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Math Assignment 345, Prof. Shi
Due: Tuesday, Sept 12 (9:30am) 1. The following data taken from the 1996 Information Please Almanac show U.S. population from 1800 to 1860, just prior to the Civil War. year 1800 1810 1820 1830 1840 1850 1860 Use linear population in million 5.31 7.24 9.64 12.87 17.07 23.19 31.44 regression to nd the best linear model describing the data. (First use Matlab program, then use the formula given in class, and compare the results.) 2. Populations may be harvested for a variety of reasons including economic gain (e.g. harvesting sh to sell), conservation (e.g. regulating wildebeest populations to prevent overgrazing), and public health (e.g. reducing mosquito abundance to reduce the incidence rate of the West Nile virus). There are many forms of harvesting models and we consider one of the simplest here. Consider a population that reproduces annually and whose population size in the beginning of the n-th year is xn . Let r denote the number of progeny produced year per per individual and h denote the number of individuals harvested per year. If harvesting occurs occurs after reproduction and before the beginning of a new year, then xn+1 = rxn h. Solve the linear dierence equation xn+1 = rxn h, x0 = K by using the following way: (a) Find the equilibrium x . (b) Make a change of variable yn = xn x , and show that yn satises yn+1 = ryn . (c) Solve the equation of yn and give the formula for xn . 3. (a) Use Matlab to generate and plot the rst 30 terms of the solutions to the dierence equation xn+1 = rxn (1 xn ), x0 = 0.1 with r = 0.9, 1.5...