Now it’s time to apply what you’ve learned to begin building simulation models that mimic many of the feedback dynamics in complex problems like the Kaibab ecosystem. This chapter will lead you through a series of model building exercises that provide building blocks that are useful in a wide variety of real-life situations, including the problems on the Kaibab plateau. Along the way, you will learn some of the basics of system dynamics model building.  For each exercise you will build a running simulator of a simple system. Your model will include appropriate graphs and simple user interface features to allow you and others to experiment with the model. We’ll use Stella Architect®, created by ISEE Systems, as the model-building platform. The sequence of models you will build over the next few weeks are listed below.

4.1. A Simple Bathtub Model: Behavior when Flows are Constant

Figure 15 shows the stock and flow diagram and an associated BOTG that you will create for your first model. The stock and flow structure is a system dynamics representation of the bathtub analogy in Figure 6. The BOTG shows the amount of water in the bathtub (left axis) and the flow rates in the faucet and drain (right axis).

4.2: Constants, Initial Values, and Equations

No matter what simulation modeling software you are using, the software will take care of the difficult math for you.  Your job is to figure out the calculations between variables and the values for constants, nonlinear functions, and the initial conditions of stocks.

Each stock needs an initial condition or starting value.  The stock in the bathtub model starts at zero.  There is no water in the bathtub when the model starts to simulate.  If I were looking at the world population and starting my model in 1950, I would go to a reliable source, perhaps the Population Division of the United Nations, and get data.  The world population in 1950 was approximately 2.5 billion people.  I would use this value as the initial condition of my population stock.  All stocks need an initial value or starting condition.  You can make the initial value of a stock a function of another or a combination of other variables in your model. We will show you some simple examples later in the text.  You will not need them now.

Constants are values that are not affected by other variables in the model.  An example would be the number of miles driven per year per vehicle.  The Federal Highway Administration data indicates that this is 14,500 miles per vehicle per year.  If you multiplied this by the number of registered vehicles, you would be able to calculate the total miles driven each year.  New information may come along, and the value of the constant could be changed.  You could also vary this value to determine what would happen if more or fewer miles were driven per vehicle each year.  The only rule with constants is that you should not add them to a model simply to make the units correct.

An auxiliary variable is calculated from other variables in the model.  In the example in Figure 16, the variable labeled Total Miles Driven per Year is an auxiliary variable and is calculated by multiplying Miles Driven per Year per Vehicle (a constant) times Vehicles (a stock).  Auxiliary variables are going to be calculated by multiplying, dividing, adding, subtracting, or some combination of these operations.  This also applies to rate (flow) equations.  Technically, all auxiliary variables are rate equations in that auxiliary variables will flow back to change rates (flows) into and out of all stocks.

The other type of equation showing up in a model is a nonlinear or graphical equation.  These are used when the output of a variable does not change proportionally to the input and this will be discussed in detail later.

4.3. A Simple Population Model: Behavior When Flows Vary Over Time

The purpose of the bathtub model exercise was to introduce you to some of the basic concepts used in building running simulators involving stocks and flows. Of course, most problems of interest (like the Kaibab, for example) do not involve bathtubs, faucets, and drains. In the case of the Kaibab, the stocks of interest are populations of animals and plants.

In fact, many system dynamics models involve stocks of populations of various types. The next several modeling exercises will introduce you to some of the fundamentals for modeling population dynamics.  We’ll begin with a simple population model: one that involves births and deaths. In later exercises we will add more complexity to the model by considering how births and deaths are impacted by limitations of the surrounding ecosystem, by the presence of predators, and by consumption (or overconsumption) of limited resources.

Imagine a population of people in a region filled with abundant food and other resources necessary for that population to flourish over the foreseeable future.  In this idyllic situation, let’s assume (for simplicity) that there is no threat of starvation, disease, war, or other factors that could alter the basic dynamics created by the processes of birth and death.

Figure 17 shows the stock and flow structure (and behavior over time) of this simple, idyllic population. This stock and flow diagram matches the structure we used for the deer population in Figure 7. As in that figure, there are two feedback loops in this system. You will shortly get a chance to identify those loops and explore their role in the system.

4.3.1. A Word About the Unit of Time

Because we are modeling a human population over time, we do not need to track the population size every second, minute, day, or probably even every month. Instead, we can probably use a time unit of one year for our simulation. Hence, all of the flows in this model will be expressed as the number of people added to or removed from the population over a given year.

4.3.2. Fraction Birth and Death Rates

Figure 17 includes two new variables that require some explanation. These are the Fractional Birth Rate and the Fractional Death Rate.  In general, these represent the fraction by which the Population changes because of births or deaths in one time unit of the simulation (in one year). For example, a Fractional Birth Rate of 0.10 means that the Population will grow by 10% each year – all because of births. This is equivalent to saying that 10% of the population will have a child born during the next year. That’s why the Fractional Birth Rate is sometimes called the per capita birth rate.  On the other hand, a Fractional Death Rate of, say, 0.05 means that the Population will shrink by 5%, from what it would have been, because of deaths (i.e., 5% of the Population will die over the next year). The Fractional Death Rate is sometimes called the per capita death rate.

The units for these quantities are not immediately obvious. One way to determine the units for these quantities is to think about how they are used to calculate the two flows in the diagram. Given the physical explanation of the Fractional Birth Rate in the previous paragraph, it is clear that, to calculate the number of births in a given year, we just multiply the Fractional Birth Rate by the Population. For example, if there are currently 1000 people in the Population and the Fractional Birth Rate is 0.05, then over the next year we would expect to see Births that increase the Population by 5%. That means there would be an inflow of 0.05*1000 = 50 people via the Births inflow. In general, this translates into the following equation for calculating the Births inflow:

Births = Fractional Birth Rate * Population

However, we also know (from the earlier discussion about unit compatibility) that, if the units for the Population stock is measured in people, then the units for the Births flow must be measured in people/year.  By substituting these units into the above equation we see that:

Units for Births flow = Units for Fractional Birth Rate * Units for Population
people/year = Units for Fractional Birth Rate * people

This means that the units for the Fractional Birth Rate must be 1/year (aka dimensionless/year, fraction/year, or 1/year) in order for the units on the right side of the equal sign to work out to be people/year.

Using the same reasoning, the units for the Fractional Death Rate will also be dimensionless/year.

Exercise 4.4: Apply Your Model to World Population Projections

Now apply your model to the world population, starting in 1950. The graph below (from the United Nations) shows the BOTG from 1950 through 2100. The values from 1950 through 2016 are official UN world population estimates. The values beyond 2016 are projections based on the UN’s population models. The three curves (labeled as the “Low,” “Medium” and “High” variants) show three different behaviors over time that could emerge between 2016 and 2100.

 

1. Of the three curves in the graph, which curve (or curves) do you think represent the UN’s “hopes” for the future? Which curve (or curves) represent the UN’s “fears” for the future. Be ready to explain your answers.
2. Now adapt your population model to see how well it matches the UN’s projections in the above graph. Do this by changing units for the Population stock to be in thousands of people (so that 5 billion people will be expressed with the number 5 million). Change the units in the two flows to be compatible with the change in the stock units. Adjust the Run Specs on the model to start in 1950 and run through 2100. Also, adjust the DT value so that the module updates the population every month. Set the Population to start at 2,536,300 thousand people: the population in 1950.
3. Run your model and adjust the Fractional Birth Rate and Fractional Death Rate for the world population to match as closely as you can the BOTG up through 2016 in the figure above. Use the actual population values in the lower right box as a check. Hint: Try setting the Precision on the input ranges for these two variables to be 0.001 to allow finer adjustment of these variables.
4. Record the Fractional Birth Rate and Fractional Death Rate that best match the UN curve up through 2016. How do you feel about these fractional values in terms of their appropriateness for a human population? Can you find data on the web to give you an idea of what these values are in practice?
5. Can you find other values for these fractional rates that will produce the same BOTG that you got in #4. Explain.
6. How does the resulting simulation BOTG compare to the UN historical trends up through 2010? How does your model’s output compare to the projected trends beyond 2016? What issues do you see?
7. What does the resulting BOTG from your model (with the Fractional Birth Rate and Fractional Death Rate you’ve chosen) suggest about the world population through 2100? How does this differ from the UN’s model? How do you feel about the accuracy of your simulation? Explain.

