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  • 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

    Ford, Modeling the environment 2009

  • 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

    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

    Ford, Modeling the environment 2009

  • DRAWING CAUSAL LOOP DIAGRAMS

    Start with stocks and flows

    Ford, Modeling the environment 2009

  • 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 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

    Dont reveal system parameters (net rates, hidden loops, non-linear relationships)

    Cant predict dynamic behaviour

    Necessity of simulation!

    System dynamics modelling involves identification, mapping-out and simulation of systems 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

    +

    -

    Ford, Modeling the environment 2009

  • 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)

    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 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

    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 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

    Ford, Modeling the environment 2009

    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)

    Cor