Lab Exercise 4:
Stock Control sub-models

The last exercise looked at the rate of growth as determined by how much of a resource flowed into a stock from a unlimited source. This exercise the focus is on the flow from the stock.

These models are pretty simple and small to be very interesting just on their own, but they play central roles as sub-models or modules of other models. For example, a population growth model may depend on a food resource that becomes available depending on the concentration of the resource. We are going to examine three forms of flow control:

1. constant release
2. linear outflow
3. Michaelis-Menten

The assignment is to create a model diagram, an example graph that shows both the stock and the flow and a list of the equations. As we go along, there will be less specific step-by-step instructions (notes). If you have trouble with any of these steps you might want to make a quick review of the step-by-step instructions in exercise 3.

1. Constant Release

The diagram looks like this:

The flow is set to a constant. When the stock runs out, the flow drops to zero. This diagram is actually simpler than it should be because STELLA sets the default for the stocks to be non-negative. As soon as the stock is depleted there is no more flow.

In this example, I set the stock to be 100 and the constant drain to be 5. That means it should run out in about 20 time units. You will need to set the time interval (from 0 to XX) to be long enough to watch this run out.

An example graph output looks like this:

The equation list for this little model can be accessed by making sure you're in the "model" mode on the "map/model" toggle and then hit the down arrow until you get the equations.

The equation window will look like this:

These are interpreted in the following way;

The stock is equal to the stock from the previous time step minus the outflow of loss_of_patience rate times the time interval.

The flow called "loss_of_patience" is set to the constant rate.

The constant rate is the set to 5.

 

2. Linear outflow

Less information is supplied for this section.

The diagram shows a specific link between the volume of a water reservoir and the rate of water flowing out. As the reservoir is more full, the rate is faster and the water flow decreases linearly until it goes to zero.

The equations are:

water_reservoir(t) = water_reservoir(t - dt) + (- water_flow) * dt
INIT water_reservoir = 100

OUTFLOWS:
water_flow = water_reservoir*wier_constant

wier_constant = .1

Create your own model, use different constants and create 2 different graphs,

graph 1 shows both the stock and flow vs. time

graph 2 shows the water flow (Y) vs. the water_reservoir (X)

To create this second graph, create a graph, double click on the graph to get the "define graph" dialog box, then click on "scatter" radio button on the top row. Set X axis to be "water_reservoir" and the Y axis to be the "water_flow".

Write a short explanation for why you get a curved relationship between flow and time but you get a linear relationship between the reservoir and the flow rate. In particular, why do the dots get closer together as they approach zero?

3. Michaelis-Menten formula

As described in class, the Michaelis-Menten equation describes the relationship between the velocity of a reaction (such as resource utilization rate) and the amount of that resource. It is particularly useful when the resource utilization requires acquisition and handling. The two parameters address these two processes mechanistically. The Km is related to the ability to acquire the resource at low concentrations and the Vmax is the maximum rate of turnover or use of the resource.

Below are the model diagram and equations:

stock(t) = stock(t - dt) + (- flow) * dt
INIT stock = 100

OUTFLOWS:
flow = Vmax*stock/(stock+Km)
Km = 5
Vmax = 4

Create a page with the model and two example graphs of stock and flow as a function of time and flow (Y) vs. stock (X) as a scatter XY plot.

Write a short explanation of how these two graphs relate to the equations.