4.2.3 Applying the Basic Population Model to Other Contexts

In the previous section, you learned that the birth and death flows in the basic population model are calculated as follows:

Births = Population * Fractional Birth Rate

Deaths = Population * Fractional Death Rate

That is, the two flows are determined by multiplying the stock by a numeric constant (the Fractional Birth Rate or the Fractional Death Rate). Hence the flows are proportional to the size of the stock because they are calculated by multiplying the stock by a constant (the fractional birth or death rate).

This proportionality relationship is a commonly occurring relationship between a stock and its flows. Whenever you are building a stock and flow model, if you believe that the flows depend on the size of the stock, a good place to start is to see if a proportional relationship makes sense.

For example, consider how the level of support for a particular political cause might be modeled. Figure 18 shows a logical stock and flow model for this phenomenon.

Notice that this model looks just like the simple population model in Figure 17. The only evident change is in the names of the stock, flows, and fractional rate constants. Furthermore, the flows in this model would be calculated the same way as in the population model: by multiplying the stock by the indicated fractional rate constant.

Now let’s return to the original population model to further explore other forms that proportional stock and flow relationships can take. Suppose, for example, that we do not know the fractional death rate in the model, but we do know that people in this population live for 80 years, on average. Can we use this to calculate the death outflow?  The answer is yes!  Since the average lifespan is 80 years, we know that every year about one-eightieth of the population will die. In other words, the death flow would be calculated by dividing the population by the 80-year lifespan. In equation form, the equation for the death outflow would be as follows.

Deaths = Population / 80 (Average Lifespan),

or equivalently

Deaths = Population * 0.0125, since 1/80 = 0.0125.

Now consider a lake that has water flowing into it (via a stream) and flowing out of it (via the extension of the same stream).  Further suppose that we know that water flows into the lake at a rate of 1000 gallons per day and that, because of mixing, each drop of water remains in the lake for, on average, 20 days. The 20-day residence time is analogous to the 80-year lifespan in the population model. That is, each day, one-twentieth of the water in the lake will flow out and downstream. The stock and flow model for this situation is given below in Figure 19.

Notice that the outflow is proportional to the stock. It is calculated by dividing the stock by the Residence time. Also notice that there is no causal link pointing from the stock back into the inflow. That’s because the inflow rate is a constant 1000 gal/day; it doesn’t depend on the amount of Water in the Lake.

4.3. Reflecting on Model Validity: When is a Model “Good Enough?”

In the previous Action Item, you used a UN-based BOTG on world population to help you choose values for the starting value of the Population stock, as well as for the Fractional birth rate and Fractional death rate. This is a common practice in modeling: using historical data to help us “tune” the parameters in the model and then checking to see how the model matches the past and whether it produces future behavior that we think is reasonable.  You probably noticed a few issues as you worked on that exercise.

1. Inappropriate behavior over time.  Your simple population model cannot produce the historical curves nor the “hopes and fears” curves produced by the UN model. In fact, the only behavior your model can produce is exponential growth or exponential decline (depending on what values you set the Fractional birth and Fraction death rates). Because the UN model is based on real historical data, and because it is produced by expert population modelers, it suggests that the simple feedback structure in your model does not capture the feedback structure used in the UN model.
2. Indeterminate parameter values.  You may have noticed that there are many combinations of values for the Fractional birth rate and Fractional death rate that produce the same BOTG. The next Action Item shows several pairs of Fractional birth and death rates, where each produces the exact same BOTG.

The two issues of inappropriate behavior over time, and indeterminate parameter values are common, but necessary challenges when we build models of dynamic systems. During every step of the modeling process, we should test our model’s validity to see if it produces behaviors that we know should be present in the system. We also check to see if the numeric quantities we are using in the model are, reasonable, given the context of the problem.  There are many other validity tests that need to be done^[1], but those are beyond the scope of this text. However, when we test model validity and see problems, we probably need to make some changes to the model!

However, we can never get a perfect model…one that provides precise output that we can quote with absolute confidence. That’s because a model is a simplified version of reality. It inevitably involves compromises. We make choices about which details from the real system to include in the model, which to ignore, and which to include as a sufficient approximation to reality.   Otherwise, if we try to build a perfect model, we’ll never finish, and our model will be too complicated to be understandable or useful. Just what constitutes a “good enough” model is a difficult question to answer, and one that requires more time and effort than is appropriate at this stage. However, we can keep in mind a profound and simple quote from one of the great modelers and statisticians of the last century, G.E.P. Box.

This statement provides a rough guide to modelers of all types. To be “good enough” a model must be useful to those who depend on it to help them make decisions. A model need not provide perfect predictions. It does need to provide predictions that are good enough to help a decision maker know what sorts of conditions could lead to system behavior that reliably mimics their hopes or their fears. Given such a model, along with a knowledge of how the feedback structure in the model works to create that behavior, a leader is in a much better position to make interventions that produce the desired results.

It is clear that a simple population model is not “good enough” to give us insights about the future of the world population. In fact, it’s not good enough to model most real-life populations, including those of plants, animals, bacteria…you name it. We know that no population of living organisms can grow exponentially forever.

Hence, we now turn our attention to ways our simple population model can be expanded to account for dynamics that constrain unfettered growth thereby causing the population size to eventually level off, or decline or maybe even oscillate over time. But before we can accomplish this, we must first address the problem of indeterminate parameter values. We will show a way to simplify the simple population model to help with this issue.

4.4. The Simple Population Model Revisited: Using a Biflow

We’ll begin by tackling the problem of the indeterminate parameter values that you saw when you tried to find values for the Fractional Birth Rate and Fractional Death Rate to match the United Nations BOTG in the previous Modeling Challenge.

One of the problems with finding the correct values for the Fractional Birth Rate and Fractional Death Rate is that many times the data we have about a population gives us only the fractional net growth rate. For example, here are a few lines of data from the United Nations^[2] used to make the World Population BOTG you saw earlier.

Notice that, for each year, this table shows the number of people (in thousands) and the Fractional Net Growth Rate from the previous year.  To get the Fractional Net Growth Rate for a given year, we calculate how many people were added since last year and express that value as a fraction of last year’s population.   For example, the fractional net growth rate from 1990 to 1991 is
(5,362,591 thousand – 5,269,760 thousand)/(55,269,760 thousand) = 0.01648 or 1.648%. That is, the population grew by 1.648% from 1990 to 1991.

With population data, we seldom know the Fractional Birth Rate and Fractional Death Rate, but we can often use historical data to calculate the Fractional Net Growth Rate. In addition, it turns out that the Fractional Net Growth Rate is the difference between the Fractional Birth Rate and the Fractional Death Rate, i.e.:

Fractional Net Growth Rate = Fractional Birth Rate – Fractional Death Rate.

You saw in in the previous Modeling Challenge that the Fractional Net Growth Rate determines the growth trajectory of the population. Hence, if we know the Fractional Net Growth Rate, we don’t need to know the Fractional Birth Rate and the Fractional Death Rate. Instead of separate birth and death flows, we can use a single Net growth flow.

Figure 20 shows the original population model with two flows and a revised version, where we’ve combined the Birth and Death flows into a single Net Flow that depends on the Fractional Net Growth Rate. Notice the arrows on both ends of this flow. This is a biflow; it can flow into or out of the stock, depending on whether the flow is positive or negative. As long as the Net Flow is greater than zero, the population will grow. If the Net Flow is negative, the population will shrink.

