Chapter 6 S Shaped Growth -- 6 Exercises

The flow diagram shows a model to simulate growth in a private college. The simulation begins with 200 students and 100 faculty, so the student to faculty ratio is two to one. Students normally require around 4 years to graduate, so the graduation rate is assumed to remain constant at 25%/yr. The drop out rate is constant at 15%/yr. A special feature of this college is that new students may only be admitted if they are nominated by an existing student. Each existing student has the right to propose two nominees per year. The existing students will fully exercise this right if they are impressed with their education.

The model assumes the existing students will be impressed with their education when the student to faculty ratio is low. If the ratio should increase, however, the existing students will become less impressed, and they will be less willing to make nominations. Assume that the nomination rate is given by the nonlinear relationship in the chart to the right. At ratios of 10 or less, the nomination rate will be at the maximum -- 2 nominees per student per year. At the oppposite extreme, the nomination rate will fall completely to zero if the student-faculty ratio climbs to 50.

1. Size of the College
How large will the existing student body become if the number of faculty is fixed at 100? (From your reading of chapters 5 and 6, you should be able to derive the ultimate size without simulating the model.)

2. Build and Simulate:
Build the model to simulate the growth in the student body over a ten year time interval with DT, the time step, set to 0.25 years. Turn in a time graph of the number of existing students to document your work. Does the model confirm your answer in the previous exercise?

3. Stable Equilibrium?
If you complete the previous exercise correctly, you should see the student body reach a dynamic equilibrium. Do you believe that the equilibrium is stable, unstable or neutral?

4. Stability Test
Introduce a "disturbance flow" (as explained in chapter 5) to remove around 25% of the students in the 10th year of the simulation and allow the simulation to run for 20 years. Turn in a time graph of the number of existing students to document your results. Does the test confirm your answer in the previous exercise?

5. Build a Larger College
The drop out rate of is 15%/yr in the previous exercises. This is relatively high rate -- 60% as large as the graduation rate. Suppose the college wants to build to a larger size,and it goes to work on the drop out problem. How much larger will the college become if the drop out rate were reduced to zero? (For example, do you think the college will eventually have 60% more students than in the 2nd exercise?)

6. Confirm Your Estimate
Set the drop out rate to zero and run the model. Does the new simulation confrim your estimate in the previous exercise?