## Buildng Tools: Growth

Growth: Exponential, Convergent, Logistic: How much of this will my students this semester know? How much of this will I have to remind them? And how much of this will I have to teach them for the first time?

The Uses of Math

• Muḥammad ibn Mūsā al-Khwārizmī (c. 780-850): Al-Kitāb al-Mukhtaṣar fī Hisāb al-Jabr wa’l-MuḳābalaThe Compendious Book on Calculation by Completion and Balancing
• Isaac Newton (1642-1727): Philosophiæ Naturalis Principia MathematicaMathematical Principles of Natural Philosophy
• Arithmetic and accounting
• Algebra and calculus
• What-if machines—ways of doing a huge number of potential calculations all at once…

Exponential Growth

• dy/dt = g(y - a)
• does nothing for a long time—stays very near a—then explodes
• And keeps on exploding…
• Rules of thumb for an annual growth rate g:
• (y-a) doubles every 0.693/g years
• (y-a) grows a thousandfold every 6.91/g years

Exponential Convergence

• dy/dt = g(k - y)
• and then stays there
• (k-y) halves in… guess what? 0.693/g
• (k-y) shrinks to a thousandth of its initial value in… guess what? 6.91/g

Combine the Two: Logistic Growth

• Math
• dy/dt = g(y-a)(k-y)/k
• y = a + (k-a)[exp(gx)]/(k-a+exp(gx)-1)
• a is the initial population
• k is the carrying capacity
• g is the unimpeded growth rate (you’ll see this called “r”)
• Pierre-Francois Verhulst in 1838, building a mathematical model of Thomas Malthus’ Essay on the Principal of Population
• Rediscovered by McKendrick, by Pearl and Reed, and by Lotka

Logistic Growth: Things to Remember

• Asymptote: a (in the negative direction, for growth and logistic)
• Asymptote: k (in the positive direction, for convergence and logistic)
• Rule of 72: 72 divided by the growth rate gives you the doubling (for growth) or halving (for convergence) time
• Rule of 720: Multiply the doubling time by 10 to get the thousand-fold time
• Why 72? Why not 0.693?