4.5. Modeling S-Shaped Growth – Population Growth Under Constraints

At this point, your population model still does not seem realistic. It suggests that the world population will keep growing exponentially up through 2100. In fact, as long as the Fractional Net Growth Rate is positive, the population will continue to grow exponentially. But this doesn’t make sense. After all, there is a finite amount of land area on the earth. There are finite resources such as water and food that can support a human population. In addition, as the population grows, the impact of humans on those resources also grows. Over time, we would expect that humans might reach a maximum sustainable population. In addition, if we are not good stewards of the earth, we might overuse or degrade its natural resources. In that case, the world population could “overshoot” the capacity of the planet to sustain it. Then the world population would peak and eventually start to decline. The decline could be gradual or catastrophic, depending on the degree to which the planet’s resources have been degraded.

These two scenarios (growth to a sustainable population vs overshoot and collapse) are the same scenarios that the United Nations is concerned with. The figure here shows the same world population BOTG referred to earlier. Notice the three different scenarios or “variants” on this BOTG.  We already know that the “High variant” is not likely. The earth’s human population cannot continue to grow without bound forever. The “Medium variant” curve corresponds to the case in which the world population reaches a stable, sustainable level. The “Low variant” represents an overshoot and collapse scenario.  From here on, we’ll refer to the “Medium variant” curve as an example of S-shaped growth and the “Low variant” curve as an example of overshoot and collapse.

Our goal in this section is to add feedback dynamics to our simple population model in order to mimic the S-shaped growth behavior over time. The resulting model will be based on what is known about the feedback dynamics affecting any population as it grows to a sustainable level.

Then we’ll turn our attention (in the next section) to adapting our S-shaped growth model to account for the case in which a population so degrades the resources on which it depends that it “overshoots” the sustainable population size and eventually starts to decline, or even collapse catastrophically.

Every population of living creatures depends on resources for its survival. Some of those resources are so abundant that we will not have to worry about over consuming them…at least for the imaginable future. Sunlight is an example of one such resource. It has taken the sun about 4.5 billion years to burn ½ of the hydrogen fuel it needs for nuclear fusion. That means that the sun will probably last for a few more billion years before its fuel is exhausted.

Other resources are not so abundant, but many of them are renewable. This means that they regenerate themselves over time, often on an annual cycle, but sometimes over much longer time periods. Food (either from animals or plants) is one such resource. Both plants and animals grow. They can be consumed by humans (or other animals), but they can also regenerate themselves. Water is another example of a renewable resource. Although the amount of water on earth has remained constant for millions of years, water is also recycled through the atmosphere and ground to replenish rivers, lakes, oceans, and even underground aquifers. Humans consume water, but they also return it to the broader ecosystem through waste and when they die.

Of course, some renewable resources can be degraded through pollution or other problems. We’ll get to that later.

For now, let’s imagine an ecosystem with a population of people who depend on the resources of that ecosystem for survival. Ecologists and population modelers use the term carrying capacity to refer to the number of people (or bears, sharks, etc.) that the ecosystem can sustainably support indefinitely. Carrying capacity is a complex concept. In practice, it is almost impossible to directly measure. The carrying capacity of an ecosystem can vary over time as the climate and landscape change. Geological changes can affect it. Natural disasters can impact it.  It also depends on who or what else is present in the ecosystem. Different species of animals that compete for the same food source will live under a different carrying capacity than if there was no competition.

Even though an ecosystem’s carrying capacity is difficult to measure, common sense tells us that there have to be limits to how much any population of people, animals, plants, etc. can grow.   So, ecologists and modelers use the concept of carrying capacity to explain how nature works to constrain the growth of a population. Let’s take a look at a system model that does just that.

The concept of carrying capacity is important beyond the study of population or dynamics or natural ecosystems.  For example, a manufacturing firm will seek to grow its business by building more factories and marketing its products. For some period of time, the growth can be quite rapid…even exponential. However, all firms eventually reach a point of “stasis” in which growth slows, levels off, and sometimes even declines. That’s because the firm is encountering some sort of limits to their growth. It may be that the market into which they sell their product is saturated, limiting sales. Or perhaps they have exhausted the supply of raw materials needed to produce their products. Maybe technology has produced a replacement that has made their product obsolete. Whatever the case, the company cannot grow without bound forever.  It will either settle for a slow, possibly flat growth or it will “reinvent itself” with new products, mergers or other strategies.

4.5.1. Effects of Carrying Capacity – The Simple S-Shaped Growth Population Model

The next figure shows a stock and flow model whose feedback dynamics produce S-shaped growth and whose structure depends on the concept of carrying capacity as the limiting size at which a population can be sustained.

Before you build this model, let’s notice a few important features. First, compare this model to the simple population you built earlier. Can you see that the S-shaped growth model includes all the features of the simpler model, but that it also has some additional features to account for the role of carrying capacity?  This illustrates an important principle in system dynamics modeling:  We can model more complex systems by using a simpler related system as a starting point and thinking about what additional feedback structure is needed to model the more complex problem.

Let’s discuss each of the features in the S-shaped Growth model.  Then you will build the model and explore its behavior over time.

• Population stock, Net Flow, and Fractional Net Growth Rate: These three model variable have the exact same meaning and as in the simple population model with a biflow. In addition, the equation for the Net Flow is the same as before: Net flow = Fractional Net Growth Rate * Population.
• Baseline Fractional Growth Rate: This variable stands for the fractional net growth rate of the Population whenever there are abundant resources on which to live. This variable is used in the equation for calculating the Fractional Net Growth Rate. Because it’s a fractional growth rate, it’s units are dimensionless/year or 1/year.  These units are interchangeable.
• Carrying Capacity: This is the population size that the ecosystem can sustain for the long term. Hence, its units are the same units as the stock.
• Ratio of Population to Carrying Capacity As the name implies, this is calculated as the ratio of the Population to the Carrying capacity. This Ratio is less than 1 when the current Population size is less than the Carrying Capacity. In this case, the Population still has “room to grow” in the ecosystem.  However, as the Population nears the Carrying capacity, the Ratio of Population to Carrying Capacity will get close to 1. As this happens, the Net Growth should slow, moving toward a value of zero.  Because both the Population and the Carrying Capacity are measured in people units, when we take the ratio, the units cancel out. Hence, this variable has units that are dimensionless.
• Adequacy of  the Ecosystem to Support Continued Growth: This variable is calculated as 1 – (Ratio of Population to Carrying Capacity). When the Population is well below the Carrying Capacity (and the Ratio of Population to Carrying Capacity is hence close to zero), the Adequacy of the Ecosystem to Support Continued Growth value will be close to 1. As the Population approaches the Carrying Capacity (and the Ratio of Population to Carrying Capacity hence approaches 1), thus the Adequacy of  the Ecosystem to Support Continued Growth will approach zero. Higher Adequacy of the Ecosystem to Support Continued Growth values imply that the Population can grow at a more rapid place. It has units that are dimensionless.
• Fractional Net Growth Rate: This represents the actual Fractional Net Growth Rate of the Population, after accounting for the Adequacy of the Ecosystem to Support Growth of the population. It is calculated by multiplying the Baseline Fractional net growth rate times the Adequacy. This, in effect, scales down the Baseline Fractional net growth, according to how close the Population is to the Carrying capacity.  The units for this variable are 1/year.
• UN Low Variant, UN Medium Variant, and UN High Variant. These three variables are included in the model and plotted on the BOTG to provide reference modes (benchmark behavior over time) for comparing your output to the three projections provided by the United Nations. These will be created using graphical functions in Stella Architect®. Because each variable shows the UN projections for world population, their units are people in thousands.

4.5.2. Applying S-Shaped Model Structure in Other Contexts

The system structure that gives rise to the S-shaped behavior over time can be used to explain many different types of phenomena that the growth of a population under the constraint of a carrying capacity.  In order to see this, look at Figure 23, which shows the general stock and flow structure behind the S-Shaped behavior over time, but this time with the variable names in the model changed to be more generic. The text boxes beside several of the variables describe the physical role of each variable in a way that is not directly dependent on the context of a population that depends on an ecosystem for survival.

