32.3: Population Dynamics
- Page ID
- 69578
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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()
Find the \(100 \times 100\) matrix A2 such that \(x(t+1) = Ax(t)\)
Plot the population distribution at year 2020.