion, tiger, wolf, etc.). If I reduce the amount of food required, the pattern of

extinction occurs sooner. If I increase the amount of food required, the pattern of

extinction occurs later. As is shown above, if the amount of food required is equal to that

required by a top carnivore, then herbivores escape extinction.

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Generation of Insights

I expect the creation and presentation of this model to generate new areas and questions

for research. I will deal with this topic in the Chapter IV, Conclusions and Significance.

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Chapter IV: Conclusions and Significance

Introduction

Like validity, conclusions are meaningless unless they relate to purpose. As I have stated

before, there are two purposes to this work: first, that the dynamics of the second-order

predation hypothesis be explained; and second, that both the overkill hypothesis and the

second-order predation hypothesis be simulated using the same assumptions and values.

Conclusions

In the base model, I addressed the first purpose by showing how the dynamic of boom-

and-bust impacts plants, herbivores, and carnivores. In the elaborations of the base

model, I showed how the same dynamic acts upon trees and grass, and browsers, grazers,

and mixed feeders. In the first chapter I said:

“The task of the model is to explore the hypothesis that second-order predation

resulted in an overpopulation of herbivores which overgrazed their environment

resulting in widespread extinction.”

The second purpose of this work was to simulate both overkill and second-order

predation hypotheses in the same modeling environment and using the same assumptions,

so that they could be evaluated in the same context. The hypothesis to be tested was:

“Second-order predation and its subsequent boom-and- bust ecological dynamic,

explains the data better than overkill alone? The alternative hypothesis is that overkill

alone explains the data better.”

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The criterion for success as stated in Chapter II was as follows:

“I will consider the modeling and simulation project a success if it indicates the

conditions under which the second-order predation hypothesis would have operated.

If these conditions are found to be less likely to produce extinction than those

required for the overkill hypothesis then the second-order predation hypothesis will

be rejected.”

Using the same assumptions and starting values, conditions imposed by the overkill

hypothesis are less likely to produce extinction than conditions imposed by the second-

order predation hypothesis. Put more briefly, second-order predation is more consistent

with extinction than is overkill. Thus, the alternative hypothesis is rejected.

Discussion

The second-order predation, as a working hypothesis, explains features of the

Pleistocene-Holocene transition that were previously unexplained. (Some of these

features are shown in Table 2 of Chapter I, p. 27) The second-order predation hypothesis

allows me to suggest some reasons for several of the differences between the Pleistocene

and the Holocene.

Climate and Vegetation

The second-order predation hypothesis, as a working hypothesis, suggests that during the

Holocene, trees radically declined due to herbivore over-browsing. When trees

repopulated, they would have filled in from the mountain refugia where they had

survived. This in-filling would have created closed-canopy forests, as there would have

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been no large herbivores, like elephants, to knock down trees and establish gaps in the

forest. This sequence of events has been shown to have occurred in recent times in

Australia, when Aboriginal fire-stick farming was eliminated in areas where there were

no large herbivores (Flannery, 1995).

At the same time that trees were filling in from the mountain ranges, bison would

have been repopulating the plains. Giant herds of new, smaller, obligate-grazing bison

would have assumed a different life style in order to cope with scarcity conditions. By

roaming and foraging over vast areas, they would have maintained their grazing lawns

and kept the forest from encroaching (McNaughton, 1979, 1984; McNaughton et al,

1986). At some point, a balance would have been reached, where the plains maintained

by bison and the closed canopy forest would have met. On the plains, because there were

very few trees, there would have been very little transpiration and hence a low relative

humidity and an increase in continentality.

As a working hypothesis, the second-order predation hypothesis, may take us further

in explaining more recent extinction events. As pointed out by Burney and MacPhee:

Although the extinctions in Madagascar occurred only a short time ago, they

resemble those that came at the end of the Pleistocene. (Burney & MacPhee,

1988)

The extinction event in Madagascar coincided with the introduction of people and is

associated with a change in the vegetation. Burney reports:

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A sediment core from Lake Kavitaha, central Madagascar, provides a stratigraphic

record of changes in pollen spectra and charcoal influx in the late Holocene. The

earliest pollen spectra distantly resemble the modern pollen rain of a vegetational

mosaic in northern Madagascar, although results of principal component analysis

suggest no close modern analog. A about 1300 yr. B.P., a marked rise in charcoal is

followed by a decline in pollen of woody taxa, culminating in a change to grass-

dominated pollen spectra within about 4 centuries. Pollen of woody taxa decline

below 15% of total terrestrial pollen and spores beginning about 600 yr. B.P. The

influx of charcoal from graminoid sources remains high until recent centuries. The

late Holocene changes in vegetation and fire ecology at the site were approximately

contemporaneous with the latest 14C dates for the extinct megafauna and the earliest

dates for human occupation. (Burney, 1988)

The model would suggest that the boom phase of the boom/bust pattern accounted for

the change from woody taxa to grass dominated pollen and the increase in aridity. Once a

new equilibrium was reached fire the increase in aridity itself would exacerbate the

tendency for fires, natural and anthropegenic to keep the landscape in grasses.

