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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 sizes—important 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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This note was uploaded on 09/24/2009 for the course AO 104 taught by Professor Jeffery during the Spring '09 term at UCLA.
 Spring '09
 Jeffery

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