Strategies and Rubrics for Teaching Chaos and Complex Systems Theories as Elaborating, Self-Organizing, and Fractionating Evolutionary Systems Fichter, Lynn S., Pyle, E.J., and Whitmeyer, S.J., 2010, Journal of Geoscience Education (in press)

The Logistic System X next

Population Growth and the Gypsy Moth

X next Next years population =r X rate of growth this years population

Population Growth and the Gypsy Moth X next Next years population =r X rate of growth this years population Human population growth curve

Population Growth and the Gypsy Moth X next =r X(1-X) Positive feedback Negative feedback The logistic function or logistic curve models the S-curve of growth of some set P. The initial stage of growth is approximately exponential; then, as competition arises, the growth slows, and at maturity, growth stops. Equilibrium state

X next :A Model of Deterministic Chaos (A.k.a. the Logistic or Verhulst Equation) X next = rX X = population size - expressed as a fraction between 0 (extinction) and 1 (greatest conceivable population). X next is what happens at the next iteration or calculation of the equation. Or, it is the next generation. r = rate of growth - that can be set higher or lower. It is the positive feedback. It is the “tuning knob” (1-X) acts like a regulator (the negative feedback), it keeps the growth within bounds since as X rises, 1-X falls. Modeling an Evolutionary System (1-X)

X next :A Model of Deterministic Chaos (A.k.a. the Logistic or Verhulst Equation) X next = rX Logistic – population ranges between 0 (extinction) and 1 (highest conceivable population) Iterated – algorithm is calculated over and over Recursive – the output of the last calculation is used as the basis of the next calculation. Modeling an Evolutionary System (1-X)

X next and Deterministic Chaos X next = rX (1- X) X =.02 and r = 2.7 X next = rX (1-X) X next = (2.7) (.02) (1-.02 =.98) X next =.0529 Iteration X Value Modeling an Evolutionary System X =.02 and r = 2.7 X next = rX (1-X) X next = (2.7) (.02) (1-.02 =.98) X next = Equilibrium state

Last run at 20 generations But, what about these irregularities? Are they just meaningless noise, or do they mean something?

Experimenting With X next and Deterministic Chaos X next = rX (1-X) A time-series diagram

r = 2.7

r = 2.9

r = 3.0

r = 3.1

r = 3.3

r = 3.4

r = 3.5

r = 3.6

r = 3.7

r = 3.8

r = 3.9

r = 4.0

r = 4.1

Learning Outcomes 1. Computational Viewpoint: In a dynamic system the only way to know the outcome of an algorithm is to actually calculate it; there is no shorter description of its behavior. 2. Positive/Negative Feedback Behavior stems from the interplay of positive and negative feedbacks. 3. ‘r’ Values Rate of Growth, or how hard the system is being pushed. High ‘r’ means the system is dissipating lots of energy and/or information. 4. Deterministic does not equal Predictable At high ‘r’ values the behavior of the system becomes inherently unpredictable.