Systems Blog

002: Dynamic Equilibrium

posted Apr 30, 2017, 12:07 PM by Jon Darkow   [ updated Sep 3, 2017, 9:44 AM ]

002: Dynamic Equilibrium


Carbon is accumulating in our atmosphere. NASA reported that the average global CO2 concentrations in the atmosphere during January 2017 were 405.34 ppm (ESRL web.) In 1958, when global CO2 concentrations in the atmosphere were first collected reliably and routinely, the concentration was 338.45 ppm (ESRL web.) Accumulations in a system with inflows and outflows often lead to misconceptions. Students tend to pattern match an accumulation in a stock with only the inflow.


Imagine the atmosphere. Beautiful right! Now carbon dioxide can be an inflow (emissions, respiration, volcanoes), and an outflow (photosynthesis, runoff, ocean absorption.) However, most people tend to pattern match how a stock changes, by the changes in either the inflow or the outflow (Stave, 2010.) For example, if emissions from fossil fuels or volcanoes increases, students assume the stock will accumulate carbon. In dynamic systems, outflows are not always constant. Rather, outflows can behave independently of inflows, or be regulated by changes in the stock or the inflows.


The explorable simulation below allows you to change the inflow rate and outflow rate over 30 years. By clicking on the graphical input by the inflow and outflow you can change the pattern of behavior. Also, in the graphical input below the stock predict how the stock will behave given your changes to the flows. This is fun. Predict the change in the stock’s behavior before you run the simulation to test your systems thinking skills.





Here is an assessment I give my students to test their systems thinking skills. Match flows A, B, and C with the stocks X, Y, and Z. The answer key is at the bottom of this post. While the system is simple, one input, one output, understanding their dynamics can be confusing. If we are to help people, appreciate the accumulations of carbon emissions, and the actions required to maintain sustainable levels of emissions, we need heuristics that can teach these systems. Having students experience systems by perturbing and creating computational models can help reduce the cognitive load necessary to become systems thinkers.


Figure 1: Assessment 1



Assessment 2:

Here is another assessment I have given to middle school students, high school students, and science teachers at a technology conference. In all cases, nearly 100% of the audience answers question 1 correctly, and around 15% of the audience answers question 2 correctly. This assessment is based on work from John Sterman and Krystyna Stave. Examine Figure 2 below. Answer the following questions to understand accumulations with dynamic sources and sinks.

  1. What year is the highest rate of carbon dioxide added to the atmosphere? (Flow question)

  2. What year is the amount of carbon in the atmosphere rising? (Stock question)


Figure 2: Stock and flows dynamic


Check the answer key yourself. Run the simulation. Play with others. Accumulations, slow and fast, our changing our world and their trajectory is extremely challenging to reverse. Understanding accumulations in the context of inflows and outflows is an important aspect of our changing world.















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Assessment 1 Answer Key:

Flows A = Stock Z, Flows B = Stock Y, Flows C = Stock X

Assessment 2 Answer Key:

Answer Key:

1 = Year 26, 2 = Year 19



Here are the major stocks and flows of the carbon cycle reported by the IPCC that I used for the simulation.

                                                                                            7.3 The Carbon Cycle and the Climate System - AR4 WGI, 2007




Works Cited

"7.3 The Carbon Cycle and the Climate System." 7.3 The Carbon Cycle and the Climate System - AR4 WGI Chapter 7: Couplings Between Changes in the Climate System and Biogeochemistry. Web. 19 Apr. 2017.

Stave, Krystyna. "Participatory System Dynamics Modeling for Sustainable Environmental Management: Observations from Four Cases." Sustainability 2.9 (2010): 2762-784. Print.

Sterman, John D. "Does Formal System Dynamics Training Improve People's Understanding of Accumulation?" System Dynamics Review 26.4 (2010): 316-34. Print.

Team, ESRL Web. "ESRL Global Monitoring Division - Global Greenhouse Gas Reference Network." ESRL Co2 Trends RSS. 01 Oct. 2005. Web. 30 Apr. 2017.


001: Dynamic Equilibrium

posted Apr 18, 2017, 12:35 PM by Jon Darkow   [ updated Sep 3, 2017, 9:44 AM ]

Think of a lake. All kinds and quantities of pollutants can be added to the lake. Similarly, the type and quantity of pollutant in the lake determine how quickly the pollutant can be removed. We can take stock of how much pollutant is in a lake at any given time. For example, I have methods in my own lab of testing quantities of atrazine, phosphates, and nitrates.

Assume in a lake 10 grams/day of atrazine can be added, and 10 grams/day of atrazine can be removed. The rates of change of a stock are inflows and outflows. 

Here is a model of this simple system. At initial conditions, the pollutant added is 25 grams/day, and the amount removed is 25 grams/day. The stock at the beginning of the simulation run is 100 grams in the lake. How much pollution will accumulate in the lake after 30 days?

Because the rate of removal of the pollutant was equal to the rate of addition of the pollutant, the stock of atrazine did not change. Even though each day 25 grams of atrazine was added to the lake during 30 days (750 grams total was added), the same amount of atrazine at the same rate was being removed. This model is at dynamic equilibrium. The lake had new pollutants added and old pollutants removed. Changes were happening, but the system was in equilibrium.

Perturb the system. What are different methods of reducing the pollutant to zero after 30 days? There are many ways. It depends on the source of the pollutant (the rate added), and the sink of the pollutant (the rate removed.)

Stocks and flows are the basis of systems dynamics. All the simulations I build are system dynamics models. So I thought I would start with the basics. 1 stock, 2 flows.  In each blog, I will include an explorable systems dynamics model. System dynamics models have helped me think of the world more definitively and operationally

Biology is littered with examples of dynamic equilibrium.
  • Birth rate and death rate in a stable population.
  • The rate of cellular respiration and photosynthesis in a plant or stream.
  • Lynx populations and moose populations.
  • Heart rate
  • Water balance
From this simple model of a single stock, I then wonder what are the effects of the inflows and outflows? Are the effects additive, multiplicative, nonlinear? What else affects the stock? What does the stock affect? Are the systems closed or open? Where is the greatest leverage point in the system? What other biological, social, economic, and physical systems similar to this one? Compare how they are being perturbed? What emerges? Are there feedback loops?

As questions emerge so does my understanding. Through modeling, I discover quickly where the gaps are in my understanding. 



This example is based on work from Krystyna Stave from the University of Nevada, and John Sterman from MIT.

Stave, K. A., A. Beck, and C. Galvan. "Improving Learners' Understanding of Environmental Accumulations through Simulation." Simulation & Gaming 46.3-4 (2014): 270-92.


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