6_CellProliferation.ipynb

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    "# Stem Cell Proliferation Growth \n",
    "\n",
    "\n",
    "## Unconstrained growth \n",
    "\n",
    "Stem Cell Proliferation Growth In general stem cells undergo a phase of proliferation that expands the population, then they start to differentiate to produce the different types of progeny. Let’s calculate how a population of stem cells grow in time. Lets assume we have an initial number of cells `N` in a population \n",
    "\n",
    "After a given time we have a number of new stem cells `∆N` (number of newly produced cells) during a given time interval, `∆t`, is proportional to the initial  number of cells `N`. (If a population of 20,000 cells produces 1200 new cells in 1 h, a 4-fold bigger population of 80,000 cells of the same type of microorganism will produce 4 times as many, viz. 4800, new cells in 1 h): \n",
    "\n",
    "$$\\frac{\\Delta N}{\\Delta t} \\sim N\\tag{1} $$\n",
    "\n",
    "Because the newly produced cells always add to the population, i.e. N increases steadily, we have to regard time intervals (and accordingly numbers of newly formed cells) that are as small as possible. Mathematically we thus have to deal with infinitesimal increases or differentials: \n",
    "\n",
    "$$\\frac{\\mathrm{d} N}{\\mathrm{d} t} \\sim N\\tag{2}$$\n",
    "\n",
    "To get from proportionality  to an equation, a proportionality factor, μ, is introduced. This is the specific growth rate or often termed simply growth rate. Sorting of variables yields: \n",
    "\n",
    "$$\\frac{\\mathrm{d} N}{N}  = \\mu \\mathrm{d} t\\tag{3}$$\n",
    "\n",
    "If we integrate this differential equation:\n",
    "\n",
    "$$\\int \\frac{\\mathrm{d} N}{N}  = \\int \\mu \\mathrm{d} t \\tag{4}$$\n",
    "\n",
    " we obtain:\n",
    " \n",
    " $$\\ln N = \\mu t + C \\tag{5}$$\n",
    " \n",
    " The undefined integration constant `C` can be fixed if we define the initial conditions: $N(t = 0) = N_0$\n",
    "\n",
    " $$\\ln N_0 = \\mu 0 + C =C \\tag{6}$$\n",
    " \n",
    " so the equation becomes:\n",
    " \n",
    "  $$\\ln N = \\mu t + \\ln N_0 \\tag{7}$$\n",
    "  \n",
    "  and after applying the properties of logatithms:\n",
    "  \n",
    "  $$\\ln N - \\ln N_0= \\mu t  \\tag{8}$$\n",
    "  $$\\ln \\frac{N}{N_0}= \\mu t  \\tag{9}$$\n",
    "  $$\\frac{N}{N_0}= e^{\\mu t}  \\tag{10}$$\n",
    "  $$N= N_0 e^{\\mu t}  \\tag{11}$$\n",
    "  \n",
    "   If the time that has passed is exactly the average length of the cell cycle of the stem cell population, we can write that in $t = T$ the population has doubled, so $N_0$ has increased to $2 N_0$:\n",
    "\n",
    "   $$2 N= N_0 e^{\\mu T}  \\tag{12}$$\n",
    "   \n",
    "   $$\\frac{2 N_0}{N_0 }=2= e^{\\mu T}  \\tag{13}$$\n",
    "   \n",
    "   $$ \\ln 2= \\mu T  \\tag{14}$$\n",
    "   \n",
    "   So, the proportionality constant is related to the cell cycle length as:\n",
    "\n",
    "   $$  \\mu = \\frac{\\ln 2}{T}  \\tag{15}$$\n",
    "\n",
    "\n"
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       "Plots.GRBackend()"
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   "source": [
    "using Plots\n",
    "gr()"
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    "N₀=200\n",
    "t=collect(0:0.1:10)\n",
    "T=10\n",
    "μ=log(2)/T\n",
    "plot(t,t->N₀*exp(μ*t),label=\"T=10\",seriestype=:line,ylims = (200,400))\n",
    "T=5\n",
    "μ=log(2)/T\n",
    "plot!(t,t->N₀*exp(μ*t),label=\"T=5\",seriestype=:line,ylims = (200,400))\n",
    "T=15\n",
    "μ=log(2)/T\n",
    "plot!(t,t->N₀*exp(μ*t),label=\"T=15\",seriestype=:line,ylims = (200,400))"
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    "# Logistic Growth \n",
    "\n",
    "We can introduce in the original equation a term that accounts for the limitation of resources \n",
    "\n",
    "$$1 -\\frac{N}{K} \\tag{16}$$\n",
    "\n",
    "This factor is close to 1 (i.e., has no effect) when N is much smaller than K, and which is close to 0 when N is close to K. The resulting differential equation is called is the logistic growth model.\n",
    "\n",
    "\n",
    "$$\\frac{\\mathrm{d} N}{\\mathrm{d} t}=\\mu N(1 -\\frac{N}{K}) \\tag{17}$$\n",
    "\n",
    "separate variables\n",
    "\n",
    "$$\\frac{\\mathrm{d} N}{N(1 -\\frac{N}{K})}=\\mu \\mathrm{d} t   \\tag{18}$$\n",
    "\n",
    "Our next goal would be to integrate both sides of this equation, but the form of the right hand side doesn't look elementary and will require a partial fractions expansion. That is, we wish to write \n",
    "\n",
    "\n",
    "$$\\frac{1}{N(1 -\\frac{N}{K})} = \\frac{A}{N}+\\frac{B}{1 -\\frac{N}{K}} \\tag{19}$$\n",
    "\n",
    "where $A$ and $B$ are unknown constants. If we multiply on the left and right hand sides by  $N \\left( 1- \\frac{N}{K} \\right)$ (which is equivalent to putting the right hand side over a common denominator) we arrive at the equation \n",
    "\n",
    "$$\n",
    "1 = A \\ \\left( 1 -\\frac{N}{K}\\right) + B \\cdot N  = A + N (B - \\frac{A}{K}) \\tag{20}\n",
    "$$\n",
    "\n",
    "Since there is no term with $N$ on the left hand side, we see that \n",
    "$$\n",
    "B - \\frac{A}{K} = 0 \\quad \\mbox{ or } \\quad B = \\frac{A}{K} \\tag{21}\n",
    "$$\n",
    "\n",
    "If we set $B = \\frac{A}{K}$ then we are left with $A=1$, and thus the partial fraction decomposition is \n",
    "\n",
    "$$\n",
    "\\frac{1}{N(1 -\\frac{N}{K})} = \\frac{1}{N}+\\frac{\\frac{1}{K}}{1 -\\frac{N}{K}} \\tag{22}$$\n",
    "\n",
    "so the integral becomes\n",
    "\n",
    "$$\\frac{\\mathrm{d} N}{N}+\\frac{\\frac{\\mathrm{d} N}{K}}{1 -\\frac{N}{K}}=\\mu \\mathrm{d} t   \\tag{23}$$\n",
    "\n",
    "the first part is simply:\n",
    "\n",
    "$$\n",
    "\\int\\frac{dN}{N} = \\ln (N) \\tag{24},\n",
    "$$\n",
    "\n",
    "For the second term, we must use a substitution $u=1-\\frac{N}{K}$, which gives a differential  $du = \\frac{-1}{K} \\ dN$. Thus we may write the second term on the right hand side as:\n",
    "\n",
    "$$\n",
    "\\int \\frac{dN/K}{\\left( 1-\\frac{N}{K} \\right)} = \\int \\frac{-du}{u} = -\\ln (u) = -\\ln (1-N/K) \\tag{25}\n",
    "$$\n",
    "\n",
    "Putting all these terms together gives us:\n",
    "\n",
    "$$\n",
    "\\mu t + c = \\ln (N)-\\ln (1-\\frac{N}{K}) = \\ln \\left[\\frac{N}{1-N/K} \\right] \\tag{26}\n",
    "$$\n",
    "\n",
    "Here we have used the property of logarithms to equate the difference of the logs with the log of the quotient. The additional term, $c$, on the left hand side is the free constant of integration, which will be determined by considering initial conditions to the differential equation. Exponentiating both sides of the equation gives \n",
    "\n",
    "$$\n",
    "e^{\\mu t + c} = \\frac{N}{1-\\frac{N}{K}} \\tag{27},\n",
    "$$\n",
    "\n",
    "so\n",
    "\n",
    "$$\n",
    " e^{\\mu t} e^c = \\frac{N}{1-\\frac{N}{K}} \\tag{28}\n",
    "$$\n",
    "\n",
    "$$\n",
    "e^{\\mu t} C = \\frac{N}{1-\\frac{N}{K}} \\tag{29}\n",
    "$$\n",
    "\n",
    "to find $C$ we use teh inital condition that $N(t=0)=N_0$, and substituting gives \n",
    "\n",
    " \n",
    "$$\n",
    "C = C e^0 = \\frac{N_0}{1-\\frac{N_0}{K}} = \\frac{N_0}{1-\\frac{N_0}{K}} \\frac{K}{K} =\\frac{K N_0}{K-N_0} \\tag{30}$$\n",
    "\n",
    "Solving now for $P$, we first cross-multiply to arrive at \n",
    "\n",
    "$$\n",
    "\\left(1-\\frac{N}{K} \\right) C e^{\\mu t} = N \\tag{31}\n",
    "$$\n",
    "\n",
    "and putting all terms including $N$ on one side of the equation, \n",
    "$$ \n",
    "C e^{ \\mu t} = N \\left[1 + \\frac{C e^{ \\mu t}}{K} \\right]  \\tag{32}\n",
    "$$\n",
    "\n",
    "Solving now for $N$, \n",
    "\n",
    " \n",
    "$$\n",
    "N = \\frac{C e^{\\mu t}}{1 + \\frac{C e^{\\mu t}}{K} } =\n",
    "\\frac{\\frac{K \\cdot N_0}{K-N_0} e^{\\mu t}}{1 + \\frac{\\frac{K\\cdot N_0}{K-N_0} e^{\\mu t}}{K} }\\tag{33}\n",
    "$$\n",
    "\n",
    "Simplifying this expression by multiplying numerator and denominator by  $(K-N_0) e^{-\\mu t}$ gives \n",
    "\n",
    "$$\n",
    "N = \\frac{K N_0}{N_0 +(K-N_0) e^{- \\mu t}} \\tag{34}\n",
    "$$"
   ]
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       "  681.022,567.81 683.047,564.854 685.072,561.91 687.097,558.98 689.122,556.062 691.147,553.158 693.171,550.267 695.196,547.389 697.221,544.524 699.246,541.673 \n",
       "  701.271,538.834 703.296,536.008 705.321,533.195 707.346,530.395 709.371,527.608 711.396,524.834 713.421,522.073 715.446,519.325 717.471,516.59 719.496,513.867 \n",
       "  721.521,511.157 723.546,508.46 725.571,505.776 727.596,503.104 729.621,500.445 731.646,497.799 733.671,495.166 735.696,492.545 737.721,489.936 739.746,487.341 \n",
       "  741.771,484.757 743.796,482.186 745.821,479.628 747.846,477.082 749.871,474.549 751.896,472.028 753.921,469.519 755.946,467.022 757.971,464.538 759.996,462.066 \n",
       "  762.021,459.607 764.046,457.159 766.071,454.724 768.096,452.301 770.121,449.889 772.146,447.49 774.171,445.103 776.196,442.728 778.221,440.365 780.246,438.014 \n",
       "  782.271,435.675 784.296,433.347 786.321,431.032 788.346,428.728 790.371,426.436 792.396,424.156 794.421,421.887 796.446,419.63 798.471,417.385 800.496,415.151 \n",
       "  802.521,412.929 804.546,410.718 806.571,408.519 808.596,406.331 810.621,404.155 812.646,401.99 814.671,399.836 816.696,397.693 818.721,395.562 820.746,393.442 \n",
       "  822.771,391.333 824.796,389.236 826.821,387.149 828.846,385.073 830.871,383.009 832.895,380.955 834.92,378.913 836.945,376.881 838.97,374.86 840.995,372.85 \n",
       "  843.02,370.851 845.045,368.862 847.07,366.885 849.095,364.918 851.12,362.961 853.145,361.015 855.17,359.08 857.195,357.155 859.22,355.241 861.245,353.337 \n",
       "  863.27,351.443 865.295,349.56 867.32,347.687 869.345,345.825 871.37,343.973 873.395,342.13 875.42,340.298 877.445,338.477 879.47,336.665 881.495,334.863 \n",
       "  883.52,333.071 885.545,331.29 887.57,329.518 889.595,327.756 891.62,326.003 893.645,324.261 895.67,322.528 897.695,320.805 899.72,319.092 901.745,317.389 \n",
       "  903.77,315.695 905.795,314.01 907.82,312.335 909.845,310.67 911.87,309.014 913.895,307.367 915.92,305.73 917.945,304.102 919.97,302.483 921.995,300.873 \n",
       "  924.02,299.273 926.045,297.682 928.07,296.1 930.095,294.527 932.12,292.963 934.145,291.408 936.17,289.862 938.195,288.324 940.22,286.796 942.245,285.277 \n",
       "  944.27,283.766 946.295,282.264 948.32,280.771 950.345,279.286 952.37,277.81 954.395,276.343 956.42,274.884 958.445,273.434 960.47,271.992 962.495,270.559 \n",
       "  964.52,269.134 966.545,267.717 968.57,266.308 970.595,264.908 972.619,263.517 974.644,262.133 976.669,260.757 978.694,259.39 980.719,258.031 982.744,256.679 \n",
       "  984.769,255.336 986.794,254.001 988.819,252.673 990.844,251.354 992.869,250.042 994.894,248.738 996.919,247.442 998.944,246.153 1000.97,244.873 1002.99,243.6 \n",
       "  1005.02,242.334 1007.04,241.076 1009.07,239.826 1011.09,238.583 1013.12,237.348 1015.14,236.12 1017.17,234.899 1019.19,233.686 1021.22,232.48 1023.24,231.281 \n",
       "  1025.27,230.09 1027.29,228.905 1029.32,227.728 1031.34,226.558 1033.37,225.396 1035.39,224.24 1037.42,223.091 1039.44,221.949 1041.47,220.814 1043.49,219.686 \n",
       "  1045.52,218.565 1047.54,217.451 1049.57,216.344 1051.59,215.243 1053.62,214.149 1055.64,213.062 1057.67,211.981 1059.69,210.907 1061.72,209.84 1063.74,208.779 \n",
       "  1065.77,207.725 1067.79,206.677 1069.82,205.635 1071.84,204.6 1073.87,203.572 1075.89,202.55 1077.92,201.534 1079.94,200.524 1081.97,199.521 1083.99,198.523 \n",
       "  1086.02,197.532 1088.04,196.547 1090.07,195.569 1092.09,194.596 1094.12,193.629 1096.14,192.669 1098.17,191.714 1100.19,190.765 1102.22,189.822 1104.24,188.885 \n",
       "  1106.27,187.954 1108.29,187.029 1110.32,186.109 1112.34,185.196 1114.37,184.288 1116.39,183.385 1118.42,182.488 1120.44,181.597 1122.47,180.712 1124.49,179.832 \n",
       "  1126.52,178.957 1128.54,178.088 1130.57,177.225 1132.59,176.367 1134.62,175.514 1136.64,174.667 1138.67,173.825 1140.69,172.989 1142.72,172.157 1144.74,171.331 \n",