This kind of system structure has two feedback loops: A reinforcing feedback loop that dominates the system as long as the value of the stock is well below its Growth ceiling. Under such conditions, the Stock will grow exponentially. However, as the Stock grows, it eventually approaches the Growth ceiling and triggers mechanisms that begin to dampen its growth. These mechanisms are not specified in the diagram (it depends on the specific context). Their collective action, however, is represented in the variable Adequacy of the system to support continued growth – a value that approaches zero as the Stock approaches the Growth ceiling. In this formulation, the Adequacy is multiplied times the Baseline fractional net growth rate to determine the actual Fractional net growth rate. This effectively drives the Fractional net growth toward zero, leveling off the growth trajectory of the Stock.

The next Team Modeling Activity gives you practice applying the S-shaped stock and flow structure to some new contexts.

4.6. Reflections on the Journey so Far – How Structure Creates Behavior

When you built the S-shaped growth model, you created a system story to explain how the feedback structure in this system creates the S-shaped growth behavior over time growth curve.  Such stories are possible because of the simple truth that structure determines behavior.  By this we mean that the feedback structure in a system is the most important determinant of how that system will behave over time.

For example, consider the original population model. The original model with separate birth and death flows models includes two feedback loops: one balancing loop and one reinforcing loop.  As you learned earlier, reinforcing feedback leads to growth, and balancing feedback leads either to movement toward equilibrium, or oscillation around an equilibrium.  Hence, a system like the simple population model can exhibit only three types of behavior, depending on which of these two feedback loops dominate.

• Exponential growth – This occurs when the reinforcing feedback in Figure 23 dominates the behavior of the system — and that happens when the Fractional Birth Rate exceeds the Fractional Death Rate, or in the net flow model, when the Fractional Net Growth Rate is positive.
• Movement toward equilibrium – This occurs when the balancing feedback in Figure 23 dominates the system — and that happens when the Fractional Birth Rate is less than the Fractional Death Rate, or in the net flow model, when the Fractional Net Growth Rate is negative.
• Equilibrium – This will occur in two situations:
  • When there are no people present. In that case, there are not births or deaths, and the population will remain at zero.
  • When both feedback loops exhibit equal influence on the system. This happens in the origianl model when the Fractional Birth Rate is equal to the Fractional Death Rate, or in the net flow model, when the Fractional Net Growth Rate is zero.

Since the Fractional Birth and Death Rates (or the Net Growth Rate) are fixed quantities in this model, the relative influence of each of the feedback loops in this system does not change over time.  That is, the system exhibits only one of the above three behaviors for the entire period we simulate. See Figure 24.

The S-shaped growth system also has only two feedback loops. Why is it that this model produces more complex behavior than we see with the simple population models? The answer is that the relative influence of the two feedback loops in the S-shaped growth model changes over time. That is, neither feedback loop dominates all the time! Early on, the reinforcing feedback dominates the system. Later, the balancing feedback dominates the system.

You can see this shift in loop dominance by examining the BOTG created by the S-shaped model, as shown in Figure 25. Notice that, early on, the graph exhibits exponential growth ― the graphical signature of reinforcing feedback!  Then later on, as the Population size approaches the system’s Carrying Capacity, the BOTG exhibits the kind of goal-seeking behavior that comes from a balancing feedback loop.

To be sure, both feedback loops are work during the entire simulation. However, their relative influence on the behavior of the Population stock varies over time.

But why does this shift actually happen? What is it about the real-life system that causes this?  Common sense would suggest that, early on, the population has abundant resources to support growth. Hence, exponential growth is the logical result. However, as the Population grows, it gets closer to the Carrying capacity of the ecosystem. This means that there are not as many resources to go around. Fewer people are born; fewer can survive with the existing resources. Hence, Population growth slows, and eventually levels off at zero growth, as in Figure 24.

The point of all of this discussion is to illustrate how an analysis of the feedback structure in a system can give us insights into why it behaves in the ways we see in a simulation or a real-life BOTG of historical trends.

Of course, real life systems can exhibit even more complex behavior than S-shaped growth. When this happens, it means that the underlying system has a more complex feedback structure than what we’ve seen so far.

Consider the United Nations’s low variant curve in the World Population BOTG graph shown earlier. This early part of this curve mimics the exponential growth of both the high variant and medium variant curves. What causes the low variant curve to decline instead of leveling off toward the global carrying capacity?  In a later section you will explore this question by examining a system structure that can mimic this behavior quite nicely – the overshoot and collapse structure.

4.7. Making Your Model Available to Others: Building a User Interface

One of the best ways to explore a system dynamics model is create a user interface that allows you and others to experiment with the variables in model and see how they the affect the behavior of the system.  This kind of experimentation gives you insights into how the model works. If the model is reasonably valid (i.e. “useful” in the sense that G.E.P. Box meant), then experimenting with it can help you understand how the real-life system works. This can be immensely helpful to anyone who has to make decisions that impact the system.

Models that are used in this way are called virtual worlds or microworlds^[3]. These models are used as a rough approximation to the system we wish to influence.

You may realize that you have already played with a microworld. In the Kaibab challenge you interacted with a user interface for a system dynamics model of the Kaibab ecosystem. The figure below shows that interface. This particular interface allowed you to adjust variables in the model corresponding to real-life decisions made back in 1907.  You made adjustments and watched them play out in the BOTG.  You didn’t have access to the underlying stock and flow model. At that time in the course, you would probably have been overwhelmed by the all the stocks flows, and causal connections in the model. Besides, we wanted your experience to feel as “real” as possible. Since the real-life decision makers back in 1907 didn’t have access to the model structure, neither did you!

An important skill of any system dynamics modeler is to be able to create useful interfaces that allow others to experiment with the system and learn about the connections between actions and their outcomes.  Most system dynamics software packages allow you to build user interfaces. Stella Architect® has an excellent utility for this.

In addition to allowing a user to experiment with the model, a good user interface can also educate them on why the system behaves the way it does. This is done by including in the interface a system story that logically unfolds the feedback structure in the system and explains how that structure creates the behavior over time that we see in the graphical output.

In the next Modeling Challenge you will build a user interface of the S-shaped growth model. Your instructor will guide you through this process. Once you finish, you will have your own microworld that you and others can use to run experiments, follow the system story, and better understand the dynamics of this system.

4.8. Overshoot and Collapse – Non-Renewable Resources and Eroding Carrying Capacity

The UN’s low variant curve corresponds to the situation in which the world population overshoots the earth’s carrying capacity and subsequently drops back to a sustainable level. This decline might be caused by conflict over resources, disease, or other factors.  This kind of decline is possible if one (or both) of the following two situations exists:

1. There is a delay in information so that the humans don’t realize that the global population is approaching (or has maybe even exceeded) the global carrying capacity. After all, no one really knows for sure what the earth’s carrying capacity is. Besides, even if someone knew what size of population could be supported on earth, how would that be communicated to everyone on the planet? Why should anyone believe it? Who would have the political authority to enforce lower birth rates to make sure we didn’t overshoot the carrying capacity? Would it even be right for someone to have that kind of authority?  All of these issues are at least part of the reason that humans could exceed the global carrying capacity and not realize it for some time…maybe for a generation or more.
2. The carrying capacity of the planet is not constant. It could increase over time because of technological advances, such as more productive agricultural practices. On the other hand, the global carrying capacity could decrease over time drop as things like pollution and deforestation degrade the capacity of the global ecosystem to support human life. Large scale air pollution, water pollution, and the challenge of human-driven climate change are known realities, and their effect on the earth’s carrying capacity is not fully understood.

These two possibilities suggest an approach to modeling the kind of overshoot and decline (or overshoot and collapse) behavior that we see in the UN’s low variant curve for human population. We can do this by adding a stock and flow structure that shows how the earth’s carrying capacity depends on some natural resources whose status changes over time. For example, if the world population depends on some non-renewable resource for survival, the degradation or consumption of that resource would jeopardize lives. Oil and other fossil fuels are resources that exist in finite quantities that are not expected to change appreciably over millions of years. And yet, much of modern society is based on the availability of those resources.  What happens when or if those resources are depleted?

