Strategies and Rubrics for Teaching Chaos and Complex

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 Xnext

Population Growth and the Gypsy Moth

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

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

Population Growth and the Gypsy Moth Positive feedback Negative feedback Xnext = r X (1 -X) Equilibrium state 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.

Modeling an Evolutionary System Xnext : A Model of Deterministic Chaos (A. k. a. the Logistic or Verhulst Equation) X next = r. X (1 -X) X = population size - expressed as a fraction between 0 (extinction) and 1 (greatest conceivable population). X is what happens at the next iteration or calculation of the equation. Or, it is the next generation. next 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 Xnext : A Model of Deterministic Chaos (A. k. a. the Logistic or Verhulst Equation) X next = r. X (1 -X) 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 Xnext and Deterministic Chaos X next = r. X (1 -X) . 65. 64 . 62 Equilibrium state . 60. 61 . 58 . 62 Iteration X Value 0 0. 0200000 1 0. 0529200 2 0. 1353226 3 0. 3159280 4 0. 5835173 5 0. 6561671 6 0. 6091519 7 0. 6428318 8 0. 6199175 9 0. 6361734 10 0. 6249333 11 0. 6328575 12 0. 6273420 13 0. 6312168 14 0. 6285118 15 0. 6304087 16 0. 6290826 17 0. 6300117 18 0. 6293618 44 0. 6296296 45 0. 6296296 46 0. 6296296 47 0. 6296296 48 0. 6296296 49 0. 6296296 50 0. 6296296 . 35 . 13. 02 . 05 X =. 02 and r = 2. 7 X next = r. X (1 -X) X next = (2. 7) (. 02) (1 -. 02 =. 98) X next =. 0529

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

Experimenting With Xnext and Deterministic Chaos X next = r. X (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 At high ‘r’ values the behavior of the system becomes inherently Predictable unpredictable.