       "  1146.77,170.51 1148.79,169.695 1150.82,168.884 1152.84,168.079 1154.87,167.278 1156.89,166.483 1158.92,165.693 1160.94,164.908 1162.97,164.128 1164.99,163.353 \n",
       "  1167.02,162.582 1169.04,161.817 1171.07,161.057 1173.09,160.301 1175.12,159.55 1177.14,158.804 1179.17,158.063 1181.19,157.326 1183.22,156.594 1185.24,155.867 \n",
       "  1187.27,155.145 1189.29,154.427 1191.32,153.713 1193.34,153.005 1195.37,152.301 1197.39,151.601 1199.42,150.906 1201.44,150.215 1203.47,149.529 1205.49,148.847 \n",
       "  1207.52,148.169 1209.54,147.496 1211.57,146.827 1213.59,146.163 1215.62,145.503 1217.64,144.847 1219.67,144.195 1221.69,143.547 1223.72,142.904 1225.74,142.265 \n",
       "  1227.77,141.63 1229.79,140.999 1231.82,140.372 1233.84,139.749 1235.87,139.13 1237.89,138.515 1239.92,137.904 1241.94,137.298 1243.97,136.695 1245.99,136.096 \n",
       "  1248.02,135.5 1250.04,134.909 1252.07,134.322 1254.09,133.738 1256.12,133.158 1258.14,132.582 1260.17,132.01 1262.19,131.441 1264.22,130.876 1266.24,130.315 \n",
       "  1268.27,129.758 1270.29,129.204 1272.32,128.653 1274.34,128.107 1276.37,127.564 1278.39,127.024 1280.42,126.488 1282.44,125.955 1284.47,125.426 1286.49,124.901 \n",
       "  1288.52,124.378 1290.54,123.86 1292.57,123.344 1294.59,122.832 1296.62,122.323 1298.64,121.818 1300.67,121.316 1302.69,120.817 1304.72,120.322 1306.74,119.83 \n",
       "  1308.77,119.341 1310.79,118.855 1312.82,118.372 1314.84,117.893 1316.87,117.417 1318.89,116.944 1320.92,116.474 1322.94,116.007 1324.97,115.543 1326.99,115.082 \n",
       "  1329.02,114.624 1331.04,114.17 1333.07,113.718 1335.09,113.269 1337.12,112.823 1339.14,112.38 1341.17,111.941 1343.19,111.503 1345.22,111.069 1347.24,110.638 \n",
       "  1349.27,110.21 1351.29,109.784 1353.32,109.361 1355.34,108.941 1357.37,108.524 1359.39,108.11 1361.42,107.698 1363.44,107.289 1365.47,106.883 1367.49,106.479 \n",
       "  1369.52,106.078 1371.54,105.68 1373.57,105.284 1375.59,104.891 1377.62,104.501 1379.64,104.113 1381.67,103.728 1383.69,103.345 1385.72,102.965 1387.74,102.588 \n",
       "  1389.77,102.213 1391.79,101.84 1393.82,101.47 1395.84,101.102 1397.87,100.737 1399.89,100.374 1401.92,100.014 1403.94,99.6561 1405.97,99.3005 1407.99,98.9473 \n",
       "  1410.02,98.5965 1412.04,98.248 1414.07,97.9018 1416.09,97.5579 1418.12,97.2163 1420.14,96.877 1422.17,96.54 1424.19,96.2052 1426.22,95.8726 1428.24,95.5423 \n",
       "  1430.27,95.2142 1432.29,94.8882 1434.32,94.5645 1436.34,94.2429 1438.37,93.9235 1440.39,93.6062 1442.42,93.291 1444.44,92.9779 1446.47,92.6669 1448.49,92.358 \n",
       "  1450.52,92.0512 1452.54,91.7464 1454.57,91.4437 1456.59,91.143 1458.62,90.8443 1460.64,90.5476 1462.67,90.2529 1464.69,89.9602 1466.72,89.6694 1468.74,89.3806 \n",
       "  1470.77,89.0937 1472.79,88.8088 1474.82,88.5257 1476.84,88.2446 1478.87,87.9653 1480.89,87.688 1482.92,87.4124 1484.94,87.1388 1486.97,86.867 1488.99,86.597 \n",
       "  1491.02,86.3288 1493.04,86.0624 1495.07,85.7978 1497.09,85.535 1499.12,85.274 1501.14,85.0147 1503.17,84.7571 1505.19,84.5013 1507.22,84.2472 1509.24,83.9949 \n",
       "  1511.27,83.7442 1513.29,83.4952 1515.32,83.2479 1517.34,83.0023 1519.37,82.7583 1521.39,82.5159 1523.42,82.2752 1525.44,82.0361 1527.47,81.7987 1529.49,81.5628 \n",
       "  1531.52,81.3285 1533.54,81.0958 1535.57,80.8647 1537.59,80.6351 1539.62,80.4071 1541.64,80.1806 1543.67,79.9556 1545.69,79.7322 1547.72,79.5103 1549.74,79.2898 \n",
       "  1551.77,79.0709 1553.79,78.8535 1555.82,78.6375 1557.84,78.4229 1559.87,78.2099 1561.89,77.9982 1563.92,77.788 1565.94,77.5792 1567.97,77.3719 1569.99,77.1659 \n",
       "  1572.02,76.9613 1574.04,76.7581 1576.07,76.5563 1578.09,76.3559 1580.12,76.1568 1582.14,75.959 1584.17,75.7626 1586.19,75.5675 1588.22,75.3738 1590.24,75.1813 \n",
       "  1592.27,74.9902 1594.29,74.8004 1596.32,74.6118 1598.34,74.4245 1600.37,74.2385 1602.39,74.0538 1604.41,73.8703 1606.44,73.688 1608.46,73.507 1610.49,73.3272 \n",
       "  1612.51,73.1486 1614.54,72.9713 1616.56,72.7951 1618.59,72.6202 1620.61,72.4464 1622.64,72.2738 1624.66,72.1024 1626.69,71.9322 1628.71,71.7631 1630.74,71.5951 \n",
       "  1632.76,71.4283 1634.79,71.2626 1636.81,71.0981 1638.84,70.9346 1640.86,70.7723 1642.89,70.6111 1644.91,70.451 1646.94,70.292 1648.96,70.134 1650.99,69.9771 \n",
       "  1653.01,69.8213 1655.04,69.6666 1657.06,69.5129 1659.09,69.3602 1661.11,69.2086 1663.14,69.058 1665.16,68.9085 1667.19,68.7599 1669.21,68.6124 1671.24,68.4659 \n",
       "  1673.26,68.3203 1675.29,68.1758 1677.31,68.0322 1679.34,67.8897 1681.36,67.7481 1683.39,67.6074 1685.41,67.4677 1687.44,67.329 1689.46,67.1912 1691.49,67.0544 \n",
       "  1693.51,66.9184 1695.54,66.7835 1697.56,66.6494 1699.59,66.5162 1701.61,66.384 1703.64,66.2526 1705.66,66.1222 1707.69,65.9926 1709.71,65.864 1711.74,65.7362 \n",
       "  1713.76,65.6092 1715.79,65.4832 1717.81,65.358 1719.84,65.2336 1721.86,65.1101 1723.89,64.9875 1725.91,64.8656 1727.94,64.7446 1729.96,64.6245 1731.99,64.5051 \n",
       "  1734.01,64.3866 1736.04,64.2689 1738.06,64.152 1740.09,64.0359 1742.11,63.9205 1744.14,63.806 1746.16,63.6922 1748.19,63.5793 1750.21,63.467 1752.24,63.3556 \n",
       "  1754.26,63.2449 1756.29,63.135 1758.31,63.0258 1760.34,62.9174 1762.36,62.8097 1764.39,62.7028 1766.41,62.5966 1768.44,62.4911 1770.46,62.3863 1772.49,62.2822 \n",
       "  1774.51,62.1789 1776.54,62.0763 1778.56,61.9743 1780.59,61.8731 1782.61,61.7725 1784.64,61.6727 1786.66,61.5735 1788.69,61.475 1790.71,61.3772 1792.74,61.28 \n",
       "  1794.76,61.1835 1796.79,61.0877 1798.81,60.9925 1800.84,60.898 1802.86,60.8041 1804.89,60.7109 1806.91,60.6183 1808.94,60.5264 1810.96,60.435 1812.99,60.3443 \n",
       "  1815.01,60.2542 1817.04,60.1648 1819.06,60.0759 1821.09,59.9877 1823.11,59.9 1825.14,59.813 1827.16,59.7265 1829.19,59.6407 1831.21,59.5554 1833.24,59.4707 \n",
       "  1835.26,59.3866 1837.29,59.3031 1839.31,59.2201 1841.34,59.1377 1843.36,59.0559 1845.39,58.9747 1847.41,58.894 1849.44,58.8138 1851.46,58.7342 1853.49,58.6551 \n",
       "  1855.51,58.5766 1857.54,58.4987 1859.56,58.4212 1861.59,58.3443 1863.61,58.2679 1865.64,58.1921 1867.66,58.1167 1869.69,58.0419 1871.71,57.9676 1873.74,57.8938 \n",
       "  1875.76,57.8205 1877.79,57.7477 1879.81,57.6754 1881.84,57.6036 1883.86,57.5323 1885.89,57.4615 1887.91,57.3911 1889.94,57.3213 1891.96,57.2519 1893.99,57.183 \n",
       "  1896.01,57.1146 1898.04,57.0467 1900.06,56.9792 1902.09,56.9122 1904.11,56.8456 1906.14,56.7795 1908.16,56.7138 1910.19,56.6486 1912.21,56.5839 1914.24,56.5196 \n",
       "  1916.26,56.4557 1918.29,56.3923 1920.31,56.3293 1922.34,56.2667 1924.36,56.2046 1926.39,56.1429 1928.41,56.0816 1930.44,56.0207 1932.46,55.9603 1934.49,55.9003 \n",
       "  1936.51,55.8406 1938.54,55.7814 1940.56,55.7226 1942.59,55.6642 1944.61,55.6063 1946.64,55.5487 1948.66,55.4915 1950.69,55.4346 1952.71,55.3782 1954.74,55.3222 \n",
       "  1956.76,55.2665 1958.79,55.2113 1960.81,55.1564 1962.84,55.1019 1964.86,55.0478 1966.89,54.994 1968.91,54.9406 1970.94,54.8876 1972.96,54.8349 1974.99,54.7826 \n",
       "  1977.01,54.7307 1979.04,54.6791 1981.06,54.6279 1983.09,54.577 1985.11,54.5265 1987.14,54.4763 1989.16,54.4265 1991.19,54.377 1993.21,54.3278 1995.24,54.279 \n",
       "  1997.26,54.2305 1999.29,54.1824 2001.31,54.1346 2003.34,54.0871 2005.36,54.0399 2007.39,53.9931 2009.41,53.9466 2011.44,53.9004 2013.46,53.8545 2015.49,53.809 \n",
       "  2017.51,53.7637 2019.54,53.7188 2021.56,53.6742 2023.59,53.6298 2025.61,53.5858 2027.64,53.5421 2029.66,53.4987 2031.69,53.4556 2033.71,53.4128 2035.74,53.3703 \n",
       "  2037.76,53.328 2039.79,53.2861 2041.81,53.2444 2043.84,53.2031 2045.86,53.162 2047.89,53.1212 2049.91,53.0807 2051.94,53.0405 2053.96,53.0005 2055.99,52.9608 \n",
       "  2058.01,52.9214 2060.04,52.8823 2062.06,52.8434 2064.09,52.8048 2066.11,52.7665 2068.14,52.7284 2070.16,52.6906 2072.19,52.653 2074.21,52.6157 2076.24,52.5787 \n",
       "  2078.26,52.5419 2080.29,52.5054 2082.31,52.4691 2084.34,52.4331 2086.36,52.3973 2088.39,52.3618 2090.41,52.3265 2092.44,52.2914 2094.46,52.2566 2096.49,52.2221 \n",
       "  2098.51,52.1877 2100.54,52.1536 2102.56,52.1198 2104.59,52.0862 2106.61,52.0528 2108.64,52.0196 2110.66,51.9867 2112.69,51.954 2114.71,51.9215 2116.74,51.8892 \n",
       "  2118.76,51.8572 2120.79,51.8254 2122.81,51.7938 2124.84,51.7624 2126.86,51.7312 2128.89,51.7003 2130.91,51.6696 2132.94,51.639 2134.96,51.6087 2136.99,51.5786 \n",
       "  2139.01,51.5487 2141.04,51.519 2143.06,51.4895 2145.09,51.4603 2147.11,51.4312 2149.14,51.4023 2151.16,51.3736 2153.19,51.3451 2155.21,51.3168 2157.24,51.2888 \n",
       "  2159.26,51.2609 2161.29,51.2331 2163.31,51.2056 2165.34,51.1783 2167.36,51.1512 2169.39,51.1242 2171.41,51.0974 2173.44,51.0709 2175.46,51.0445 2177.49,51.0182 \n",
       "  2179.51,50.9922 2181.54,50.9664 2183.56,50.9407 2185.59,50.9152 2187.61,50.8898 2189.64,50.8647 2191.66,50.8397 2193.69,50.8149 2195.71,50.7903 2197.74,50.7658 \n",
       "  2199.76,50.7415 2201.79,50.7174 2203.81,50.6934 2205.84,50.6696 2207.86,50.646 2209.89,50.6225 2211.91,50.5992 2213.94,50.5761 2215.96,50.5531 2217.99,50.5302 \n",
       "  2220.01,50.5076 2222.04,50.485 2224.06,50.4627 2226.09,50.4405 2228.11,50.4184 2230.14,50.3965 2232.16,50.3748 2234.19,50.3532 2236.21,50.3317 2238.24,50.3104 \n",
       "  2240.26,50.2892 2242.29,50.2682 2244.31,50.2474 2246.34,50.2266 2248.36,50.2061 2250.39,50.1856 2252.41,50.1653 2254.44,50.1452 2256.46,50.1251 2258.49,50.1053 \n",
       "  2260.51,50.0855 \n",
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       "  255.775,1402.7 257.8,1392.66 259.825,1382.63 261.85,1372.61 263.875,1362.6 265.9,1352.61 267.925,1342.63 269.95,1332.67 271.975,1322.72 273.999,1312.79 \n",
       "  276.024,1302.88 278.049,1292.99 280.074,1283.12 282.099,1273.26 284.124,1263.43 286.149,1253.63 288.174,1243.84 290.199,1234.08 292.224,1224.34 294.249,1214.63 \n",
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       "  316.524,1109.76 318.549,1100.42 320.574,1091.12 322.599,1081.85 324.624,1072.62 326.649,1063.43 328.674,1054.28 330.699,1045.17 332.724,1036.09 334.749,1027.06 \n",
       "  336.774,1018.06 338.799,1009.11 340.824,1000.2 342.849,991.337 344.874,982.513 346.899,973.733 348.924,964.997 350.949,956.306 352.974,947.66 354.999,939.059 \n",
       "  357.024,930.505 359.049,921.997 361.074,913.536 363.099,905.123 365.124,896.757 367.149,888.439 369.174,880.17 371.199,871.95 373.224,863.78 375.249,855.658 \n",
       "  377.274,847.587 379.299,839.566 381.324,831.596 383.349,823.676 385.374,815.807 387.399,807.99 389.424,800.225 391.449,792.511 393.474,784.849 395.499,777.239 \n",
       "  397.524,769.682 399.549,762.178 401.574,754.726 403.599,747.327 405.624,739.981 407.649,732.689 409.674,725.45 411.699,718.264 413.723,711.131 415.748,704.053 \n",
       "  417.773,697.028 419.798,690.056 421.823,683.139 423.848,676.275 425.873,669.465 427.898,662.709 429.923,656.007 431.948,649.358 433.973,642.764 435.998,636.223 \n",
       "  438.023,629.736 440.048,623.303 442.073,616.924 444.098,610.598 446.123,604.325 448.148,598.107 450.173,591.941 452.198,585.829 454.223,579.77 456.248,573.764 \n",
       "  458.273,567.81 460.298,561.91 462.323,556.062 464.348,550.267 466.373,544.524 468.398,538.834 470.423,533.195 472.448,527.608 474.473,522.073 476.498,516.59 \n",
       "  478.523,511.157 480.548,505.776 482.573,500.445 484.598,495.166 486.623,489.936 488.648,484.757 490.673,479.628 492.698,474.549 494.723,469.519 496.748,464.538 \n",
       "  498.773,459.607 500.798,454.724 502.823,449.889 504.848,445.103 506.873,440.365 508.898,435.675 510.923,431.032 512.948,426.436 514.973,421.887 516.998,417.385 \n",
       "  519.023,412.929 521.048,408.519 523.073,404.155 525.098,399.836 527.123,395.562 529.148,391.333 531.173,387.149 533.198,383.009 535.223,378.913 537.248,374.86 \n",
       "  539.273,370.851 541.298,366.885 543.323,362.961 545.348,359.08 547.373,355.241 549.398,351.443 551.423,347.687 553.447,343.973 555.472,340.298 557.497,336.665 \n",