Figure 25 shows a stock and flow diagram corresponding to this kind of situation. We call this model an Overshoot and Collapse Model because it identifies a generic system structure by which a population might overshoot its carrying capacity and subsequently collapse to a much smaller size.  Notice that this diagram looks very similar to the S-shaped growth model, but with some important changes. The next two Modeling Challenges give you a chance to explore the structure of this model and to build a running simulator.

The explanation of the variables in Figure 27 follow:

• Population, Net growth, Fractional Net Growth Rate, Baseline Fractional Growth Rate, and Adequacy of the Ecosystem to Support Continued Growth ― See the discussion on the S-shaped growth model and use the same model setup, values, and equations for these variables, as outlined in the Modeling Challenge: Build the S-shaped model with one exception. The Baseline Fractional Growth Rate should be set to 0.05. Note that, unlike in the S-shaped model, the Carrying Capacity in the overshoot and collapse model is no longer a constant value. Details on this variable are given below.
• Ratio of Population to Carrying Capacity: This has the same physical interpretation and units as in the original S-shaped growth model. However, in this model, the Carrying Capacity will decline as the Actual Resource Base is consumed by the Population. Hence, the Carrying Capacity may reach zero if all the resources are consumed. To avoid dividing by zero in this ratio, we use the built-in function that prevents division by zero. The built-in is simply a double division in the equation.  Carrying Capacity is now Population//Carrying Capacity.
• Desired Years of Resources: The global average lifespan of humans (71 years in 2017). This is used to calculate the Desired Years of Consumption (described shortly). This modeling structure assumes that Carrying Capacity is determined by the amount of resources needed to last for the average human lifetime. Anything less than that will cause widespread awareness of resource shortages, leading to lower consumption rates (this might be an optimistic assumption so feel free to experiment with different values for this constant).
• Desired Years of Consumption: This variable captures the concept of whether enough resources are available to continue operating as we have in the past. We are being explicit about consuming non-renewable resources, but the consequences of running out are in the future, and our behavior may not be changed until the scarcity becomes evident in our lifetime.
• Baseline Annual Consumption per Person: The amount of resources consumed per person per year whenever resources are plentiful. This variable also has a ghost version in the model. A ghost is an exact copy of the original variable. We use ghosts to avoid drawing long causal links that cross over other parts of the diagram.  This is purely for cosmetic purposes to make the diagram easier to read.  In the diagram, we’ve used the convention of showing it with an italic font and different colors.    Measured in units of Resources/(People * years).
• Actual Resource Base: The stock of nonrenewable resources that the population depends on for survival. Set the initial value of this stock to be equal to the Initial Resource Base (next variable).  Measured in units of Resources.
• Initial Resource Base: An exogenous variable used to set the starting value for the Actual Resource Base. Measured in units of Resources.
• Resource Consumption: The consumption of the resource by the human Population. The amount of resources consumed per year is determined by the number of people in the Population, the Baseline Annual Consumption per Person, and the Effect of the Resources Ratio on Consumption (explained below). Units are Resources/Years.
• Ratio of Actual to Initial Resource Base: The comparison of the Actual Resource Base to the Initial Resource Base. This comparison measures the fraction of the Actual Resource Base to the Initial Resource Base.  A value close to 1 indicates that resources are plentiful and near their original condition.  A value close or approaching 0 indicates that resources are becoming depleted.  Measured in units that are dimensionless.
• Effect of the Resources Ratio on Consumption: This is a nonlinear function that captures the idea that people will continue to consume resources at a regular rate until the resources fall below 50 percent. At this point, resources are more difficult to extract and less is consumed.  The nonlinear function falls to zero when no resources are left as it is physically impossible to consume non-renewable resources once they are gone.  The exercise below will provide you with information to build this nonlinear function.  The units are dimensionless.
• Carrying Capacity: Same as before – the number of people that the system can support. In this case, it is calculated as the number of people that the Actual Resource Base can support, assuming that people consume the resource at the annual rate defined by the Baseline Annual Consumption per Person, and assuming that enough resources are needed to sustain the human population for one lifetime. Units are People.
• UN Low Variant, UN Medium Variant, and UN High Variant: These are the same graphical function variables used in the S-shaped model. You can simply copy those variables from that model and paste them into this model. However, do not copy and paste these variables until you’ve set the model settings to their proper values.

4.9. Population Growth with Renewable Resources: Achieving Sustainability

In the previous overshoot and collapse model, the Population exceeded the carrying capacity of the system partly because that carrying capacity depended on a nonrenewable resource that was consumed by the Population. The fact that the Actual resource stock in that model has only an outflow means that the resources will never be replenished. They will only decline. As a result, the carrying capacity in the system erodes over time as the resource base is consumed, and the population eventually collapses.

However, it may be that the resources are renewable. Critical resources such as water or food can be replenished with the seasons of the year through agriculture and other methods. The next Action Item gives you an opportunity to enhance the original overshoot and collapse model by adding an inflow into the Resource stock, making it a renewable resource.

4.10. Oscillation: A Predator-Prey Case Study

4.10.1. Introduction to Oscillatory Behavior

Up to this point in your modeling journey, you have modeled the basic dynamics that dictate the future of a population, be it a population of humans, animals, or plants. You have also learned how the feedback structure affecting population dynamics can serve as a framework for modeling system behavior from a variety of problems that, on the surface, seem far afield from the problem of population growth.

That’s part of the beauty of using system dynamics as a lens for looking at the world. The dynamics affecting the support of political cause or the sale of smartphones may appear on one level to be quite different from those that affect the growth of a population. However, through the lens of system dynamics, we find that the underlying feedback structure of these seemingly different phenomena can explain the behavior of a widely varying collection of real-life phenomena.

We now turn our attention to another type of behavior over time that is common in nature and human activities: oscillatory behavior.  We will use the dynamics related to the predator-prey relationship as a case study to help us explore some ways this type of behavior can arise. In particular, we will return to the Kaibab Plateau to model the dynamic relationship between the Kaibab mule deer population and the population of predators that depend on the mule deer for food.

An important theme to remember as we examine the dynamics of oscillatory behavior is the following.

In predator-prey systems, the stocks of predators and their prey often oscillate around equilibria levels over time (see Figure 28).  Initially, both populations exhibit a steady state (constant) pattern at two different equilibrium levels: one equilibrium for the prey population, and another for the predator population. The equilibrium levels are those population sizes at which the predators and prey co-exist in a steady-state balance. The predators kill just enough prey to maintain their population size, and the prey would grow at a rate to offset the deaths through predation (and other natural causes).

Notice, however, that about ⅓ of the way into the simulation, something happens to “kick” the system out of equilibrium. At this point, both populations start to oscillate around their respective equilibria.  This is how such systems work. If, for any reason, either population moves away from its equilibrium point, both populations will begin to oscillate…sometimes indefinitely, but often for a limited time until the system settles back into equilibrium (as seen in Figure 28).

Why does this oscillation happen?  Imagine, for example, that a disease strikes the prey population, so that many prey die.  You can see this in Figure 28 by the sharp drop in the prey population ⅓ of the way into the BOTG.  This drop in the prey population creates a food shortage for the predators, so that a predator population can no longer be supported at the equilibrium size; the predators will decline in numbers. This allows the prey population to rebound (since there are fewer predators in the system to hunt and kill prey).  As the prey population rebounds, the predator population will follow suit as it becomes easier for them to find food.  The process can continue for some time as the predator and prey populations “seek out” their respective equilibria.

What sort of feedback structure creates the kind of behavior we see in Figure 28?  There are many different formulations of how the predator-prey system works^[4], but all of those formulations involve a feedback structure that is similar to the one shown in Figure 29.  Let’s examine that figure to see how the structure in this model makes sense for the predator-prey problem.

