Time Paths

System dynamics modeling is concerned with the dynamic behavior of systems - that is, the behavior of systems over time. In system dynamics modeling, the modeler attempts to identify the patterns of behavior being exhibited by important system variables, and then build a model that can mimic the patterns. Once a model has this capability, it can be used as a laboratory for testing policies aimed at altering a system's behavior in desired ways.

Figure 1 is a straight line with a positive slope and an example of a dynamic time path. Although actual systems can exhibit a variety of time paths (often simultaneously), the good news is that the number of distinct time path "families" that exists is relatively small.

Figures 2 through 5 present a collection of fifteen distinct time paths which occur frequently in the real world. These paths have been grouped into five distinct families, one of which is shown in each of the figures. Other, more complicated, time paths are observable, but they are almost always combinations of the paths presented here.

Linear Family

The first identifiable family of time paths is the linear family, which is presented in Figure 2. The linear family of paths includes equilibrium, linear growth, and linear decline. Given the simplicity and intuitive appeal of these paths, it is important to point out some facts which can place them in the proper perspective, vis-à-vis system dynamics modeling.

The first thing to note is that the average person, not trained in dynamic modeling, tends to think that actual systems grow or decline in a linear fashion - that is, as shown in second and third time paths of Figure 2.The truth, however, is that most systems do not grow and decline along linear time paths, but rather along exponential time paths (as shown in Figure 3). Pure linear time paths are usually generated by systems devoid of feedback - a crucial building block of both actual systems and system dynamics models.

A second thing to note is that the equilibrium time path presented in Figure 2 is a dynamic behavior that few actual systems exhibit. It is a state of perfect balance in which there are no pressures for change. In fact, from a system dynamics point of view, equilibrium implies that all of a system's state variables reach their desired values simultaneously - a very artificial situation. Curiously, much of modern economics and management science uses models based on the concept of equilibrium. System dynamicists, on the other hand, believe that the most interesting system behaviors (i.e., problems) are disequilibrium behaviors and that the most effective system dynamics models exhibit disequilibrium time paths. This is not to say, however, that the concept of equilibrium is useless. On the contrary, system dynamics models are often placed in an initial state of equilibrium to study their "pure" response behavior to policy shocks.

Exponential Family

The second distinct family of time paths is the exponential family, shown in Figure 3. The exponential family consists of exponential growth and exponential decay. As previously mentioned, real systems tend to grow and decline along exponential time paths, as opposed to linear time paths.

Goal-seeking Family

The third distinct family of time paths is the goal-seeking family as shown in Figure 4. All living systems (and many nonliving systems) exhibit goal-seeking behavior. Goal-seeking behavior is related to exponential decay. This can be seen by comparing the second time path in Figure 3 with the second time path in Figure 4. The only difference between the two is that in Figure 3, the time path is seeking a goal of zero, whereas in Figure 4, the time path is seeking a nonzero goal.

Oscillation Family

The fourth distinct family of time paths is the oscillation family. Oscillation is one of the most common dynamic behaviors in the world and is characterized by many distinct patterns. Four of these distinct patterns are shown in Figure 5: sustained, damped, exploding, and chaos.

Oscillations that are sustained can have periodicities (the number of peaks that occur before the cycle repeats) of any number. Sustained oscillations are characterized by a periodicity of one. Damped oscillations are exhibited by systems that utilize dissipation or relaxation processes. Examples of dissipation or relaxation processes include friction in physical systems and information smoothing in social systems. Exploding oscillations either grow until they settle down into a sustained pattern or grow until the system is torn apart. . As a result, exploding oscillations usually do not occur very frequently in the real world, nor last very long when they do. Chaotic behavior is an oscillatory time path that unfolds irregularly and never repeats (i.e., its period is essentially infinite). Chaos is a unique type of time path because it is an essentially random pattern that is generated by a system devoid of randomness.

S-shaped Family

The fifth distinct family of time paths is the s-shaped family, shown in Figure 6. Close inspection of the s-shaped growth time path pattern reveals that it is really a combination of two time paths - exponential growth and goal-seeking behavior. More precisely, in the case of s-shaped growth, exponential growth gives way to goal-seeking behavior as the system approaches its limit or carrying capacity (indicated by the green line).

Sometimes, however, a system can overshoot its carrying capacity. If this occurs and the system's carrying capacity is not completely destroyed, the system tends to oscillate around its carrying capacity. On the other hand, if the system overshoots and its carrying capacity is damaged, the system will eventually collapse. This is referred to as an "overshoot and collapse" system response. A third possible outcome from an overshoot event is that a system will simply reverse direction and approach the system capacity in a reverse s-shaped pattern. As with a "normal" s-shaped pattern, a reverse s-shaped pattern is a combination of two time paths -- exponential decay and a self-reinforcing spiral of decline.