Chapter 17. Introduction to Oscillations -- 5 Exercises

1. Expand the Student Body Model from Chapter 6

Review your work with the model of a private college from the exercises for chapter 6. The previous model assumed that a change in the student-faculty ratio leads instantly to a change in the students' willingness to nominate new students for admission.

The model shown below assumes that there is a delay in the students' behavior. It uses a SMTH1, a first order smooth function, to calculate the "effective ratio." The "effective ratio" is then used to find the nomination rate using the same nonlinear relationship in the previous exercises.

Build this model to show the impact of the delay in the student's assessment of the student-faculty ratio. Assume that all parameters are the same as the exercise from chapter 6 and the "delay in student reaction" is 1 year. Do you believe the new model will show S shaped growth? Do you think the college will eventually end up with the same number of students as in the exercise from chapter 6?

2. Simulate the New College Model

Simulate the new model to confirm the oscillatory behavior shown here. The number of existing students is simulated to grow to over 8,000 within the first four years. It then drops rapidly reaching a low of under 3,000 by the 7th year. The simulation then reveals that the college would experience damped oscillations before arriving at an eventual size of around 3,500.

Are These Results Numically Accurate?
These results were generated with DT set at 0.25 years. When you have verified these results, cut DT in half and repeat the simulation. Do you get the same results? If not, cut DT in half again, and repeat the simulation again. You will eventually arrive at accurate results. Are the accurate results more volatile or less volatile than the results shown above?

3. Avoid the Wild Beginning
Suppose the college wishes to avoid the "roller coaster" behavior shown in first ten yers of the previous exercise. Assume your only point of intervention is the maximum number of new students that existing students may nominate per year. The current policy sets this maximum at 2. What value would you suggest to allow the college to avoid the wild beginning?

Test your policy by changing the values of the nomination rate when the effective value of the student-faculty ratio is low. Use the model to find a nominations policy that allows the college to grow to its natural size in a more controlled manner. Document your results with a compartive time graph showing the number of existing students in the base case simulation (shown above) and in your policy simulation.

4 Add Random, Annual Disturbances to the Flower Model

Many of the examples in chapter 17 in the book ignore the effects of random disturbances. This makes classification of the oscillatory behavior easier. We simply watch the oscillations over time to learn if they dampen out. But random disturbances are always present in environmental systems, so it is useful to introduce randomness to learn how it might alter the cyclical tendencies of the system. The diagram below shows how annual randomness might be introduced in the intrinsic growth rate of the flower model.

Several statistical functions may be used to sample from a normal, uniform, Poisson and exponential distribution. Each of these functions assumes that a new number is drawn at random during each time step of the simulation. For example, if we thought the intrinsic growth rate could vary between 50%/yr and 150%/yr, the equation would be:

intrinsic_growth_rate = RANDOM(.5,1.5,123)

The RANDOM function is appropriate if we feel that the growth rate should be drawn from a uniform distribution. The "123" is included as a "seed." It guarantees that Stella will select the same stream of random numbers each time we repeat the simulation. The flower simulations in Figures17.5-9 were generated with a DT of 0.125 years. With DT at 1/8th of a year, Stella would select a new value for the intrinsic growth rate eight times during a single year of the simulation. This would suggest that we have a change in growth conditions for each new time interval of 45 days duration.

Now, suppose we would rather not see such frequent variations in the growth conditions. Suppose the flower system is more likely to experience a "good year" or a "bad year," for example. If we want to maintain a random number over the entire year, we must sample from the random distribution and "hold" the value for a one year interval. The diagram shows a conveyor and a hold interval to ensure that the random number will be maintained for an entire year. The model measures time in months and sets the DT to 1 month. The random number changes every month draws randomly from the uniform distribution over the interval from 50%/yr to 150%/yr. The is it time? variable is set to 1 whenever a new year has arrived, and the pick new number will be deposited into the conveyor stock. The new equations are shown below:

random_number_changes_every_month = random (.5,1.5,123)
is_it_time_to_select_a_new_number? = if (mod(time,hold_interval) = 0) then 1 else 0
pick_new_number = random_number_changes_every_month*is_it_time_to_select_a_new_number?

The transit time for the conveyor matches the hold interval, so the conveyor holds the new number for a 12 month period.

Build this model to verify that it generates the behavior shown below:


5. Impact of the Disturbances on the Oscillations

With the annual changes in the previous exercise, we can learn how the randomness in the intrinsic growth rate influences the flower model. Let's consider the flower model with the lag time set at two years. Recall from Figure 17.7 that we expect damped oscillations in the absence of random disturbances. The area would eventually reach 800 acres after a couple of oscillations, but no further oscillations would be visible after 40 years. Now, run the model in the previous exercise to verify the results shown below.

The flowered area seems headed toward 800 acres, just as in the simulation in the book without disturbances. But the oscillations shown here continue year after year. It appears that the disturbances act to keep the oscillatory behavior "alive" throughout the simulation.

Remember this result when you encounter a system that exhibits damped oscillations. Perhaps random disturbances will keep the oscillations "alive" in your system. Also, keep this result in mind when viewing time series data on oscillatory systems. If the oscillations persist over time, you might be able to explain the persistent oscillations with parameters that would generate either "sustained" or "damped" oscillations in the absence of random disturbances.