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Basic behavior of past population and consumption can be reproduced by adding the results of four simulations, each graphed below along with the result of combining them:

Here, F is the average per-capita ecological footprint in Earths/person/year and Fmin is the minimum ecological footprint. Points on the graph represent a value of global footprint (Fe = P * F, where P is population), separated in each simulation by a difference of 0.01 Earths/year. Simulations and the result all start at year 10000 B.C.E., when P = 1E+06 people and f₀ = F₀ / Fmin = 8.5E-01.

The simulations are calculated as follows:

Simulation 1

f₁ = f₀

P₁ = Fe / (f₁ * Fmin)

Maximum Limits:

f₁	(Felow / Fmin) / P₁ - 2
P₁	(Felow / Fmin) / (f₁ + 2)
Fe	Fehigh

Simulation 2

f₂ = (Fe / P₂) / Fmin

P₂ = P₀

Maximum Limits:

f₂	(Felow / Fmin) / P₂ - 2
P₂	(Felow / Fmin) / (f₂ + 2)
Fe	Fehigh

Simulation 3

f₃ = f₀

P₃ = Fe / (f₃ * Fmin)

Maximum Limits:

f₃	(Fhigh / Fmin) / P₃ - 2
P₃	(Fhigh / Fmin) / (f₃ + 2)
Fe	Fehigh

Simulation 4

f₄ = (Fe / P₂) / Fmin

P₄ = P₀

Maximum Limits:

f₄	(Fhigh / Fmin) / P₄ - 2
P₄	(Fhigh / Fmin) / (f₄ + 2)
Fe	Fehigh

The result is calculated using two equations:

P_result = (a₁ * P₁ + a₂ * P₂ + a₃ * P₃ + a₄ * P₄) * (f_result > 1)

+ (a₁ * P₁ + a₂ * P₂ + a₃ * P₃ + a₄ * P₄) * f_result * (f_result <= 1)

f_result = a₁ * f₁ + a₂ * f₂ + a₃ * f₃ + a₄ * f₄

The weights ( a_n ) and maximum global footprints (Felow and Fehigh) are chosen to best fit the result to historical data:

a₁ = 1.66546947E-01

a₂ = 7.14486E-04

a₃ = 8.32734736E-01

a₄ = 3.83058E-06

Felow = 2.6E-01

Fehigh = 2.972069653

The resulting values of f , P, and Fe are shown below along with historical data:

Time t_n - t_n-1 in years between two values of Fe can be estimated using the following relationships:

t_n - t_n-1 = (Feratio_n - Feratio_n-1) / (0.01 * Fe_n)

Feratio = Fe / Fehigh

The validity of this estimate can be seen in the following graph of population (normalized to the value in 2015, the last historical value). Roll over the image to see the entire history.

Normalizing global variables, the future is projected as follows: