The Mathematics of Sustainability
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. • But for most major US cities, Google Earth deals in resolutions that can resolve objects as small as half a meter or less, at least four thousand times denser, or sixteen million times more storage than the above example. We‘re talking about images that would (and do) literally take many terabytes to store. There is no way that such a thing could ever be drawn on today‘s PCs, especially not in real-time. And yet it happens every time you run Google Earth.
An Inquiry into Google Maps
Citation: http://www.realityprime.com/articles/how-google-earth-really-work Michael Goldberg Google‘s Solution:
• As one would surely guess, Google does not serve their entire multi- terabyte world image to every user who wants to see a map of BU campus. • Instead, the entire globe is broken up into 512x512pixel tiles at 20 different levels of ―zoom,‖ in such a way that only the most relevant tiles are served to the user, transforming an implausible task into a plausible one. • A zoom factor ranges from 0 to 19. At 0, the entire Earth fits on a single tile, at 1, the area is divided into quadrants such that four tiles cover the earth, and at 2, each of these four tiles is further subdivided into four more. • In this way, each successive decrease in the zoom level quadruples the number of titles, such that there could conceivably be 419 ≈ 275 billion of the highest resolution (zoom 19) tiles. • Although this number is exceptionally large, it is within the bounds of what would be reasonable for Google to store on their servers, as long as the average user isn‘t asking to view every single tile.
An Inquiry into Google Maps
Citation: http://www.codeproject.com/KB/scrapbook/googlemap.aspx?display=Print Michael Goldberg Background Information: Understanding Types and Purposes for Mapping Projections
• Unless the product map in question is spherical in nature (i.e., a globe) some sort of projection is required to transform the original target subject into something mappable in two dimensions; the world, for instance, is not itself flat. • Google uses the Mercator projection for this task. The Mercator projection is popular for all kinds of navigation, dating back to the days of sailing with compasses, as it reproduces rhumb lines, or lines of constant course as one would follow using a compass, as straight lines on the resulting map. • Not all projections result in straight rhumb lines. In fact the Mercator and Lambert Conformal Conic Projections are two types of map notoriously good for navigation, specifically because they maintain the shape of features and angular relationships between them, with straight rhumb lines as a result. • There are at least two sub-types of Mercator projection depending on which line is used as a starting point, the transverse and oblique. The primary function of Google Maps is, of course, ―getting directions,‖ making the Mercator projection an obvious choice.
The following government website appears to be a good source on mapping projections: http://www.nationalatlas.gov/articles/mapping/a_projections.html
Background Information for Mapping
Michael Goldberg Background Information: Computer Jargon
• A Pixel is the smallest element of an image that can be individually manipulated or processed. If you consider an image as a grid of tiny squares of color, each smallest possible, intelligible square is a pixel. Modern computer screens typically display over one million pixels at once. • A megapixel is precisely one million pixels, and remains a common unit used for measuring the capability of modern digital cameras. For example, the best consumer-grade digital cameras available (as of May 2011) can produce images with 16 megapixels. • The resolution of an image (or screen display) describes the number of pixels it contains. For example a Full High Definition TV or computer monitor has a resolution of 1920x1080, that is, 1920 pixels wide and 1080 pixels high, displaying a total of 2073600 pixels, approximately 2 megapixels. • A bit is a unit for representing data commonly expressed as either the character 1 or 0. • A byte is eight consecutive bits, such that a byte can store any one of 28 = 256 different possible values (since there are two choices for each of the eight bits). • A gigabyte is a collection of 230 (approximately 1 billion) bytes, while a terabyte is 240 (approximately 1 trillion). For example, a DVD can store approximately 4 gigabytes, while a new computer hard drive may store 1 terabyte. Background Information for Mapping
Michael Goldberg Problem: • Corporations are now under substantial social pressure to adopt a ―triple-bottom line‖ (people, planet, profits) approach to making business decisions • Without questioning the importance of the triple-bottom- line (TBL) approach, corporations are aware that profitability is essential to their continued growth, and consequently their survival, as an organization. • Therefore, corporations must consider a method for determining which environmental sustaining decisions are beneficial to the company, versus those which are prohibitively detrimental to the company's profitability
Determining the Profitability of Environmental Sustainability
Jason Abed Potential Solution:
• Basic techniques can be used to develop a method to determine the profitability of any potential decision: 1) Identify Key Performance Indicators that affect profitability (whether directly or indirectly) 2) Develop a forecasting model and determine worst case, most likely case, and best case scenarios. 3) Determine the initial investment (initial costs) 4) Determine the estimated future cash flows, based on the probabilities and results of the forecast 5) Perform Discounted Cash Flow (DCF) Analysis, to determine the Net Present Value (NPV) of the project. 6) If NPV > 0, make the investment
Determining the Profitability of Environmental Sustainability
Jason Abed Background Information: NPV and DCF
• Initial Investment costs are considered as ‗Time Zero‖ and are not discounted. NPV is then calculated by adding the summation of all future values as such: 푁 퐹푉 푁푃푉 = 푡 1 + 푖 푡 푡=0 • NPV is thus the interest adjusted amount of all cash flows of a project—and is thus an ideal method for determining profitability
Determining the Profitability of Environmental Sustainability
Jason Abed Background Information: Finance Terminology 푁 퐹푉 푁푃푉 = 푡 • NPV = The Sum of the discounted cash flows: 1 + 푖 푡 푡=0 • Where N is the number of time periods (t) and i is the cost of capital or the interest rate at which the company acquires money (capital) to be used in projects • APR: Annualized Percentage Rate: The interest rate for a whole year (the rate, for a payment period, multiplied by the number of payment periods in a year) • EAR: Effective Annual Rate: The actual rate of interest when compounding occurs more often then once a year: = [1+ (i/n) ^ n ] -1 where i = the APR and n = the number of compounding periods • Balance Sheet Equation: Assets= Debt + Shareholders Equity Determining the Profitability of Environmental Sustainability
Jason Abed Background / Further Readings
• Banerjee, Preetha M., and Vanita Shastri. Social Responsibility and Environmental Sustainability in Business: How Organizations Handle Profits and Social Duties. Los Angeles: Response, 2010. Print. • "Corporate Social Responsibility." Wikipedia, the Free Encyclopedia. Web. 18 April 2011.
Determining the Profitability of Environmental Sustainability
Jason Abed