Animals

I said in the first chapter:

…In order to be an improvement, any new hypothesis must address and explain: 1)

the extinction of horses in North America; 2) the extinction of the ground sloth; 3) the

bias in favor of ruminants; and 4) the bias in favor of small mammal size.

If one accepts the second-order predation hypothesis, as a working hypothesis, all

four of these issues can be explained. The extinction of horses would have been due to

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starvation during the boom-and-bust dynamic. The fact that they currently survive in a

Holocene climate is no longer an anomaly. The period of boom-and-bust was a sort of

ecological bottleneck, through which animals like horses were not able to pass. But when

they were eventually reintroduced in North America, they were able to successfully adapt

to the warmer, more continental climate.

In addition, mixed feeders went extinct as quickly as did browsers and did not

experience as large a boom period. In the model, mixed feeders respond to the boom in

browser and grazer populations more than they respond to the reduction in carnivores.

This suggests reasons why the North American camel, which was a mixed feeder, was

not able to survive the “bottleneck.”

Similar forces at work can also explain the extinction of the ground sloth. Hansen’s

(1978) study of ground sloth dung deposited between 13,000 BP and 10,000 BP found

that over time, the percentage of dietary supplement plants found in the dung declined.

More and more frequently, the sloth’s diet consisted less of its staple food, globemallow,

and more of a plant known as Mormon tea. Mormon tea is shunned by most herbivores,

probably because it contains more deleterious antiherbivory compounds than other plants

with similar protein and energy compositions (Guthrie,1989). At first, the ground sloth to

was able to process these toxins, although Mormon tea was not its preferred food. But

slowly the percentage of Mormon tea surpassed the staple, which was becoming

increasingly scarce. Thus, one sees a pattern of ever-increasing consumption of an

undesirable food, until the sloth finally disappeared. If one accepts the second-order

predation hypothesis, this is a picture of an animal that was increasingly unable to find its

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preferred food. Initially, it simply ate more and more of a food it could tolerate, but one

that was not eaten by other animals. Over time, the less desirable food came to dominate

the sloth’s entire diet. Ultimately, the sloth was unable to process a steady diet of

antiherbivory toxins and died out as a result (Phillips, 1988).

Although the model is a vast simplification of the real world, accepting the second-

order predation hypothesis makes the bias in favor of ruminants and smaller size animals

comprehensible. In a condition of environmental exhaustion, animals that are more

efficient will be favored. All things being equal, ruminants will extract more from the

same quantity of forage than non-ruminants. Similarly, within a species, smaller animals

will be competitively favored because they require less food to attain reproductive

maturity and will therefore leave more offspring.

Thus, the new hypothesis of second-order predation explains such observations better

than do the existing hypotheses.

Archaeological Evidence

Applying the implications of second-order predation to a perspective on emerging human

populations, one might consider the phenomenon of Paleo-Indian big-game hunting as a

response to the overpopulation of herbivores. Yellen and Harpending (1972), Yellen

(1976), and Harpending and Davis (1976) suggest that an environment in which

resources are widely scattered and only locally abundant tends to support group mobility

and a networked social structure. During the boom phase or the boom-and-bust cycle,

people would have found gathering their food more difficult because of the scarcity of

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plants. Scavenging for or hunting animals would have been easier because it would have

taken them less time to locate an overabundant animal population. If, as suggested by

Haynes (1995), many animals were starving because of a lack of forage, it would have

made sense to scavenge for dead ones and kill off weakened ones. The more complex of

my models suggests that there would have been an abundance of one kind of herbivore

while another kind was going extinct. This would have exacerbated local abundance and

local scarcity and would have made a networked, highly mobile lifestyle more likely.

Following this line of reasoning, the homogeneity of Paleo-Indian artifacts can be

explained in two ways. First, people who did not adopt the mobile, networked lifestyle

that maximized the use of weakened and dying animals would have had a hard time

finding sufficient resources. Second, those people who followed herbivores would have

met and exchanged information, they probably also exchanged cultural practices, which

resulted in a certain uniformity of artifacts.

With the extinction of megafauna, two changes would have occurred. The first would

have been a crash in the size of human populations that continued to rely mainly on

hunting and scavenging. The second change would have taken place in the lifestyle of

those that managed to survive the extinction event. Living in a new ecosystem with a

depauperate fauna and re-established vegetation, these people would have turned to

gathering plants for their food and consequently would have adopted a more sedentary

lifestyle (Yellen and Harpending, 1972; Yellen, 1976; and Harpending and Davis, 1976).

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With the increase in reliable resources, the world view of the survivors might also

have changed. They would have seen or heard about the massive starvation and therefore

might have been hesitant to share anything that they saw as necessary to the survival of

their own small group. This change in perspective might have resulted in a kind of Paleo-

Dark Ages. As in any dark age, knowledge and skills would have been lost. Such a

scenario would account for the drop in the quality of artifacts, as well as the loss of

homogeneity.

If one assumes that the second-order predation hypothesis holds more generally, then

the rest of the Pleistocene needs to be re-examined in light of this assumption. It may

have implications for the invention of agriculture and for the invention of war (Whitney-

Smith, 1995).

