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32.3: Population Dynamics

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    69578

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    %matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import sympy as sym
sym.init_printing()

    In this section, we consider the distribution of a population at different ages. Let \(x(t)\) be a 100-vector with \(x_i(t)\) denoting the number of people with agent \(i−1\).

    The birth rate is given in the vector \(b\), where \(b_i\) is the average number of births per person with age \(i−1\). \(b_i=0\) for \(i<13\) and \(i>50\).

    The death rate is given by the vector \(d\), where \(d_i\) is the portion of those aged \(i−1\) who dies this year.

    Then we can model the population with the dynamic systems. We consider the 0-year old first. It includes all newborn from all ages, so we have \(x_1(t+1) = b^{\top}x(t)\). Then we consider all other ages for \(i > 1\). \(x_i(t+1) = (1-d_i)x_{i-1}(t)\)

    population = np.array([
       3.94415,       3.97807,       4.09693,       4.11904,       4.06317,       4.05686,       4.06638,       4.03058,
       4.04649,       4.14835,       4.17254,       4.11442,       4.10624,       4.11801,       4.16598,       4.24282,
       4.31614,       4.39529,       4.50085,       4.58523,       4.51913,       4.35429,       4.26464,       4.19857,
       4.24936,       4.26235,       4.15231,       4.24887,       4.21525,       4.22308,       4.28567,       3.97022,
       3.98685,       3.88015,       3.83922,       3.95643,       3.80209,       3.93445,       4.12188,       4.36480,
       4.38327,       4.11498,       4.07610,       4.10511,       4.21150,       4.50887,       4.51976,       4.53526,
       4.53880,       4.60590,       4.66029,       4.46463,       4.50085,       4.38035,       4.29200,       4.25471,
       4.03751,       3.93639,       3.79493,       3.64127,       3.62113,       3.49260,       3.56318,       3.48388,
       2.65713,       2.68076,       2.63914,       2.64936,       2.32367,       2.14232,       2.04312,       1.94932,
       1.86427,       1.73696,       1.68449,       1.62008,       1.47107,       1.45533,       1.40012,       1.37119,
       1.30851,       1.21287,       1.16142,       1.07481,       0.98572,       0.91472,       0.81421,       0.71291,
       0.64062,       0.53800,       0.43556,       0.34499,       0.28139,       0.21698,       0.16944,       0.12972,
       0.09522,       0.06814,       0.04590,       0.03227])
d = np.array([
       0.00623,       0.00044,       0.00027,       0.00020,       0.00016,       0.00012,       0.00011,       0.00011,
       0.00012,       0.00011,       0.00010,       0.00013,       0.00013,       0.00015,       0.00020,       0.00025,
       0.00037,       0.00047,       0.00064,       0.00071,       0.00076,       0.00087,       0.00087,       0.00088,
       0.00094,       0.00092,       0.00095,       0.00093,       0.00099,       0.00101,       0.00103,       0.00109,
       0.00110,       0.00114,       0.00115,       0.00120,       0.00131,       0.00137,       0.00146,       0.00156,
       0.00162,       0.00185,       0.00201,       0.00216,       0.00243,       0.00258,       0.00298,       0.00325,
       0.00351,       0.00387,       0.00413,       0.00454,       0.00494,       0.00533,       0.00571,       0.00602,
       0.00670,       0.00710,       0.00769,       0.00828,       0.00860,       0.00932,       0.00998,       0.01101,
       0.01250,       0.01282,       0.01404,       0.01515,       0.01687,       0.01830,       0.01967,       0.02133,
       0.02347,       0.02562,       0.02800,       0.03083,       0.03441,       0.03711,       0.04126,       0.04448,
       0.04964,       0.05539,       0.06149,       0.06803,       0.07673,       0.08561,       0.09540,       0.10636,
       0.11802,       0.13385,       0.15250,       0.16491,       0.18738,       0.20757,       0.22688,       0.25196,
       0.27422,       0.29239,       0.32560,       0.34157])
b = np.array([
       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,
       0.00000,       0.00000,       0.00020,       0.00020,       0.00020,       0.00020,       0.00020,       0.01710,
       0.01710,       0.01710,       0.01710,       0.01710,       0.04500,       0.04500,       0.04500,       0.04500,
       0.04500,       0.05415,       0.05415,       0.05415,       0.05415,       0.05415,       0.04825,       0.04825,
       0.04825,       0.04825,       0.04825,       0.02250,       0.02250,       0.02250,       0.02250,       0.02250,
       0.00510,       0.00510,       0.00510,       0.00510,       0.00510,       0.00035,       0.00035,       0.00035,
       0.00035,       0.00035,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,
       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,
       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,
       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,
       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,
       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,       0.00000,
       0.00000,       0.00000,       0.00000,       0.00000])

plt.subplot(1,3,1)
plt.plot(population)
plt.title('population in 2010')
plt.subplot(1,3,2)
plt.plot(b)
plt.title('birth rate in 2010')
plt.subplot(1,3,3)
plt.plot(d)
plt.title('death rate in 2010')

plt.show()
    Do This

    Find the \(100 \times 100\) matrix A2 such that \(x(t+1) = Ax(t)\)

    A2 = 
print(A2)
    Do This

    Plot the population distribution at year 2020.

    This page titled 32.3: Population Dynamics was last modified on Sat, 17 Sep 2022 15:14:36 GMT and is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Dirk Colbry via source content that was edited to the style and standards of the LibreTexts platform.