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The Logistic System X next

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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

this years population

Next years population

Population Growth and the Gypsy Moth

rate of growth

Xnext

=

r

X

this years population

Next years population

Human population growth curve

Population Growth and the Gypsy Moth

Negative

feedback

Positive

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)

(1-X)

X next = rX

X = population size- expressed as a fraction between 0 (extinction) and 1 (greatest conceivable population).

X nextis 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

Xnext :A Model of Deterministic Chaos

(A.k.a. the Logistic or Verhulst Equation)

(1-X)

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.

X = .02 and r = 2.7

X next = rX (1-X)

X next = (2.7) (.02) (1-.02 = .98)

X next = .0529

Modeling an Evolutionary System

Xnext and Deterministic Chaos

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

X next = rX (1-X)

Equilibrium state

.65

.64

.62

.62

.61

.60

.58

.35

X = .02 and r = 2.7

X next = rX (1-X)

X next = (2.7) (.02) (1-.02 = .98)

X next = .0529

.13

.05

.02

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 = 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.