       "  559.522,333.071 561.547,329.518 563.572,326.003 565.597,322.528 567.622,319.092 569.647,315.695 571.672,312.335 573.697,309.014 575.722,305.73 577.747,302.483 \n",
       "  579.772,299.273 581.797,296.1 583.822,292.963 585.847,289.862 587.872,286.796 589.897,283.766 591.922,280.771 593.947,277.81 595.972,274.884 597.997,271.992 \n",
       "  600.022,269.134 602.047,266.308 604.072,263.517 606.097,260.757 608.122,258.031 610.147,255.336 612.172,252.673 614.197,250.042 616.222,247.442 618.247,244.873 \n",
       "  620.272,242.334 622.297,239.826 624.322,237.348 626.347,234.899 628.372,232.48 630.397,230.09 632.422,227.728 634.447,225.396 636.472,223.091 638.497,220.814 \n",
       "  640.522,218.565 642.547,216.344 644.572,214.149 646.597,211.981 648.622,209.84 650.647,207.725 652.672,205.635 654.697,203.572 656.722,201.534 658.747,199.521 \n",
       "  660.772,197.532 662.797,195.569 664.822,193.629 666.847,191.714 668.872,189.822 670.897,187.954 672.922,186.109 674.947,184.288 676.972,182.488 678.997,180.712 \n",
       "  681.022,178.957 683.047,177.225 685.072,175.514 687.097,173.825 689.122,172.157 691.147,170.51 693.171,168.884 695.196,167.278 697.221,165.693 699.246,164.128 \n",
       "  701.271,162.582 703.296,161.057 705.321,159.55 707.346,158.063 709.371,156.594 711.396,155.145 713.421,153.713 715.446,152.301 717.471,150.906 719.496,149.529 \n",
       "  721.521,148.169 723.546,146.827 725.571,145.503 727.596,144.195 729.621,142.904 731.646,141.63 733.671,140.372 735.696,139.13 737.721,137.904 739.746,136.695 \n",
       "  741.771,135.5 743.796,134.322 745.821,133.158 747.846,132.01 749.871,130.876 751.896,129.758 753.921,128.653 755.946,127.564 757.971,126.488 759.996,125.426 \n",
       "  762.021,124.378 764.046,123.344 766.071,122.323 768.096,121.316 770.121,120.322 772.146,119.341 774.171,118.372 776.196,117.417 778.221,116.474 780.246,115.543 \n",
       "  782.271,114.624 784.296,113.718 786.321,112.823 788.346,111.941 790.371,111.069 792.396,110.21 794.421,109.361 796.446,108.524 798.471,107.698 800.496,106.883 \n",
       "  802.521,106.078 804.546,105.284 806.571,104.501 808.596,103.728 810.621,102.965 812.646,102.213 814.671,101.47 816.696,100.737 818.721,100.014 820.746,99.3005 \n",
       "  822.771,98.5965 824.796,97.9018 826.821,97.2163 828.846,96.54 830.871,95.8726 832.895,95.2142 834.92,94.5645 836.945,93.9235 838.97,93.291 840.995,92.6669 \n",
       "  843.02,92.0512 845.045,91.4437 847.07,90.8443 849.095,90.2529 851.12,89.6694 853.145,89.0937 855.17,88.5257 857.195,87.9653 859.22,87.4124 861.245,86.867 \n",
       "  863.27,86.3288 865.295,85.7978 867.32,85.274 869.345,84.7571 871.37,84.2472 873.395,83.7442 875.42,83.2479 877.445,82.7583 879.47,82.2752 881.495,81.7987 \n",
       "  883.52,81.3285 885.545,80.8647 887.57,80.4071 889.595,79.9556 891.62,79.5103 893.645,79.0709 895.67,78.6375 897.695,78.2099 899.72,77.788 901.745,77.3719 \n",
       "  903.77,76.9613 905.795,76.5563 907.82,76.1568 909.845,75.7626 911.87,75.3738 913.895,74.9902 915.92,74.6118 917.945,74.2385 919.97,73.8703 921.995,73.507 \n",
       "  924.02,73.1486 926.045,72.7951 928.07,72.4464 930.095,72.1024 932.12,71.7631 934.145,71.4283 936.17,71.0981 938.195,70.7723 940.22,70.451 942.245,70.134 \n",
       "  944.27,69.8213 946.295,69.5129 948.32,69.2086 950.345,68.9085 952.37,68.6124 954.395,68.3203 956.42,68.0322 958.445,67.7481 960.47,67.4677 962.495,67.1912 \n",
       "  964.52,66.9184 966.545,66.6494 968.57,66.384 970.595,66.1222 972.619,65.864 974.644,65.6092 976.669,65.358 978.694,65.1101 980.719,64.8656 982.744,64.6245 \n",
       "  984.769,64.3866 986.794,64.152 988.819,63.9205 990.844,63.6922 992.869,63.467 994.894,63.2449 996.919,63.0258 998.944,62.8097 1000.97,62.5966 1002.99,62.3863 \n",
       "  1005.02,62.1789 1007.04,61.9743 1009.07,61.7725 1011.09,61.5735 1013.12,61.3772 1015.14,61.1835 1017.17,60.9925 1019.19,60.8041 1021.22,60.6183 1023.24,60.435 \n",
       "  1025.27,60.2542 1027.29,60.0759 1029.32,59.9 1031.34,59.7265 1033.37,59.5554 1035.39,59.3866 1037.42,59.2201 1039.44,59.0559 1041.47,58.894 1043.49,58.7342 \n",
       "  1045.52,58.5766 1047.54,58.4212 1049.57,58.2679 1051.59,58.1167 1053.62,57.9676 1055.64,57.8205 1057.67,57.6754 1059.69,57.5323 1061.72,57.3911 1063.74,57.2519 \n",
       "  1065.77,57.1146 1067.79,56.9792 1069.82,56.8456 1071.84,56.7138 1073.87,56.5839 1075.89,56.4557 1077.92,56.3293 1079.94,56.2046 1081.97,56.0816 1083.99,55.9603 \n",
       "  1086.02,55.8406 1088.04,55.7226 1090.07,55.6063 1092.09,55.4915 1094.12,55.3782 1096.14,55.2665 1098.17,55.1564 1100.19,55.0478 1102.22,54.9406 1104.24,54.8349 \n",
       "  1106.27,54.7307 1108.29,54.6279 1110.32,54.5265 1112.34,54.4265 1114.37,54.3278 1116.39,54.2305 1118.42,54.1346 1120.44,54.0399 1122.47,53.9466 1124.49,53.8545 \n",
       "  1126.52,53.7637 1128.54,53.6742 1130.57,53.5858 1132.59,53.4987 1134.62,53.4128 1136.64,53.328 1138.67,53.2444 1140.69,53.162 1142.72,53.0807 1144.74,53.0005 \n",
       "  1146.77,52.9214 1148.79,52.8434 1150.82,52.7665 1152.84,52.6906 1154.87,52.6157 1156.89,52.5419 1158.92,52.4691 1160.94,52.3973 1162.97,52.3265 1164.99,52.2566 \n",
       "  1167.02,52.1877 1169.04,52.1198 1171.07,52.0528 1173.09,51.9867 1175.12,51.9215 1177.14,51.8572 1179.17,51.7938 1181.19,51.7312 1183.22,51.6696 1185.24,51.6087 \n",
       "  1187.27,51.5487 1189.29,51.4895 1191.32,51.4312 1193.34,51.3736 1195.37,51.3168 1197.39,51.2609 1199.42,51.2056 1201.44,51.1512 1203.47,51.0974 1205.49,51.0445 \n",
       "  1207.52,50.9922 1209.54,50.9407 1211.57,50.8898 1213.59,50.8397 1215.62,50.7903 1217.64,50.7415 1219.67,50.6934 1221.69,50.646 1223.72,50.5992 1225.74,50.5531 \n",
       "  1227.77,50.5076 1229.79,50.4627 1231.82,50.4184 1233.84,50.3748 1235.87,50.3317 1237.89,50.2892 1239.92,50.2474 1241.94,50.2061 1243.97,50.1653 1245.99,50.1251 \n",
       "  1248.02,50.0855 1250.04,50.0464 1252.07,50.0079 1254.09,49.9699 1256.12,49.9324 1258.14,49.8954 1260.17,49.8589 1262.19,49.823 1264.22,49.7875 1266.24,49.7525 \n",
       "  1268.27,49.718 1270.29,49.684 1272.32,49.6504 1274.34,49.6173 1276.37,49.5847 1278.39,49.5525 1280.42,49.5207 1282.44,49.4894 1284.47,49.4585 1286.49,49.428 \n",
       "  1288.52,49.398 1290.54,49.3684 1292.57,49.3391 1294.59,49.3103 1296.62,49.2819 1298.64,49.2539 1300.67,49.2262 1302.69,49.1989 1304.72,49.172 1306.74,49.1455 \n",
       "  1308.77,49.1194 1310.79,49.0936 1312.82,49.0681 1314.84,49.043 1316.87,49.0183 1318.89,48.9939 1320.92,48.9698 1322.94,48.946 1324.97,48.9226 1326.99,48.8995 \n",
       "  1329.02,48.8767 1331.04,48.8543 1333.07,48.8321 1335.09,48.8103 1337.12,48.7887 1339.14,48.7675 1341.17,48.7465 1343.19,48.7258 1345.22,48.7054 1347.24,48.6853 \n",
       "  1349.27,48.6655 1351.29,48.6459 1353.32,48.6267 1355.34,48.6076 1357.37,48.5889 1359.39,48.5704 1361.42,48.5521 1363.44,48.5341 1365.47,48.5164 1367.49,48.4988 \n",
       "  1369.52,48.4816 1371.54,48.4645 1373.57,48.4478 1375.59,48.4312 1377.62,48.4149 1379.64,48.3987 1381.67,48.3829 1383.69,48.3672 1385.72,48.3517 1387.74,48.3365 \n",
       "  1389.77,48.3214 1391.79,48.3066 1393.82,48.292 1395.84,48.2776 1397.87,48.2634 1399.89,48.2493 1401.92,48.2355 1403.94,48.2218 1405.97,48.2084 1407.99,48.1951 \n",
       "  1410.02,48.182 1412.04,48.1691 1414.07,48.1564 1416.09,48.1438 1418.12,48.1314 1420.14,48.1192 1422.17,48.1072 1424.19,48.0953 1426.22,48.0836 1428.24,48.072 \n",
       "  1430.27,48.0606 1432.29,48.0494 1434.32,48.0383 1436.34,48.0274 1438.37,48.0166 1440.39,48.006 1442.42,47.9955 1444.44,47.9852 1446.47,47.975 1448.49,47.9649 \n",
       "  1450.52,47.955 1452.54,47.9452 1454.57,47.9355 1456.59,47.926 1458.62,47.9166 1460.64,47.9074 1462.67,47.8982 1464.69,47.8892 1466.72,47.8804 1468.74,47.8716 \n",
       "  1470.77,47.863 1472.79,47.8544 1474.82,47.846 1476.84,47.8378 1478.87,47.8296 1480.89,47.8215 1482.92,47.8136 1484.94,47.8057 1486.97,47.798 1488.99,47.7904 \n",
       "  1491.02,47.7829 1493.04,47.7755 1495.07,47.7681 1497.09,47.7609 1499.12,47.7538 1501.14,47.7468 1503.17,47.7399 1505.19,47.7331 1507.22,47.7263 1509.24,47.7197 \n",
       "  1511.27,47.7131 1513.29,47.7067 1515.32,47.7003 1517.34,47.694 1519.37,47.6878 1521.39,47.6817 1523.42,47.6757 1525.44,47.6698 1527.47,47.6639 1529.49,47.6581 \n",
       "  1531.52,47.6524 1533.54,47.6468 1535.57,47.6413 1537.59,47.6358 1539.62,47.6304 1541.64,47.6251 1543.67,47.6198 1545.69,47.6147 1547.72,47.6096 1549.74,47.6045 \n",
       "  1551.77,47.5996 1553.79,47.5947 1555.82,47.5899 1557.84,47.5851 1559.87,47.5804 1561.89,47.5758 1563.92,47.5712 1565.94,47.5667 1567.97,47.5623 1569.99,47.5579 \n",
       "  1572.02,47.5536 1574.04,47.5493 1576.07,47.5451 1578.09,47.541 1580.12,47.5369 1582.14,47.5328 1584.17,47.5289 1586.19,47.5249 1588.22,47.5211 1590.24,47.5173 \n",
       "  1592.27,47.5135 1594.29,47.5098 1596.32,47.5061 1598.34,47.5025 1600.37,47.499 1602.39,47.4955 1604.41,47.492 1606.44,47.4886 1608.46,47.4852 1610.49,47.4819 \n",
       "  1612.51,47.4786 1614.54,47.4754 1616.56,47.4722 1618.59,47.4691 1620.61,47.466 1622.64,47.4629 1624.66,47.4599 1626.69,47.4569 1628.71,47.454 1630.74,47.4511 \n",
       "  1632.76,47.4483 1634.79,47.4455 1636.81,47.4427 1638.84,47.44 1640.86,47.4373 1642.89,47.4346 1644.91,47.432 1646.94,47.4294 1648.96,47.4268 1650.99,47.4243 \n",
       "  1653.01,47.4218 1655.04,47.4194 1657.06,47.417 1659.09,47.4146 1661.11,47.4123 1663.14,47.4099 1665.16,47.4077 1667.19,47.4054 1669.21,47.4032 1671.24,47.401 \n",
       "  1673.26,47.3988 1675.29,47.3967 1677.31,47.3946 1679.34,47.3925 1681.36,47.3905 1683.39,47.3885 1685.41,47.3865 1687.44,47.3845 1689.46,47.3826 1691.49,47.3807 \n",
       "  1693.51,47.3788 1695.54,47.377 1697.56,47.3751 1699.59,47.3733 1701.61,47.3715 1703.64,47.3698 1705.66,47.3681 1707.69,47.3663 1709.71,47.3647 1711.74,47.363 \n",
       "  1713.76,47.3614 1715.79,47.3598 1717.81,47.3582 1719.84,47.3566 1721.86,47.355 1723.89,47.3535 1725.91,47.352 1727.94,47.3505 1729.96,47.3491 1731.99,47.3476 \n",
       "  1734.01,47.3462 1736.04,47.3448 1738.06,47.3434 1740.09,47.342 1742.11,47.3407 1744.14,47.3394 1746.16,47.338 1748.19,47.3367 1750.21,47.3355 1752.24,47.3342 \n",
       "  1754.26,47.333 1756.29,47.3317 1758.31,47.3305 1760.34,47.3294 1762.36,47.3282 1764.39,47.327 1766.41,47.3259 1768.44,47.3248 1770.46,47.3236 1772.49,47.3225 \n",
       "  1774.51,47.3215 1776.54,47.3204 1778.56,47.3194 1780.59,47.3183 1782.61,47.3173 1784.64,47.3163 1786.66,47.3153 1788.69,47.3143 1790.71,47.3133 1792.74,47.3124 \n",
       "  1794.76,47.3115 1796.79,47.3105 1798.81,47.3096 1800.84,47.3087 1802.86,47.3078 1804.89,47.3069 1806.91,47.3061 1808.94,47.3052 1810.96,47.3044 1812.99,47.3036 \n",
       "  1815.01,47.3027 1817.04,47.3019 1819.06,47.3011 1821.09,47.3003 1823.11,47.2996 1825.14,47.2988 1827.16,47.2981 1829.19,47.2973 1831.21,47.2966 1833.24,47.2959 \n",
       "  1835.26,47.2951 1837.29,47.2944 1839.31,47.2937 1841.34,47.2931 1843.36,47.2924 1845.39,47.2917 1847.41,47.2911 1849.44,47.2904 1851.46,47.2898 1853.49,47.2892 \n",
       "  1855.51,47.2885 1857.54,47.2879 1859.56,47.2873 1861.59,47.2867 1863.61,47.2861 1865.64,47.2856 1867.66,47.285 1869.69,47.2844 1871.71,47.2839 1873.74,47.2833 \n",
       "  1875.76,47.2828 1877.79,47.2822 1879.81,47.2817 1881.84,47.2812 1883.86,47.2807 1885.89,47.2802 1887.91,47.2797 1889.94,47.2792 1891.96,47.2787 1893.99,47.2782 \n",
       "  1896.01,47.2778 1898.04,47.2773 1900.06,47.2769 1902.09,47.2764 1904.11,47.276 1906.14,47.2755 1908.16,47.2751 1910.19,47.2747 1912.21,47.2742 1914.24,47.2738 \n",
       "  1916.26,47.2734 1918.29,47.273 1920.31,47.2726 1922.34,47.2722 1924.36,47.2718 1926.39,47.2715 1928.41,47.2711 1930.44,47.2707 1932.46,47.2703 1934.49,47.27 \n",
       "  1936.51,47.2696 1938.54,47.2693 1940.56,47.2689 1942.59,47.2686 1944.61,47.2682 1946.64,47.2679 1948.66,47.2676 1950.69,47.2673 1952.71,47.2669 1954.74,47.2666 \n",
       "  1956.76,47.2663 1958.79,47.266 1960.81,47.2657 1962.84,47.2654 1964.86,47.2651 1966.89,47.2648 1968.91,47.2645 1970.94,47.2643 1972.96,47.264 1974.99,47.2637 \n",
