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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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