4.10.2. Telling the Story: How Feedback Creates Oscillations in the Predator-Prey System

To understand how the feedback structure in the Predator-Prey model can create oscillations, we must first understand the role played by each of the five feedback loops in this model.

 

 

• Loops R1 and B1: S-Shaped Growth Dynamics.  The upper left portion of the model corresponds to the usual S-shaped growth structure we have seen several times. That is, one set of dynamics affecting the status of the deer population are those dynamics associated with natural unconstrained growth (R1 loop) and the constrained growth as the deer population approaches the carrying capacity of the system (B1 loop).
• Loop R2: Simple Predator Population Dynamics. The R2 feedback loop represents the force of unconstrained growth on the Predator Population, just as in the simple population model.  This kind of growth would be possible in the presence of an infinite food supply. Of course, this is not the case – a reality that is captured in some other feedback loops in the system.  The Net Predator Growth Rate is a biflow indicating that negative values will allow the flow the reverse course and flow out of the stock.  This captures the concept that predators could die of starvation if there is not enough prey.
• The Dynamics Behind Oscillation: One Major Feedback Loop.
  We noted earlier that a major balancing feedback loop (a feedback loop with two stocks) oscillatory behavior is possible. The B3 feedback loop is the major loop and contains a stock for deer and one for predators.  The B2 feedback loop is a minor negative loop.  The combination of the the B3 and B2 feedback loop generate the dampened oscillation observed in Figure 28.  You will notice in Figure 29 that there is a double hash mark on the causal links running into the Effect of Deer Density on Predator Births and into the Effect of Deer Density on Predator Births.  This hash mark is used to indicate delays in the feedback loop.
• In this case, there is a delay in the effect of Deer density on the two Effect variables. By this we mean that minor changes in Deer density will not result in immediate sizable changes in either Effect variable. Instead, Deer density must consistently increase (or decrease) for a period of time before it has an appreciable effect. The two bullets below explain how these two feedback loops work and why the indicated delays make sense.
  • Loop B2: The role of predation. Feedback loop B2 represents balancing effect of predation to keep the size of the Deer Population from getting too large or too small. Through this feedback loop, a decline in the Deer Population through predation is constrained by the effect of Deer Density on the Effect of Deer Density on Predator Hunting Efficiency.  In particular, if there are fewer deer (so that Deer Density decreases), predators will be less efficient in finding deer for food and less deer will die through predation, allowing the Deer Population to rebound. If the Deer population grows, then Effect of Deer Density on Predator Hunting Efficiency will also increase, leading to more Deer Predation and a subsequent decline in the population.  The Effect of Deer Density on Predator Hunting Efficiency is a nonlinear function with a positive slope.  The slope flattens out as the Deer Density doubles or declines towards zero.  An increased Deer Density indicates an abundance of deer and predators will kill more deer up to a certain point.  Predators kill to eat and reducing hunting when satiated.  The same logic applies when Deer Density declines. Once Deer Density drops to critical levels, the Effect of Deer Density on Predator Hunting Efficiency is evident; increased hunting activity by predators will be insufficient to compensate for the loss in food supply. 
  • Loop B3: The mutual interaction between the Deer Population and the Predator Population. This feedback loop represents the dynamics by which changes in the Deer Population and the Predator Population puts in motion forces that generate the oscillation. This loop represents a mutual effect that the Deer and Predator Populations have on one another. As the Deer Population increases, more deer are killed and eaten by Predators, fueling growth in the Predator Population. This in turn leads to over-hunting of Deer and a subsequent decline in the Deer Population. Each stock represents a delay as the inflows and outflows are decoupled and take time to react to changes.  The B3 feedback loop also contains a nonlinearity captured in the Effect of Deer Density on Predator Births.  The slope of this nonlinear function is positive and when Deer Density is one, the Effect of Deer Density on Predator Births will be zero.  That is births and deaths are equal and the predator population will not change.  When the Deer Density is above one, the Effect of Deer Density on Predator Births will be positive and the Predator Population will growth.  When the Deer Density is less than one, the Effect of Deer Density on Predator Births will be negative, meaning that deaths are greater than births, and the Predator Population will decline.

4.10.3. Another Look at the Oscillations in the Predator-Prey BOTG

Recall that in Chapter 2 we discussed how balancing feedback works to correct against a stock’s movement away from an equilibrium or steady state level. Anytime a stock in a balancing feedback loop moves away from its natural equilibrium point, the balancing feedback will work to create a balancing force to move the stock back toward the equilibrium state.

Figure 28 shows both the equilibrium levels for the Predator and Prey Populations, along with how the system responds when one of those stocks is “kicked” out of equilibrium by some external force or event.  When that happens, each population oscillates around its equilibrium point because the B2 and B3 balancing feedback loops constantly work to push the stocks back toward equilibrium.  The B2 loops provides this compensatory action on only the Deer Population. The B3 loops provides compensatory forces on both the Deer and Predator Populations. Moreover, the delays in the causal structure in the B2 and B3 loops is what causes their action on the system to create oscillatory behavior. That’s because the delayed effect in each loop causes its balancing force to be exerted for too long. The loop overcompensates when the stocks move out of equilibrium, swinging the stocks toward the opposite extreme.

Figure 30 further elaborates on how this works in the predator prey model. That figure shows the behavior generated by the structure in Figure 29.  Deer and predators start out in equilibrium in Figure 30.  At time 1925 the Baseline Fractional Deer Growth Rate is reduced by 25 percent for 2 years.  This captures an exogenous shock to the system and could represent a prolonged drought or a disease that reduces the fecundity of the deer for 2 years.  The decline in the Deer Population results in scarcity (a drop in Deer Density) and a lagged decline in the Predator Population.  The reduced Predator Population results in a drop in Deer Predation and an increase in the Deer Population, which, in turn, increases Deer Density.  The increased Deer Density results in an increase in the Net Growth of Predators and an increase in the Predator Population.  The B3 feedback loop is a second-order balancing feedback loop and generates the oscillation.

The B2 feedback loop is a minor balancing feedback loop.  It only has one stock and it works to dampen the oscillations.  When the Deer Population declines the Deer Density also declines.  A lower Deer Density results in the efficiency of predator hunting declining (Effect of Deer Density on Predator Hunting Efficiency).  This results in fewer deer being taken through Deer Predation and an increase in Deer Population.  Carefully study this figure, while keeping the stock and flow diagram (Figure 29) in front of you for reference.

 

4.10.4. Building the Predator-Prey Model: Explanation of the Variables

The table below gives a detailed explanation of the variables in the predator-prey model in Figure 29 and specifies numeric values and ranges for variables. You will use this information to build the model in the next Action Item.

Explanation of the Variables int eh Predator-Prey Model in Figure 29, with Suggested Values Corresponding to the Mule Deer on the Kaibab in 1907

