Question

The pier in Santa Monica, CA, is a popular destination for both tourists and locals. Visitors ride the Ferris wheel (F), eat ice cream (C), or just walk around on the pier (W). Write a dynamical model for the numbers of people engaged in these activities given the following assumptions. (Hint: Start by drawing a diagram of this system and labeling the stocks and flows. People entering the pier always start by just walking around. E people enter the pier each minute. Visitors leave at a constant per capita rate d. They can leave only when they are walking around. Due to fear of nausea, people do not go directly from eating ice cream to riding the Ferris wheel. Visitors prefer to go on the Ferris wheel with friends. Thus, the probability that any one individual will go on the Ferris wheel is proportional to the number of people walking around, with proportionality constant b. Riders leave the Ferris wheel at per capita rate n. When visitors leave the Ferris wheel, a fraction z of them go directly to eating ice cream. The others walk around. Visitors who are walking around prefer to avoid long lines for ice cream. Thus, the per capita rate at which they get ice cream is proportional to the inverse of the number of people already doing so, with proportionality constant m. People who are eating ice cream stop doing so at a constant per capita rate k.

Answer 1

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Please note that for this problem, we will not add the signs to the contributions. We will add the signs when evaluating the inflow/outflow at the very end. Also, we will number the arrows according to the bullets numbering mentioned in the text. Since not all bullets will yield an equation, we will go by the arrows instead and obtain all necessary equations to set up our model.

Arrow #2: E people enter the pier each minute.

We can see that:

Arrow #3: Visitors leave at a constant per capita rate d. They can leave only when they are walking around.

We can see that:

(once again, the label does not have any sign associated with it)

Arrow #5: Visitors prefer to go on the Ferris wheel with friends. Thus, the probability that any one individual will go on the Ferris wheel is proportional to the number of people walking around, with proportionality constant b.

We can see that, the probability for one walking person to go to the Ferris wheel is:

As a result, the rate at which the walkers (W) will move to the Ferris wheel (F), will be:

Arrow #6: Riders leave the Ferris wheel at per capita rate n.

We can see that:

Arrow #7: When visitors leave the Ferris wheel, a fraction z of them go directly to eating ice cream. The others walk around.

We know that the number of people leaving the Ferris wheel per unit of time is: n · F

A fraction of them will go to eating ice cream. As a result, the arrow (7a) will be labeled as:

The rest of them will go back to walking. As a result, the arrow (7b) will be labeled as:

Arrow #8: Visitors who are walking around prefer to avoid long lines for ice cream. Thus, the per capita rate at which they get ice cream is proportional to the inverse of the number of people already doing so, with proportionality constant m.

We have:

Arrow #9: People who are eating ice cream stop doing so at a constant per capita rate k.

We have:

Putting Things Together:

Using the equations/labels obtained from above, we will label our diagram as follows:

We know that:

From the diagram, we have the following equations: