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Unformatted text preview: Mathematics of Growth and Human Population 1 Growth and Populaiton Growth Rate ! Exponential Growth ! Halflife and Doubling Times ! Disaggregated Growth Resource Consumption Logistic and Gaussian Growth Models Human Population Growth ! Birth, Death, Fertility Rates ! Age Structures 2 Growth Rate Growth rate enables prediction of future sizesimportant for decisionmaking ! Fuel usage and air pollution ! Improvements in energy efficiency ! Population growth and water demand ! Deforestation rates and global effects ! Cost and cleanup time of accidental contamination 3 Exponential Growth Growth rates are proportional to the present quantity of people, resources, etc. Example: Number of students in a school increases by 2% each year. N = starting number of students N t = number of students in t years r = annual growth rate Year zero = N Year one = N 1 = N + rN = N (1 + r ) Year two = N 2 = N 1 + rN 1 = N 1 (1 + r ) = N (1 + r ) 2 Year three = N 3 = N 2 (1 + r ) = N (1 + r ) 3 ...Year t = N t = N (1 + r ) t 4 Year t = N t = N (1 + r ) t Exponential law for periodic increments of growth discrete increases at the end of each time period Example If the school has 1500 students now and the Board of Education decides to increase the student body by 2% every fall, how many students will there be in 7 years? 5 Year t = N t = N (1 + r ) t Year 7 = N 7 = 1500(1 + 0.02) 7 t = 7 yr N = 1500 students r = 0.02 = 1723 students A more realistic model uses continuous growth over time, with the growth rate again proportional to population size N ... 6 r is the growth rate with units 1/time dN dt = r ! N N = N e rt 7 Example From 1990 to 1997, installed wind power in the U.S. grew by about 0.15% per year, resulting in a wind capacity of 1700 MW in 1997. If these rates are sustained , what will the wind energy capacity be in 2008? N = 1700 MW, r = 0.0015 yr 1 , t = 11 years N = N e rt N = 1700 MW ! e 0.0015 yr " 1 i 10 yr ( ) = 1728 MW 8 Actual growth of wind energy from 1998 to 2003 was about 23.1%, with total capacity in 2003 of 6374 MW. What is the new estimate of wind energy capacity in 2008? 19549 MW this June; 9022 MW under construction N = 6374 MW ! e 0.231 yr " 1 i 5 yr ( ) = 20231 MW 9 Total Energy Consumption (quadrillion BTUs) 2004 2030 20062030 growth rate Petroleum 40.06 43.99 0.4% Natural Gas 22.30 23.39 0.2% Coal 22.50 29.90 1.2% Nuclear 8.21 9.50 0.6% Hydropower 2.89 3.00 0.2% Biomass 2.50 5.51 3.3% Other Renewable 0.88 2.45 4.4% Other 0.19 0.21 0.3% Total 99.52 118.01 0.7% Source: www.eia.doe.gov/oiaf/pdf/tables.pdf 10 Example Decay instead of growth If the present mountain lion habitat in California is 8400 km 2 , and is shrinking by 5% per year, when will it reach 5000 km 2 ?...
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 Spring '09
 Jeffery

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