EXPONENTIAL DECAY  Wikispacessystems2010.wikispaces.com/file/view/LV05.pdf · EXPONENTIAL DECAY...
Transcript of EXPONENTIAL DECAY  Wikispacessystems2010.wikispaces.com/file/view/LV05.pdf · EXPONENTIAL DECAY...
EXPONENTIAL DECAY
Rule for the halflife:
decay rate(%/yr) * halflife (years) = 70
Exponential decay at 6.9%/yr
t.... timer.... rate of decay
Y(T) = ert
halflife
EXPONENTIAL GROWTH
t.... timer.... growth rate
Y(T) = ert
doubling time
Exponential growth at 6.9%/yr
Rule for doubling time:
growth rate(%/yr) * doubling time (years) = 70
SYSTEMS THEORY BASICS11.03.2010
Christina Morgenstern, PhD
INFORMATION FEEDBACKAND
CAUSAL LOOP DIAGRAMS
CAUSAL LOOP DIAGRAMS
Maps of cause and effect relationships
Causal loop diagrams portray feedback at work in a system
Words = variables
Arrows = causal connections
Reinforcing (positive, +) loop
Balancing (negative, ) loop
Ford, Modeling the environment 2009
closed chain of cause and effect
POSITIVE FEEDBACK
Originates in control engineering
+ labelling: 2 variables change in the SAME direction
Can lead to growth in the system
If there are no negative arrows
If there is an even number of negative arrows
Ford, Modeling the environment 2009
NEGATIVE FEEDBACK
Originates in control engineering (stable control of electrochemical systems)
 labelling: 2 variables change in the opposite direction
Goalseeking process
If there is an odd number of negative signs
Ford, Modeling the environment 2009
FEEDBACK CONTROL IN A HOME HEATING SYSTEM
Ford, Modeling the environment 2009
Two coupled negative feedback loops are striving to reach different goals
COUPLED LOOPS
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DRAWING CAUSAL LOOP DIAGRAMS
Start with stocks and flows
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REVEALING FEEDBACK LOOPS
Ford, Modeling the environment 2009
Add arrows to explain
flows
Flow
Stock
CREATING CAUSAL LOOP DIAGRAMS IN STELLA  I
Use modules and connectors to draw loop diagram
Place text box in middle of loop and
label
Add polarity to connecting arrows (right click)
Ford, Modeling the environment 2009
CREATING CLD FROM EXISTING MODELS
Loop pad tool on Interface
WHY WE DRAW CAUSAL LOOP DIAGRAMS
To see feedback loops that determine dynamic behaviour
Same array of feedback loops creates same behaviour (archetypes)
Diagrams for communication NOT for simulation
A is for Acquainted: Get acquainted with the system and the problem
B is for Be Specific: Be specific about the dynamic problem
C is for Construct: Construct the stockandflow diagram
D is for Draw: Draw the causal loop diagram
E is for Estimate: Estimate the parameter values
R is for Run: Run the model to get the reference mode
S is for Sensitivity: Conduct a sensitivity analysis
T is for Test: Test the impact of policies
THE DOWNTURN OF CAUSAL LOOP DIAGRAMS
Do not distinguish between information and noninformation flows
Don’t reveal system parameters (net rates, hidden loops, nonlinear relationships)
Can’t predict dynamic behaviour
Necessity of simulation!
System dynamics modelling involves identification, mappingout and simulation of system’s stocks,
flows, feedback loops and nonlinearities.
Exercises 2
THE IMPACT OF FEEDBACK
Sshaped growth: positive and negative structures fight for dominance leading to long term equilibrium.
Loop dominance
Early years positive loop drives exponential growth
As systems fills space available dominance shifts
Equilibrium
Flowers model
Epidemic model
+

Ford, Modeling the environment 2009
IDENTIFYING LOOP DOMINANCE
Mathematical methods for identifying loop dominance (uncovering structurebehaviour relationships)
Eigenvalue elasticity analysis (EEA)
Pathway participation metric (PPM)
Statistical screening
EIGENVALUE ELASTICITY ANALYSIS
Eigenvalue elasticity
JW ForresterA measure of sensitivity of behaviour to parameter values
A large elasticity meansThat causal link is important for dynamics
Causal links with large elasticities may form loops (dominant feedback loops)
Downturns:Rigor mathematical foundation
Requires identification of all loops and links that pass through model
Fails to relate dominant structure to variable of interest
PATHWAY PARTICIPATION METRIC
Majtahedzadeh 1997
Incorporated in software Digest
Pathways between two state variables are considered as the primary building blocks of influential structure
Combination of pathways define influential system structure
Downturns:Identifies only a single feedback loop at any time (but many generate behaviour)
Does not capture modelwide dynamics
STATISTICAL SCREENING
Ford and Flynn, 2005
Identification of parameters most strongly correlated with model outputs at different times of simulation
Relies on efficient sampling methods to learn behavioural tendencies in a limited number of simulations
Simulations are exported to a spreadsheet to learn the most important inputs to the model
SSHAPED GROWTH
Flowered area: 10 acres
Empty area: 990 acres
Total area: use summer function
Decay rate: 20%/year
Intrinsic growth rate: 100%/year (no resource limit)
Growth rate multiplier a graphical function with fraction occupied
Simulate for 20 years
Graph to display: flowered area (01000 acres) and growth and decline (0400 acres)
Ford, Modeling the environment 2009
Why did the flowers not expand to the entire area?
