North Carolina State University Department of Mathematics An Introduction to Applied Mathematics MA 325 Assignment #2 DUE Date: April 12, 2017 1. Consider the following logistic model for population dynamics: dP P = bP (1 ) dt K with initial condition P (0) = P0 . Let the growth rate b be equal to 0.2 and the carrying capacity K be equal to 1.0E10. The initial world population is 2,557,628,654, which is the total population in 1950. (a) Use SimBiology, perform a \"run scan with sensitivities\" on this model with the following attributes: For For For For Use Full Normalization. Generate Plots After Run Discuss the sensitivity of the population with respect to b and K. Repeat the above exercise, with StopTime set to be 65 (65 years from 1950). Simulation Settings, set StopTime to be 10 (10 years from 1950). the values to Scan, use \"multivariate normal distribution\". the Number of Iterations, use 8. the covariance matrix, use \u0014 \u0015 0.005 0 0 1000 For this experiment, turn in the screen captures of the Task Editor and Figures screens. (b) Use the world population data provided together with the assignment. Use the data from 1950-1960 (first 11 data points), estimate the parameters b and K. Turn in the screen captures of Fit Summary and Observation versus Prediction. Are the estimated values of b and K reasonable. Repeat the above exercise with the data from 1950-2015 (the whole data set). How are the parameters b and K different from the case of 1950-1960? Are the parameter values reasonable? Year P 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 14 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 2,557,628,654 2,594,939,877 2,636,772,306 2,682,053,389 2,730,228,104 2,782,098,943 2,835,299,673 2,891,349,717 2,948,137,248 3,000,716,593 3,043,001,508 3,083,966,929 3,140,093,217 3,209,827,882 3,281,201,306 3,350,425,793 3,420,677,923 3,490,333,715 3,562,313,822 3,637,159,050 3,712,697,742 3,790,326,948 3,866,568,653 3,942,096,442 4,016,608,813 4,089,083,233 4,160,185,010 4,232,084,578 4,304,105,753 4,379,013,942 4,451,362,735 4,534,410,125 4,614,566,561 4,695,736,743 4,774,569,391 4,856,462,699 4,940,571,232 37 38 39 40 41 42 43 44 45 46 47 47 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 5,027,200,492 5,114,557,167 5,201,440,110 5,288,955,934 5,371,585,922 5,456,136,278 5,538,268,316 5,618,682,132 5,699,202,985 5,779,440,593 5,857,972,543 5,935,213,248 6,012,074,922 6,088,571,383 6,165,219,247 6,242,016,348 6,318,590,956 6,395,699,509 6,473,044,732 6,551,263,534 6,629,913,759 6,709,049,780 6,788,214,394 6,866,332,358 7,006,907,989 7,089,451,551 7,176,092,192 7,260,780,278 7,346,633,037