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[edit]DescriptionAssumed development of population of precolumbian tiwanaku 1.png |
English: Assumed development of population of Pre-Columbian Tiwanaku |
Date | |
Source | Own work |
Author | Merikanto |
Sources of asumptions: Peak population, final population.
Initial population assumed same than final
Journal of Theoretical Biology
Volume 291, 21 December 2011, Pages 29-32
Journal of Theoretical Biology
A mathematical model for the Andean Tiwanaku civilization collapse: Climate variations
JC Flores, Mauro Bologna, Deterlino Urzagasti
Dates of origin, peak and collapse
The center cannot hold: A Bayesian chronology for the collapse of Tiwanaku
Erik J. Marsh ,
Alexei Vranich,
Deborah Blom,
Maria Bruno,
Katharine Davis,
Jonah Augustine,
Nicole C. Couture,
Santiago Ancapichún,
Kelly J. Knudson,
Danijela Popović,
Gianni Cunietti
PLOS
Published: November 22, 2023
https://doi.org/10.1371/journal.pone.0288798
- 3
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- Tiwanaku population growth , python3, based on simple assumptions
- 31.12.2023 0000.0000
-
- A mathematical model for the Andean Tiwanaku civilization collapse: Climate variations
import math
import numpy as np
import scipy
import matplotlib.pyplot as plt
from matplotlib.offsetbox import AnchoredText
def calcu_t2_r(t1,t2,p1,p2):
td1=abs(t2-t1)
pd1=p2/p1
r1=math.log(pd1)/td1
tt2=math.log(2)/r1
return(r1,tt2)
def calcu_population(pop0, r0, deltat, tas1, tas2,populimit1,shrinkt1, shrinkr1, collapset1, collapsefrac1):
rangelen1=int((tas2-tas1)/deltat)
taa1=tas1
tb1=0
r1=r0
p1=pop0
taa1=tas1
tees=[]
pops=[]
nuna=0
for n in range(0,rangelen1):
tb1=tb1+deltat
taa1=taa1+deltat
p2=p1*math.exp(tb1*r1)
#print(taa1,p2)
if(p2>populimit1):
p2=populimit1
r1=1e-4
if(taa1>shrinkt1):
r1=shrinkr1
tb1=0
p1=p2
#print(r1,p1)
nuna=nuna+1
#print(shrinkt1, collapset1)
print(taa1,p2,r1)
#if(nuna>2): quit(-1)
if(taa1>collapset1):
#print("Collapse")
p2=p2*collapsefrac1
r1=-1
tees.append(taa1)
pops.append(p2)
#print(taa1,tb1,p2) print(taa1,p2)
tees=np.array(tees)
pops=np.array(pops)
return(tees, pops)
- population growth calcu times
p1=2000 ##
p2=45000 ## tiwanaku
ta1=150 ##
ta2=800
- rendder times
te1=150
te2=1200
deltat=5 ## time step
- growth rate
- r1=0.0031
- r1=0.00225
- r1=1e-3 ## often in ancient cultures
rc1,tc_two1=calcu_t2_r(ta1,ta2,p1,p2)
print(" r %",rc1)
print(" T2 ",int(tc_two1))
tee2=900
deetee2=tee2-ta1
r1=rc1
t_two=tc_two1
pee2=math.exp(r1*deetee2)*p1
print(pee2)
collapsepros1=2000/45000
- assumed pop collapse to 5%
tees0, pops0=calcu_population(p1/4, r1, deltat, te1, te2, 100000, 800, -r1, 1020,collapsepros1)
tees1, pops1=calcu_population(p1/2, r1, deltat, te1, te2, 100000, 800, -r1, 1020,collapsepros1)
tees2, pops2=calcu_population(p1, r1, deltat, te1, te2, 100000, 800, -r1, 1020,collapsepros1)
tees3, pops3=calcu_population(p1*2, r1, deltat, te1, te2, 100000, 800, -r1, 1020,collapsepros1)
tees4, pops4=calcu_population(p1*3, r1, deltat, te1, te2, 100000, 800, -r1, 1020,collapsepros1)
plt.plot(tees0,pops0, lw=2, color="violet", linestyle=":", label="500")
plt.plot(tees1,pops1, lw=3, color="blue", linestyle="--", label="1000")
plt.plot(tees2,pops2, lw=5, color="green", label="2000")
plt.plot(tees3,pops3, lw=3, linestyle=":", color="orange", label="4000")
plt.legend(title="Initial population")
SMALL_SIZE = 16
MEDIUM_SIZE = 18
BIGGER_SIZE = 20
plt.rcParams.update({'font.size': MEDIUM_SIZE,})
plt.tick_params(labelsize=MEDIUM_SIZE)
plt.rc('font', size=SMALL_SIZE) # controls default text sizes
plt.rc('xtick', labelsize=MEDIUM_SIZE) # fontsize of the tick labels plt.rc('ytick', labelsize=MEDIUM_SIZE)
plt.suptitle("Assumed Tiwanaku population")
plt.title("Constant growth rate, shrink, sudden collapse")
plt.xlabel("Year BC/AD", size=MEDIUM_SIZE)
plt.ylabel("Population, estimated", size=MEDIUM_SIZE)
plt.grid()
- annot1="Initial population at "+str(ta1)+ " BC "+str(p1)
- annot2="Population at "+str(ta2)+ " AD "+str((p2+10)/1000000)+" million"
annot3="Growth rate "+str(round(r1*10000)/100)+ " %"
annot4="Duplication time "+str(int(round(t_two)))+ " yr"
- plt.text(-10000, 4.0e6, "Assumptions", fontsize=12)
- plt.text(-10000, 3.5e6, annot1, fontsize=12)
- plt.text(-10000, 3.25e6, annot2, fontsize=12)
plt.text(200, 8.0e4, annot3, fontsize=20)
plt.text(200, 7.0e4, annot4, fontsize=20)
plt.show()
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