# Introduction to Models of CO2 Concentration

The concentration of carbon-dioxide (CO2) in lakes and rivers plays a central role in the health of those aquatic ecosystems. Plants take CO2 out of the water during the day for photosynthesis and add CO2 to the water at night. Animals always exhale carbon-dioxide into the water as they breathe. The ecosystem will be in a healthy equilibrium when there is roughly the same amount of CO2 entering the water as leaving it. When CO2 concentrations rise too high, algae growth dramatically increases: the body of water "blooms" and becomes covered by a dense layer of algae. At this point, the underlying water becomes oxygen and light deprived, and is no longer capable of sustaining a diversity of life forms. In effect, the ecosystem may die.

### Figure 6: A healthy, happy lake.

Limnologists (scientists who study freshwater environments) monitor carbon-dioxide levels in lakes and rivers in order to intervene, should the ecosystem become threatened. (For example, by running a water cannon, such as those one often sees in the lakes in front of corporate headquarters, more oxygen is mixed into the water.) However, one type of testing yields measurements of CO2 production rates, as opposed to concentrations.

The rate at which carbon-dioxide is produced is a function of time. To obtain information about the concentration of CO2, we must add up all of the instantaneous changes. In other words, we must integrate the rate of change.

Since the scientist only has the data points he or she collected, it is necessary to choose a model function for the data. The model function can then be integrated exactly to produce a numerical estimate for the total CO2 concentration.

In order to know when to intervene in an endangered lake, a limnologist must also have an idea of the accuracy of the estimate.

## Question 6

Suppose a model function R(t) representing the rate of change of carbon-dioxide in a river over a 24 hour period has been chosen. That is, R(t) represents the rate at which CO2 is entering or leaving the river at time t.
• By knowing the model function and the concentration of CO2 in the river at the beginning of the 24 hour period, describe how you could estimate the concentration at the end of the period.

• Use mathematical notation to describe the "concentration function", C(n), that gives an approximate CO2 concentration after n hours have passed. Explain your notation clearly enough for another student to easily decipher it. (Hint: express C(n) in terms of an integral involving R(t).)

In the next pages, we put these ideas into practice by numerically integrating actual CO2 rate data collected over a 24 hour period starting at dawn, and analyzing the results.
Next:Choosing a Good Model Function
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