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Graphical Analysis

One of the objects of systems analysis is to learn which processes, components, and variable are most important to gaining an understanding of a system and its functions. Flow charts are one tool that provides a visual description of the system; and if accurately constructed, the flow chart can provide important insight about which processes are important and which variables should be measured. A second tool that can be used is graphical analysis. The first practice session presented some basic ideas about graphing and analyzing graphs. You already know how to read a graph as an indication of a causal relationship and you know the difference between linear and exponential graphs and between positive and negative relationships. You also are aware of the power of graphs as a way of allowing you to estimate values that were not measured although you also know that extrapolation can lead to serious problems if the assumptions inherent in the extrapolation are not correct. Graphical analysis has other uses as well. I think two of the most important are recognizing limits or limiting factors within a system and describing the effects of feedback on the behavior of the system.

An example of a limiting factor is oxygen in an aquatic ecosystem. All organism except anaerobic bacteria within the ecosystem are dependent on the availability of oxygen that has dissolved in the water. One could easily see this in a graphical presentation of the two variables dissolved oxygen and biodiversity (number of species) of the system. As oxygen content of the water increases the numbers of species found in the ecosystem increases exponentially. Or put another way as the amount of oxygen in the environment decreases, the number of species decreases exponentially. A biologist interested in aquatic ecosystems would want to learn about the factors that control movement of oxygen into and out of the ecosystem. The simple flow chart and a graph are shown here. The graph also suggests that some factor other than dissolved oxygen limits biodiversity in rich aquatic ecosystems. To better understand the system, the investigator would expand the flow chart to show variables that impact on the dissolved oxygen content as well as to try to understand what additional factors are important in controlling biodiversity.

Another example of limits is the diameter of the trees sampled and described in the first practice session. The growth curve of the trees suggested that diameters could not increase above about 40 inches for that species of tree. Recognizing this limit might lead one to ask why and then begin a series of investigations to determine the nature of the factors that limit the growth of the trees. What is important to me in each of these examples, is that the recognition of limits implies an underlying process that controls some aspect of system function. In other words finding a variable that has a discrete and recognizable limit or that imposes limits on other variables can provide a researcher with direction for future thought and experimentation.

Graphing feedback can provide even more information about a system because it the feedback within a system that provides internal controls of system functions. For example if we were interested in the earth's climate system we might begin by constructing a flow chart that relates energy input and output to average temperature.

Input of energy (from the sun) is not shown in the flow chart to be controlled by feedback. However, energy output is because as the planet gets hotter it will radiate more energy into space and as it radiates more energy into space, it will get colder. Therefore, this aspect of the system is controlled by negative feedback. It is possible to construct schematic graphs (graphs that illustrate the relationships without actual data). The effect of temperature on energy output is exponential. As temperature increases, energy radiated to space increases exponentially. The effect of energy loss on temperature is linear. Loss of one calorie or one watt of energy by a system will reduce temperature by so many degrees irregardless of the temperature. The two graphs are shown below.

Both graphs show the same variables. They differ in the representation of dependent and independent variables. Be sure you recognize which graph is showing temperature as the dependent variable. However, because both graphs describe the same variables, it should be possible to combine the relationships in a single graph. A convention is needed to identify the independent and dependent variables for each curve (line). We will use T/EO to represent temperature controlled by energy output and EO/T for energy output controlled by temperature.

One of the interesting aspects of graphs is that only a few of all possible values are "allowed" or predicted. In a simple graph there should be only one allowed value of Y for any one value of X. The allowed values are represented by the curve. Only those values that fall on the curve are consistent with the relationship predicted or shown by the graph. But what happens when we combine the two relationships? How many "allowed" values are there in the graph shown above? Yes, there is only one set of X and Y values (T and EO values) that is consistent with the relationships shown, the point of intersections. All other positions on the graph must change because either the value of T shown does not predict the EO, or the value of EO does not predict the T or neither predicts the other according to the graph. (You might take a few minutes to be sure that you understand this concept. It is very important to the ideas that are presented later.) On the other hand, the point where T predicts EO and where EO predicts T, the point where the lines intersect is pretty special. In fact it is a steady state. If the system is at that intersection, it will remain there until some force acts to shift one of the variables. If it is a stable steady state, what will happen then?

Another example: Star size is controlled by a variety of variables. For a particular star, the size, or radius, can be shown to be derived from an interaction of internal pressure (pressure pushing the gases outward) and gravitational force acting on the gases pulling them in. Considering only the relationship between size and gravity, as star size increases, gravitational force drops and as gravitational force increases, star size decreases. Before reading on draw a flow chart that illustrates this feedback. Now draw two graphs illustrating the two relationships indicated in your feedback loop. Assume both relationships are linear. When you have completed this exercise, click here to move on to the next page.

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