Introduction to Systems Thinking

Stocks, Flows and GraphsHere we qualitatively introduce some key features of system dynamics before becoming more quantitative in the following chapter. We begin here with an introductory discussion of stocks, flows, equilibrium, and graphs. Time delays, and feedback processes are also central to system thinking but will be discussed at a later time.A stock is a system component that expresses the amount of something stored in a system. Examples of stocks include bank balance ($), water volume in a tank or lake (ft3), energy content (Joules), population size (pure number), and atmospheric carbon loading (GT or GegaTonn). The symbol for a stock used by the Stella modeling environment is a box see Figure 1. Whenever possible, as in this case, we will adopt the Stella II icons for our qualitative discussions as well.

Figure 1. The symbols used by Stella II and by us to represent a stock (left) and flow(right).Flows are what changes a stock and are expressed in units of stock per unit time. The symbol that well use for a flow is designed to look like a valve and is shown in Figure 1. Monthly deposits ($/month), river discharge into a lake (ft3/min read as cubic feet per minute), rate of solar radiation capture by a collector in kWatts (1000 J/s), net birth rate (1/yr), and carbon emissions in (Gtonn/yr) are all examples of flows that could be linked to the respective stocks bank balance, water volume in a tank or lake, energy content, population size, or atmospheric carbon loading.Q1: Come up with your own example of a stock and a corresponding flow for that stock. Include units.

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Figure 2. Water flow and accumulation into a tank.

Stocks and flows go together. Flows represent how fast a stock changes or the rate of change of the stock per unit time. For example, assume that the stock of interest is water accumulation in a tank and the flow is the flow rate of water from the inlet pipe; see Figure 2. To give us a feel for the graphical relationship between stocks and flows , four graphs of water inflow and the corresponding water volume are shown in Figure 3 a-d. Graphs are one of the most prevalent tools for a systems thinker. They compile large amounts of information in a visual format so that trends and patterns may be easily recognized.Idea 1: Graphically, the slope or steepness of the stock vs. time graph equals the net flow. If the water volume stays constant, as in Figure 3a (right), then the net flow is zero; Figure 3a (left). If the water volume steadily increases, as in Figure 3b (right), then the net flow is constant; Figure 3b (left). By steadily increasing we mean that the slope of the water volume vs. time graph is constant so the water volume vs. time graph is a straight line. If the water volume increases at a slower and slower rate, as in Figure 3c (right), then the net flow is getting smaller; Figure 3c (left). Increasing at a slower and slower rate means that the slope of the water volume vs. time graph is uphill but getting less and less steep. If the water volume increases at a faster and faster rate, as in Figure 3d (right), then the net flow is getting larger; Figure 3d (left). Increasing at a faster and faster rate means that the slope of the water volume is uphill but getting less and less steep.

Figure 3 a-d. Four graphs of water inflow (left) and the corresponding water volume (right).Q2: Describe in your own words how the water accumulates in the tank for the figure below.

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On the axes provided, sketch a plausible flow curve for the behavior shown in the water volume (stock graph) at right.

Figure for Q2.Another way of looking at the relationship between stocks and flows is that a stock stores the flow history. Using this idea we can describe the relationship between the stock and flow graphs in Figure 3 from a different perspective. When the net flow is zero as in Figure 3a (left), then the water volume remains constant as in Figure 3a (right).

When the net flow is constant, as in Figure 3b (left), then the water volume steadily increases (increases linearly), as in Figure 3b (right). If the net flow is getting smaller (Figure 3c, left) then the water volume increases at a slower and slower rate (Figure 3c, right).

If the flow is getting larger (Figure 3d, left) then the water volume increases at a faster and faster rate (Figure 3d, right).

The two graphs of Figure 3b help illustrate a more general graphical relation between a flow and its stock. The change in stock over a time interval represents the accumulation of the flow for that time interval.Idea 2: Graphically, the Area underneath the flow vs. time graph between two times is equal to the change in stock over that time interval.In Figure 4 we show that the area under the flow graph (for constant flow) from 5 to 10 seconds is the same as the volume change from 5 to 10 seconds. Here area is essentially the height of the rectangle in units of L/s multiplied by the width in s and is highlighted in Figure 4. Although we show this area change in stock relation for this simple case of a constant flow graph, this relationship holds no matter how complicated the flow graph is.

Figure 4. The relationship between area underneath flow graph (left) and the corresponding change in stock.Idea 1 and idea 2 seem like two different ideas but, from a mathematical perspective, they are really just two different ways of saying the same thing. Our goal here is not to become overwhelmed with details, but to build some intuition regarding the relationship between stocks and flows.Q3: Describe in your own words how the water accumulates in the tank for the figure below.

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Q3a: Does the tank water volume always increase for the flow given in the figure below?Ans:

Q3b: How much water is in the tank after 20 minutes? Assume that the tank starts out with 100L (at t=0)

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On the axes provided, sketch a plausible water volume curve for the behavior shown in the water flow at left. Assume that the tank starts out with 100L (at t=0)

Often two flows compete against each other as is the case for a water tank with inflow and out flow shown in Figure 5, or a bank balance with deposits, interest gained, and withdrawals.

Figure 5. A tank with an inflow and outflow influencing the water volume in different directions.In the case of the water tank with both inflow and outflow the net inflow must be considered not just the inflow.

