The Mathematics of Sustainability GUTS Project Report -- Spring 2011 • GUTS Team: • Mary Farina ―A Sampling of Mathematical Models‖ (CAS: Biology & Environmental Analysis and Policy) • Michael Goldberg ―An Inquiry into Google Maps‖ (CAS: Computer Science and Mathematics) • Jason Abed ―Determining the Profitability of Environmental Sustainability‖ (SMG and CAS: Computer Science) • GUTS Advisor: • Emma Previato, Department of Mathematics and Statistics • Our team's project consists of exploring and designing an untraditional course experience based on mathematical techniques devoted to issues of environmental sustainability. • The course is interdisciplinary; the common threads are mathematical models and methods for quantifying, discussing and resolving sustainability issues. The experience is participatory: students learn through case studies, by working in teams. • Our project's goal is to explore some of these cases and research the mathematical methods that contribute to the solution. We stopped short of developing thorough mathematical backgrounds but provided a guide for implementing these cases in the course. Introduction Photo credit: Mary Farina How can exponential and geometric growth be used to model the population size of different species? Citation: Owen-Smith, Norman. ―Descriptive Models.‖ Introduction to Modeling in Wildlife and Resource Conservation. Blackwell Publishing, 2007. Pages 14 to 43. Mary Farina • Geometric growth is used to project the population size of species which reproduce a discrete number n times per year. nt • Nt is the population size after t years: Nt = N0(1 + r/n) • Example: Species reproduces once per year • n = 1 t • Nt = N0(1 + r) • Annual population growth rate = r • Nt ∙ r = ΔN/Δt Photo credit: Mary Farina • ΔN represents the change in population size over some discrete time interval Δt (in this case, the time interval is one year). • As the time step is made infinitesimally small, the geometric growth becomes exponential growth. Geometric Growth Mary Farina • Exponential growth is used to model populations which reproduce continuously throughout each year (for example, humans). • There are an infinite amount of growth cycles per year • The length of each generation is infinitesimally small • Rate of population growth: dN/dt = N ∙ r • Instantaneous population growth rate = r (slightly different than the r used for geometric growth) • Integrating this equation gives Nt, the future population size at any particular time t: rt Nt = N0 e Exponential Growth Mary Farina • In nature, populations rarely grow unconditionally. • They are usually limited by resource availability, predators, and other environmental factors. • The carrying capacity, K, is the maximum number of individuals that can be supported by the environment. • This variable can be factored into both exponential and geometric growth equations. • Adding the carrying capacity variable adds density dependence to the growth. This is called logistic growth. • As a population approaches its carrying capacity, its growth slows. When a population exceeds its carrying capacity, its growth becomes negative. Carrying Capacity Image source: http://wallpapers-diq.com/wp/15__Sumatran_Tiger.html Mary Farina • Example: The integrated discrete equation: • Nt = N0 (1 + r0 [1-N0/K]) • This equation models population growth in species that reproduce a discrete number of times per year. • There are Nt individuals after t generations. • This equation incorporates K, the carrying capacity. • This equation gives particularly interesting results as r, the rate of population growth, is increased. Carrying Capacity Mary Farina • When r < 1, the population increases smoothly to K in sigmoid graph. Population over time, r = 0.5 57 52 K = 50 47 42 37 32 Population 27 22 17 12 7 0 2 4 6 8 10 12 14 Time Mary Farina • When 1< r < 2, the graph fluctuates before it reaches K, overshooting and then undershooting K until it gradually reaches equilibrium. • As r increases, the population can reach the carrying capacity in fewer steps; it also means that the population can more easily exceed the carrying capacity. Population over time, r = 1.8 57 52 K = 50 47 42 37 32 Population 27 22 17 12 7 0 2 4 6 8 10 12 14 Time Mary Farina • As r increases further still, the fluctuations become a permanent, stable limit cycle; the graph never reaches equilibrium. Population over time, r = 2.5 67 57 47 37 Population 27 17 7 0 2 4 6 8 10 12 14 Time Mary Farina • When r > 2.7, the oscillations become chaotic! Chaotic patterns produced by the discrete time logistic equation are common in different branches of science, including ecology and meteorology. Population over time, r = 3 77 67 57 47 37 Population 27 17 7 0 2 4 6 8 10 12 14 Time Mary Farina • Dr. Kenneth Golden is a professor at the University of Utah who researches the microstructure of sea ice in order to better model water flows through ice; these water flow models are required build further models which estimate the melting rates of Arctic and Antarctic ice. • Because ice melt is so closely linked to climate change, Dr. Golden‘s work will ultimately improve global climate change models. Current models tend to underestimate the rate of glacier melt in the polar regions. Using Fractals to Model Sea Ice Microstructure Citation: Golden, Kenneth. ―Modeling Sea Ice in the Changing Climate.