David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Review - STELLA Model Elements• Reservoirs – These are the default stock type– Think of a reservoir as an undifferentiated pile of stuff (many instances of the same stuff)– Reservoirs passively accumulate inflows minus outflows (they are simply containers)– Any units which flow into a Reservoir lose their individual identity - Reservoirs mix together all units into an undifferentiated mass as they accumulate• Flows – Function is to fill and drain stocks–To bend a flow pipe, depress the shift key and change the direction of mouse movement as you drag the flow. Each time you depress the shift key, a 90 degree bend will be put in the flow pipeDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Review - STELLA Model Elements•Flows Cont.– To draw an inflow to a stock, make sure that your cursor makes contact with the stock before you release the mouse button. The stock will turn gray on contact to let you know that it will receive the flow. If you release the mouse button prematurely, a cloud will appear at the destination end of the flow pipe –To replace a cloud with a stock, select the stock with the Hand tool. Drag the stock over the cloud. When the cursor (the tip of the index finger on the hand) is directly atop of the cloud, the cloud will turn gray. Release the mouse button, and the flow will be connected to the stock (the cloud will disappear)David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Review - STELLA Model Elements• Converters – These serve a utilitarian role in the software– They hold values for constants, define external inputs to the model, calculate algebraic relationships, and serve as repositories for graphical functions– In general, they convert inputs into outputs, hence the name "converter" • Connectors – These connect elements– There are two types of connectors available in STELLA– Action connectors are shown as solid, directed wires– Information connectors (which we most likely will not be using) are signified by a dashed wireDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Rules for Building Systems Models 1. Make systems diagrams as simple as possible2. Relationships between elements should be defined mathematically3. If a mathematical expression is not available, define relationships using graphs4. Observe the conservation law and maintain consistency in units5. Reservoir values can only be changed by inflows and outflowsDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Model Structures and Behavior Patterns• The systems modeler believes that the behavior of the system is a function of the system itself• Translating that idea to the model realm, this means that certain structures of elements should produce certain types of behavior patterns• We are going to look at five common behavior patterns and their associated structures:LinearGrowthor DecayExponentialGrowthor DecayLogisticGrowthOvershootandCollapseOscillationDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Linear Growth or Decay - Example• Consider the following example of a system:– An oil reserve contains 10,000,000 barrels of oil– The oil is consumed at a rate of 10,000 barrels per day• What would the system diagram look like here?•What entity changes with time here?•What is/are the process(es) that cause that change?•What determines the rate of change?•Because it is constant, no converter is requiredDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Linear Growth or Decay - Example• What is the difference equation for this system?Oil Reserves tomorrow = Oil Reserves Today – 10,000 barrelsor more generally:Oil Reserves in t days = Present Oil Reserves – 10,000 * t daysand shown mathematically:R(t+∆t) = R(t) – (10,000 * ∆t)• How will this system behave?– Reserve begins with 107barrels– Decrease 10,000 barrels per day– After 1000 days, reserve is empty– NOTE: Once empty, eqn. not validDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Linear Growth or Decay – System Features, Diagrams, and Equations• For a reservoir to exhibit linear growth or decay, the sum of all inflows minus the sum of all outflows to the reservoir must be constant–A positive constant indicates growth–A negative constant indicates decay– If the constant is zero, the reservoir content remains constantGeneric Linear System• System can have any number of inflows and outflow• Not all flows need be constant• It is necessary for the differencebetween the sums to be constantDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Linear Growth or Decay – System Features, Diagrams, and Equations• For our generic example, the difference equation is:R(t+∆t) = R(t) + {(Inflow 1 + Inflow 2 + Inflow 3) – (Outflow 1 + Outflow 2)} * ∆tor in general:R(t+∆t) = R(t) + {(Inflow 1 + … + Inflow n) – (Outflow 1 + … + Outflow n)} * ∆t• This can be rearranged as:Σi = 1nInflowi-Σj = 1nOutflowjR(t+∆t) - R(t) =* ∆tand divided by ∆t to give:nnΣi = 1Inflowi-Σj = 1OutflowjR(t+∆t) - R(t) =∆tDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Linear Growth or Decay – System Features, Diagrams, and Equations• We are now set up to find the instantaneous rate of change of the reservoir with respect to time by taking the derivative of the expression:R(t+∆t) - R(t) ∆tlim∆t Æ 0dR(t)dt== constant = knnΣi = 1Inflowi-Σj = 1Outflowj=• In a linear system, the value kis the slope of the line• We have positive values of kfor growth and negativevalues for decay• What would k be in the Oil Reserve example?David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Linear Growth or Decay – Feedbacks and Steady State ConditionsGeneric Linear System• The generic linear system contains no loops, therefore it cannot have any feedbacks• This system changes as a constant rate and has linear chain of cause and effect• For a system to be in a steady state, the rate of changeof the contents of the reservoir must be equal to zero• We have shown that the rate of change here is constant•In the Oil Reserve example, this is true until we run out of oil on day 1000, then the system is in steady stateDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Exponential Growth or Decay - Examples• Consider the following example of a system:– A pair of white mice escape from their cage– They mate and have offspring, which mature and then do the same, generation after generation …• What would the system