Here, F is the average percapita 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 f_{0} = F_{0} / Fmin = 7.97E01.
The simulations are calculated as follows:
Simulation 1 
f_{1} = f_{0}
P_{1} = Fe / (f_{1} * Fmin)
Maximum Limits:
f_{1} 
(Felow / Fmin) / P_{1}  2 
P_{1} 
(Felow / Fmin) / (f_{1} + 2) 
Fe 
Fehigh 

Simulation 2 
f_{2} = (Fe / _{}P_{2}) / Fmin
P_{2} = P_{0}
Maximum Limits:
f_{2} 
(Felow / Fmin) / P_{2}  2 
P_{2} 
(Felow / Fmin) / (f_{2} + 2) 
Fe 
Fehigh 

Simulation 3 
f_{3} = f_{0}
P_{3} = Fe / (f_{3} * Fmin)
Maximum Limits:
f_{3} 
(Fhigh / Fmin) / P_{3}  2 
P_{3} 
(Fhigh / Fmin) / (f_{3} + 2) 
Fe 
Fehigh 

Simulation 4 
f_{4} = (Fe / _{}P_{2}) / Fmin
P_{4} = P_{0}
Maximum Limits:
f_{4} 
(Fhigh / Fmin) / P_{4}  2 
P_{4} 
(Fhigh / Fmin) / (f_{4} + 2) 
Fe 
Fehigh 

The result is calculated using two equations:
P_{result} = (a_{1} * P_{1} + a_{2} * P_{2} + a_{3} * P_{3} + a_{4} * P_{4}) * (f_{result} > 1)
+ (a_{1} * P_{1} + a_{2} * P_{2} + a_{3} * P_{3} + a_{4} * P_{4}) * f_{result} * (f_{result} <= 1)
f_{result} = a_{1} * f_{1} + a_{2} * f_{2} + a_{3} * f_{3} + a_{4} * f_{4}
The weights ( a_{n} ) and maximum global footprints (Felow and Fehigh) are chosen to best fit the result to historical data:
a_{1} = 1.66546503E01
a_{2} = 7.14484E04
a_{3} = 8.32732517E01
a_{4} = 6.49531E06
Felow = 2.5E01
Fehigh = 2.972069653
The resulting values of f , P, and Fe are shown below along with historical data and includes a futurecast with no global warming: 