Implications for Further Research

The first challenge for further research would be to find field evidence to support or

falsify the theory. Dating evidence, which might show that the extinctions took place

over a longer period of time, would throw serious doubt on the hypothesis. It might also

be falsified if the synchronicity of extinctions were found to be in error.

Evidence that H. sapiens hunted carnivores in the New World would be helpful in

supporting the theory. According to Price (1986), remains of wild cats have been found in

late Pleistocene sites in Belgium. Soffer (1985) suggests that the large number of foot

bones of fur-bearing animals found in late Pleistocene archaeological sites in Siberia

show that humans hunted carnivores for their fur.

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More implications for further research are rooted in the shortcomings pointed out in

the validity chapter. The first would be to model a larger system that would include the

feedback loop between vegetation and relative humidity. To do this, it would be

necessary to determine the transpiration rate and moisture requirements of a mixed

parkland. With this information, one could create a feedback between the two tree stocks

and relative humidity that would be tied back to tree production.

Another research project would be to create a model of the Pleistocene, so that the

dynamics could work within geographic cells. Such a model could test whether or not

trees would behave as the hypothesis has suggested.

Yet another project would be to model the impact of fire on the plant-herbivore

system. Flannery (1995) suggests that the use of fire by H. sapiens is a result of the

extinction of large herbivores rather than a cause. He argues that once the large

herbivores of Australia were extinct, Aboriginal people used fire to keep the landscape

open. When Europeans excluded the small, anthropogenic fires, the landscape filled in,

and the newly created closed-canopy forests, with their dense understoreys, led to larger,

more disastrous fires. Thus, the model could be used to simulate fire before, during, and

after the extinction of herbivores. The results of this modeling effort could then be

compared with what exists in the archaeological record.

Significance of the Model for Science and Research

The significance of this model is that it makes an attempt to specify a mechanism

whereby climate and the introduction of H. sapiens could have combined to result in

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megafaunal extinction during the Pleistocene-Holocene transition. Furthermore, the

modeling environment that has been created for this project can be adapted quite easily to

simulate other hypotheses of extinction.

The models created for this project have been given interfaces to allow users to

specify the values they want for many variables. In the interfaces, some variables are

controlled by sliders, as shown in Figure 69A. Moving slider one (1) lets users change the

number of H. sapiens entering the New World. The number of carnivores killed by H.

sapiens is changed by moving slider two (2). Slider three (3) controls the number of

pounds of herbivores required to support a pound of H. sapiens per year. The

EquilibriumSwitch runs the model in dynamic equilibrium mode. It inserts a one-time

pulse reduction in carnivore population, which shows up in all the other sectors.

By changing the slopes of the various graphs, as shown in Figure 69B, users can

change the hunting rates for carnivores or H. sapiens, the birth and efficiency rates of the

various herbivores, and the limit allocated to trees.

Thus, with the help of the interfaces, students of the Pleistocene can use the model as

an experimental space. When they input their own values, the dynamics will become

clearer, and it will be easier for them to see that the result is due less to specific values

than to the dynamics of the system itself.

The other implication for research is the use of a model as an exploratory device to

test the consistency of one hypothesis against another. By creating a standard in which

hypotheses can be tested using the same assumptions and starting values, models can

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Fig. 69. Interface to the model:A, using the slider; and B, changing the curve

A.

.B

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become laboratories. They will prove to be useful to archaeologists, paleontologists,

paleo-ecologists, and any other scientists in fields where field observations are likely to

yield incomplete evidence. By using models, they will be able to test which hypotheses

are most consistent with the observations that are possible.

A Broader Significance for the Modern World

The model shows that extinctions are more likely to occur because of overpopulation and

the resultant destruction of habitat than through overhunting. It suggests that we need to

be careful ourselves that we do not so overpopulate the earth that we destroy our own

habitat.

We hear every day in the news that more and more habitats are being destroyed and

more and more species are being threatened. Are the endangered species, like canaries in

the mine shaft, warning us that we are also on the brink of another great extinction event,

one that could have as dire an implication for us as conditions at the end of the

Pleistocene had for the megaherbivores? The frightening thing is that just as we do not

know what the threshold of destruction was in ages past, we also do not know what the

future threshold of our own destruction might be. Nor do we know whether we might be

able, through chance or intention, to pull back from it.

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Appendix A: Equations

This appendix lists all the equations in the final, four-herbivore model. It is presented to

make replication of the model easier.

Continent

Area = 30000

GrassK = K*(1-WoodMix)

K = Area*Kmult

Kmult = 25

Plants = BigTrees + SmallTrees + GrassHigh + GrassLow

TreeK = K*WoodMix

TreePrC = 1-AllTrees/Plants

WoodMix = GRAPH(TreePrC)

(0.4, 0.697), (0.46, 0.438), (0.52, 0.354), (0.58, 0.319), (0.64, 0.301), (0.7, 0.27),

(0.76, 0.249), (0.82, 0.231), (0.88, 0.196), (0.94, 0.144), (1.00, 0.00)DOCUMENT: Controller for the amount of carrying capacity allotted to trees. The rest is

allocated to grass.

