exponential decay - wikispacessystems2010.wikispaces.com/file/view/lv05.pdf · exponential decay...

EXPONENTIAL DECAY

Rule for the half-life:

decay rate(%/yr) * half-life (years) = 70

Exponential decay at 6.9%/yr

t.... timer.... rate of decay

Y(T) = e-rt

half-life

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


NEGATIVE FEEDBACK

Originates in control engineering (stable control of electrochemical systems)

- labelling: 2 variables change in the opposite direction

Goal-seeking process

If there is an odd number of negative signs


FEEDBACK CONTROL IN A HOME HEATING SYSTEM


Two coupled negative feedback loops are striving to reach different goals

COUPLED LOOPS


DRAWING CAUSAL LOOP DIAGRAMS

Start with stocks and flows


REVEALING FEEDBACK LOOPS


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)


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 stock-and-flow 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 non-information flows

Don’t reveal system parameters (net rates, hidden loops, non-linear relationships)

Can’t predict dynamic behaviour

Necessity of simulation!

System dynamics modelling involves identification, mapping-out and simulation of system’s stocks,

flows, feedback loops and non-linearities.

Exercises 2

THE IMPACT OF FEEDBACK

S-shaped 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

+

-


IDENTIFYING LOOP DOMINANCE

Mathematical methods for identifying loop dominance (uncovering structure-behaviour 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 model-wide 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

S-SHAPED 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 (0-1000 acres) and growth and decline (0-400 acres)


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


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


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 S-shaped growth

Problem of defining K (a.k.a. carrying capacity)

OTHER OUTCOMES...

Overshoot and collapse

Oscillations

Reverse S-shape 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 steady-states: glucose concentrations, body temperature and acid-base)

2. Steady-state 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 self-government.

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


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 host-parasite 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


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?


1: temp: 20-30°C, average intrinsic growth rate: 0.67/yr2: temp: 22-28°C, average intrinsic growth rate: 1.0/yr3: temp: 23-27°C, average intrinsic growth rate: 1.2/yr

exponential decay - wikispacessystems2010.wikispaces.com/file/view/lv05.pdf · exponential decay...

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