       "  1977.01,47.2634 1979.04,47.2632 1981.06,47.2629 1983.09,47.2627 1985.11,47.2624 1987.14,47.2621 1989.16,47.2619 1991.19,47.2616 1993.21,47.2614 1995.24,47.2612 \n",
       "  1997.26,47.2609 1999.29,47.2607 2001.31,47.2605 2003.34,47.2602 2005.36,47.26 2007.39,47.2598 2009.41,47.2596 2011.44,47.2594 2013.46,47.2592 2015.49,47.259 \n",
       "  2017.51,47.2588 2019.54,47.2586 2021.56,47.2584 2023.59,47.2582 2025.61,47.258 2027.64,47.2578 2029.66,47.2576 2031.69,47.2574 2033.71,47.2572 2035.74,47.257 \n",
       "  2037.76,47.2569 2039.79,47.2567 2041.81,47.2565 2043.84,47.2563 2045.86,47.2562 2047.89,47.256 2049.91,47.2558 2051.94,47.2557 2053.96,47.2555 2055.99,47.2554 \n",
       "  2058.01,47.2552 2060.04,47.2551 2062.06,47.2549 2064.09,47.2548 2066.11,47.2546 2068.14,47.2545 2070.16,47.2543 2072.19,47.2542 2074.21,47.254 2076.24,47.2539 \n",
       "  2078.26,47.2538 2080.29,47.2536 2082.31,47.2535 2084.34,47.2534 2086.36,47.2532 2088.39,47.2531 2090.41,47.253 2092.44,47.2529 2094.46,47.2528 2096.49,47.2526 \n",
       "  2098.51,47.2525 2100.54,47.2524 2102.56,47.2523 2104.59,47.2522 2106.61,47.2521 2108.64,47.252 2110.66,47.2518 2112.69,47.2517 2114.71,47.2516 2116.74,47.2515 \n",
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   "source": [
    "N₀=200\n",
    "t=collect(0:0.1:100)\n",
    "T=10\n",
    "μ=log(2)/T\n",
    "K=400\n",
    "plot(t,t->N₀*K/(N₀ +(K-N₀)*exp(-μ*t)),label=\"T=10\",seriestype=:line,ylims = (200,400))\n",
    "T=5\n",
    "μ=log(2)/T\n",
    "plot!(t,t->N₀*K/(N₀ +(K-N₀)*exp(-μ*t)),label=\"T=5\",seriestype=:line,ylims = (200,400))\n",
    "T=15\n",
    "μ=log(2)/T\n",
    "plot!(t,t->N₀*K/(N₀ +(K-N₀)*exp(-μ*t)),label=\"T=15\",seriestype=:line,ylims = (200,400))"
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   "source": [
    "# Differentiation dynamics\n",
    "\n",
    "We assume a developing organ as a population of cycling progenitors 'P' that cycle with an average cell cycle $T$. Some of these cells terminally differentiate, exit the cell cycle and acquire a given specialized phenotype 'D'. We start from a initial population of progenitors $P_0$ and differentiated $D_0$ cells. A common approac is top characterize the dynamics of the population focusing on the outcome of the cell division of the progenitors. In principle, each division of a 'P' cell can give two progenitors ('pp' division), two differentiated cells ('dd' division) and also an assymetric mode of divisin where a progenitor and a differentiated cell is generated ('pd' division). If we calculate the average amount of 'P' and 'D' generated after a single cell cycle (n=1) we can write the number of progenitor and differentiated cells as:\n",
    "\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "P_1&=&P_{0}(2pp+pd) \\tag{35}\\\\\n",
    "D_1&=&D_{0}+P_{0}(2dd+pd)\\tag{36}\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "where using the condition $pp+pd+dd=1$, \n",
    "\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "P_1&=&P_{0} (1+pp-dd)\\tag{37}\\\\\n",
    "D_1&=&D_{0}+P_{0}(1+dd-pp)\\tag{38}\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "Therfore, for n=2,\n",
    "\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "P_2&=&P_{1} (1+pp-dd)\\tag{39}\\\\\n",
    "D_2&=&D_{1}+P_{1}(1+dd-pp)\\tag{40}\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "applying eqs. 37 and 38, we obtain\n",
    "\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "P_2&=&P_{0}(1+pp-dd)(1+pp-dd)=P_{0} (1+pp-dd)^2\\tag{41}\\\\\n",
    "D_2&=&D_{0}+P_{0}(1+dd-pp)+P_{0}(1+pp-dd)(1+dd-pp)\\tag{42}\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "and rearranging terms in eq. 42:\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "D_2&=&D_{0}+P_{0}(1+dd-pp)(1+(1+pp-dd))\\tag{43}\\\\\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "Subsequently, for n=3\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "P_3&=&P_{2} (1+pp-dd)\\tag{44}\\\\\n",
    "D_3&=&D_{2}+P_{2}(1+dd-pp) \\tag{45}\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "\n",
    "applying eqs. 41 and 42, we obtain\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "P_3&=&P_{0} (1+pp-dd)^2 (1+pp-dd)= P_{0} (1+pp-dd)^3 \\tag{46}\\\\\n",
    "D_3&=&D_{0}+P_{0}(1+dd-pp)(1+(1+pp-dd) + (1+pp-dd)^2) \\tag{47}\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "therefore, for $n$ steps, we obtain, \n",
    "\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "P_n&=&P_{0} (1+pp-dd)^n \\tag{48}\\\\\n",
    "D_n&=&D_{0}+P_{0}(1+dd-pp)(1+(1+pp-dd)+(1+pp-dd)^2+...+(1+pp-dd)^{n-1})\\tag{49}\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "where the second term in eq. 49 can be written as\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "1+(1+pp-dd)+(1+pp-dd)^2+...+(pp-dd)^{n-1}=\\displaystyle\\sum_{i=0}^{n-1} (1+pp-dd)^i\\tag{50}\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "which renaming $r=1+pp-dd$ is equivalent to \n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "\\displaystyle\\sum_{i=0}^{n-1} r^i=\\frac{1-r^n}{1-r}\\tag{51}\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "therefore, eq. 50 can be written as\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "D_n&=&D_{0}+P_{0}(1+dd-pp)\\frac{1-(1+pp-dd))^n}{1-(1+pp-dd)} \\tag{51}\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "which, after simplifying terms, can be rewritten as\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "D_n&=&D_{0}+P_{0}(1-(1+pp-dd)^n)\\frac{1+dd-pp}{dd-pp}\\tag{52}\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "and taking into account eq. 48, we obtain the final equation for the number of progenitors 'P' and differentiated 'D' cells in a stem cell population that is growing and differentiating:\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "D_n&=&D_{0}+(P_{0}-P_{n})\\frac{1+dd-pp}{dd-pp}=D_{0}+(P_{n}-P_{0})\\frac{1+dd-pp}{pp-dd} \\tag{53}\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "Interestingly, both equations depend on the iteration step $n$ only via the number of progenitors at a given time in the system $P_n$. \n",
    "\n",
    "For simplicity, the system of equations has been derived for a situation of discrete $n=\\Delta t/T$, (n=1,2,3...), i.e., with the time step equal to the average cell cycle $\\Delta t=T$. If we instead consider the time step as half of the cell cycle ($\\Delta t=T/2$), then $n=2 \\Delta t/T$, (n=1,2,3...), and eq. 37 is now:\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "P_1&=&P_{0} (1+pp-dd)^{\\frac{1}{2}}\\tag{54}\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "and following identical iteration steps we arrive at\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "P_n&=&P_{0} (1+pp-dd)^{\\frac{n}{2}}\\tag{55}\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "while eqs. for 'D' cells remain the same, since it does not depend explicitly on the iteration step. This way, for $\\delta t=1$, and following the same process we obtain\n",
    "$$\n",
    "\n",
    "\\begin{eqnarray}\n",
    "P_n&=&P_{0} (1+pp-dd)^{\\frac{n}{T}}\\tag{56}\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "which can be generalize for the continuum limit $n=t$,\n",
    "\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "P_{t}&=&P_0 (1+pp-dd)^{\\frac{t}{T}}\\tag{57}\\\\\n",
    "D_{t}&=&D_{0}+(P_{t}-P_{0})\\frac{1+dd-pp}{pp-dd}\\tag{58}\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "Finally, f we rewrite  $\\Delta$$P=P_{t}-P_{0}$, we obtain the expression. \n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "P_{t}&=&P_0 (1+pp-dd)^{\\frac{t}{T}}\\tag{59}\\\\\n",
    "D_{t}&=&D_{0}+\\Delta P\\frac{1+dd-pp}{pp-dd}\\tag{60}\n",
    "\\end{eqnarray}\n",
    "$$"
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       "  1333.5,1408.11 1334.26,1407.74 1335.03,1407.36 1335.8,1406.98 1336.57,1406.6 1337.33,1406.22 1338.1,1405.84 1338.87,1405.46 1339.63,1405.08 1340.4,1404.7 \n",
       "  1341.17,1404.32 1341.94,1403.94 1342.7,1403.56 1343.47,1403.18 1344.24,1402.8 1345,1402.41 1345.77,1402.03 1346.54,1401.65 1347.3,1401.27 1348.07,1400.88 \n",
       "  1348.84,1400.5 1349.61,1400.11 1350.37,1399.73 1351.14,1399.35 1351.91,1398.96 1352.67,1398.58 1353.44,1398.19 1354.21,1397.81 1354.98,1397.42 1355.74,1397.03 \n",
       "  1356.51,1396.65 1357.28,1396.26 1358.04,1395.87 1358.81,1395.49 1359.58,1395.1 1360.35,1394.71 1361.11,1394.32 1361.88,1393.93 1362.65,1393.54 1363.41,1393.16 \n",
       "  1364.18,1392.77 1364.95,1392.38 1365.72,1391.99 1366.48,1391.6 1367.25,1391.21 1368.02,1390.81 1368.78,1390.42 1369.55,1390.03 1370.32,1389.64 1371.09,1389.25 \n",
       "  1371.85,1388.86 1372.62,1388.46 1373.39,1388.07 1374.15,1387.68 1374.92,1387.28 1375.69,1386.89 1376.46,1386.49 1377.22,1386.1 1377.99,1385.71 1378.76,1385.31 \n",
       "  1379.52,1384.92 1380.29,1384.52 1381.06,1384.12 1381.83,1383.73 1382.59,1383.33 1383.36,1382.93 1384.13,1382.54 1384.89,1382.14 1385.66,1381.74 1386.43,1381.34 \n",
       "  1387.2,1380.94 1387.96,1380.55 1388.73,1380.15 1389.5,1379.75 1390.26,1379.35 1391.03,1378.95 1391.8,1378.55 1392.57,1378.15 1393.33,1377.75 1394.1,1377.35 \n",
       "  1394.87,1376.94 1395.63,1376.54 1396.4,1376.14 1397.17,1375.74 1397.94,1375.34 1398.7,1374.93 1399.47,1374.53 1400.24,1374.13 1401,1373.72 1401.77,1373.32 \n",
       "  1402.54,1372.91 1403.31,1372.51 1404.07,1372.1 1404.84,1371.7 1405.61,1371.29 1406.37,1370.89 1407.14,1370.48 1407.91,1370.07 1408.67,1369.67 1409.44,1369.26 \n",
       "  1410.21,1368.85 1410.98,1368.44 1411.74,1368.04 1412.51,1367.63 1413.28,1367.22 1414.04,1366.81 1414.81,1366.4 1415.58,1365.99 1416.35,1365.58 1417.11,1365.17 \n",
       "  1417.88,1364.76 1418.65,1364.35 1419.41,1363.94 1420.18,1363.52 1420.95,1363.11 1421.72,1362.7 1422.48,1362.29 1423.25,1361.87 1424.02,1361.46 1424.78,1361.05 \n",
       "  1425.55,1360.63 1426.32,1360.22 1427.09,1359.81 1427.85,1359.39 1428.62,1358.98 1429.39,1358.56 1430.15,1358.14 1430.92,1357.73 1431.69,1357.31 1432.46,1356.9 \n",
       "  1433.22,1356.48 1433.99,1356.06 1434.76,1355.64 1435.52,1355.23 1436.29,1354.81 1437.06,1354.39 1437.83,1353.97 1438.59,1353.55 1439.36,1353.13 1440.13,1352.71 \n",
       "  1440.89,1352.29 1441.66,1351.87 1442.43,1351.45 1443.2,1351.03 1443.96,1350.61 1444.73,1350.19 1445.5,1349.76 1446.26,1349.34 1447.03,1348.92 1447.8,1348.5 \n",
       "  1448.57,1348.07 1449.33,1347.65 1450.1,1347.22 1450.87,1346.8 1451.63,1346.38 1452.4,1345.95 1453.17,1345.53 1453.94,1345.1 1454.7,1344.67 1455.47,1344.25 \n",
       "  1456.24,1343.82 1457,1343.39 1457.77,1342.97 1458.54,1342.54 1459.31,1342.11 1460.07,1341.68 1460.84,1341.26 1461.61,1340.83 1462.37,1340.4 1463.14,1339.97 \n",
       "  1463.91,1339.54 1464.67,1339.11 1465.44,1338.68 1466.21,1338.25 1466.98,1337.81 1467.74,1337.38 1468.51,1336.95 1469.28,1336.52 1470.04,1336.09 1470.81,1335.65 \n",
       "  1471.58,1335.22 1472.35,1334.79 1473.11,1334.35 1473.88,1333.92 1474.65,1333.49 1475.41,1333.05 1476.18,1332.62 1476.95,1332.18 1477.72,1331.74 1478.48,1331.31 \n",
       "  1479.25,1330.87 1480.02,1330.44 1480.78,1330 1481.55,1329.56 1482.32,1329.12 1483.09,1328.68 1483.85,1328.25 1484.62,1327.81 1485.39,1327.37 1486.15,1326.93 \n",
       "  1486.92,1326.49 1487.69,1326.05 1488.46,1325.61 1489.22,1325.17 1489.99,1324.73 1490.76,1324.29 1491.52,1323.84 1492.29,1323.4 1493.06,1322.96 1493.83,1322.52 \n",
       "  1494.59,1322.07 1495.36,1321.63 1496.13,1321.19 1496.89,1320.74 1497.66,1320.3 1498.43,1319.85 1499.2,1319.41 1499.96,1318.96 1500.73,1318.52 1501.5,1318.07 \n",
       "  1502.26,1317.62 1503.03,1317.18 1503.8,1316.73 1504.57,1316.28 1505.33,1315.83 1506.1,1315.39 1506.87,1314.94 1507.63,1314.49 1508.4,1314.04 1509.17,1313.59 \n",
       "  1509.94,1313.14 1510.7,1312.69 1511.47,1312.24 1512.24,1311.79 1513,1311.34 1513.77,1310.89 1514.54,1310.43 1515.31,1309.98 1516.07,1309.53 1516.84,1309.08 \n",
       "  1517.61,1308.62 1518.37,1308.17 1519.14,1307.72 1519.91,1307.26 1520.67,1306.81 1521.44,1306.35 1522.21,1305.9 1522.98,1305.44 1523.74,1304.98 1524.51,1304.53 \n",
       "  1525.28,1304.07 1526.04,1303.61 1526.81,1303.16 1527.58,1302.7 1528.35,1302.24 1529.11,1301.78 1529.88,1301.32 1530.65,1300.86 1531.41,1300.4 1532.18,1299.94 \n",