Trees

BigTrees(t) = BigTrees(t - dt) + (Maturity - OutBt) * dt

INIT BigTrees = 93584.5900DOCUMENT: The stock of Big Trees (measured in a.u.(animal units))

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Maturity = SmallTrees*MatRate

OutBt = BtEating+(BtDeath*BigTrees)

SmallTrees(t) = SmallTrees(t - dt) + (InTrees - Maturity - OutSt) * dt

INIT SmallTrees = 18716.9200

DOCUMENT: The Stock of Small Trees (measured in a.u.(animal units))

InTrees = If AllTrees<.001 then Reseed else TreeRepo*(TreeRepoRate*(1-

AllTrees/TreeK))

Maturity = SmallTrees*MatRate

OutSt = StEating+(SmallTrees*StDeath)

AllTrees = BigTrees + SmallTrees

AmtEatBt = (BzMxTreeNeed-SmallTrees)*BtRatio

BtDeath = .03

BtEating = If BtNeed > BigTrees then BigTrees else BtNeed

BtNeed = If BzMxTreeNeed > SmallTrees then AmtEatBt else 0

BtRatio = 1.5

BtRepo = BigTrees*.4

BzMxTreeNeed = TreesNeedBz + TreesNeedMx

MatRate = .15

203

Reseed = 1

StDeath = .05

StEating = If BzMxTreeNeed>SmallTrees then SmallTrees else BzMxTreeNeed

StRepo = SmallTrees*.6

TreeRepo = BtRepo+StRepo

TreeRepoRate = if SmallTrees=0 then 0 else .25

Grass

GrassHigh(t) = GrassHigh(t - dt) + (InHi - OutHi - HitoLo) * dt

INIT GrassHigh = 25624.7500DOCUMENT: The stock of high quality grass (measured in a.u.(animal units))

InHi = GrRepo*(RateGr*(1-AllGrass/GrassK))

OutHi = If GrassHigh > GrassNeed then GrassNeed + (GrassHigh*DRateH) else

GrassHigh

HitoLo = GrassHigh*HiLoRate

GrassLow(t) = GrassLow(t - dt) + (HitoLo - OutLo) * dt

INIT GrassLow = 169123.3600DOCUMENT: The stock of low quality grass (measured in a.u.(animal units))

HitoLo = GrassHigh*HiLoRate

OutLo = (GrassLow*DRateLo)+Eatlo

204

AllGrass = GrassLow + GrassHigh

AmtLo = (GrasNeedGz[Ruminant]-AvailHi[Ruminant])*LoGRatio

AvailHi[Ruminant] = GrassHigh*RatiosGz[Ruminant]

AvailHi[NonRuminant] = GrassHigh*RatiosGz[NonRuminant]

DRateH = .01

DRateLo = .1

Eatlo = If GrassNeed >GrassHigh THEN AmtLo ELSE 0

GrassNeed = ARRAYSUM(GrasNeedGz[*]) + GrassNeedMx

GrRepo = HiRepo+LoRepo

HiLoRate = .66

HiRepo = GrassHigh*.5

LoGRatio = 1.5

LoRepo = GrassLow*.5

RateGr = .33

Browsers

Browsers(t) = Browsers(t - dt) + (InBz - OutBz) * dt

INIT Browsers = 1619.180DOCUMENT: The stock of browsers n thousand pounds

205

InBz = Browsers*(BzBirth*(1-Browsers/AllTrees))

OutBz = Browsers*(DeathRateBz)+HuntBz

BzBirth = BzBirthRate*BzDesnityEffectGrf

BzBirthRate = .6

BzEfficiency = 1

DeathRateBz = .4

HuntBz = delay((BzHHs+BzHCrn),HuntDelay)

TreesNeedBz = Browsers*BzEfficiency

BzDesnityEffectGrf = GRAPH(BzDensity)

(0.00, 0.00), (0.00273, 0.00), (0.00545, 0.03), (0.00818, 0.105), (0.0109, 0.28),

(0.0136, 0.825), (0.0164, 1.00), (0.0191, 1.00), (0.0218, 1.00), (0.0245, 1.00),

(0.0273, 1.00), (0.03, 1.00)

Grazers

Grazers[Ruminant](t) = Grazers[Ruminant](t - dt) + (InGz[Ruminant] -

OutGz[Ruminant]) * dt

INIT Grazers[Ruminant] = 1043.130DOCUMENT: The stock of ruminant grazers in thousand pounds

Grazers[NonRuminant](t) = Grazers[NonRuminant](t - dt) + (InGz[NonRuminant] -

OutGz[NonRuminant]) * dt

206

INIT Grazers[NonRuminant] = 747.750

DOCUMENT: The stock of non-ruminant grazers in thousand pounds

InGz[Ruminant] =

Grazers[Ruminant]*(GzBirth[Ruminant]*(1-Grazers[Ruminant]/AllGrass))

InGz[NonRuminant] =

Grazers[NonRuminant]*(GzBirth[NonRuminant]*(1-Grazers[NonRuminant]/AllGrass))

OutGz[Digestion] = Grazers[Digestion]*(DeathRateGz[Digestion])+HuntGz[Digestion]

DeathRateGz[Digestion] = .4

FoodNeed = (Grazers[Ruminant]* SetEffGz[Ruminant])+(Grazers[NonRuminant]*

SetEffGz[NonRuminant])

