Viewing a problem as a "system"

Definition of the "systems approach"

For the purposes of this course, we are going to use a specific definition of how to think of a problem as a system. This isn't the only definition of a "system" and might be different than the one (or ones) that you are familiar with. The purpose of our limited definition is to provide a short list of characteristics of a system and then try to describe many different structures and behaviors using just this short list. This intellectual process is both a good way to start looking at problems and will help us see similarities between different systems.

Because this "systems" approach is useful for simplifying problems, looking for significant processes and identifying controls, it is a good way to generate environmental information. But the systems approach can also be used to create simulations of future conditions and to communicate these to other people who are making decisions. One of the benefits of this approach is that it clearly identifies the assumptions on which simulations are based. A good "systems" model is both a valuable research tool and a platform for decision making discussions.

Tools

There are five objects that we will use to represent the structure and behavior of our chosen system. For example we will look at the growth of a population of rabbits (see Figure 1). This figure contains the five main tools that we will use, stocks, flows, information flows, convertors/constants and a source/sink.

Figure 1. A simple systems diagram for the increase in a population of rabbits.

Stocks are a quantity of something. Water in a tank is a good example of a stock. Sometimes these are called reservoirs. All the stocks that are connected with flows will have the same units, that is all the stocks will be a quantity of water, or an amount of carbon, or the number of people, etc. In our example, the stock is the number of rabbits in the population. We represent this in a systems diagram with a box.

A source or sink is a special case of a stock, these either have an unlimited, unchanging concentration or they are outside of the system that we are studying. In our example, the source of new matter for rabbits is not being considered. You can easily imagine another model where the amount of food available to the rabbit population limited the amount of new rabbits being born. In this case, we would probably model the system to include the nutrients as a stock rather than a source/sink. A source/sink is represented as a cloud in our diagrams.

Flows connect stocks or source/sinks. The flow will increase any stock that it flows into or decrease a stock that it flows out of. All the flows that are connected to a stock will have the same units which will be, whatever the units of the stocks are per time. For example this could be liters of water per hour, tons of carbon per year, or in our example, rabbits per month.

When we have information that is needed in the model as a constant or we need to make a calculation, we show that as a "converter/constant". In our example, the growth rate constant for the rabbits was given as a constant. In the diagram, this is circle.

Information connectors illustrate the flow of information, not material from other components to either flows or converters. Information cannot flow to a stock because the stocks can't do anything with that information. In the simplest form, an information flow simply notifies an action of the concentration of a stock, the rate of flow, or the value in a converter/constant. In our example, information flows brought in the values of the growth rate constant and the number of rabbits to the "birth of new rabbits" flow. The flow is calculated as the growth rate constant * the number of rabbits.

These five tools can be used in very flexible ways to describe the structure of different systems. An important value of this systems approach is that the structure indicates particular types of behavior of the system. In our example of rabbit growth with unlimited resources (indicated by the source/sink tool), the population would grow exponentially. As there are more rabbits, the increase in rabbits will increase, leading to an even higher population of rabbits, and so on. A mathematical model of this population growth would give the following type growth shown in Figure 2. (of course the population can't continue to grow like this forever.)

Figure 2. Rabbit population growth predicted from the model in Figure 1. The initial rabbit stock was set to 10 and the growth rate constant was set to 0.1 per month.

The structure of the model is a positive-feedback system. As the stock increases, that increase positively affects that flow that is leading to that stock. Many biological systems have this structure and function. Sometimes this is a good thing, such as in the growth of food crops and forests, the more crops or forests the faster they grow. Sometimes this is a bad feature for humans such as the spread of a disease (the more infected people, the faster the disease will spread) or growth of invasive species.

We will examine several "simple" structures that are very common. These simple structures can be combined in larger models to describe very complex and busy processes. For example, if we were to create a model for global warming it would have positive and negative feedback components, open and closed systems and steady state structures embedded in the larger model. These "simple" structures that we are starting with are like the sentences in a larger document. You might be able to understand the individual sentences but not understand the entire document, but it is very likely that if you don't at least understand the sentences, you won't understand the total document.

Model structures and behaviors

The following structures and behaviors can be found in many larger systems models. The analysis of a system should start with determining the extent or boundaries of the system as you plan to study it, and then look for smaller structures and then how these smaller units are related.

