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Unformatted text preview: Homework 3 AMATH 383, Autumn 2009 Due: Monday, November 9, after class 1. (4+2+2+3+2 points) Carrying Capacity: Consider the following census data t 1790 1820 2000 N 3 . 9 10 6 9 . 6 10 6 281 10 6 of the United States. We assume these data follow the population model (logistics equation) dN dt = N (1 N K ) and use the data to compute and K . (a) Show that the general solution of the ODE with initial condition N ( t ) = N is given by N ( t ) = K 1 + ( K N 1) exp ( ( t t )) (b) Lets take t = 1790 and N accordingly. To compute from the data we assume that between 1790 and 1820 exponential growth dominates dN dt = N. Calculate from the solution of this ODE with initial condition N ( t ) = N and the data for 1820. (c) Use the solution of the full logistics equation and the data for the year 2000 to calculate the carrying capacity K . (d) Find as much census data for additional years as possible from the internet and plot the data points together with the full solution and pure exponential growth. What could be reasons for the deviations?...
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