       "  1532.95,1299.48 1533.72,1299.02 1534.48,1298.56 1535.25,1298.1 1536.02,1297.64 1536.78,1297.18 1537.55,1296.72 1538.32,1296.25 1539.09,1295.79 1539.85,1295.33 \n",
       "  1540.62,1294.86 1541.39,1294.4 1542.15,1293.93 1542.92,1293.47 1543.69,1293 1544.46,1292.54 1545.22,1292.07 1545.99,1291.61 1546.76,1291.14 1547.52,1290.67 \n",
       "  1548.29,1290.21 1549.06,1289.74 1549.83,1289.27 1550.59,1288.8 1551.36,1288.33 1552.13,1287.86 1552.89,1287.39 1553.66,1286.92 1554.43,1286.45 1555.2,1285.98 \n",
       "  1555.96,1285.51 1556.73,1285.04 1557.5,1284.57 1558.26,1284.1 1559.03,1283.63 1559.8,1283.15 1560.57,1282.68 1561.33,1282.21 1562.1,1281.73 1562.87,1281.26 \n",
       "  1563.63,1280.78 1564.4,1280.31 1565.17,1279.83 1565.94,1279.36 1566.7,1278.88 1567.47,1278.41 1568.24,1277.93 1569,1277.45 1569.77,1276.98 1570.54,1276.5 \n",
       "  1571.31,1276.02 1572.07,1275.54 1572.84,1275.06 1573.61,1274.58 1574.37,1274.1 1575.14,1273.62 1575.91,1273.14 1576.68,1272.66 1577.44,1272.18 1578.21,1271.7 \n",
       "  1578.98,1271.22 1579.74,1270.74 1580.51,1270.25 1581.28,1269.77 1582.04,1269.29 1582.81,1268.8 1583.58,1268.32 1584.35,1267.84 1585.11,1267.35 1585.88,1266.87 \n",
       "  1586.65,1266.38 1587.41,1265.9 1588.18,1265.41 1588.95,1264.92 1589.72,1264.44 1590.48,1263.95 1591.25,1263.46 1592.02,1262.97 1592.78,1262.49 1593.55,1262 \n",
       "  1594.32,1261.51 1595.09,1261.02 1595.85,1260.53 1596.62,1260.04 1597.39,1259.55 1598.15,1259.06 1598.92,1258.57 1599.69,1258.07 1600.46,1257.58 1601.22,1257.09 \n",
       "  1601.99,1256.6 1602.76,1256.1 1603.52,1255.61 1604.29,1255.12 1605.06,1254.62 1605.83,1254.13 1606.59,1253.63 1607.36,1253.14 1608.13,1252.64 1608.89,1252.14 \n",
       "  1609.66,1251.65 1610.43,1251.15 1611.2,1250.65 1611.96,1250.16 1612.73,1249.66 1613.5,1249.16 1614.26,1248.66 1615.03,1248.16 1615.8,1247.66 1616.57,1247.16 \n",
       "  1617.33,1246.66 1618.1,1246.16 1618.87,1245.66 1619.63,1245.16 1620.4,1244.66 1621.17,1244.15 1621.94,1243.65 1622.7,1243.15 1623.47,1242.64 1624.24,1242.14 \n",
       "  1625,1241.64 1625.77,1241.13 1626.54,1240.63 1627.31,1240.12 1628.07,1239.62 1628.84,1239.11 1629.61,1238.6 1630.37,1238.1 1631.14,1237.59 1631.91,1237.08 \n",
       "  1632.68,1236.57 1633.44,1236.06 1634.21,1235.56 1634.98,1235.05 1635.74,1234.54 1636.51,1234.03 1637.28,1233.52 1638.04,1233.01 1638.81,1232.49 1639.58,1231.98 \n",
       "  1640.35,1231.47 1641.11,1230.96 1641.88,1230.45 1642.65,1229.93 1643.41,1229.42 1644.18,1228.91 1644.95,1228.39 1645.72,1227.88 1646.48,1227.36 1647.25,1226.85 \n",
       "  1648.02,1226.33 1648.78,1225.81 1649.55,1225.3 1650.32,1224.78 1651.09,1224.26 1651.85,1223.75 1652.62,1223.23 1653.39,1222.71 1654.15,1222.19 1654.92,1221.67 \n",
       "  1655.69,1221.15 1656.46,1220.63 1657.22,1220.11 1657.99,1219.59 1658.76,1219.07 1659.52,1218.55 1660.29,1218.02 1661.06,1217.5 1661.83,1216.98 1662.59,1216.46 \n",
       "  1663.36,1215.93 1664.13,1215.41 1664.89,1214.88 1665.66,1214.36 1666.43,1213.83 1667.2,1213.31 1667.96,1212.78 1668.73,1212.26 1669.5,1211.73 1670.26,1211.2 \n",
       "  1671.03,1210.67 1671.8,1210.15 1672.57,1209.62 1673.33,1209.09 1674.1,1208.56 1674.87,1208.03 1675.63,1207.5 1676.4,1206.97 1677.17,1206.44 1677.94,1205.91 \n",
       "  1678.7,1205.37 1679.47,1204.84 1680.24,1204.31 1681,1203.78 1681.77,1203.24 1682.54,1202.71 1683.31,1202.18 1684.07,1201.64 1684.84,1201.11 1685.61,1200.57 \n",
       "  1686.37,1200.03 1687.14,1199.5 1687.91,1198.96 1688.68,1198.43 1689.44,1197.89 1690.21,1197.35 1690.98,1196.81 1691.74,1196.27 1692.51,1195.73 1693.28,1195.19 \n",
       "  1694.04,1194.65 1694.81,1194.11 1695.58,1193.57 1696.35,1193.03 1697.11,1192.49 1697.88,1191.95 1698.65,1191.41 1699.41,1190.86 1700.18,1190.32 1700.95,1189.78 \n",
       "  1701.72,1189.23 1702.48,1188.69 1703.25,1188.14 1704.02,1187.6 1704.78,1187.05 1705.55,1186.51 1706.32,1185.96 1707.09,1185.41 1707.85,1184.87 1708.62,1184.32 \n",
       "  1709.39,1183.77 1710.15,1183.22 1710.92,1182.67 1711.69,1182.12 1712.46,1181.57 1713.22,1181.02 1713.99,1180.47 1714.76,1179.92 1715.52,1179.37 1716.29,1178.82 \n",
       "  1717.06,1178.27 1717.83,1177.71 1718.59,1177.16 1719.36,1176.61 1720.13,1176.05 1720.89,1175.5 1721.66,1174.94 1722.43,1174.39 1723.2,1173.83 1723.96,1173.28 \n",
       "  1724.73,1172.72 1725.5,1172.16 1726.26,1171.61 1727.03,1171.05 1727.8,1170.49 1728.57,1169.93 1729.33,1169.37 1730.1,1168.81 1730.87,1168.25 1731.63,1167.69 \n",
       "  1732.4,1167.13 1733.17,1166.57 1733.94,1166.01 1734.7,1165.45 1735.47,1164.88 1736.24,1164.32 1737,1163.76 1737.77,1163.19 1738.54,1162.63 1739.31,1162.07 \n",
       "  1740.07,1161.5 1740.84,1160.94 1741.61,1160.37 1742.37,1159.8 1743.14,1159.24 1743.91,1158.67 1744.68,1158.1 1745.44,1157.53 1746.21,1156.97 1746.98,1156.4 \n",
       "  1747.74,1155.83 1748.51,1155.26 1749.28,1154.69 1750.05,1154.12 1750.81,1153.55 1751.58,1152.97 1752.35,1152.4 1753.11,1151.83 1753.88,1151.26 1754.65,1150.68 \n",
       "  1755.41,1150.11 1756.18,1149.54 1756.95,1148.96 1757.72,1148.39 1758.48,1147.81 1759.25,1147.23 1760.02,1146.66 1760.78,1146.08 1761.55,1145.5 1762.32,1144.93 \n",
       "  1763.09,1144.35 1763.85,1143.77 1764.62,1143.19 1765.39,1142.61 1766.15,1142.03 1766.92,1141.45 1767.69,1140.87 1768.46,1140.29 1769.22,1139.71 1769.99,1139.13 \n",
       "  1770.76,1138.54 1771.52,1137.96 1772.29,1137.38 1773.06,1136.79 1773.83,1136.21 1774.59,1135.62 1775.36,1135.04 1776.13,1134.45 1776.89,1133.87 1777.66,1133.28 \n",
       "  1778.43,1132.69 1779.2,1132.11 1779.96,1131.52 1780.73,1130.93 1781.5,1130.34 1782.26,1129.75 1783.03,1129.16 1783.8,1128.57 1784.57,1127.98 1785.33,1127.39 \n",
       "  1786.1,1126.8 1786.87,1126.21 1787.63,1125.62 1788.4,1125.02 1789.17,1124.43 1789.94,1123.84 1790.7,1123.24 1791.47,1122.65 1792.24,1122.05 1793,1121.46 \n",
       "  1793.77,1120.86 1794.54,1120.27 1795.31,1119.67 1796.07,1119.07 1796.84,1118.47 1797.61,1117.88 1798.37,1117.28 1799.14,1116.68 1799.91,1116.08 1800.68,1115.48 \n",
       "  1801.44,1114.88 1802.21,1114.28 1802.98,1113.68 1803.74,1113.07 1804.51,1112.47 1805.28,1111.87 1806.05,1111.27 1806.81,1110.66 1807.58,1110.06 1808.35,1109.45 \n",
       "  1809.11,1108.85 1809.88,1108.24 1810.65,1107.64 1811.41,1107.03 1812.18,1106.42 1812.95,1105.82 1813.72,1105.21 1814.48,1104.6 1815.25,1103.99 1816.02,1103.38 \n",
       "  1816.78,1102.77 1817.55,1102.16 1818.32,1101.55 1819.09,1100.94 1819.85,1100.33 1820.62,1099.72 1821.39,1099.11 1822.15,1098.49 1822.92,1097.88 1823.69,1097.27 \n",
       "  1824.46,1096.65 1825.22,1096.04 1825.99,1095.42 1826.76,1094.81 1827.52,1094.19 1828.29,1093.57 1829.06,1092.96 1829.83,1092.34 1830.59,1091.72 1831.36,1091.1 \n",
       "  1832.13,1090.48 1832.89,1089.86 1833.66,1089.24 1834.43,1088.62 1835.2,1088 1835.96,1087.38 1836.73,1086.76 1837.5,1086.14 1838.26,1085.51 1839.03,1084.89 \n",
       "  1839.8,1084.27 1840.57,1083.64 1841.33,1083.02 1842.1,1082.39 1842.87,1081.77 1843.63,1081.14 1844.4,1080.52 1845.17,1079.89 1845.94,1079.26 1846.7,1078.63 \n",
       "  1847.47,1078 1848.24,1077.38 1849,1076.75 1849.77,1076.12 1850.54,1075.49 1851.31,1074.86 1852.07,1074.22 1852.84,1073.59 1853.61,1072.96 1854.37,1072.33 \n",
       "  1855.14,1071.69 1855.91,1071.06 1856.68,1070.43 1857.44,1069.79 1858.21,1069.16 1858.98,1068.52 1859.74,1067.88 1860.51,1067.25 1861.28,1066.61 1862.05,1065.97 \n",
       "  1862.81,1065.33 1863.58,1064.7 1864.35,1064.06 1865.11,1063.42 1865.88,1062.78 1866.65,1062.14 1867.41,1061.5 1868.18,1060.85 1868.95,1060.21 1869.72,1059.57 \n",
       "  1870.48,1058.93 1871.25,1058.28 1872.02,1057.64 1872.78,1057 1873.55,1056.35 1874.32,1055.71 1875.09,1055.06 1875.85,1054.41 1876.62,1053.77 1877.39,1053.12 \n",
       "  1878.15,1052.47 1878.92,1051.82 1879.69,1051.17 1880.46,1050.52 1881.22,1049.88 1881.99,1049.22 1882.76,1048.57 1883.52,1047.92 1884.29,1047.27 1885.06,1046.62 \n",
       "  1885.83,1045.97 1886.59,1045.31 1887.36,1044.66 1888.13,1044 1888.89,1043.35 1889.66,1042.69 1890.43,1042.04 1891.2,1041.38 1891.96,1040.73 1892.73,1040.07 \n",
       "  1893.5,1039.41 1894.26,1038.75 1895.03,1038.09 1895.8,1037.43 1896.57,1036.77 1897.33,1036.11 1898.1,1035.45 1898.87,1034.79 1899.63,1034.13 1900.4,1033.47 \n",
       "  1901.17,1032.81 1901.94,1032.14 1902.7,1031.48 1903.47,1030.81 1904.24,1030.15 1905,1029.48 1905.77,1028.82 1906.54,1028.15 1907.31,1027.49 1908.07,1026.82 \n",
       "  1908.84,1026.15 1909.61,1025.48 1910.37,1024.81 1911.14,1024.14 1911.91,1023.47 1912.68,1022.8 1913.44,1022.13 1914.21,1021.46 1914.98,1020.79 1915.74,1020.12 \n",
       "  1916.51,1019.44 1917.28,1018.77 1918.05,1018.1 1918.81,1017.42 1919.58,1016.75 1920.35,1016.07 1921.11,1015.4 1921.88,1014.72 1922.65,1014.04 1923.41,1013.37 \n",
       "  1924.18,1012.69 1924.95,1012.01 1925.72,1011.33 1926.48,1010.65 1927.25,1009.97 1928.02,1009.29 1928.78,1008.61 1929.55,1007.93 1930.32,1007.24 1931.09,1006.56 \n",
       "  1931.85,1005.88 1932.62,1005.19 1933.39,1004.51 1934.15,1003.83 1934.92,1003.14 1935.69,1002.45 1936.46,1001.77 1937.22,1001.08 1937.99,1000.39 1938.76,999.706 \n",
       "  1939.52,999.017 1940.29,998.328 1941.06,997.639 1941.83,996.949 1942.59,996.259 1943.36,995.567 1944.13,994.876 1944.89,994.184 1945.66,993.491 1946.43,992.798 \n",
       "  1947.2,992.104 1947.96,991.41 1948.73,990.715 1949.5,990.02 1950.26,989.324 1951.03,988.628 1951.8,987.931 1952.57,987.234 1953.33,986.536 1954.1,985.838 \n",
       "  1954.87,985.139 1955.63,984.439 1956.4,983.739 1957.17,983.039 1957.94,982.337 1958.7,981.636 1959.47,980.934 1960.24,980.231 1961,979.528 1961.77,978.824 \n",
       "  1962.54,978.12 1963.31,977.415 1964.07,976.709 1964.84,976.004 1965.61,975.297 1966.37,974.59 1967.14,973.883 1967.91,973.175 1968.68,972.466 1969.44,971.757 \n",
       "  1970.21,971.047 1970.98,970.337 1971.74,969.626 1972.51,968.915 1973.28,968.203 1974.05,967.491 1974.81,966.778 1975.58,966.064 1976.35,965.351 1977.11,964.636 \n",
       "  1977.88,963.921 1978.65,963.205 1979.42,962.489 1980.18,961.772 1980.95,961.055 1981.72,960.337 1982.48,959.619 1983.25,958.9 1984.02,958.181 1984.78,957.461 \n",
       "  1985.55,956.74 1986.32,956.019 1987.09,955.298 1987.85,954.575 1988.62,953.853 1989.39,953.129 1990.15,952.406 1990.92,951.681 1991.69,950.956 1992.46,950.231 \n",
       "  1993.22,949.505 1993.99,948.778 1994.76,948.051 1995.52,947.323 1996.29,946.595 1997.06,945.866 1997.83,945.137 1998.59,944.407 1999.36,943.677 2000.13,942.946 \n",
       "  2000.89,942.214 2001.66,941.482 2002.43,940.749 2003.2,940.016 2003.96,939.282 2004.73,938.548 2005.5,937.813 2006.26,937.078 2007.03,936.342 2007.8,935.605 \n",
       "  2008.57,934.868 2009.33,934.13 2010.1,933.392 2010.87,932.653 2011.63,931.914 2012.4,931.174 2013.17,930.433 2013.94,929.692 2014.7,928.951 2015.47,928.208 \n",
       "  2016.24,927.466 2017,926.722 2017.77,925.978 2018.54,925.234 2019.31,924.489 2020.07,923.743 2020.84,922.997 2021.61,922.251 2022.37,921.503 2023.14,920.755 \n",
       "  2023.91,920.007 2024.68,919.258 2025.44,918.508 2026.21,917.758 2026.98,917.008 2027.74,916.256 2028.51,915.504 2029.28,914.752 2030.05,913.999 2030.81,913.245 \n",
       "  2031.58,912.491 \n",
       "  \"/>\n",
       "<polyline clip-path=\"url(#clip1203)\" style=\"stroke:#e26f46; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
       "  1264.46,1673.28 1265.22,1671.87 1265.99,1670.45 1266.76,1669.04 1267.52,1667.62 1268.29,1666.21 1269.06,1664.79 1269.83,1663.37 1270.59,1661.95 1271.36,1660.53 \n",
       "  1272.13,1659.11 1272.89,1657.69 1273.66,1656.26 1274.43,1654.84 1275.2,1653.41 1275.96,1651.98 1276.73,1650.55 1277.5,1649.12 1278.26,1647.69 1279.03,1646.26 \n",
       "  1279.8,1644.83 1280.57,1643.4 1281.33,1641.96 1282.1,1640.52 1282.87,1639.09 1283.63,1637.65 1284.4,1636.21 1285.17,1634.77 1285.94,1633.33 1286.7,1631.89 \n",
       "  1287.47,1630.44 1288.24,1629 1289,1627.55 1289.77,1626.1 1290.54,1624.66 1291.3,1623.21 1292.07,1621.76 1292.84,1620.31 1293.61,1618.85 1294.37,1617.4 \n",
       "  1295.14,1615.94 1295.91,1614.49 1296.67,1613.03 1297.44,1611.57 1298.21,1610.11 1298.98,1608.65 1299.74,1607.19 1300.51,1605.73 1301.28,1604.27 1302.04,1602.8 \n",
