Make sure to use Matlab,well labeled graphs, written explanations, and Matlab code.
(1) In the early stages of a population's growth, it can often be assumed that the population grows at a rate proportional to its current size. This assumption is reasonable as long we assume that growth due to new births dominates the growth rate due to immigration and that deaths due to resource limitations are negligible, and leads to what is called the Malthusian Law of Population Growth. The table below shows some values for the US population according to census measurements from 1790 to 1890 Year Population 1890 62,947,714 1880 50,155,783 1870 38,558,371 1860 31,443,321 1850 23,191,876 1840 17,063,353 1830 12,860,702 1820 9,638,453 1810 7,239,881 1800 5,308,483 1790 3,929,214 (a) Translate the Malthusian Law into a differential equation for p(t), the US population in millions t years after 1790. (b) Determine the general solution to your differential equation from part (a) and use the US population measurements from 1790 and 1800 to estimate any unknown parameter values. (c) Use your equation from (b) to estimate the doubling time of the US population (that is, the time it took the population to double its 1790 value). How does the model's estimate compare with what the data suggests? (d) What happens if you try to use your equation to estimate the 1870 population? Why do you think the answer is so far off (think about what happened in the 1860s)? (1) In the early stages of a population's growth, it can often be assumed that the population grows at a rate proportional to its current size. This assumption is reasonable as long we assume that growth due to new births dominates the growth rate due to immigration and that deaths due to resource limitations are negligible, and leads to what is called the Malthusian Law of Population Growth. The table below shows some values for the US population according to census measurements from 1790 to 1890 Year Population 1890 62,947,714 1880 50,155,783 1870 38,558,371 1860 31,443,321 1850 23,191,876 1840 17,063,353 1830 12,860,702 1820 9,638,453 1810 7,239,881 1800 5,308,483 1790 3,929,214 (a) Translate the Malthusian Law into a differential equation for p(t), the US population in millions t years after 1790. (b) Determine the general solution to your differential equation from part (a) and use the US population measurements from 1790 and 1800 to estimate any unknown parameter values. (c) Use your equation from (b) to estimate the doubling time of the US population (that is, the time it took the population to double its 1790 value). How does the model's estimate compare with what the data suggests? (d) What happens if you try to use your equation to estimate the 1870 population? Why do you think the answer is so far off (think about what happened in the 1860s)
