Appendix B - Equations

Variables and Controllers for both models (see Appendix A - Diagrams, for diagrams of the relationships)

Area = The total area of the continent

M25 = The number of animal units per unit area

Regen K & Rate K = control of the carrying capacity for herbivores

K limit = Herbivore carrying capacity

B rate H = Birth rate of herbivores

Regen H = Regeneration rate of herbivores

Herb AU = Herbivores in animal units

Out H = Out flow of herbivores from the population

M1295 = The number of Homo sapiens per unit area

Regen HsL & Rate HsL = control of the carrying capacity for Homo sapiens

Hs Limit = Homo sapiens carrying capacity

B rate Hs = Birth rate of Homo sapiens

Migration Pulse Hs = start time of Homo sapiens migration

Regen Hs = Regeneration rate of Homo sapiens

H. sapiens = Homo sapiens population level in individuals

H kill rate and Kill = the number of animals killed per unit of H. sapiens.

Model 1 - Values and relationships based on Whittington and Dyke (1989)

Herbivore Equations

Herb_AU(t) = Herb_AU(t - dt) + (Regen_H - Out_H) * dt

INIT Herb_AU = 25000000

Regen_H = Herb_AU*(B_rate_H*(1-Herb_AU/K_limit))

Out_H = H_kill_rate

Homo sapiens Equations

H_sapiens(t) = H_sapiens(t - dt) + (Regen_Hs) * dt

INIT H_sapiens = 0

Regen_Hs = Migration_Pulse_Hs+H_sapiens*(B_rate_Hs*(1-H_sapiens/Hs_limit))

Limit Equations Herbivore and Homo sapiens

Regen_HsL = Hs_Limit*(R_rate_HsL*(1-Hs_Limit/M_1295))

Hs_Limit(t) = Hs_Limit(t - dt) + (Regen_HsL) * dt

INIT Hs_Limit = 30000

K_limit(t) = K_limit(t - dt) + (Regen_K) * dt

INIT K_limit = 50000000

Regen_K = K_limit*(R_rate_K*(1-K_limit/M_25))

Auxiliary Equations and Values

Area = 3000000

B_rate_H = .25

B_rate_Hs = .045

H_kill_rate = H_sapiens*Kill

Kill = 3.862/3

Migration_Pulse_Hs = PULSE(200, 0, 1000)

M_1295 = Area*1.295

M_25 = Area*25

R_rate_HsL = .525

R_rate_K = .8

Model 2 - Interactive Carrying Capacity

The major difference between model 1 and two is the addition of equations and values that reflect the relationship between the death of herbivores and the degradation of the carrying capacity of the environment for Homo sapiens.