• Initial Deer Population: This is an exogenous variable that can be used to set (and change) the initial size of the Deer Population. Set this variable to be equal to 4000, with an input range running from 2000 to 8000 in increments of 100 deer (i.e. precision = 100) and set the scale = none.
• Deer Population, Deer Net Growth Rate, Baseline Fractional Deer Growth Rate, Actual Fractional Net Growth Rate, Kaibab Adequacy to Support the Deer Population, Ratio of the Deer Population to Carrying Capacity – See the discussion on the S-shaped growth model.
  • Set the Deer Population to be equal to the variable Initial Deer Population.
  • The Baseline Fractional Deer Growth Rate is 1.0 (meaning that there is, on average one deer born each year per each deer in the population). Set the input range to run from 0.1 to 1.5 in increments of 0.01.
  • The Actual Fractional Net Growth Rate, Kaibab Adequacy to Support the Deer Population, and the Ratio of the Deer Population to Carrying Capacity variables are calculated the same way as in the original S-shaped growth model.
  • Set the Kaibab Deer Carrying Capacity to be 40000, with an input range running from 4000 to 60000 deer in increments of 1000 deer.
• Deer Predation: The number of deer killed by predators each year. This is calculated as the product of the Baseline Annual Kills Per Predator, the number of animals in the Predator Population, and the Effect of Deer Density on Predator Efficiency (all explained below).
• Baseline Annual Kills Per Predator: The number of deer that would be killed per predator each year, assuming that there was a natural balance between the predator and deer populations. This variable is set to a value of 40 deer per predator per year (units are Deer/(Predators*Year)). Set the input range to run from 20 to 60 with the precision set to 1.
• Deer Density: A measure of how many deer there are in a given area on the Kaibab Plateau. This is measured as the number of deer per 1000 acres, and is calculated by dividing the Deer Population by the Kaibab Area in Thousands of Acres.  This is then divided by the Normal Deer Density per Acre.
• Normal Deer Density per Acre: The number of deer per acre under ideal conditions.  This value is set to 5 and has units of Deer/Thousands of Acres.
• Effect of Deer Density on Predator Efficiency: This is a dimensionless value used as a multiplier to increase or decrease the Deer predation, according to the Deer density level. This value varies between 0 and 1.5. Its value is 0 when there are no deer present (i.e. when deer density = 0 deer per 1000 acres) and it is 1.5 when there is an abundance of deer (i.e. when the deer density reaches 10 deer per 1000 acres or more). Hence, where there are no deer present, the Deer predation will fall to zero. If there is an overabundance of deer, the Deer predation will be 1.5 times as great as it would be a equilibrium.  This variable was formatted in the diagram to use a boldface, underline font to highlight that its value is not set by an equation, but by a graphical function showing how the Effect is calculated from the Deer Density.  that graphical (nonlinear) function is shown below.

• Kaibab Area in Thousands of Acres: The size of the Kaibab Plateau; units are thousands of acres. Since the Kaibab plateau covers 800,000 acres, this variable is set to the value 800.
• Initial Predator Population: This specifies how many predators are in the ecosystem at the start of the simulation. Set this to be 90 predators. Set the input range to run from 20 to 130 in increments of 1 predator.
• Predator Population: The stock of predators in the ecosystem. Set the initial value to be equal to the variable Initial Predators Population.
• Net Predator Growth Rate: The net number of predators added to (or removed from) the Predator Population each year through the natural birth and death processes. This value is calculated as the product of the Predator Population, the Baseline Fractional Predator Growth Rate, and the Effect of Deer Density on Predator Births.
• Baseline Fractional Predator Growth Rate: This represents the fraction by which the Predator Population will grow each year, assuming there are an abundance of deer available for food. Set this value to 0.7 with an input range from 0.2 to 1.5 with a precision of 0.01.
• Effect of Deer Density on Predator Births: This is a dimensionless value used to represent how the Deer Density on the Kaibab Plateau affects the growth rates of the predator herd. This variable is used as a multiplier to increase or decrease the Net Predator Growth Rate, according to the Deer Density value (with greater density implying more food for predators, and hence a greater reproductive rate). This Effect  of Deer Density on Predator Births variable is also determined by a graphical relationship with Deer Density. When the Deer density is 0, the Effect is -1.0, meaning that the Predator population will experience negative growth (from a lack of food). When the Deer density is 1, the effect is 0, meaning that the Predator Population will experience flat growth (neither growing nor shrinking). When the Deer Density is more than 2 , the Effect of Deer Density on Predator Births is 1.5, meaning that the Predator Population will grow at 1.5 times the Baseline Fractional Net Growth Rate (because there are abundant deer available for food).  Below is the graphical function defining how the Effect of Deer Density on Predator Births is determined by the Deer Density.

4.11. Putting it all Together: A kaibab Ecosystem Model

4.11.1. You’ve Been Here Before

It’s now time to build model of the Kaibab ecosystem.  This model will be very similar to the simulator you used in the Kaibab Challenge at the beginning of this Learning Guide.  Creating this model will require you to combine many of the feedback structures you’ve already learned from previous models.

In the Kaibab Challenge, you were placed back in the early 1900’s and asked to make decisions so that the mule deer population on the Kaibab Plateau would grow and flourish for generations to come. You quickly discovered that this was no easy task. In fact, (just like those who actually worked on this problem in the early 20th century), you found that the most obvious approaches to growing the deer population (such as reducing the number of predators) would lead to disaster. In many ways, this unfortunate outcome was inevitable for the following reasons:

• You had misleading information: Many thought that the Kaibab could support up to 30,000-40,000 deer. Given that there were currently only 4000 deer on the Kaibab, it seemed obvious that measures could be taken to increase the population while remaining within the natural constraints of the ecosystem.
• Stakeholders muddied the decision making process: Your job was at stake! The President of the United States and your own boss had expectations that made it difficult for you to objectively choose the best course of action.
• The ecosystem behaved in counterintuitive ways: The deer population did not follow the trajectory you had hoped for while it seemed that all was going well, but then (for most of you) the deer population exploded and consumed all of the forage. When the forage disappeared, the deer population collapsed. For others of you, the deer population was on what looked like a promising growth trajectory, but you probably noticed that was still growing. This led to an uneasy feeling that the deer population might eventually grow to outstrip the capacity of the ecosystem.

By now you have learned that these challenges come from an underlying system with multiple competing feedback loops involving deer, predators, livestock, and the forest on the plateau. In the models you have built and explored, you’ve seen that even fairly simple systems can behave in ways that defy our common sense.

You now have practice with the tools of system dynamics to help you analyze problems involving feedback systems like the Kaibab Plateau. You can build simulators that explore how the important feedback dynamics in such systems affect their behavior over time. Armed with that kind of understanding, you are in much better position to make good decisions.

And so we return to the Kaibab Plateau and the mule deer population in the early 1900’s. Our purpose in doing so is to revisit the challenging problem armed with the tools of system dynamics and to see how simulating such a system can lead to better decisions in the future. You will produce a simulator and user interface, along with a system story to explain how the feedback structure in the system works to create the irruption and subsequent collapse of the Kaibab mule deer population when the predator population is reduced by too much. Your interface will allow the user to experiment with the system and find conditions which lead to a healthy, sustainable Kaibab ecosystem with a healthy balance between the deer, the predators, and the forest.

4.11.2. Building the Kaibab Ecosystem Model

Figure 31 shows the stock and flow diagram for the Kaibab ecosystem model, and Exhibit 4.11 provides a detailed description of all of the variables in the model, along with guidance on values, equations, units, and input ranges.  Take a few moments to study Figure 29 and notice that this model is comprised from building blocks that correspond to the models from previous sections of this learning guide. These building blocks are:

• S-shaped growth. This structure is the basic building block for modeling the deer population. The carrying capacity in the model is determined by the ability of the Kaibab forest to generate enough forage to support the deer.
• A renewable resource that the population needs for survival. Because the deer population depends on the forage generated by the Kaibab forest, we augment the S-shaped growth structure by letting the carrying capacity be determined by the amount of forage in the forest. The amount of forage available is represented by the Forage biomass stock in this part of the model. This stock is “drained” by an outflow representing the Forage consumed by deer.  As the trees and plants on the Kaibab propagate and grow, this stock of biomass is replenished. This is represented by the inflow labeled Net annual forage growth.
• Predator-prey dynamics. The predators are the wolves and cougars on the Kaibab. The prey are the mule deer. The same feedback structure is used here as in the original predator prey model.

In addition to these building blocks, the model in Figure 33 shows an additional outflow from the Predator population to account for removal of predators by hunting. In the early 1900’s it was believed that a reduction in the number of predators on the Kaibab would allow the deer population to grow from its current level of 4000 deer to higher levels. At the time, it was believed that the carrying capacity was somewhere between 30,000 and 40,000 deer. By allowing for predators to be removed by hunting, the model will let us evaluate the impact of that policy.

Following Figure 33, Exhibit 4.11 explains the physical meaning and role of each variable in the model. That table provides guidance that you can use to specify the units for each variable, along with the numeric values or equations used to calculate each variable, as appropriate.

