Balancing Loops: Growth and Decline

Early on in a system dynamics (SD) course, it is usual for a learner to be introduced to two types of first-order balancing loops: one where the stock grows, and one where the stock declines.

Death Process

Take the decline situation first. If there are no births or migration, a population will decline over time through deaths, figure 1:

Figure 1: Model of Death Process

The population declines exponentially, figure 2. I have sometimes heard it said that the stock declines because of the balancing loop. But this is not true. The role of the balancing loop is to SLOW the decline. The stock declines because deaths are an outflow subtracting people from the stock! If there were no loop, the decline would be straight. By the way, there are issues with this model, which I will return to at the end.

Figure 2: Graph of the Death Process – Negative Exponential Decline

In figure 1 note that loop B has one positive polarity and one negative polarity. The plus is a same way link, and the minus is a subtract from link because it is attached to a stock. It is the subtract from link that ensures the stock declines. You could read this as: the fewer in the population, the fewer deaths, thus fewer are subtracted from the population. The “fewer subtracted” is the control that slows the decline. Do not say: the more population, the more deaths, thus the less population! Firstly, in this case, there cannot be “more population” as there is only an outflow. Secondly, even if there were an inflow that could allow more population, the last part should then read: “the more is subtracted from the stock.” It is then clear the action of the loop is opposing the initial intention, thus is giving control.

Goal-Seeking Process

The other example of a balancing loop SD learners usually see is the goal-seeking process. Employees are recruited with the rate of recruitment proportional to the vacancies. There is a target number of employees. Thus, the vacancies are the target minus the shortfall, figure 3.

Figure 3: Goal-Seeking Model

The population grows as new employees are recruited, but the growth slows down as the vacancies are reduced, figure 4. The more employees, the fewer vacancies, thus less recruitment, therefore the fewer employees added. As in the death process, the role of the balancing loop is to slow the changes, but this time it slows the growth of the stock. If there were no loop, then growth would be linear. The curve is negative exponential, but it approaches the limit from below, unlike figure 2, where it is approached from above.

Figure 4: Graph of the Goal Seeking Process – Negative Exponential from Below

The negative polarity in figure 3 is an opposite way link. The plus on the connector into the recruitment flow is a same way link. However, the plus on the flow into the stock is an add to link. Thus positive polarities are interpreted differently if the plus is on a connector or a flow, same way, add to. Likewise with negative polarities: opposite way for a connector, subtract from for a flow.

First-order balancing loops are decelerating forces. They oppose uniform changes in a stock. Contrast this with first-order reinforcing loops which are accelerating forces, which enhance the changes in the stock. Put another way, first-order balancing loops are dissipation forces whereas first-order reinforcing loops are generative forces, a bit like a heat source. This is the sociomechanics interpretation.

Postscript

I said earlier there were some issues with the death model of figure 1. I set a death rate of 10%, which gives a typical lifetime of 10 years (or thereabouts as there are logs involved.) Yet there are still many individuals left after 10 years. Indeed there are still some after 40 years! Why? The type of SD used here has continuous time. The stock is an aggregation of individuals, and it is assumed the individuals are homogeneously mixed in all their properties – including age – at all times. But if deaths are the only process, then this assumption will be broken, because as time passes, the remaining individuals get older. In fact, the stock would decline as a straight line as the death rate would increase as the average age of the stock goes up. This type of SD is not meant to capture this type of situation – instead, it needs incremental time and conveyors. In most models, births accompany the deaths so that the stock stays reasonably mixed in age structure. In these cases, the death model works fine.

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