The Tragedy of Sahel

CASE STUDY: Tragedy of the Commons
Overview:
The phenomenon known as the "tragedy of the Commons" is a classic environmental event that can be applied to a number of different situations and locations throughout the world. It is best know as the "tragedy of the Sahel", referring to the area in Africa below the Sahara Desert.
The Tragedy of the Sahel is an instance of a more general phenomenon, The Tragedy of the Commons as articulated by Garrett Hardin. The Tragedy of the Commons occurs when people individually perform rational acts intended to further their immediate self-interest, but the combination of every ones behavior hurts the long-term, collective interest. Hardin offers the following example to illustrate the problem. Imagine a cattle pasture, or common, open to all. If any single herdsman increases his herds size, then he benefits directly from that increase. It is the individual who profits from the extra milk and meat produced by additional animals.
Yet increasing the number of cattle also yields negative consequences: additional food, water, and space -- all provided by the common pasture -- are required to support the extra animals. Should these resources become strained, overgrazing will occur. However, the individual herdsman himself does not pay the costs of overgrazing. Instead, such costs affect all herdsmen as well as the community at large. For example, if all of a pastures grass gets eaten, then no food remains for the cattle and they will die, creating a famine. Such famines affect not only the herdsmen who caused the destruction, but also the people who depend on the cattle for food. Each individual in the community, including the herdsman, pays the costs of overgrazing. Thus it is in the herdsmans self-interest to increase his cattle holdings to the point of overgrazing because he receives the benefits of the increase and pays only a fraction of the costs.
In this discussion, the term tragedy is used in a precise and philosophical manner. The Tragedy of the Commons is not tragic because overgrazing and famine occur due to factors beyond human control. The situation is tragic because famine results from people acting in accordance with the incentives presented to them by their community. The very process of receiving the benefits of additional cattle and the division of their costs lies at the root of the Tragedy of the Commons. Thus, a link is created from human desires and motivations to unpleasant, physical consequences. It is the inevitableness of these consequences that most embodies the term tragic.
Building the Model:
For purposes of this model, we're only going to concern ourselves with three main components: 1. Cattle population 2. Grass available 3. People population
Building the Cattle model:
Cattle population changes from simple births and deaths. The average birth rate for cattle is 30% of the cattle population. Initially in this model there are 3,000 head of cattle. Cattle can die in one of two ways: they are either killed by people (known as the "offtake" percentage), or die of natural causes. The total death rate depends on three factors: cattle mortality rate (which depends on the amount of grass eaten), the cattle offtake percentage, and a "medical factor". The medical factor, a dimensionless number between 1 and 2, takes into account improvements in technology, which can lower the cattle mortality rate (cattle mortality rate = cattle mortality/medical factor). Cattle mortality (a separate variable from cattle mortality rate) depends directly on the amount of grass eaten per cow. The table below shows this relationship: Grass eaten per cow | Cattle Mortality (percentage) | 0 | 100 | 200 | 50 | 400 | 30 | 600 | 20 | 800 | 12 | 1000 | 8 | 1200 | 6.5 | 1400 | 5.75 | 1600 | 5.0 | 1800 | 4.5 | 2000 | 4.0 |
REMINDER: don't forget to chance percentages to decimal numbers (100% = 1.0). Also remember that grass eaten per cow is the independent variable on the X axis, and cattle mortality is the dependent variable on the Y axis.