GrasNeedGz[Digestion] = Grazers[Digestion]*GzEfficiency[Digestion]

GzBirth[Digestion] = GzDensityEffectGrf[Digestion]*GzBirthGrf[Digestion]

GzBirthGrf[Digestion] = GRAPH(GzEfficiency[Digestion])

GzBirthGrf [Ruminants]

(0.600, 0.591); (0.690, 0.569); (0.780, 0.559); (0.870, 0.551); (0.960, 0.548);

(1.050, 0.545); (1.140, 0.539); (1.230, 0.535); (1.320, 0.528); (1.410, 0.521);

(1.500, 0.508)

GzBirthGrf [NonRuminants]

(0.825, 0.906), (0.900, 0.816), (0.975, 0.730), (1.050, 0.640), (1.250, 0.550),

(1.200, 0.460), (1.275, 0.370), (1.350, 0.276), (1.425, 0.190), (1.500, 0.100 )DOCUMENT: The curve describing the birth rate of Ruminants is flatter than that of

Non-Ruminants. Ruminants are able to withstand a greater degree of environmental

stress.

207

GzEffGrf[Digestion] = GRAPH(GzGrass)

GzEffGrf[Ruminant]

(0.990, 2.000); (0.991, 1.858); (0.992, 1.662); (0.993, 1.415); (0.994, 1.190);

(0.995, 1.010); (0.996, 0.853); (0.997, 0.725); (0.998, 0.620); (0.999, 0.538);

(1.000, 0.515)

GzEffGrf[NonRuminant]

(0.990, 2.000); (0.991, 1.603); (0.992, 1.355); (0.993, 1.168); (0.994, 1.025);

(0.995, 0.890); (0.996, 0.785); (0.997, 0.688); (0.998, 0.613); (0.999, 0.560);

(1.000, 0.523)DOCUMENT: Efficiency curve for NonRuminants is more concave than that of

Ruminants. The curve for Ruminants is more sigmoid. Ruminants are assumed to be

better buffered from fluctuations in the environment than NonRuminants.

GzEfficiency[Digestion] = SetEffGz[Digestion]*GzEffGrf[Digestion]

GzGrass = If AllGrass< 1 then 0 else 1-FoodNeed/AllGrass

HuntDelay = 1.5

HuntGz[Digestion] = Delay((GzHHs[Digestion]+GzHCrn[Digestion]),HuntDelay)

RatiosGz[Digestion] = Grazers[Digestion]/SumGz

SetEffGz[Ruminant] = .8

SetEffGz[NonRuminant] = .9

SumGz = ARRAYSUM(Grazers[*])

208

GzBirthGrf[Digestion] = GRAPH(GzEfficiency[Digestion])

GzBirthGrf[Ruminant] (0.600, 0.591); (0.690, 0.569); (0.780, 0.559); (0.870,

0.551); (0.960, 0.548); (1.050, 0.545); (1.140, 0.539); (1.230, 0.535); (1.320,

0.528); (1.410, 0.521); (1.500, 0.508)

GzBirthGrf[NonRuminant] (0.825, 0.906), (0.900, 0.816), (0.975, 0.730), (1.050,

0.640), (1.250, 0.550), (1.200, 0.460), (1.275, 0.370), (1.350, 0.276), (1.425,

0.190), (1.500, 0.100 )

DOCUMENT: The curve describing the birth rate of Ruminants is flatter than that of

Non-Ruminants. Ruminants are able to withstand a greater degree of environmental

stress.

GzDensityEffectGrf[Digestion] = GRAPH(GzDensity[Digestion])

(0.00, 0.00), (0.00273, 0.00), (0.00545, 0.03), (0.00818, 0.105), (0.0109, 0.28),

(0.0136, 0.825), (0.0164, 1.00), (0.0191, 1.00), (0.0218, 1.00), (0.0245, 1.00),

(0.0273, 1.00), (0.03, 1.00)

GzEffGrf[Digestion] = GRAPH(GzGrass)

GzEffGrf[Ruminant]

(0.990, 2.000); (0.991, 1.858); (0.992, 1.662); (0.993, 1.415); (0.994, 1.190);

(0.995, 1.010); (0.996, 0.853); (0.997, 0.725); (0.998, 0.620); (0.999, 0.538);

(1.000, 0.515)

GzEffGrf[NonRuminant]

(0.990, 2.000); (0.991, 1.603); (0.992, 1.355); (0.993, 1.168); (0.994, 1.025);

(0.995, 0.890); (0.996, 0.785); (0.997, 0.688); (0.998, 0.613); (0.999, 0.560);

(1.000, 0.523)

209

DOCUMENT: Efficiency curve for NonRuminants is more concave than that of

Ruminants. The curve for Ruminants is more sigmoid. Ruminants are assumed to be

better buffered from fluctuations in the environment than NonRuminants.