‖ Brown University. Providence, RI. 13 Mar. 2010. Lecture. Mary Farina • Sea ice is a composite material, composed of pure ice, brine, and salt. Dr. Golden has developed a computerized tomography of the brine microstructure. • The tomography is based on the idea of fractals, or structures which repeat on several scales. Dr. Golden has found that ice forms a multi-scale microstructure, and these scales differ by ten orders of magnitude. Sea ice microstructures resemble the pattern formed by the Sierpiński Triangle. Using Fractals to Model Sea Ice Microstructure Mary Farina • The Sierpiński Triangle is formed by applying the following pattern to an equilateral triangle with a side length of 1: Image source: http://mathworld.wolfram.com/SierpinskiSieve.html • If n is the number times the pattern is iterated, (with n =1 corresponding to the second triangle), the following formulas are true: n o Nn = the number of black triangles = 3 n -n o Ln = the length of a side of a triangle = (½) = 2 o An = the ratio of black triangles to total triangles = the fractional 2 n area that is black = Ln Nn = (¾) Using Fractals to Model Sea Ice Microstructure Citation: Sierpiński Sieve. Wolfram MathWorld. Accessed online, 19 May 2011. Available http://mathworld.wolfram.com/SierpinskiSieve.html. Mary Farina • The enormity of human consumption can be astounding. • Doubling times reveal rapidly increasing rates of resource depletion. • For example, the most recent doubling time of fossil fuel consumption is estimated at 20 years; in the past 20 years, humans doubled the total amount of fossil fuels ever burned! During this small 20 year window, humans used as much fossil fuel as had been used during all preceding millennia. Modeling Cumulative Consumption Citation: Herendeen, Robert A. Ecological Numeracy. New York: John Wiley & Sons, 1998. Print. Mary Farina • Cumulative consumption problems can be approached algebraically. • The sum of a geometric series can be used to calculate how long a finite resource will last. • Suppose that M tons of fossil fuel are available; during the first year, the fuel is used at a rate of C0 tons per year. However, annual consumption is growing geometrically at a rate r per year. How long will the resource last? • During the first year, C1 tons are used. • During the second year, C2 = C1(1 +r) tons are used. 2 • During the third year, C3 = C2(1 +r) = C1(1 +r) tons are used. • Total consumption for years 1 through t is the sum St = C1 + C2 + C3 + ∙ ∙ ∙ + Ct Modeling Cumulative Consumption Mary Farina • Assuming geometric growth: 2 3 t • St = C1 ∙ (1 + x + x + x + …+ x ) (where x = 1 + r) • The goal now is to find t such that the available stock M equals the cumulative consumption: 2 3 t-1 • M = St = 1 + x + x + x + …+ x 2 3 t • St+1 = 1 + x + x + x + …+ x t • St+1 = St + x = x∙St + 1 t • St = (1-x )/(1-x) Image source: http://www.etftrends.com/wp-content/uploads/2009/09/oil.jpg Modeling Cumulative Consumption Mary Farina • Substituting x with (r + 1), and multiplying by C0 tons per year, the rate at which we are starting, M will equal: t • M = C0 ∙([1+r] – 1)/r • Finally, t, the number of years that this resource will last, can be calculated: • t = ln(1 + rM/C0)/ln(1+r) Modeling Cumulative Consumption Mary Farina Mapping is an essential component in the study and practice of sustainability. When the disastrous earthquake struck, a delegation of BU students led by Professor Sucharita Gopal (CAS Geography and Environment) traveled to Haiti to help. They carried neither food nor water for the relief effort, but instead they helped to organize the relief efforts with carefully designed up-to-date relevant maps that they had created at BU. (See BU Today Article: http://www.bu.edu/today/node/10194 ) Mapping Michael Goldberg Problem: • The space required to store even a relatively low resolution image of the Earth is daunting, even on today‘s computers. • The Earth is approximately 40,000 km in circumference. With only one pixel of color data for every square kilometer of surface, a whole-earth image would be about 40,000 pixels wide and roughly half as tall—an image of 800 megapixels and 2.4 gigabytes at least. And this is just your basic run-of-the-mill one-kilometer-per-pixel whole-earth image. The smallest feature you could resolve with such an image is about 2 kilometers wide … no buildings, rivers, roads, or people would be apparent.