EQUILIBRIUM DIAGRAM
Keep model  will be extended
Numerical display
A snapshot of the system at one
point in simulation
Ford, Modeling the environment 2009
EQUILIBRIUM
State of a system in which competing influences are balanced
Conditions remain constant over time (equilibrium)
Stability of equilibrium?
Test for stability using computer simulation
Stable equilibrium
Unstable equilibrium
Neutral equilibrium
Ford, Modeling the environment 2009
UNDERLYING MATHEMATICAL EXPLANATION
Logistic equation
A(t) = A0ert/ (1+(A0/K) * (ert 1))
A(t).... area of flowers as a function of timet.... time in yearsA0.... 10 acres at the start of the simulationr.... net growth rate at the start (intrinsic growth rate  decay rate = 0.8%)K = 800 acres, the area shown at the end of the simulation
Widely used mathematical expression in ecology and population biology
One of many versions of Sshaped growth
Problem of defining K (a.k.a. carrying capacity)
OTHER OUTCOMES...
Overshoot and collapse
Oscillations
Reverse Sshape behaviour
CC... carrying capacity
HOMEOSTASIS
State of equilibrium in organism
Walter Cannon (The Wisdom of the Body, 1932)
1. Constancy in an open system, such as our bodies represent, requires mechanisms that act to maintain this constancy. (Regulation of steadystates: glucose concentrations, body temperature and acidbase)
2. Steadystate conditions require that any tendency toward change automatically meets with factors that resist change. (An increase in blood sugar results in thirst as the body attempts to dilute the concentration of sugar in the extracellular fluid).
3. The regulating system that determines the homeostatic state consists of a number of cooperating mechanisms acting simultaneously or successively. (Blood sugar is regulated by insulin, glucagons, and other hormones that control its release from the liver or its uptake by the tissues).
4. Homeostasis does not occur by chance, but is the result of organized selfgovernment.
CONTROL MECHANISM
Receptor
Control center (brain)
Effector
Biology, Campbell
RESPONSE
Blood platelet accumulation
Oxytocin release during child birth
Blood pressureTemperature control
Biology, Campbell
SIMILAR SYSTEM STRUCTURE
Biology, Campbell
BLOOD PRESSURE CONTROL
Ford, Modeling the environment 2009
THE IMPACT OF HOMEOSTASIS
Principles of stabilisation apply to systems beyond physiology
Ecology (Howard T. Odum, 1954  2002)
Environmental systems will likely arise from a combination of negative feedback loops working in tandem
Consider BOTH positive and negative feedback to build understanding of (environmental, social, economic) systems
“Really good homeostatic control comes only after a period of evolutionary adjustment. New ecosystems (new type of agriculture) or
new hostparasite assemblages tend to oscillate more violently and to be less able to resist outside perturbation as compared with mature systems
in which the components have had a chance to make mutual adjustments to each other” Odum
HOMEOSTATIC PLATEAU/ SPAN OF CONTROL
Input within span of control > homeostatic process maintain control
Negative feedback responsible for control
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Bo
dy c
ore
te
mp
ambient temp
Shivering Sweating
EXAMPLES FOR SPAN OF CONTROL
Example External factor Internal variable (y axis)
Outside the span of control
Body temperature
Ambient temperature (two sided)
Core temperature Runaway behaviour
Home heating Outdoor temperature (one
sided)
Temperature inside the house
No control
Blood loss Size of wound (one sided)
Blood pressure Runaway behaviour
INTRODUCING RANDOMNESS
Environmental systems are exposed to external inputs that vary in an unpredictable fashion (random)
Randomness implies lack of predictability
Creates irregularities which appear to be superimposed on the underlying trend
Stochastic simulations (with values drawn from statistical distributions)
Good test for model behaviour
STOCHASTIC SIMULATION OF FLOWER MODEL
Low temp: 20 degCHigh temp: 30 degC
Seed: any value
Degrees C Growth rate
20 0
21 0.2
22 0.7
23 1
24 1.2
25 1.2
26 1.2
27 128 0.7
29 0.2
30 0
WHAT IS THE DIFFERENCE
TO THE PREVIOUS
SIMULATION?
Lower equilibrium due to temperature
variation across a wider range and reduction of intrinsic growth rate
WHAT WOULD HAPPEN IF RANDOMNESS IS DECREASED?
Ford, Modeling the environment 2009
1: temp: 2030°C, average intrinsic growth rate: 0.67/yr2: temp: 2228°C, average intrinsic growth rate: 1.0/yr3: temp: 2327°C, average intrinsic growth rate: 1.2/yr