MixedFeeders

MixedFeeders(t) = MixedFeeders(t - dt) + (InMx - OutMx) * dt

INIT MixedFeeders = 1405.070

DOCUMENT: The stock of mixed feeders in thousand pounds

InMx = MixedFeeders*MxBirth

OutMx = MixedFeeders*(MxDeathGrf)+HuntMx

GrassFactorMx = If AllGrass< 1 then 0 else 1- GrassNeedMx/AllGrass

GrassNeedMx = MixedFeeders*(MxEfficiency*(1-MxTreePerC))

HuntMx = delay((MxHHs+MxHCrn),HuntDelay)

MxBirth = MxBirthGrf*MxDensityEffectGrf

MxTreePerC = .5

TreeFactorMx = If AllTrees< 10 then 0 else 1-TreesNeedMx/AllTrees

TreeRatio = If AllTrees < 1 then 0 else AllTrees/(AllTrees+AllGrass)

TreesNeedMx = MixedFeeders*(MxEfficiency*MxTreePerC)

210

MxBirthGrf = GRAPH(TreeFactorMx)

(0.945, 0.00), (0.95, 0.00), (0.955, 0.07), (0.96, 0.137), (0.965, 0.207), (0.97,

0.277), (0.975, 0.347), (0.98, 0.417), (0.985, 0.483), (0.99, 0.557), (0.995, 0.626),

(1.00, 0.697), (1.00, 0.697)DOCUMENT: As trees increase birth rates increase.

MxDeathGrf = GRAPH(GrassFactorMx)

(0.989, 1.00), (0.99, 1.00), (0.991, 0.915), (0.992, 0.825), (0.993, 0.73), (0.994,

0.62), (0.995, 0.525), (0.996, 0.43), (0.997, 0.32), (0.998, 0.225), (0.999, 0.11),

(1.00, 0.015), (1.00, 0.015)

DOCUMENT: As grass increases death decreases.

MxDensityEffectGrf = GRAPH(MxDensity)

(0.00, 0.00), (0.00273, 0.00), (0.00545, 0.03), (0.00818, 0.105), (0.0109, 0.28),

(0.0136, 0.825), (0.0164, 1.00), (0.0191, 1.00), (0.0218, 1.00), (0.0245, 1.00),

(0.0273, 1.00), (0.03, 1.00)

MxEfficiency = GRAPH(TreeRatio)

(-0.1, 4.00), (1.39e-017, 4.00), (0.1, 3.70), (0.2, 2.21), (0.3, 1.25), (0.4, 1.05), (0.5,

1.00), (0.6, 1.05), (0.7, 1.25), (0.8, 2.21), (0.9, 3.67), (1.00, 4.00), (1.10, 4.00)

DOCUMENT: As mixed feeders move away from their optimum mix of grass and browse

they become less efficient. The optimum efficiency is at the base of the 'U'.

Carnivores

Carnivores(t) = Carnivores(t - dt) + (InCrn - OutCrn - OutStepCrn) * dt

INIT Carnivores = 842.650DOCUMENT: The stock of carnivores in thousand pounds

InCrn = Carnivores*(BRateCrn*(1-Carnivores/CarnivoreK))DOCUMENT: Intflow to the Carnivore stock.

211

OutCrn = Carnivores*(DRateCrn)

DOCUMENT: Outflow from the Carnivore stock.

OutStepCrn = If EquilibriumTest = 1 then PULSE((Carnivores*0.2), -11000, 0) else

Hsapiens*(CrnDensityGrf*AmtHsKillCrn)DOCUMENT: Secondary outflow from the carnivore stock.

AmtHsKillCrn = 0.025DOCUMENT: The rate at which H. sapiens kills predators.

BRateCrn = .4DOCUMENT: Assigned death rate of carnivores based loosely on wolf data.

CarnivoreK = DELAY(Herbivores,HerbivoreDelay)

DRateCrn = .33

DOCUMENT: Assigned death rate of carnivores based loosely on wolf data.

EquilibriumTest = 0DOCUMENT: Switch on (1) to demonstrate dynamic equilibrium, with a 0.2 one-time

reduction in Carnivores at -11000 years. Overrides settings for AmtMigrateHs and

AmtHsKillCrn.

HerbivoreDelay = 1

CrnDensityGrf = GRAPH(CarnivoreDensity)

(0.00, 0.00), (0.00526, 0.00), (0.0105, 0.105), (0.0158, 0.435), (0.0211, 0.775),

(0.0263, 0.94), (0.0316, 1.00), (0.0368, 1.00), (0.0421, 1.00), (0.0474, 1.00),

(0.0526, 1.00), (0.0579, 1.00), (0.0632, 1.00), (0.0684, 1.00), (0.0737, 1.00),

(0.0789, 1.00), (0.0842, 1.00), (0.0895, 1.00), (0.0947, 1.00), (0.1, 1.00)

Hsapiens

212

Hsapiens(t) = Hsapiens(t - dt) + (InHs + HsMigration - OutHs) * dt

INIT Hsapiens = 0DOCUMENT: The stock of carnivores in thousand pounds. It is initialized at 0 because

the model starts before the migration of Hsapiens to the new world.

InHs = If Hsapiens=0 then 0 else Hsapiens*(BrateHs*(1-Hsapiens/Herbivores))

HsMigration = If EquilibriumTest = 1 then 0 else STEP((AmtMigrateHs),TimeMigrateHs)DOCUMENT: Input to Hsapiens stock. The amount of input is AmtMigrateHs. The time

of input is TimeMigrateHs

OutHs = Hsapiens*(DRateHs)DOCUMENT: Outflow from the stock of Hsapiens.