Exercise 17: Variable Description for the Kaibab Ecosystem Model
Variables from the S-Shaped Growth Building Block, As Applied tot eh Kaibab Deer Population
Variable	Physical Description	Numeric Value or Equation	Input Limits (for Interface)	Units
Deer Population	Left as exercise	Make equal to Initial Deer Population	Do not adjust	Left as exercise
Initial Deer Population	Left as exercise	4000	Do not adjust	Deer
Baseline Fractional Deer Growth Rate	Left as Exercise	1.0	0.5 to 1.5 (precision = 0.01)	Left as exercise
Actual Fractional Net Growth Rate	Left as exercise
Kaibab Adequacy to Support the Deer Population	Left as exercise
Ratio of the Deer Population to Carrying Capacity	Left as exercise
Kaibab Deer Carrying Capacity	Left as exercise	Forage Biomass/(Desired Consumption per Deer*Deer Carrying Capacity in Years)	Nothing to adjust	Left as exercise
Variables from the S-Shaped Growth Building Blocks, as Applied tot he Forage Biomass on the Kaibab
Variable	Physical Description	Numeric Value or Equation	Input Limits (for Interface)	Units
Forage Biomass	Forage available for deer to eat	Initial Value = 317000	Do not adjust	Metric Tons
Baseline Biomass Growth Fraction	Left to exercise	0.10	0.05 to 0.25 (precision = 0.01)	Left as exercise
Actual Biomass Growth Fraction	Left as exercise	Left as exercise	Do not adjust	Left as exercise
Net Annual Forage Growth	Net change in the forage biomass on the Kaibab via natural growth and decay, apart from consumption by animals.	Left as exercise	Do not adjust	Left as exercise
Kaibab Adequacy for Forage	Left as exercise	Left as exercise	Do not adjust	Left as exercise
Ratio of Forage to Forage Carrying Capacity	Left as exercise	Left as exercise	Do not adjust	Left as exercise
Kaibab Forage Carrying Capacity	Amount of forage biomass that the Kaibab can support on an ongoing basis.	350000	200000 to 600000 (precision = 1000)	Left as exercise
Forage Consumed per Deer	Left as exercise	Left as exercise	Do not adjust	Left as exercise
Variables dictating the interdependence between the deer population on the forage ( a renewable resource). Use the model from section 4.10 as a building block for this section.  Things make look slightly different, but the concepts should hold.
Variable	Physical Description	Numeric Value or Equation	Input Limits (for Interface)	Units
Actual Consumption per Deer	Left as exercise	Left as exercise	Left as exercise	Left as exercise
Desired Consumption per Deer	The amount of forage consumed by a single deer in a year, when abundant forage is available.	0.75	0.4 to 0.8 (precision 0.01)	Left as exercise
Ratio of Actual to Required Biomass	Measures how much forage is available, relative to what is need to sustain the current deer population.  Values below 1 indicate a shortage of forge biomass.
Effect of Forage Availability on Consumption	This nonlinear or graphical function captures how Actual Consumption per Deer will change given the Ratio of Actual to Required Biomass.  Deer will continue to eat a normal amount of forage until only about 30 percent of what is need is available.  Deer will then start to eat less as foraging becomes more difficult.			Dimensionless
Deer Carrying Capacity in Years	Number of years of forage needed to sustain the deer population.	2	1 to 7 (precision = .1)	Left as exercise
Required Forage Biomass	Forage biomass needed to sustain the existing deer population for the number of years indicated by the value of the Deer Carrying Capacity in Years.	Left as exercise	Left as exercise	Left as exercise
Variable from the Predator-Prey building block (as applied to the deer and predators on the Kaibab).
Variable	Physical Description	Numeric Value or Equation	Input Limits (for Interface)	Units
Predator Population	Number of predators on the Kaibab Plateau	Initial Value = Initial Predator Population	Do not adjust	Predators
Initial Predator Population	Used to set the initial size of the Predator Population.  Notice that this is different from the original predator-prey model you built.	100	Provided above	Predators
Net Predator Growth Rate	Left as exercise	Left as exercise	Do not adjust	Left as exercise
Baseline Fractional Predator Growth	Left as exercise	.5	.5	to 1.5 (precision = 0.01)
Effect of Deer Density on Predator Births		See the Explanation of the Variables int eh Predator-Prey Model in Figure 29, with Suggested Values Corresponding to the Mule Deer on the Kaibab in 1907		Left as exercise
Effect of Deer Density on Predator Hunting Efficiency		See the Explanation of the Variables int eh Predator-Prey Model in Figure 29, with Suggested Values Corresponding to the Mule Deer on the Kaibab in 1907		Left as exercise
Deer Density	The ratio of deer to area all of which is divided by a normal.  Results in a scale where one is normal and anything above 1 indicates that you have too many deer and anything below one indicates that you have to few deer compared to what is normal deer per acres.	Deer Population / Kaibab Area in Thousands of Acres / Normal Deer Density per Acre	Do not adjust	Left as exercise
Kaibab Area in Thousands of Acres	Land area of the Kaibab Plateau.  This is approximately 800,000 acres.  We are measuring this in thousands of acres so that is why we use 800 when setting the value.	800	Do not adjust	Left as exercise
Normal Deer Density per Acre	Number of deer per 1000 acres on the Kaibab	5	Do not adjust	Deer/Thousands of Acres
Deer Predation	Left as exercise	Left as exercise	Left as exercise	Left as exercise
Baseline Annual Kills per Predator	The average number of deer killed per predator in a given year.	40	20 to 60 (precision = 0.1)	Deer/(Predator*Years)
Variables related to the predator removal by hunting.
Variable	Physical Description	Numeric Value or Equation	Input Limits (for Interface)	Units
Predators Hunted	Number of predators removed each year through hunting.	Left as exercise		Left as exercise
Fraction Predators Killed per Year	Fraction of predators killed each year.	0	0 to .5 (precision = .01)	Left as exercise
Program Start Year	The year the predator hunting policy is implemented.  From that year on, a fraction of the remaining predators will be hunted and killed each year.	1907	1905 to 1920 (precision = 1)	Left as exercise
Other Ancillary Variables
Variable	Physical Description	Numeric Value or Equation	Input Limits (for Interface)	Units
Historical Deer Behavior Over Time Graph	A reference mode showing the actual deer population estimates from 1900 through 1950.  This is defined by the graphical function shown here.  You can display this BOTG with the Deer Population to help us tune the model parameters to match the historical record.			Deer

Exercises 4:18 Modeling Challenge – Build Kaibab Model and User Interface

1. Build the Kaibab ecosystem model shown here. Use the descriptions of the variables in Exhibit 4.11 to guide your efforts. As you build the model, pay attention to the following:
   1. Document the physical meanings and units of every variable
   2. Label the polarities of all causal links using the +/- notation.
   3. Identify all feedback loops and label/number the loops with the circular identifiers and numbered labels (R1, R2, B1, etc)
   4. Make sure that all the links and variables involved in a given feedback loop are highlighted whenever you pass your cursor over the circular loop identifier.
   5. Use the following model settings
      1. Start time: 1900
      2. Stop time: 1950
      3. DT = 1/20 year
      4. Integration method: RK2
      5. Time units: years

2. Check your model to make sure it is running correctly by making sure you get the following BOTGs.

3. Build a user interface to match this online working version and also shown below.

4. Adjust the model parameters to match as closely as you can the historical BOTG. As you do this, make sure that your adjustments have some connection to reality by keeping in mind the following facts:

• 1. Somewhere between 800 and 1500 predators were removed via hunting from 1907 through 1932.
  2. The annual net growth fraction for deer is somewhere between 0.7/years and 1.2/years.
  3. The annual net growth fraction for the predators is somewhere between 0.5/years and 1.2/years.
  4. In the 1920’s the Kaibab forest started showing signs of severe overconsumption from the deer population.