Boundaries of the system - When we decide to study or communicate information about a system, we first have to explicitly define the boundaries to our description and what flows in and out. A "closed system" is one in which the boundary includes all stocks and flows and for which there are no source/sink components. Often the decision of whether or not a systems is open or closed requires a judgment based on the significance of some of the smaller losses or gains and a decision on the time scale of your study. For example, you might model a forest as a closed system for nutrients ignoring initially small amounts of nitrogen that comes in from rain or lost through streams. The time scale question is apparent if, for example, you are studying the gain and loss of species in a city park but are ignoring evolution. The description and diagramming of a systems model should attempt to make these boundaries very clear.

Figure 3: Several examples of open and closed systems. a and b are open, c is closed.

Positive and negative feedback - A stock that controls the flow into that stock can be described as having a negative or positive feedback. Sometimes we will talk about positive or negative feedback "loops" which are when stock A controls controls stock B which in turn eventually controls the flow into A. These feedback loops are crucial characteristics of systems control. Figure 1 was an example of a positive feedback and the example behavior given in Figure 2. Figure 4 shows a system that contains a negative feedback system with an example output.

Figure 4. A system that contains a negative feedback control (shown in red). The system wouldn't work without the other components. The number of barnacles continues to increase until it hits a maximum and then it levels off due to lack of any more space.

Stock limitation - One of the powerful applications of the systems approach is to examine the constraints over extended periods of time. Some of these are mitigated by feedback inhibition and others are exacerbated by positive feedback. Stock limitation is an absolute limitation on the amount of a stock that can flow to other stocks or an ultimate sink. Examples of stock limitation might be the seasonal availability of nitrogen in the soil, the space trees to grow, or the amount of fossil fuels available for human consumption. Figure 5 presents two variations on a model for bacterial growth, one with and on without stock limitation.

Figure 5. Stock limitation model for bacterial growth. The stock is the amount of nutrients in the container. In model "a" there is no limiting stock, in model "b" when the limiting stock runs out, the new bacteria production rate just stops.

a.

 

b.

Steady state - The inflow to and outflow from a stock can create a situation where steady state is possible. If the input and output are equal then the value of the stock will not change with time. A slight increase in the input create or a slight decrease in the output rate can lead to an increasing stock. Figure 6 illustrates a familiar example that relates to body weight. Other examples of steady state conditions are the CO2 concentration in the atmosphere (currently not in steady state), use and replenishment of natural capital, or the human population at zero population growth.

The conditions that lead to steady state are important to understand because the steady state may be the consequence of a very slow input and very slow output, in which case not much will ever happen very quickly. Conversely, the steady state could be a very tenuous balance between rapid input and output. With rapid fluxes, slight disturbance in one rate could have dramatic consequences. A good example of this delicate balance is a pond in which a large amount of algae growth is growing and contributing oxygen to the water, but then with a slight change in temperature the large amount of algae turn from a net oxygen producer to a net oxygen consumer. These ponds crash into a scummy mass very quickly and then start to stink. The simpler the system and the controls on the system, the more likely that these rapid fluxes can flip flop. We will examine these environments more with the network view.

Figure 6. An example of a familiar steady state problem. If the input equals the output for a stock, the stock will remain constant with time, no matter how fast the input and output are. If the input exceeds the output, then the stock will increase.

"Simple" and "busier" models

We have referred to "simple" models above. These models have a few components or strings of components and all the units for stocks and flows are related. Some models might contain two parallel paths, for example for energy and nitrogen, through the ecosystem. In this course we would treat these as two simple models that had some interacting control points.

The point of using systems in this course is to take a complex and busy natural or human system, simplify it to just a few components, describe the control over the behavior and make predictions about the behavior of the system if the same controls stay in place for the future.

Several examples of simple and slightly busy models are given below. For each of these examples an analysis is provided that serves to demonstrate how you can use this to understand environmental problems.

Example 1: Changes in human population in a country. The population of a country is determined by the current population, additions from births or immigration and losses from death or emigration. If the birth rate is higher than the death rate even by a little bit, the population can experience an exponential growth rate. In many countries, industrialization has lead to a decreased death rate followed by a decreased birth rate. The overall side-effect of industrialization on the population has been the stabilization of population size. Some countries however, stalled at a level of industrialization that only realized a decrease in the death rate but left the birth rate high. These countries are experiencing rapid population growth rates.

