Growth & Decay Dynamics in Environmental Science: Linear, Exponential, & Logistic Models, Lab Reports of Geography

Download Growth & Decay Dynamics in Environmental Science: Linear, Exponential, & Logistic Models and more Lab Reports Geography in PDF only on Docsity! GEOG 110 – Lab #2 – Modeling Growth and Decay Dynamics Due Date: 11:59 pm September 30, 2005 Objectives: Become familiar with three behavior patterns that are present in many dynamic systems; linear growth or decay, exponential growth or decay, and logistic growth. Learn how to translate the mathematical representations of these patterns into STELLA models. Background: As you should have learned from Lab #1, identifying the components of a system and the linkages between them is just one part of constructing a model. An equally critical activity is defining the rules that govern the system. This is usually done by specifying a set of equations that state what the system will do under a given set of conditions. In the graphical language of STELLA, this is the act of replacing those question marks in the Model view with equations (or sometimes a graph element, which still dictates an x - y relationship, and is used in the instance when a credible mathematical expression of the relationship cannot be found). In environmental science, we tend to learn of these equations or relationships through observations: We measure some variables of interest, look at the sort of plot they produce when graphed on two axes, if possible hypothesize an equation that describes the observed behavior, and then test it to see if it is a useful model of the phenomenon. There are five common behavior patterns that you should become familiar with to help you construct environmental models in STELLA:  Linear growth or decay  Exponential growth or decay  Logistic growth  Overshoot and collapse  Oscillation In this exercise, we will examine the first three of these common behavior patterns in detail. Resources: This lab exercise does not include the background and theory required for you to understand the origins and applicability of these behavior patterns. For that information, you should look to material from the lectures and Chapter 2 (pp. 28-65) of the course text. At this point, you should have already have successfully completed Lab #1, and you should have the basic familiarity with STELLA required to build the simple models for this exercise. If you need to refer to resources to help you with the details of using STELLA, refer back to those suggested for Lab #1. From this point onward, the lab exercises will not be provided in step-by-step detail, 1 because the focus from this point onward will be on what the models represent, not on the nuts and bolts of constructing them. Procedure: This second exercise (and subsequent exercises) features slightly less specific instructions. If you cannot recall how to before some basic operation in STELLA, refer back to lab exercise #1 or refer to the resources referred to therein. Linear Growth or Decay 1. Get STELLA started and open up the lake model you constructed for Lab #1. Conveniently, that lake model is a system that features a stock that displays linear behavior using the equations and values we specified. You can compare our system to the generic linear system diagram (Figure 2.3, p. 34) from the course text and see the structural similarity. Recall that your system should look like this: 2. Run the model using the values we specified last time (Precipitation=25 m3/time, Evaporation=20 m3/time, initial lake volume=100 m3) and have a look at the output. Is the system’s stock (the lake) displaying linear growth or decay with these values? 3. Change just one of the two processes’ values such that the system displays the opposite behavior (i.e. either growing or decaying at the opposite rate to which it did previously). There are two possible ways you can accomplish this. 4. You may recall that the Run Specs for our model were set so that the model would run for 12 time steps. Experiment with changing the length of the simulation (i.e. number of time steps in the model run) to see what the final value of the stock is at the end of model runs of different lengths. Exponential Growth or Decay In order to experiment with exponential behavior, we are going to add a further process to our lake model. It was always a little odd that the only way water left our lake was through evaporation. Let’s assume that our lake also has a dam on it, and that it has an open gate to release water from the lake in proportion to how full the lake is. 2 To delete a connector entirely, click on the non-arrow end, and use the Clear menu item. 13. We will now fill in the required values, setting the units as we do so:  Your generator pond’s initial value = 100 m3  Unconstrained Flow Rate = 0.5/time  Pond Capacity = 4000 m3  Inflow = your__generator_pond*Unconstrained_Flow_Rate  Outflow = your__generator_pond*Unconstrained_Flow_Rate* your__generator_pond/ Pond Capacity 14. Create a Graph and label it Pond Graph, and set it up to show your generator pond’s volume. 15. Set the Run Specs so the model runs for 24 time steps, and set the Sim Speed so 0.25 real secs = 1 time step. 16. Maximize the Pond Graph and set it up so you can watch it as the model runs. Run the model. Your resulting graph should look like this: 17. Save this model as youronyen-lab2-log.stm in your course directory at this point, so I can see you’ve successfully built this model and run it. 18. Try doubling and halving the Pond Capacity (using values of 8000 m3 and 2000 m3), and see what effect this has on the shape of the graph. 19. Return the Pond Capacity to 4000 m3 and try doubling and halving the Unconstrained Flow Rate (using values of 1 and 0.25) to see what impact this has on the performance of the system. Exercises: Linear Growth and Decay 1) Did the lake model, with its original values (precipitation=25 m3/time, evaporation=20 m3/time, initial lake volume=100 m3) display linear growth or decay? 2) There are two ways you can change one of the two process rates to get the system to display the opposite behavior (with the same magnitude of 5 change in volume per time). List both pairs of process rates than can produce this result. 3) How many time steps would it take before the lake stock is completely drained using the requested process rates that produce linear decay in the lake model? How many time steps would it take for the lake stock’s volume to be triple its initial value? Exponential Growth and Decay 4) What was your lake’s final volume when you ran the model with Flow Rate = 0.5 m3/time? When you doubled the number of time steps from 12 to 24 and ran the model, did your lake reach the same volume at the end of the model run? Explain what’s going on here. 5) When you doubled and halved the Flow Rate (trying 1 /time and 0.25 /time), what were your lake’s final volumes with those parameters? Logistic Growth 6) When you doubled and halved the Pond Capacity, what effect did this have on the shape of the Pond Graph’s volume versus time curve? Was there any change in how long it took to approach the steady state? 7) When you doubled and halved the Unconstrained Flow Rate, what effect did this have on the shape of the Pond Graph’s volume versus time curve? Was there any change in how long it took for the system to approach the steady state? What to do: Follow the procedures described above, saving the specified versions of your lake model and generator pond models in your course directory (e.g. mine would be davidten-lab2-exp.stm and davidten-lab2-log.stm). The following should be included in your lab report: 1. Objectives 2. Models 2.1 A brief description of the model 2.2 The systems diagram with each component properly labeled 2.3 The system components: a) Reservoirs: b) Processes: c) Converters: d) Relationships: Define them mathematically, e.g. Births=Birth_rate*Population 2.4 Rate Equation for each reservoir: identify the impact of each converter on the rate equation. 3. Results: Put the model results in your lab report and provide a brief explanation for the model behavior as see in the graph. 4. Discussion: Answer the questions in the exercise. 6