The medical factor is a time-dependent dimensionless number used to represent advances in modern medical technology. This number, always greater than one, is used to lower the death rate. Lowering the death rate results in longer life spans and increased population. Maximum medical factor is 2 -- the death rate is cut in half, the life expectancy is doubled. The minimum value is one, representing normal, traditional medical levels. The following graph shows this relationship: Medical factor (dimensionless) | TIME (in years) | 1 | 0 | 1 | 10 | 1 | 20 | 1 | 30 | 1 | 40 | 1 | 50 | 1 | 60 | 1 | 70 | 1 | 80 | 1 | 90 | 1 | 100 | 1 | 110 | 1 | 120 |
The cattle offtake percentage is the percentage of cattle per year that are taken from the herd each year for food. As the number of cattle rises in comparison with the number of people, the percentage decreases; as the ratio fall, the percentage increases. The graph below shows this relationship: Cattle per person | Cattle offtake (percent) | 0 | 60 | 1 | 45 | 2 | 34 | 3 | 26.5 | 4 | 20 | 5 | 15.5 | 6 | 11.5 | 7 | 10 | 8 | 8.5 | 9 | 7.0 | 10 | 6.5 |
Building the Grass model:
Two things can happen to the grass. It can grow, dependent on the amount of rainfall that occurs, or it can be eaten, dependent on the number of cattle there are in the area. Grass grows at a rate that depends on a number of factors. An equation for grass growth is given below: grass growth = max(1.7 - normalized grass, 0)* amount of grass * grass growth rate * effect of rainfall
"Normalized" refers to a dimensionless number of the ratio between the amount that exists now and the amount that existed at the beginning of the experiment. Normalized grass is therefore defined as: normalized grass = amount of grass / INIT(amount of grass) where INIT is a built-in function that gives the initial amount of grass. The initial amount of grass for this simulation is defined by two factors: the area times the initial range condition. The initial range condition is 100 kg per acre. Area is computed as follows: area = number of living cattle * the area needed per cow.
"Area needed per cow" is the amount of land needed for one cow to survive. An average value for this area is 20 acres. While this may seem to be a high number, keep in mind that the cow wanders over a large area to find the grass it needs.
The "effect of the rainfall" multiplier is a dimensionless number used to represent the effects of rainfall on the commons. This number influences the grass growth rate by modifying the grass growth rate. The rainfall multiplier has a maximum of 2 (double the normal rainfall), and a minimum of 0 (no rain at all). This multiplier is dependent on time: Rainfall multiplier (dimensionless) | TIME (in years) | 1 | 0 | 1 | 10 | 1 | 20 | 1 | 30 | 1 | 40 | 1 | 50 | 1 | 60 | 1 | 70 | 1 | 80 | 1 | 90 | 1 | 100 | 1 | 110 | 1 | 120 |
Grass eaten (in units of kilograms, or kg) can be calculated by multiplying the number of cattle by the amount of grass eaten per cow. The amount of grass eaten per cow depends on the density of the grass (the more grass there is in a given area, the more the cow will eat). Grass density is defined as amount of grass divided by area. The following is a graph of grass eaten per cow as a function of grass density. Grass density (grass/acre) | Grass eaten per cow (kg/cow) | 0 | 0 | 20 | 320 | 40 | 600 | 60 | 800 | 80 | 920 | 100 | 1000 | 120 | 1070 | 140 | 1140 | 160 | 1180 | 180 | 1230 | 200 | 1250 |
Building the People model:
Like most living creatures, people are born and people die. We are assuming that the death rate is directly dependent on the number of cattle that are available per person and the medical factor (i.e. technology improvements).
The initial population of people is 75,000. Births are dependent on a average birth rate (4% per year) and the existing population.
Deaths are defined as: deaths = number of people * (people mortality rate/medical factor)
People mortality is, as mentioned above, directly tied to number of cattle per person. The chart below shows this relationship: Cattle per person | People mortality (percent) | 0 | 100 | 1 | 60 | 2 | 29.5 | 3 | 7.5 | 4 | 2.5 | 5 | 2.15 | 6 | 2.0 | 7 | 1.85 | 8 | 1.7 | 9 | 1.55 | 10 | 1.45 |
This chart shows the obvious: no cattle, everyone dies.
Cattle per person is calculated based on the number of people and the number of cattle. Running the model:
Once you have completed your model, set up the time specs to run this model for 120 years. Over this period of time, you should see increasingly large changes in the grass, cattle, and people populations. It is highly recommended that you create and graph these three populations using a "normalized" value (normalized grass, cattle, and people). To normalize a value, you simply divide the current value by the initial value. The normalized values will fluctuate around one. By calculating and graphing the normalized values rather than the actual values, you will get a clearer picture of the changes in the three populations.

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