Figure 7. Population change. The population increases from birth or immigration and decreases due to emigration or death.

Analysis - The population is the only stock in this system. All of the inputs and exports are out of the system, which only means they are not being studied in this model, not that they aren't important. The population is a possible steady state situation. Notice that this version of the model has left out the control of births or deaths by the population size itself. (See Figure 1 for how it should be written.) This diagram illustrates clearly that the we need to understand the relative rates of all of these processes to predict what will happen with this population.

 

Figure 8. Busier model of population change. Economic growth in a country (which can be the result of industrialization) creates wealth. The economic wealth per capita is the total economic wealth divided by the population at any time. In models of population growth, the a decrease in death rate is correlated (I'm not stating that it is directly caused by) an initial increase in per capita wealth. If the economic wealth per capita continues to increase, families may choose to have smaller families and thus decrease the birth rate.

Analysis: This model contains two simple models that are connect through the "per capita wealth". Economic growth will increase the per capita wealth and increases in population will decrease the per capita wealth. The critical point of this analysis is that if the population grows more slowly than the population, it may result in a decreased birth rate and a steady state population. However, if the economy grows just enough to decrease the death rate but the per capita wealth doesn't increase after that point, the population will continue to grow exponentially. We will study this situation later in ESR102 when we study the demographic transition. The systems analysis of this problem can be combined with other frameworks to provide further help in describing and making decisions.

 

Example 2: Global warming and CO2 in the atmosphere. Global temperatures and the CO2 in the atmosphere are linked at multiple layers. The "busy" model diagram below shows how several simple models are linked.

Figure 9. A busy model of atmospheric temperature and the geochemical cycle for carbon. The analysis, below, identifies the simple model parts and the linkages between these sub-models.

Analysis: This model is missing many important stocks and flows. Even with this flaw, it is useful to analyze the structure and potential behavior of the model.

The top part of the model shows that the atmosphere is a potentially steady state for heat energy. The sun energy comes in and the heat is radiated back out. The amount of CO2 in the atmosphere makes the net efficiency of irradiation back into space less efficient, requiring a slightly higher atmospheric temperature to reach a steady state for the energy (heat) in the atmosphere.

The bottom part of the model shows two major fates for CO2 from the atmosphere, either going into ocean or terrestrial biomass. In this version, the only controls that are shown are the increase in respiration rates of the terrestrial and oceanic plants from higher temperature. The bottom part of the model is tracking carbon.

Notice that the top part of the model is tracking energy and the bottom part of the model is tracking carbon. There are no flows between these two halves, only an information connection and converter. The linkage of these two sub-models leads to a potentially very important behavior, run-away positive feedback of the temperature. The scenario is the following:

1. the atmospheric temperature increases,
2. which increases respiration from terrestrial and aquatic biota,
3. which leads a higher steady state of CO2 in the atmosphere
4. which, in turn, leads to higher temperature
5. and it continues

These two examples provide studies in how the systems view is valuable. Example 1 shows how to take a simple model and combine it with another simple model to study the potential interactions between processes. Example 2 shows how to dissect a model into the simple sub-models, analyze them and then put these all back together to study the overall behavior and look for potential problems.

Simulations

An important extension of the use of systems models is to create simulations that demonstrate overall system behavior given certain input conditions and constants. In this course, we will look at the components of the system, such as positive or negative feedback to look for very general system behavior. There are software applications that are useful for turning these systems diagrams into mathematical dynamic models (the diagrams and charts in this page were generated with STELLA from High Performance Systems, http://www.hps-inc.com).

Simulations of this type are extremely useful in modern decision making. For example, scientists could create a complicated and very busy model that contained information on fish, dams, river flows and electricity. This model could be run under different climate conditions and demands for energy to show which parameters affect fish survival most. In addition, the model can be shown to people who work in this arena of fish and rivers to see if the model behaves in a way they think it should; does it show low fish years when expected or high fish years following particular events? The simulation model and the accessible interface are powerful tools in addressing complicated social, economic and ecological issues.