       "  1302.81,1601.34 1303.58,1599.87 1304.35,1598.4 1305.11,1596.93 1305.88,1595.46 1306.65,1593.99 1307.41,1592.52 1308.18,1591.04 1308.95,1589.57 1309.72,1588.09 \n",
       "  1310.48,1586.62 1311.25,1585.14 1312.02,1583.66 1312.78,1582.18 1313.55,1580.7 1314.32,1579.21 1315.09,1577.73 1315.85,1576.25 1316.62,1574.76 1317.39,1573.27 \n",
       "  1318.15,1571.78 1318.92,1570.3 1319.69,1568.8 1320.46,1567.31 1321.22,1565.82 1321.99,1564.33 1322.76,1562.83 1323.52,1561.34 1324.29,1559.84 1325.06,1558.34 \n",
       "  1325.83,1556.84 1326.59,1555.34 1327.36,1553.84 1328.13,1552.33 1328.89,1550.83 1329.66,1549.32 1330.43,1547.82 1331.2,1546.31 1331.96,1544.8 1332.73,1543.29 \n",
       "  1333.5,1541.78 1334.26,1540.27 1335.03,1538.76 1335.8,1537.24 1336.57,1535.73 1337.33,1534.21 1338.1,1532.69 1338.87,1531.17 1339.63,1529.65 1340.4,1528.13 \n",
       "  1341.17,1526.61 1341.94,1525.08 1342.7,1523.56 1343.47,1522.03 1344.24,1520.51 1345,1518.98 1345.77,1517.45 1346.54,1515.92 1347.3,1514.38 1348.07,1512.85 \n",
       "  1348.84,1511.32 1349.61,1509.78 1350.37,1508.25 1351.14,1506.71 1351.91,1505.17 1352.67,1503.63 1353.44,1502.09 1354.21,1500.55 1354.98,1499 1355.74,1497.46 \n",
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       "  1364.18,1480.39 1364.95,1478.83 1365.72,1477.27 1366.48,1475.71 1367.25,1474.15 1368.02,1472.58 1368.78,1471.02 1369.55,1469.45 1370.32,1467.88 1371.09,1466.32 \n",
       "  1371.85,1464.75 1372.62,1463.18 1373.39,1461.6 1374.15,1460.03 1374.92,1458.46 1375.69,1456.88 1376.46,1455.3 1377.22,1453.73 1377.99,1452.15 1378.76,1450.57 \n",
       "  1379.52,1448.98 1380.29,1447.4 1381.06,1445.82 1381.83,1444.23 1382.59,1442.65 1383.36,1441.06 1384.13,1439.47 1384.89,1437.88 1385.66,1436.29 1386.43,1434.7 \n",
       "  1387.2,1433.1 1387.96,1431.51 1388.73,1429.91 1389.5,1428.32 1390.26,1426.72 1391.03,1425.12 1391.8,1423.52 1392.57,1421.91 1393.33,1420.31 1394.1,1418.71 \n",
       "  1394.87,1417.1 1395.63,1415.49 1396.4,1413.89 1397.17,1412.28 1397.94,1410.67 1398.7,1409.05 1399.47,1407.44 1400.24,1405.83 1401,1404.21 1401.77,1402.59 \n",
       "  1402.54,1400.98 1403.31,1399.36 1404.07,1397.74 1404.84,1396.12 1405.61,1394.49 1406.37,1392.87 1407.14,1391.24 1407.91,1389.62 1408.67,1387.99 1409.44,1386.36 \n",
       "  1410.21,1384.73 1410.98,1383.1 1411.74,1381.46 1412.51,1379.83 1413.28,1378.2 1414.04,1376.56 1414.81,1374.92 1415.58,1373.28 1416.35,1371.64 1417.11,1370 \n",
       "  1417.88,1368.36 1418.65,1366.71 1419.41,1365.07 1420.18,1363.42 1420.95,1361.77 1421.72,1360.13 1422.48,1358.48 1423.25,1356.82 1424.02,1355.17 1424.78,1353.52 \n",
       "  1425.55,1351.86 1426.32,1350.21 1427.09,1348.55 1427.85,1346.89 1428.62,1345.23 1429.39,1343.57 1430.15,1341.9 1430.92,1340.24 1431.69,1338.57 1432.46,1336.91 \n",
       "  1433.22,1335.24 1433.99,1333.57 1434.76,1331.9 1435.52,1330.23 1436.29,1328.56 1437.06,1326.88 1437.83,1325.21 1438.59,1323.53 1439.36,1321.85 1440.13,1320.17 \n",
       "  1440.89,1318.49 1441.66,1316.81 1442.43,1315.13 1443.2,1313.44 1443.96,1311.76 1444.73,1310.07 1445.5,1308.38 1446.26,1306.69 1447.03,1305 1447.8,1303.31 \n",
       "  1448.57,1301.61 1449.33,1299.92 1450.1,1298.22 1450.87,1296.53 1451.63,1294.83 1452.4,1293.13 1453.17,1291.43 1453.94,1289.72 1454.7,1288.02 1455.47,1286.32 \n",
       "  1456.24,1284.61 1457,1282.9 1457.77,1281.19 1458.54,1279.48 1459.31,1277.77 1460.07,1276.06 1460.84,1274.34 1461.61,1272.63 1462.37,1270.91 1463.14,1269.19 \n",
       "  1463.91,1267.47 1464.67,1265.75 1465.44,1264.03 1466.21,1262.31 1466.98,1260.58 1467.74,1258.86 1468.51,1257.13 1469.28,1255.4 1470.04,1253.67 1470.81,1251.94 \n",
       "  1471.58,1250.21 1472.35,1248.48 1473.11,1246.74 1473.88,1245 1474.65,1243.27 1475.41,1241.53 1476.18,1239.79 1476.95,1238.04 1477.72,1236.3 1478.48,1234.56 \n",
       "  1479.25,1232.81 1480.02,1231.06 1480.78,1229.32 1481.55,1227.57 1482.32,1225.82 1483.09,1224.06 1483.85,1222.31 1484.62,1220.55 1485.39,1218.8 1486.15,1217.04 \n",
       "  1486.92,1215.28 1487.69,1213.52 1488.46,1211.76 1489.22,1210 1489.99,1208.23 1490.76,1206.47 1491.52,1204.7 1492.29,1202.93 1493.06,1201.16 1493.83,1199.39 \n",
       "  1494.59,1197.62 1495.36,1195.84 1496.13,1194.07 1496.89,1192.29 1497.66,1190.52 1498.43,1188.74 1499.2,1186.96 1499.96,1185.17 1500.73,1183.39 1501.5,1181.61 \n",
       "  1502.26,1179.82 1503.03,1178.03 1503.8,1176.24 1504.57,1174.45 1505.33,1172.66 1506.1,1170.87 1506.87,1169.08 1507.63,1167.28 1508.4,1165.48 1509.17,1163.69 \n",
       "  1509.94,1161.89 1510.7,1160.09 1511.47,1158.28 1512.24,1156.48 1513,1154.67 1513.77,1152.87 1514.54,1151.06 1515.31,1149.25 1516.07,1147.44 1516.84,1145.63 \n",
       "  1517.61,1143.82 1518.37,1142 1519.14,1140.19 1519.91,1138.37 1520.67,1136.55 1521.44,1134.73 1522.21,1132.91 1522.98,1131.08 1523.74,1129.26 1524.51,1127.43 \n",
       "  1525.28,1125.61 1526.04,1123.78 1526.81,1121.95 1527.58,1120.12 1528.35,1118.28 1529.11,1116.45 1529.88,1114.62 1530.65,1112.78 1531.41,1110.94 1532.18,1109.1 \n",
       "  1532.95,1107.26 1533.72,1105.42 1534.48,1103.57 1535.25,1101.73 1536.02,1099.88 1536.78,1098.03 1537.55,1096.18 1538.32,1094.33 1539.09,1092.48 1539.85,1090.63 \n",
       "  1540.62,1088.77 1541.39,1086.92 1542.15,1085.06 1542.92,1083.2 1543.69,1081.34 1544.46,1079.48 1545.22,1077.61 1545.99,1075.75 1546.76,1073.88 1547.52,1072.01 \n",
       "  1548.29,1070.14 1549.06,1068.27 1549.83,1066.4 1550.59,1064.53 1551.36,1062.65 1552.13,1060.78 1552.89,1058.9 1553.66,1057.02 1554.43,1055.14 1555.2,1053.26 \n",
       "  1555.96,1051.37 1556.73,1049.49 1557.5,1047.6 1558.26,1045.72 1559.03,1043.83 1559.8,1041.94 1560.57,1040.04 1561.33,1038.15 1562.1,1036.25 1562.87,1034.36 \n",
       "  1563.63,1032.46 1564.4,1030.56 1565.17,1028.66 1565.94,1026.76 1566.7,1024.86 1567.47,1022.95 1568.24,1021.04 1569,1019.14 1569.77,1017.23 1570.54,1015.32 \n",
       "  1571.31,1013.4 1572.07,1011.49 1572.84,1009.57 1573.61,1007.66 1574.37,1005.74 1575.14,1003.82 1575.91,1001.9 1576.68,999.976 1577.44,998.052 1578.21,996.127 \n",
       "  1578.98,994.2 1579.74,992.272 1580.51,990.342 1581.28,988.411 1582.04,986.478 1582.81,984.544 1583.58,982.608 1584.35,980.671 1585.11,978.732 1585.88,976.792 \n",
       "  1586.65,974.851 1587.41,972.908 1588.18,970.963 1588.95,969.017 1589.72,967.07 1590.48,965.121 1591.25,963.17 1592.02,961.218 1592.78,959.265 1593.55,957.31 \n",
       "  1594.32,955.354 1595.09,953.396 1595.85,951.437 1596.62,949.476 1597.39,947.513 1598.15,945.55 1598.92,943.584 1599.69,941.618 1600.46,939.649 1601.22,937.679 \n",
       "  1601.99,935.708 1602.76,933.735 1603.52,931.761 1604.29,929.785 1605.06,927.808 1605.83,925.829 1606.59,923.849 1607.36,921.867 1608.13,919.884 1608.89,917.899 \n",
       "  1609.66,915.913 1610.43,913.925 1611.2,911.936 1611.96,909.945 1612.73,907.952 1613.5,905.958 1614.26,903.963 1615.03,901.966 1615.8,899.968 1616.57,897.968 \n",
       "  1617.33,895.966 1618.1,893.963 1618.87,891.959 1619.63,889.953 1620.4,887.945 1621.17,885.936 1621.94,883.926 1622.7,881.913 1623.47,879.9 1624.24,877.885 \n",
       "  1625,875.868 1625.77,873.85 1626.54,871.83 1627.31,869.808 1628.07,867.786 1628.84,865.761 1629.61,863.735 1630.37,861.708 1631.14,859.679 1631.91,857.648 \n",
       "  1632.68,855.616 1633.44,853.582 1634.21,851.547 1634.98,849.511 1635.74,847.472 1636.51,845.432 1637.28,843.391 1638.04,841.348 1638.81,839.304 1639.58,837.258 \n",
       "  1640.35,835.21 1641.11,833.161 1641.88,831.11 1642.65,829.058 1643.41,827.004 1644.18,824.949 1644.95,822.892 1645.72,820.833 1646.48,818.773 1647.25,816.711 \n",
       "  1648.02,814.648 1648.78,812.583 1649.55,810.517 1650.32,808.449 1651.09,806.38 1651.85,804.308 1652.62,802.236 1653.39,800.162 1654.15,798.086 1654.92,796.008 \n",
       "  1655.69,793.93 1656.46,791.849 1657.22,789.767 1657.99,787.683 1658.76,785.598 1659.52,783.511 1660.29,781.423 1661.06,779.333 1661.83,777.241 1662.59,775.148 \n",
       "  1663.36,773.053 1664.13,770.957 1664.89,768.859 1665.66,766.759 1666.43,764.658 1667.2,762.555 1667.96,760.451 1668.73,758.345 1669.5,756.237 1670.26,754.128 \n",
       "  1671.03,752.017 1671.8,749.905 1672.57,747.791 1673.33,745.675 1674.1,743.558 1674.87,741.439 1675.63,739.319 1676.4,737.197 1677.17,735.073 1677.94,732.948 \n",
       "  1678.7,730.821 1679.47,728.692 1680.24,726.562 1681,724.431 1681.77,722.297 1682.54,720.162 1683.31,718.026 1684.07,715.887 1684.84,713.748 1685.61,711.606 \n",
       "  1686.37,709.463 1687.14,707.318 1687.91,705.172 1688.68,703.024 1689.44,700.874 1690.21,698.723 1690.98,696.57 1691.74,694.416 1692.51,692.26 1693.28,690.102 \n",
       "  1694.04,687.942 1694.81,685.781 1695.58,683.619 1696.35,681.454 1697.11,679.288 1697.88,677.121 1698.65,674.951 1699.41,672.78 1700.18,670.608 1700.95,668.433 \n",
       "  1701.72,666.258 1702.48,664.08 1703.25,661.901 1704.02,659.72 1704.78,657.537 1705.55,655.353 1706.32,653.167 1707.09,650.98 1707.85,648.791 1708.62,646.6 \n",
       "  1709.39,644.407 1710.15,642.213 1710.92,640.017 1711.69,637.82 1712.46,635.621 1713.22,633.42 1713.99,631.217 1714.76,629.013 1715.52,626.807 1716.29,624.6 \n",
       "  1717.06,622.391 1717.83,620.18 1718.59,617.967 1719.36,615.753 1720.13,613.537 1720.89,611.319 1721.66,609.1 1722.43,606.879 1723.2,604.656 1723.96,602.432 \n",
       "  1724.73,600.206 1725.5,597.978 1726.26,595.749 1727.03,593.518 1727.8,591.285 1728.57,589.05 1729.33,586.814 1730.1,584.576 1730.87,582.336 1731.63,580.095 \n",
       "  1732.4,577.852 1733.17,575.607 1733.94,573.361 1734.7,571.113 1735.47,568.863 1736.24,566.611 1737,564.358 1737.77,562.103 1738.54,559.846 1739.31,557.588 \n",
       "  1740.07,555.328 1740.84,553.066 1741.61,550.802 1742.37,548.537 1743.14,546.27 1743.91,544.001 1744.68,541.731 1745.44,539.459 1746.21,537.185 1746.98,534.909 \n",
       "  1747.74,532.632 1748.51,530.353 1749.28,528.072 1750.05,525.789 1750.81,523.505 1751.58,521.219 1752.35,518.931 1753.11,516.641 1753.88,514.35 1754.65,512.057 \n",
       "  1755.41,509.762 1756.18,507.466 1756.95,505.168 1757.72,502.868 1758.48,500.566 1759.25,498.263 1760.02,495.957 1760.78,493.65 1761.55,491.342 1762.32,489.031 \n",
       "  1763.09,486.719 1763.85,484.405 1764.62,482.089 1765.39,479.772 1766.15,477.452 1766.92,475.131 1767.69,472.809 1768.46,470.484 1769.22,468.158 1769.99,465.83 \n",
       "  1770.76,463.5 1771.52,461.168 1772.29,458.835 1773.06,456.499 1773.83,454.162 1774.59,451.824 1775.36,449.483 1776.13,447.141 1776.89,444.797 1777.66,442.451 \n",
       "  1778.43,440.103 1779.2,437.754 1779.96,435.403 1780.73,433.05 1781.5,430.695 1782.26,428.338 1783.03,425.98 1783.8,423.62 1784.57,421.258 1785.33,418.894 \n",
       "  1786.1,416.529 1786.87,414.161 1787.63,411.792 1788.4,409.421 1789.17,407.048 1789.94,404.674 1790.7,402.298 1791.47,399.919 1792.24,397.539 1793,395.158 \n",
       "  1793.77,392.774 1794.54,390.389 1795.31,388.001 1796.07,385.612 1796.84,383.222 1797.61,380.829 1798.37,378.434 1799.14,376.038 1799.91,373.64 1800.68,371.24 \n",
       "  1801.44,368.838 1802.21,366.435 1802.98,364.029 1803.74,361.622 1804.51,359.213 1805.28,356.802 1806.05,354.389 1806.81,351.975 1807.58,349.558 1808.35,347.14 \n",
       "  1809.11,344.72 1809.88,342.298 1810.65,339.875 1811.41,337.449 1812.18,335.021 1812.95,332.592 1813.72,330.161 1814.48,327.728 1815.25,325.293 1816.02,322.857 \n",
       "  1816.78,320.418 1817.55,317.978 1818.32,315.535 1819.09,313.091 1819.85,310.645 1820.62,308.198 1821.39,305.748 1822.15,303.296 1822.92,300.843 1823.69,298.388 \n",
       "  1824.46,295.931 1825.22,293.472 1825.99,291.011 1826.76,288.548 1827.52,286.083 1828.29,283.617 1829.06,281.149 1829.83,278.678 1830.59,276.206 1831.36,273.732 \n",
       "  1832.13,271.256 1832.89,268.779 1833.66,266.299 1834.43,263.818 1835.2,261.334 1835.96,258.849 1836.73,256.362 1837.5,253.873 1838.26,251.382 1839.03,248.889 \n",
       "  1839.8,246.394 1840.57,243.898 1841.33,241.399 1842.1,238.899 1842.87,236.396 1843.63,233.892 1844.4,231.386 1845.17,228.878 1845.94,226.368 1846.7,223.856 \n",