AmtMigrateHs = .2DOCUMENT: Initial H. sapiens population in thousands.

BrateHs = .04DOCUMENT: Birth rate of Hsapiens as per Whittington & Dyke (1984) modified by an

adjustment for the death rate.

DRateHs = .033DOCUMENT: Death rate of Hsapiens.

TimeMigrateHs = -11500DOCUMENT: The time Hsapiens enters the New World

Density

DOCUMENT: Equations controlling hunting gathered together for ease in programming.

BzDensity = Browsers/Area

BzHCrn = BzHuntingCrn*PrCCrnBz

213

BzHHs = BzHuntingHs*PrCHsBz

BzHuntingCrn = BzHuntGrfCrn*FoodNeedCrn

BzHuntingHs = FoodNeedHs*BzHuntGrfHs

CarnivoreDensity = Carnivores/Area

FoodNeedCrn = 20DOCUMENT: The amount of food needed per year to support one pound of Carnivore. A

50 pound Carnivore would need 50 *20 or 1000 pounds per year.

FoodNeedHs = 10DOCUMENT: The amount of food needed per year to support one pound of H. sapiens. A

100 pound H. sapiens would need 100 *10 or 1000 pounds per year.

GzDensity[Digestion] = Grazers[Digestion]/Area

GzHCrn[Digestion] = GzHuntingCrn[Digestion]*PrCCrnGz[Digestion]

GzHHs[Digestion] = GzHuntingHs[Digestion]*PrCHsGz[Digestion]

GzHuntGrfHs[Digestion] = GRAPH (GzDensity[Digestion])

(0, 0); (0.03, 0.008); (0.06, 0.021); (0.09, 0.039); (0.12, 0.062); (0.15, 0.095);

(0.18, 0.127); (0.21, 0.159); (0.24, 0.183); (0.27, 0.193); (0.3, 0.199)

DOCUMENT: Controls the rate at which Hsapiens hunt Grazers.

GzHuntingCrn[Digestion] = FoodNeedCrn*GzHuntGrfCrn[Digestion]

GzHuntingHs[Digestion] = GzHuntGrfHs[Digestion]*FoodNeedHs

Herbivores = Browsers + MixedFeeders + ARRAYSUM(Grazers[*])

MxDensity = MixedFeeders/Area

214

MxHCrn = MxHuntingCrn*PrCCrnMx

MxHHs = MxHuntingHs*PrCHsMx

MxHuntingCrn = FoodNeedCrn*MxHuntGrfCrn

MxHuntingHs = MxHuntGrfHs*FoodNeedHs

PrCBz = Browsers/Herbivores

PrCCrnBz = Carnivores*PrCBz

PrCCrnGz[Digestion] = Carnivores*PrCGz[Digestion]

PrCCrnMx = Carnivores*PrCMx

PrCGz[Digestion] = Grazers[Digestion]/Herbivores

PrCHsBz = Hsapiens*PrCBz

PrCHsGz[Digestion] = Hsapiens*PrCGz[Digestion]

PrCHsMx = Hsapiens*PrCMx

PrCMx = MixedFeeders/Herbivores

BzHuntGrfCrn = GRAPH(BzDensity)

(-0.02, 0.00), (0.00, 0.00), (0.02, 0.02), (0.04, 0.04), (0.06, 0.061), (0.08, 0.081),

(0.1, 0.099), (0.12, 0.12), (0.14, 0.139), (0.16, 0.16), (0.18, 0.178), (0.2, 0.196),

(0.22, 0.199)

DOCUMENT: Controls the rate at which Carnivores hunt Browsers.

215

BzHuntGrfHs = GRAPH(BzDensity)

(0.00, 0.00), (0.03, 0.003), (0.06, 0.021), (0.09, 0.039), (0.12, 0.062), (0.15,

0.095), (0.18, 0.127), (0.21, 0.159), (0.24, 0.183), (0.27, 0.195), (0.3, 0.199)

DOCUMENT: Controls the rate at which Hsapiens hunt Browsers.

GzHuntGrfCrn[Digestion] = GRAPH(GzDensity[Digestion])

(-0.02, 0.00), (0.00, 0.00), (0.02, 0.02), (0.04, 0.04), (0.06, 0.061), (0.08, 0.081),

(0.1, 0.099), (0.12, 0.12), (0.14, 0.139), (0.16, 0.16), (0.18, 0.178), (0.2, 0.196),

(0.22, 0.199)

DOCUMENT: Controls the rate at which Carnivores hunt Grazers.

GzHuntGrfHs[Digestion] = GRAPH(GzDensity[Digestion])

(0, 0), (0.03, 0.008), (0.06, 0.021), (0.09, 0.039), (0.12, 0.062), (0.15, 0.095),

(0.18, 0.127), (0.21, 0.159), (0.24, 0.183), (0.27, 0.193), (0.3, 0.199)

DOCUMENT: Controls the rate at which Hsapiens hunt Grazers.

