Homework3Sol

# Homework3Sol - Homework 3 AMATH 383 Autumn 2009 Due Monday...

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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 = αN (1 - N K ) and use the data to compute α and K . (a) Show that the general solution of the ODE with initial condition N ( t 0 ) = N 0 is given by N ( t ) = K 1 + ( K N 0 - 1)exp( - α ( t - t 0 )) (b) Let’s take t 0 = 1790 and N 0 accordingly. To compute α from the data we assume that between 1790 and 1820 exponential growth dominates dN dt = αN. Calculate α from the solution of this ODE with initial condition N ( t 0 ) = N 0 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? (e) Show that the model predicts a population of 300 million for the year 2022. Solution: (a) As before N ( t ) = K Ce αt 1 + Ce αt = K 1 + ˜ Ce - αt N ( t 0 ) = K 1 + ˜ Ce - αt 0 ! = N 0 K N 0 - 1 = ˜ Ce - αt 0 ˜ C = e αt 0 ( K N 0 - 1) N ( t ) = K 1 + ( K N 0 - 1) e - α ( t - t 0 ) 1

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Homework, AMATH 383, Autumn 2009 (b) N ( t ) = N 0 e α ( t - t 0 ) ( t 0 ,N 0 ) = (1790 , 3 . 9 × 10 6 ) N ( t 1 ) = N 0 e α ( t 1 - t 0 ) ! = N 1 t 1 - t 0 = 30 N 1 N 0 = 9 . 6 3 . 9 = 2 . 46 e α · 30 = 2 . 46 α = 1 30 ln2 . 46 = 0 . 03 (c) N ( t 2 ) = K 1 + ( K N 0 - 1) e - α ( t 2 - t 0 ) ! = N 2 K N 2 = 1 + ( K N 0 - 1) e - α ( t 2 - t 0 ) K ± 1 N 2 - 1 N 0 e - α ( t 2 - t 0 ) = 1 - e - α ( t 2 - t 0 ) t 2 - t 0 = 210 K = 1 - e - α ( t 2 - t 0 ) 1 N 2 - 1 N 0 e - α ( t 2 - t 0 ) = 3 . 232 × 10 8 (d) Deviation for example due to World War II. See the plots in appendix. (e) N ( t 3 ) = K 1 + ( K N 0 - 1) e - α ( t 3 - t 0 ) ! = N 3 with N 3 = 3 . 0 × 10 8 ﬁnd t 3 =? K N 3 - 1 = ( K N 0 - 1) e - α ( t 3 - t 0 ) t 3 - t 0 = - 1 α ln K N 3 - 1 K N 0 - 1 t 3 = 2022 . 4 2. (6+2+10+4+2 points) Regulated Harvesting: A simple model for ﬁsh harvesting reads dN dt = αN (1 - N K ) - H where N ( t ) is the ﬁsh population density and H is the harvest rate. (a) Assume the harvest is
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## This note was uploaded on 01/10/2010 for the course MATH 124 taught by Professor Walker during the Spring '08 term at University of Washington.

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Homework3Sol - Homework 3 AMATH 383 Autumn 2009 Due Monday...

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