       "  1847.47,221.342 1848.24,218.827 1849,216.309 1849.77,213.79 1850.54,211.268 1851.31,208.745 1852.07,206.22 1852.84,203.692 1853.61,201.163 1854.37,198.632 \n",
       "  1855.14,196.099 1855.91,193.564 1856.68,191.028 1857.44,188.489 1858.21,185.948 1858.98,183.406 1859.74,180.861 1860.51,178.315 1861.28,175.766 1862.05,173.216 \n",
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       "  2530.57,1440.54 2531.34,1440.53 2532.11,1440.53 2532.88,1440.53 2533.64,1440.53 2534.41,1440.53 2535.18,1440.53 2535.94,1440.52 2536.71,1440.52 2537.48,1440.52 \n",
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       "  2568.93,1440.45 2569.7,1440.45 2570.46,1440.45 2571.23,1440.44 2572,1440.44 2572.77,1440.44 2573.53,1440.44 2574.3,1440.44 2575.07,1440.43 2575.83,1440.43 \n",
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       "  2645.64,1440.27 2646.41,1440.27 2647.18,1440.27 2647.94,1440.27 2648.71,1440.27 2649.48,1440.27 2650.25,1440.26 2651.01,1440.26 2651.78,1440.26 2652.55,1440.26 \n",
       "  2653.31,1440.26 2654.08,1440.26 2654.85,1440.25 2655.62,1440.25 2656.38,1440.25 2657.15,1440.25 2657.92,1440.25 2658.68,1440.24 2659.45,1440.24 2660.22,1440.24 \n",
       "  2660.99,1440.24 2661.75,1440.24 2662.52,1440.24 2663.29,1440.23 2664.05,1440.23 2664.82,1440.23 2665.59,1440.23 2666.36,1440.23 2667.12,1440.23 2667.89,1440.22 \n",
       "  2668.66,1440.22 2669.42,1440.22 2670.19,1440.22 2670.96,1440.22 2671.73,1440.22 2672.49,1440.21 2673.26,1440.21 2674.03,1440.21 2674.79,1440.21 2675.56,1440.21 \n",
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       "  2684,1440.19 2684.77,1440.19 2685.53,1440.18 2686.3,1440.18 2687.07,1440.18 2687.83,1440.18 2688.6,1440.18 2689.37,1440.17 2690.14,1440.17 2690.9,1440.17 \n",
       "  2691.67,1440.17 2692.44,1440.17 2693.2,1440.17 2693.97,1440.16 2694.74,1440.16 2695.51,1440.16 2696.27,1440.16 2697.04,1440.16 2697.81,1440.16 2698.57,1440.15 \n",
       "  2699.34,1440.15 2700.11,1440.15 2700.88,1440.15 2701.64,1440.15 2702.41,1440.15 2703.18,1440.14 2703.94,1440.14 2704.71,1440.14 2705.48,1440.14 2706.25,1440.14 \n",
       "  2707.01,1440.13 2707.78,1440.13 2708.55,1440.13 2709.31,1440.13 2710.08,1440.13 2710.85,1440.13 2711.62,1440.12 2712.38,1440.12 2713.15,1440.12 2713.92,1440.12 \n",
       "  2714.68,1440.12 2715.45,1440.12 2716.22,1440.11 2716.99,1440.11 2717.75,1440.11 2718.52,1440.11 2719.29,1440.11 2720.05,1440.11 2720.82,1440.1 2721.59,1440.1 \n",
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       "  2737.7,1440.07 2738.47,1440.06 2739.23,1440.06 2740,1440.06 2740.77,1440.06 2741.53,1440.06 2742.3,1440.05 2743.07,1440.05 2743.83,1440.05 2744.6,1440.05 \n",
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       "  2799.07,1439.93 2799.83,1439.92 2800.6,1439.92 2801.37,1439.92 2802.14,1439.92 2802.9,1439.92 2803.67,1439.91 2804.44,1439.91 2805.2,1439.91 2805.97,1439.91 \n",
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       "  2814.41,1439.89 2815.18,1439.89 2815.94,1439.89 2816.71,1439.89 2817.48,1439.88 2818.25,1439.88 2819.01,1439.88 2819.78,1439.88 2820.55,1439.88 2821.31,1439.87 \n",
       "  2822.08,1439.87 2822.85,1439.87 2823.62,1439.87 2824.38,1439.87 2825.15,1439.87 2825.92,1439.86 2826.68,1439.86 2827.45,1439.86 2828.22,1439.86 2828.99,1439.86 \n",
       "  2829.75,1439.86 2830.52,1439.85 2831.29,1439.85 2832.05,1439.85 2832.82,1439.85 2833.59,1439.85 2834.36,1439.85 2835.12,1439.84 2835.89,1439.84 2836.66,1439.84 \n",
       "  2837.42,1439.84 2838.19,1439.84 2838.96,1439.83 2839.73,1439.83 2840.49,1439.83 2841.26,1439.83 2842.03,1439.83 2842.79,1439.83 2843.56,1439.82 2844.33,1439.82 \n",
       "  2845.1,1439.82 2845.86,1439.82 2846.63,1439.82 2847.4,1439.82 2848.16,1439.81 2848.93,1439.81 2849.7,1439.81 2850.47,1439.81 2851.23,1439.81 2852,1439.81 \n",
       "  2852.77,1439.8 2853.53,1439.8 2854.3,1439.8 2855.07,1439.8 2855.83,1439.8 2856.6,1439.79 2857.37,1439.79 2858.14,1439.79 2858.9,1439.79 2859.67,1439.79 \n",
       "  2860.44,1439.79 2861.2,1439.78 2861.97,1439.78 2862.74,1439.78 2863.51,1439.78 2864.27,1439.78 2865.04,1439.78 2865.81,1439.77 2866.57,1439.77 2867.34,1439.77 \n",
       "  2868.11,1439.77 2868.88,1439.77 2869.64,1439.76 2870.41,1439.76 2871.18,1439.76 2871.94,1439.76 2872.71,1439.76 2873.48,1439.76 2874.25,1439.75 2875.01,1439.75 \n",
       "  2875.78,1439.75 2876.55,1439.75 2877.31,1439.75 2878.08,1439.75 2878.85,1439.74 2879.62,1439.74 2880.38,1439.74 2881.15,1439.74 2881.92,1439.74 2882.68,1439.74 \n",
       "  2883.45,1439.73 2884.22,1439.73 2884.99,1439.73 2885.75,1439.73 2886.52,1439.73 2887.29,1439.72 2888.05,1439.72 2888.82,1439.72 2889.59,1439.72 2890.36,1439.72 \n",
       "  2891.12,1439.72 2891.89,1439.71 2892.66,1439.71 2893.42,1439.71 2894.19,1439.71 2894.96,1439.71 2895.73,1439.71 2896.49,1439.7 2897.26,1439.7 2898.03,1439.7 \n",
       "  2898.79,1439.7 2899.56,1439.7 2900.33,1439.7 2901.1,1439.69 2901.86,1439.69 2902.63,1439.69 2903.4,1439.69 2904.16,1439.69 2904.93,1439.68 2905.7,1439.68 \n",
       "  2906.47,1439.68 2907.23,1439.68 2908,1439.68 2908.77,1439.68 2909.53,1439.67 2910.3,1439.67 2911.07,1439.67 2911.84,1439.67 2912.6,1439.67 2913.37,1439.67 \n",
       "  2914.14,1439.66 2914.9,1439.66 2915.67,1439.66 2916.44,1439.66 2917.2,1439.66 2917.97,1439.65 2918.74,1439.65 2919.51,1439.65 2920.27,1439.65 2921.04,1439.65 \n",
       "  2921.81,1439.65 2922.57,1439.64 2923.34,1439.64 2924.11,1439.64 2924.88,1439.64 2925.64,1439.64 2926.41,1439.64 2927.18,1439.63 2927.94,1439.63 2928.71,1439.63 \n",
       "  2929.48,1439.63 2930.25,1439.63 2931.01,1439.63 2931.78,1439.62 2932.55,1439.62 2933.31,1439.62 2934.08,1439.62 2934.85,1439.62 2935.62,1439.61 2936.38,1439.61 \n",
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       "  2944.82,1439.59 2945.59,1439.59 2946.36,1439.59 2947.12,1439.59 2947.89,1439.59 2948.66,1439.58 2949.42,1439.58 2950.19,1439.58 2950.96,1439.58 2951.73,1439.58 \n",
       "  2952.49,1439.58 2953.26,1439.57 2954.03,1439.57 2954.79,1439.57 2955.56,1439.57 2956.33,1439.57 2957.1,1439.57 2957.86,1439.56 2958.63,1439.56 2959.4,1439.56 \n",
       "  2960.16,1439.56 2960.93,1439.56 2961.7,1439.56 2962.47,1439.55 2963.23,1439.55 2964,1439.55 2964.77,1439.55 2965.53,1439.55 2966.3,1439.54 2967.07,1439.54 \n",
       "  2967.84,1439.54 2968.6,1439.54 2969.37,1439.54 2970.14,1439.54 2970.9,1439.53 2971.67,1439.53 2972.44,1439.53 2973.2,1439.53 2973.97,1439.53 2974.74,1439.53 \n",
       "  2975.51,1439.52 2976.27,1439.52 2977.04,1439.52 2977.81,1439.52 2978.57,1439.52 2979.34,1439.52 2980.11,1439.51 2980.88,1439.51 2981.64,1439.51 2982.41,1439.51 \n",
       "  2983.18,1439.51 2983.94,1439.5 2984.71,1439.5 2985.48,1439.5 2986.25,1439.5 2987.01,1439.5 2987.78,1439.5 2988.55,1439.49 2989.31,1439.49 2990.08,1439.49 \n",
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       "  2998.52,1439.47 2999.29,1439.47 3000.05,1439.47 3000.82,1439.47 3001.59,1439.46 3002.36,1439.46 3003.12,1439.46 3003.89,1439.46 3004.66,1439.46 3005.42,1439.46 \n",
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       "  3013.86,1439.44 3014.63,1439.43 3015.4,1439.43 3016.16,1439.43 3016.93,1439.43 3017.7,1439.43 3018.47,1439.43 3019.23,1439.42 3020,1439.42 3020.77,1439.42 \n",
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       "  3052.22,1439.35 3052.99,1439.35 3053.75,1439.35 3054.52,1439.34 3055.29,1439.34 3056.05,1439.34 3056.82,1439.34 3057.59,1439.34 3058.36,1439.34 3059.12,1439.33 \n",
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       "  3067.56,1439.31 3068.33,1439.31 3069.1,1439.31 3069.86,1439.31 3070.63,1439.31 3071.4,1439.31 3072.16,1439.3 3072.93,1439.3 3073.7,1439.3 3074.47,1439.3 \n",
       "  3075.23,1439.3 3076,1439.29 3076.77,1439.29 3077.53,1439.29 3078.3,1439.29 3079.07,1439.29 3079.84,1439.29 3080.6,1439.28 3081.37,1439.28 3082.14,1439.28 \n",
       "  3082.9,1439.28 3083.67,1439.28 3084.44,1439.28 3085.21,1439.27 3085.97,1439.27 3086.74,1439.27 3087.51,1439.27 3088.27,1439.27 3089.04,1439.27 3089.81,1439.26 \n",
       "  3090.57,1439.26 3091.34,1439.26 3092.11,1439.26 3092.88,1439.26 3093.64,1439.25 3094.41,1439.25 3095.18,1439.25 3095.94,1439.25 3096.71,1439.25 3097.48,1439.25 \n",
       "  3098.25,1439.24 \n",
       "  \"/>\n",
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       "  2369.48,1576.61 2370.25,1574.68 2371.01,1572.75 2371.78,1570.81 2372.55,1568.88 2373.31,1566.95 2374.08,1565.01 2374.85,1563.08 2375.62,1561.15 2376.38,1559.21 \n",
       "  2377.15,1557.28 2377.92,1555.34 2378.68,1553.41 2379.45,1551.48 2380.22,1549.54 2380.99,1547.61 2381.75,1545.68 2382.52,1543.74 2383.29,1541.81 2384.05,1539.88 \n",
       "  2384.82,1537.94 2385.59,1536.01 2386.36,1534.07 2387.12,1532.14 2387.89,1530.21 2388.66,1528.27 2389.42,1526.34 2390.19,1524.41 2390.96,1522.47 2391.73,1520.54 \n",
       "  2392.49,1518.6 2393.26,1516.67 2394.03,1514.74 2394.79,1512.8 2395.56,1510.87 2396.33,1508.94 2397.09,1507 2397.86,1505.07 2398.63,1503.13 2399.4,1501.2 \n",
       "  2400.16,1499.27 2400.93,1497.33 2401.7,1495.4 2402.46,1493.47 2403.23,1491.53 2404,1489.6 2404.77,1487.66 2405.53,1485.73 2406.3,1483.8 2407.07,1481.86 \n",
       "  2407.83,1479.93 2408.6,1478 2409.37,1476.06 2410.14,1474.13 2410.9,1472.19 2411.67,1470.26 2412.44,1468.33 2413.2,1466.39 2413.97,1464.46 2414.74,1462.52 \n",
       "  2415.51,1460.59 2416.27,1458.66 2417.04,1456.72 2417.81,1454.79 2418.57,1452.85 2419.34,1450.92 2420.11,1448.99 2420.88,1447.05 2421.64,1445.12 2422.41,1443.18 \n",
       "  2423.18,1441.25 2423.94,1439.32 2424.71,1437.38 2425.48,1435.45 2426.25,1433.51 2427.01,1431.58 2427.78,1429.65 2428.55,1427.71 2429.31,1425.78 2430.08,1423.84 \n",
       "  2430.85,1421.91 2431.62,1419.98 2432.38,1418.04 2433.15,1416.11 2433.92,1414.17 2434.68,1412.24 2435.45,1410.31 2436.22,1408.37 2436.99,1406.44 2437.75,1404.5 \n",
       "  2438.52,1402.57 2439.29,1400.63 2440.05,1398.7 2440.82,1396.77 2441.59,1394.83 2442.36,1392.9 2443.12,1390.96 2443.89,1389.03 2444.66,1387.09 2445.42,1385.16 \n",
       "  2446.19,1383.23 2446.96,1381.29 2447.73,1379.36 2448.49,1377.42 2449.26,1375.49 2450.03,1373.56 2450.79,1371.62 2451.56,1369.69 2452.33,1367.75 2453.09,1365.82 \n",
       "  2453.86,1363.88 2454.63,1361.95 2455.4,1360.02 2456.16,1358.08 2456.93,1356.15 2457.7,1354.21 2458.46,1352.28 2459.23,1350.34 2460,1348.41 2460.77,1346.47 \n",
       "  2461.53,1344.54 2462.3,1342.61 2463.07,1340.67 2463.83,1338.74 2464.6,1336.8 2465.37,1334.87 2466.14,1332.93 2466.9,1331 2467.67,1329.07 2468.44,1327.13 \n",
       "  2469.2,1325.2 2469.97,1323.26 2470.74,1321.33 2471.51,1319.39 2472.27,1317.46 2473.04,1315.52 2473.81,1313.59 2474.57,1311.65 2475.34,1309.72 2476.11,1307.79 \n",
       "  2476.88,1305.85 2477.64,1303.92 2478.41,1301.98 2479.18,1300.05 2479.94,1298.11 2480.71,1296.18 2481.48,1294.24 2482.25,1292.31 2483.01,1290.38 2483.78,1288.44 \n",
       "  2484.55,1286.51 2485.31,1284.57 2486.08,1282.64 2486.85,1280.7 2487.62,1278.77 2488.38,1276.83 2489.15,1274.9 2489.92,1272.96 2490.68,1271.03 2491.45,1269.09 \n",
       "  2492.22,1267.16 2492.99,1265.22 2493.75,1263.29 2494.52,1261.36 2495.29,1259.42 2496.05,1257.49 2496.82,1255.55 2497.59,1253.62 2498.36,1251.68 2499.12,1249.75 \n",
       "  2499.89,1247.81 2500.66,1245.88 2501.42,1243.94 2502.19,1242.01 2502.96,1240.07 2503.73,1238.14 2504.49,1236.2 2505.26,1234.27 2506.03,1232.33 2506.79,1230.4 \n",
       "  2507.56,1228.46 2508.33,1226.53 2509.09,1224.6 2509.86,1222.66 2510.63,1220.73 2511.4,1218.79 2512.16,1216.86 2512.93,1214.92 2513.7,1212.99 2514.46,1211.05 \n",
       "  2515.23,1209.12 2516,1207.18 2516.77,1205.25 2517.53,1203.31 2518.3,1201.38 2519.07,1199.44 2519.83,1197.51 2520.6,1195.57 2521.37,1193.64 2522.14,1191.7 \n",
       "  2522.9,1189.77 2523.67,1187.83 2524.44,1185.9 2525.2,1183.96 2525.97,1182.03 2526.74,1180.09 2527.51,1178.16 2528.27,1176.22 2529.04,1174.29 2529.81,1172.35 \n",
       "  2530.57,1170.42 2531.34,1168.48 2532.11,1166.55 2532.88,1164.61 2533.64,1162.68 2534.41,1160.74 2535.18,1158.81 2535.94,1156.87 2536.71,1154.94 2537.48,1153 \n",