Net inflow=(inflow-outflow)When the net inflow is:

Positive : the tank volume increases; upward positive slope volume vs time graph

Zero: the tank volume stays the same; flat zero slope volume vs time graph. Negative: tank volume decreases; downward negative slope vs time graph.The generic structure of a stock with competing inflows and outflows is a basic description of many processes in the natural world. For example we may be trying to model the phytoplankton abundance as a sub-model to a model of the sardine industry and how it is influenced by El Nino cycles in the equatorial pacific. The structure in figure 6 would be a good place to start. Of course the birth rate and death rates will fluctuate over the course of the year and also depend on a variety of environmental parameters such as solar intensity, water temperature, zooplankton abundances, etc. but the basic structure shown in Figure 6 will be at the core of your modeling effort.

Figure 6. The core of a simple phytoplankton sub-model.Another example is a possible sub-model for a forest management and tree harvest model. Figure 7 shows the mature tree component of this model. An additional outflow could be added to this structure to account for the possibility of tree loss by disease. The trees cut per year will depend on specific management practices and decisions that are likely related to the availability of mature trees. The maturation rate will depend on how many healthy young trees are available which also depend on forest management decisions.

Figure 7. A sub-model for a larger forest parcel management model.

A simplified model of Earths surface energy balance is shown in figure 8. Earths surface energy content is directly linked to Earths surface temperature so this core sub-model would be at the heart of a climate simulation model or weather forecast model. An additional outflow should be added in a more detailed version of this sub-model to account for vertical convection as warm air rises from the surface. However this basic model structure can help us understand several basic observations in our everyday life.

Figure 8. The flow of energy to and from Earths surface.

Figure 9 below provides us with supporting data for each component of Figure 8. The importance of data in model development and in testing model performance cannot be over emphasized.

Figure 9. (top) diurnal temperature fluctuation. (bottom) incoming solar radiation and outgoing terrestrial radiation. The NASA link: http://earthobservatory.nasa.gov/Experiments/PlanetEarthScience/GlobalWarming/GW.php provides some basic background in two videos 1) radiation from the Earth and Sun and 2) Earths Energy Balance.Because the sun rises and sets at 6 Am and 6 PM the values shown in figure 9 must be for a day near Fall or spring equinox or a location near the equator where day and night are 12 hours long all year. Like a cooling cup of coffee, the outgoing radiation depends on how hot the surface is so it follows the upper temperature graph quit closely. The question posed in the figure why lag? essentially asks why is it not hottest when the incoming solar radiation is maximum? Notice that the temperature continues to rise until about 4 PM. Thats when the inflow just equals the outflow and thats when the day becomes hottest. From about 6 AM to 4 PM the inflow of solar energy is larger than the outflow of terrestrial radiation so the temperature increases. From about 4 PM to 7 AM the outflow of energy is greater than the inflow so the temperature decreases. AT 7 AM (and 4 PM) the temperature momentarily stops changing as the inflow matches the outflow. Whenever any stock (in this case energy) is either maximum or minimum there is a balance of inflow and outflow. One way of looking at this is that just before the maximum temperature the inflow was larger than the outflow (increasing temperature) and just after the maximum the inflow is less than the outflow (decreasing temperature). So right at the maximum (or minimum) there is an energy balance.

Q4: When (at what specific times or during what time interval) is the slope of the temperature graph in Figure 9:Positive?Zero?Negative?Q5: What is the net inflow of energy to earths surface when the temperature is maximum?

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Q6: What is the net inflow of energy to earths surface when the temperature is minimum?

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Q7: The graph below shows the number of people entering and leaving a department store over a 30 minute period.

1. During which minute did the most people enter the store?______________2. During which minute did the most people leave the store? ______________3. During which minute were the most people in the store? ______________4. During which minute were the fewest people in the store? ______________When conditions are such that the net inflow to a stock is zero, then the stock is in equilibrium. Equilibrium may be sustained, if the net inflow remains zero for some length of time, or fleeting, if the net inflow is zero only for an instant; see figure 10. Fleeting equilibrium points are often associated with maximum and minimum stock values as the net inflow changes from positive to negative or negative to positive. Whenever the slope of the stock vs time graph is zero the net inflow is also zero and the stock is in equilibrium.

Figure 10. Points A and B correspond to when the stock is temporarily in equilibrium. The stock is in sustained equilibrium after point C.

The sub-model shown in Figure 8 can also help explain why the difference between the daily high temperature and daily low temperature is larger for clear cloud-free days than for cloudy days. On a clear day the solar energy reaches Earths surface without interference from clouds so the surface gains relatively large amounts of solar energy during the day. Also on a clear night, after the sun goes down, the terrestrial radiation leaves Earths surface unimpeded by clouds and so Earths surface looses energy very efficiently at night. This conceptual model suggests why it gets relatively hot during the day and cold during the night when skies are clear compared to overcast.

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Q4: When (at what specific times or during what time interval) is the slope of the temperature graph in Figure 9:

Positive? From 7AM to 4 PMZero? 7AM, 4 PMNegative? From 4 PM to 7 AMQ5: What is the net inflow of energy to earths surface when the temperature is maximum? ZeroQ6: What is the net inflow of energy to earths surface when the temperature is minimum? Zero