MxHuntGrfCrn = GRAPH(MxDensity)

(-0.02, 0.00), (0.00, 0.00), (0.02, 0.02), (0.04, 0.04), (0.06, 0.061), (0.08, 0.081),

(0.1, 0.099), (0.12, 0.12), (0.14, 0.139), (0.16, 0.16), (0.18, 0.178), (0.2, 0.196),

(0.22, 0.199)

DOCUMENT: Controls the rate at which Carnivores hunt Mixed Feeders.

MxHuntGrfHs = GRAPH(MxDensity)

(0.00, 0.00), (0.03, 0.003), (0.06, 0.021), (0.09, 0.039), (0.12, 0.062), (0.15,

0.095), (0.18, 0.127), (0.21, 0.159), (0.24, 0.183), (0.27, 0.195), (0.3, 0.199)

DOCUMENT: Controls the rate at which Hsapiens hunt Mixed Feeders.

216

Normalization Sector

DOCUMENT: Graphs are shown normalized to 100. This sector is where the

multiplication takes place.

BigTreesNormalized = BigTrees*normBT

BrowsersNormalized = Browsers*normBz

CarnivoresNormalized = Carnivores*normCandHs

GrassHighNormalized = GrassHigh*normGH

GrassLowNormalized = GrassLow*normGL

GrazersNormalized[Ruminant] = Grazers[Ruminant]*normGz[Ruminant]

GrazersNormalized[NonRuminant] = Grazers[NonRuminant]* normGz[NonRuminant]

HerbivoresNormalized = Herbivores * normHrb

HsapiensNormalized = Hsapiens*normCandHs

MixedFeedersNormalized = MixedFeeders* normMx

normBT = 0.001068552

normBz = 0.061759656

normCandHs = 0.118673233

normGH = 0.003902477

normGL = 0.000591284

normGz[Ruminant] = 0.095865328

217

normGz[NonRuminant] = 0.133734537

normHrb = 0.020767871

normMx = 0.071170831

normPl = 0.00032568

normST = 0.005342759

PlantsNormalized = Plants*normPl

SmallTreesNormalized = SmallTrees*normST

218

Appendix B - Summary Graphs

Running the model at various values produces different results. Graphs of the various

runs are below. Graphs are arranged in the following manner. The graphs for various

migration values are on each page: Thus, the graphs on page 1 are all graphs where the

migration of H. sapiens is set at 0.2K, on page 2 H. sapiens migration is set at 10K and

page 3 H. sapiens migration is set at 100K. The columns of graphs have the same value

for FoodNeedHs. The graphs in the first column all have FoodNeedHs set to 10 pounds

of herbivore per year per pound of H. sapiens, second column graphs all have

FoodNeedHs set to 20 pounds of herbivore per year per pound of H. sapiens, second

column graphs all have FoodNeedHs set to 40 pounds of herbivore per year per pound of

H. sapiens. The rows on each page all have the AmtHsKillCrn set to the same value. The

first row on each page AmtHsKillCrn is set to 0 (the position of the overkill hypothesis),

in the second row AmtHsKillCrn is set to 0.01, in the third row AmtHsKillCrn is set to

0.02, in the second row AmtHsKillCrn is set to 0.07. The key to the graphs is below.

0.2 10 0KFM

xtmema37

0.2 20 0KFM

xtmema38

0.2 40 0KFM

xtmema39

0.2 10 0.01KFM

xtmema42

0.2 20 0.01KFM

xtmema43

0.2 40 0.01KFM

xtmema44

0.2 10 0.02KFM

xtmema52

0.2 20 0.02KFM

xtmema53

0.2 40 0.02KFM

xtmema54

0.2 10 0.07KFM

xtii17

0.2 20 0.07KFM

xtii18

0.2 40 0.07KFM

xtii19

219

H. sapiens migrates into the New World 0.2K

10 10 0KFM

xtmemc37

10 20 0KFM

xtmemc38

10 40 0KFM

xtmemc39

10 10 0.01KFM

xtmemc42

10 20 0.01KFM

xtmemc43

10 40 0.01KFM

xtmemc44

10 10 0.02KFM

xtmemc52

10 20 0.02KFM

xtmemc53

10 40 0.02KFM

xtmemc54

10 10 0.07KFM

xtii57

10 20 0.07KFM

xtii58

10 40 0.07KFM

xtii59

220

H. sapiens migrates into the New World 10 K

100 10 0KFM

xtii62

100 20 0KFM

xtii63

100 40 0KFM

xtii64

100 10 0.01KFM

xtii67

100 20 0.01KFM

xtii68

100 40 0.01KFM

xtii69

100 10 0.02KFM

xtii72

100 20 0.02KFM

xtii73

100 40 0.02KFM

xtii74

100 10 0.07KFM

xtii87

100 20 0.07KFM

xtii88

100 40 0.07KFM

xtii89

221

H. sapiens migrates into the New World 100 K

222

Appendix C – Model and Runtime Software on CD Rom

The envelope below contains a CD rom with the four herbivore model and runtime

software. It is formatted for a Windows or Windows NT machine. It also contains the

text of the work in PDF format for printing.

All the above can also be downloaded from my website at

http://quaternary.net/xtinct2000

223

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