       "  2538.25,1151.07 2539.01,1149.13 2539.78,1147.2 2540.55,1145.26 2541.31,1143.33 2542.08,1141.39 2542.85,1139.46 2543.62,1137.52 2544.38,1135.59 2545.15,1133.65 \n",
       "  2545.92,1131.72 2546.68,1129.78 2547.45,1127.84 2548.22,1125.91 2548.99,1123.97 2549.75,1122.04 2550.52,1120.1 2551.29,1118.17 2552.05,1116.23 2552.82,1114.3 \n",
       "  2553.59,1112.36 2554.36,1110.43 2555.12,1108.49 2555.89,1106.56 2556.66,1104.62 2557.42,1102.69 2558.19,1100.75 2558.96,1098.82 2559.73,1096.88 2560.49,1094.95 \n",
       "  2561.26,1093.01 2562.03,1091.07 2562.79,1089.14 2563.56,1087.2 2564.33,1085.27 2565.1,1083.33 2565.86,1081.4 2566.63,1079.46 2567.4,1077.53 2568.16,1075.59 \n",
       "  2568.93,1073.66 2569.7,1071.72 2570.46,1069.79 2571.23,1067.85 2572,1065.91 2572.77,1063.98 2573.53,1062.04 2574.3,1060.11 2575.07,1058.17 2575.83,1056.24 \n",
       "  2576.6,1054.3 2577.37,1052.37 2578.14,1050.43 2578.9,1048.5 2579.67,1046.56 2580.44,1044.62 2581.2,1042.69 2581.97,1040.75 2582.74,1038.82 2583.51,1036.88 \n",
       "  2584.27,1034.95 2585.04,1033.01 2585.81,1031.08 2586.57,1029.14 2587.34,1027.21 2588.11,1025.27 2588.88,1023.33 2589.64,1021.4 2590.41,1019.46 2591.18,1017.53 \n",
       "  2591.94,1015.59 2592.71,1013.66 2593.48,1011.72 2594.25,1009.78 2595.01,1007.85 2595.78,1005.91 2596.55,1003.98 2597.31,1002.04 2598.08,1000.11 2598.85,998.171 \n",
       "  2599.62,996.235 2600.38,994.299 2601.15,992.364 2601.92,990.428 2602.68,988.492 2603.45,986.557 2604.22,984.621 2604.99,982.685 2605.75,980.749 2606.52,978.814 \n",
       "  2607.29,976.878 2608.05,974.942 2608.82,973.006 2609.59,971.071 2610.36,969.135 2611.12,967.199 2611.89,965.263 2612.66,963.327 2613.42,961.392 2614.19,959.456 \n",
       "  2614.96,957.52 2615.73,955.584 2616.49,953.648 2617.26,951.712 2618.03,949.777 2618.79,947.841 2619.56,945.905 2620.33,943.969 2621.1,942.033 2621.86,940.097 \n",
       "  2622.63,938.161 2623.4,936.225 2624.16,934.29 2624.93,932.354 2625.7,930.418 2626.46,928.482 2627.23,926.546 2628,924.61 2628.77,922.674 2629.53,920.738 \n",
       "  2630.3,918.802 2631.07,916.866 2631.83,914.93 2632.6,912.994 2633.37,911.058 2634.14,909.122 2634.9,907.186 2635.67,905.25 2636.44,903.314 2637.2,901.378 \n",
       "  2637.97,899.442 2638.74,897.506 2639.51,895.57 2640.27,893.634 2641.04,891.698 2641.81,889.762 2642.57,887.825 2643.34,885.889 2644.11,883.953 2644.88,882.017 \n",
       "  2645.64,880.081 2646.41,878.145 2647.18,876.209 2647.94,874.273 2648.71,872.337 2649.48,870.4 2650.25,868.464 2651.01,866.528 2651.78,864.592 2652.55,862.656 \n",
       "  2653.31,860.72 2654.08,858.783 2654.85,856.847 2655.62,854.911 2656.38,852.975 2657.15,851.038 2657.92,849.102 2658.68,847.166 2659.45,845.23 2660.22,843.293 \n",
       "  2660.99,841.357 2661.75,839.421 2662.52,837.485 2663.29,835.548 2664.05,833.612 2664.82,831.676 2665.59,829.74 2666.36,827.803 2667.12,825.867 2667.89,823.931 \n",
       "  2668.66,821.994 2669.42,820.058 2670.19,818.122 2670.96,816.185 2671.73,814.249 2672.49,812.312 2673.26,810.376 2674.03,808.44 2674.79,806.503 2675.56,804.567 \n",
       "  2676.33,802.631 2677.1,800.694 2677.86,798.758 2678.63,796.821 2679.4,794.885 2680.16,792.948 2680.93,791.012 2681.7,789.075 2682.46,787.139 2683.23,785.203 \n",
       "  2684,783.266 2684.77,781.33 2685.53,779.393 2686.3,777.457 2687.07,775.52 2687.83,773.584 2688.6,771.647 2689.37,769.711 2690.14,767.774 2690.9,765.837 \n",
       "  2691.67,763.901 2692.44,761.964 2693.2,760.028 2693.97,758.091 2694.74,756.155 2695.51,754.218 2696.27,752.281 2697.04,750.345 2697.81,748.408 2698.57,746.472 \n",
       "  2699.34,744.535 2700.11,742.598 2700.88,740.662 2701.64,738.725 2702.41,736.788 2703.18,734.852 2703.94,732.915 2704.71,730.978 2705.48,729.042 2706.25,727.105 \n",
       "  2707.01,725.168 2707.78,723.232 2708.55,721.295 2709.31,719.358 2710.08,717.422 2710.85,715.485 2711.62,713.548 2712.38,711.611 2713.15,709.675 2713.92,707.738 \n",
       "  2714.68,705.801 2715.45,703.864 2716.22,701.927 2716.99,699.991 2717.75,698.054 2718.52,696.117 2719.29,694.18 2720.05,692.243 2720.82,690.307 2721.59,688.37 \n",
       "  2722.36,686.433 2723.12,684.496 2723.89,682.559 2724.66,680.622 2725.42,678.686 2726.19,676.749 2726.96,674.812 2727.73,672.875 2728.49,670.938 2729.26,669.001 \n",
       "  2730.03,667.064 2730.79,665.127 2731.56,663.19 2732.33,661.253 2733.1,659.316 2733.86,657.379 2734.63,655.443 2735.4,653.506 2736.16,651.569 2736.93,649.632 \n",
       "  2737.7,647.695 2738.47,645.758 2739.23,643.821 2740,641.884 2740.77,639.947 2741.53,638.01 2742.3,636.073 2743.07,634.136 2743.83,632.198 2744.6,630.261 \n",
       "  2745.37,628.324 2746.14,626.387 2746.9,624.45 2747.67,622.513 2748.44,620.576 2749.2,618.639 2749.97,616.702 2750.74,614.765 2751.51,612.828 2752.27,610.891 \n",
       "  2753.04,608.953 2753.81,607.016 2754.57,605.079 2755.34,603.142 2756.11,601.205 2756.88,599.268 2757.64,597.33 2758.41,595.393 2759.18,593.456 2759.94,591.519 \n",
       "  2760.71,589.582 2761.48,587.644 2762.25,585.707 2763.01,583.77 2763.78,581.833 2764.55,579.896 2765.31,577.958 2766.08,576.021 2766.85,574.084 2767.62,572.147 \n",
       "  2768.38,570.209 2769.15,568.272 2769.92,566.335 2770.68,564.397 2771.45,562.46 2772.22,560.523 2772.99,558.585 2773.75,556.648 2774.52,554.711 2775.29,552.773 \n",
       "  2776.05,550.836 2776.82,548.899 2777.59,546.961 2778.36,545.024 2779.12,543.087 2779.89,541.149 2780.66,539.212 2781.42,537.274 2782.19,535.337 2782.96,533.4 \n",
       "  2783.73,531.462 2784.49,529.525 2785.26,527.587 2786.03,525.65 2786.79,523.712 2787.56,521.775 2788.33,519.838 2789.1,517.9 2789.86,515.963 2790.63,514.025 \n",
       "  2791.4,512.088 2792.16,510.15 2792.93,508.213 2793.7,506.275 2794.47,504.338 2795.23,502.4 2796,500.463 2796.77,498.525 2797.53,496.587 2798.3,494.65 \n",
       "  2799.07,492.712 2799.83,490.775 2800.6,488.837 2801.37,486.9 2802.14,484.962 2802.9,483.024 2803.67,481.087 2804.44,479.149 2805.2,477.211 2805.97,475.274 \n",
       "  2806.74,473.336 2807.51,471.399 2808.27,469.461 2809.04,467.523 2809.81,465.586 2810.57,463.648 2811.34,461.71 2812.11,459.773 2812.88,457.835 2813.64,455.897 \n",
       "  2814.41,453.959 2815.18,452.022 2815.94,450.084 2816.71,448.146 2817.48,446.209 2818.25,444.271 2819.01,442.333 2819.78,440.395 2820.55,438.457 2821.31,436.52 \n",
       "  2822.08,434.582 2822.85,432.644 2823.62,430.706 2824.38,428.769 2825.15,426.831 2825.92,424.893 2826.68,422.955 2827.45,421.017 2828.22,419.079 2828.99,417.142 \n",
       "  2829.75,415.204 2830.52,413.266 2831.29,411.328 2832.05,409.39 2832.82,407.452 2833.59,405.514 2834.36,403.576 2835.12,401.638 2835.89,399.701 2836.66,397.763 \n",
       "  2837.42,395.825 2838.19,393.887 2838.96,391.949 2839.73,390.011 2840.49,388.073 2841.26,386.135 2842.03,384.197 2842.79,382.259 2843.56,380.321 2844.33,378.383 \n",
       "  2845.1,376.445 2845.86,374.507 2846.63,372.569 2847.4,370.631 2848.16,368.693 2848.93,366.755 2849.7,364.817 2850.47,362.879 2851.23,360.941 2852,359.003 \n",
       "  2852.77,357.065 2853.53,355.127 2854.3,353.188 2855.07,351.25 2855.83,349.312 2856.6,347.374 2857.37,345.436 2858.14,343.498 2858.9,341.56 2859.67,339.622 \n",
       "  2860.44,337.683 2861.2,335.745 2861.97,333.807 2862.74,331.869 2863.51,329.931 2864.27,327.993 2865.04,326.054 2865.81,324.116 2866.57,322.178 2867.34,320.24 \n",
       "  2868.11,318.302 2868.88,316.363 2869.64,314.425 2870.41,312.487 2871.18,310.549 2871.94,308.61 2872.71,306.672 2873.48,304.734 2874.25,302.796 2875.01,300.857 \n",
       "  2875.78,298.919 2876.55,296.981 2877.31,295.042 2878.08,293.104 2878.85,291.166 2879.62,289.227 2880.38,287.289 2881.15,285.351 2881.92,283.412 2882.68,281.474 \n",
       "  2883.45,279.536 2884.22,277.597 2884.99,275.659 2885.75,273.721 2886.52,271.782 2887.29,269.844 2888.05,267.905 2888.82,265.967 2889.59,264.028 2890.36,262.09 \n",
       "  2891.12,260.152 2891.89,258.213 2892.66,256.275 2893.42,254.336 2894.19,252.398 2894.96,250.459 2895.73,248.521 2896.49,246.582 2897.26,244.644 2898.03,242.705 \n",
       "  2898.79,240.767 2899.56,238.828 2900.33,236.89 2901.1,234.951 2901.86,233.013 2902.63,231.074 2903.4,229.136 2904.16,227.197 2904.93,225.258 2905.7,223.32 \n",
       "  2906.47,221.381 2907.23,219.443 2908,217.504 2908.77,215.566 2909.53,213.627 2910.3,211.688 2911.07,209.75 2911.84,207.811 2912.6,205.872 2913.37,203.934 \n",
       "  2914.14,201.995 2914.9,200.056 2915.67,198.118 2916.44,196.179 2917.2,194.24 2917.97,192.302 2918.74,190.363 2919.51,188.424 2920.27,186.486 2921.04,184.547 \n",
       "  2921.81,182.608 2922.57,180.669 2923.34,178.731 2924.11,176.792 2924.88,174.853 2925.64,172.914 2926.41,170.976 2927.18,169.037 2927.94,167.098 2928.71,165.159 \n",
       "  2929.48,163.22 2930.25,161.282 2931.01,159.343 2931.78,157.404 2932.55,155.465 2933.31,153.526 2934.08,151.587 2934.85,149.649 2935.62,147.71 2936.38,145.771 \n",
       "  2937.15,143.832 2937.92,141.893 2938.68,139.954 2939.45,138.015 2940.22,136.076 2940.99,134.138 2941.75,132.199 2942.52,130.26 2943.29,128.321 2944.05,126.382 \n",
       "  2944.82,124.443 2945.59,122.504 2946.36,120.565 2947.12,118.626 2947.89,116.687 2948.66,114.748 2949.42,112.809 2950.19,110.87 2950.96,108.931 2951.73,106.992 \n",
       "  2952.49,105.053 2953.26,103.114 2954.03,101.175 2954.79,99.2359 2955.56,97.2969 2956.33,95.3578 2957.1,93.4188 2957.86,91.4797 2958.63,89.5406 2959.4,87.6015 \n",
       "  2960.16,85.6624 2960.93,83.7233 2961.7,81.7842 2962.47,79.8451 2963.23,77.906 2964,75.9669 2964.77,74.0277 2965.53,72.0886 2966.3,70.1494 2967.07,68.2103 \n",
       "  2967.84,66.2711 2968.6,64.3319 2969.37,62.3928 2970.14,60.4536 2970.9,58.5144 2971.67,56.5752 2972.44,54.636 2973.2,52.6968 2973.97,50.7575 2974.74,48.8183 \n",
       "  2975.51,46.8791 2976.27,44.9398 2977.04,43.0006 2977.81,41.0613 2978.57,39.122 2979.34,37.1828 2980.11,35.2435 2980.88,33.3042 2981.64,31.3649 2982.41,29.4256 \n",
       "  2983.18,27.4863 2983.94,25.547 2984.71,23.6076 2985.48,21.6683 2986.25,19.729 2987.01,17.7896 2987.78,15.8503 2988.55,13.9109 2989.31,11.9715 2990.08,10.0322 \n",
       "  2990.85,8.09279 2991.62,6.1534 2992.38,4.214 2993.15,2.27459 2993.92,0.335182 2994.68,-1.60424 2995.45,-3.54366 2996.22,-5.4831 2996.99,-7.42254 2997.75,-9.36199 \n",
       "  2998.52,-11.3014 2999.29,-13.2409 3000.05,-15.1804 3000.82,-17.1199 3001.59,-19.0593 3002.36,-20.9988 3003.12,-22.9383 3003.89,-24.8778 3004.66,-26.8173 3005.42,-28.7569 \n",
       "  3006.19,-30.6964 3006.96,-32.6359 3007.73,-34.5755 3008.49,-36.515 3009.26,-38.4546 3010.03,-40.3941 3010.79,-42.3337 3011.56,-44.2733 3012.33,-46.2129 3013.1,-48.1525 \n",
       "  3013.86,-50.0921 3014.63,-52.0317 3015.4,-53.9713 3016.16,-55.9109 3016.93,-57.8506 3017.7,-59.7902 3018.47,-61.7298 3019.23,-63.6695 3020,-65.6091 3020.77,-67.5488 \n",
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     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pp=0.6\n",
    "dd=0.2\n",
    "P₀=100\n",
    "D₀=50\n",
    "T=24\n",
    "t=collect(0:0.1:100)\n",
    "P1=plot(t,t->P₀*(1+pp-dd)^(t/T),label=\"P\",seriestype=:line,ylims = (0,4300))\n",
    "plot!(t,t->D₀+P₀*(((1+pp-dd)^(t/T))-1)*((1-pp+dd)/(pp-dd)),label=\"D\",seriestype=:line,ylims = (0,400))\n",
    "\n",
    "pp=0.5\n",
    "dd=0.3\n",
    "P2=plot(t,t->P₀*(1+pp-dd)^(t/T),label=\"P\",seriestype=:line,ylims = (0,4300))\n",
    "plot!(t,t->D₀+P₀*(((1+pp-dd)^(t/T))-1)*((1-pp+dd)/(pp-dd)),label=\"D\",seriestype=:line,ylims = (0,400))\n",
    "\n",
    "pp=0.001\n",
    "dd=0.0001\n",
    "P3=plot(t,t->P₀*(1+pp-dd)^(t/T),label=\"P\",seriestype=:line,ylims = (0,4300))\n",
    "plot!(t,t->D₀+P₀*(((1+pp-dd)^(t/T))-1)*((1-pp+dd)/(pp-dd)),label=\"D\",seriestype=:line,ylims = (0,400))\n",
    "\n",
    "plot(P1,P2,P3,layout=(1,3),legend=true,size = (800, 500))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "to do:\n",
    "    \n",
    "    it will be cool to include a term of saturations as the logistic equation does. \n",
    "    \n",
    "    We will have to use the derivative and integrate\n",
    "    \n",
    "    "
   ]
  }
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