Force

In a system dynamics model, the influence of one stock on another may be treated like a force. Activity in one stock, the source, causes another stock, the target, to accelerate. In mechanics, forces are said to “do work” on a system, either injecting or removing energy. Likewise, in the Newtonian Interpretative Framework, a source stock may inject energy into a target stock, causing it to increase in activity. The activity of a stock is measured by its kinetic energy.

Consider an example. Let stock y influence stock x (see Figure 1). This influence is like a force. If y is constant, that is, g(t) = 0, then x increases uniformly. If g(t) is constant, y changes, and x accelerates.

Figure 1: Stock y exerts a force on stock x
Figure 1: Stock y exerts a force on stock x

Let g(t) = -1, then y declines (Figure 2). While y is positive, x is increasing but decelerating. When y becomes negative, x accelerates and is not declining. g(t) = -1 is a negative force of y on x.

Figure 2: Stock Behaviour

Impact

This force has “impact,” which measures force as the ratio of the acceleration imparted on x with the rate of change of x. “Impact” is negative while x slows down and becomes positive as x accelerates. The polarity of this stock-to-stock impact determines whether x accelerates or decelerates. This negative force slows down a growing stock and turns it around so that it declines.

Impact compared with stock behaviour
Figure 3: Impact compared with stock behaviour

Kinetic Energy

The stock x has a “kinetic energy,” which represents its rate of change. Kinetic energy measures how “change” occurs in the stock, a level of its “activity.” Figure 3 shows the change in x‘s kinetic energy over time. Initially, the kinetic energy falls until x becomes stationary. That is, it is momentarily at rest, the maximum of (Figure 1). After this point, the kinetic energy of x increases (Figure 3).

Kinetic energy of y on x and work done by y on x
Figure 3: Kinetic energy of x and work done on x by y.

As x is losing kinetic energy, where does that energy go? Forces do work. The force of y on x, measured by impact, does work on x. In the first phase, negative impact, the force removes energy from x. In the second phase, positive impact, the force injects energy into x. Because this “dynamic” energy is conserved in a stock-flow system, the graph of the work done by y on x is identical to the kinetic energy (Figure 3).

Energy Sink

Modify the model of Figure 1 to include a drain on x (Figure 4). The force is now made positive, g(t) = 1, so that y now accelerates x. The balancing loop, B, opposes this force. Thus, x grows, initially accelerating but tending towards uniform change (Figure 5). The curve of x is becoming straighter. The impact of y on x is declining, indicating a reduction in acceleration.

SD model with external force and draining process
Figure 4: Driving force on x with resisting force, balancing loop B.
Figure 5: Stock behaviour of model in Figure 4 with impact of y on x

From the energy perspective, stock y is injecting energy into stock x, with loop B acting as an energy sink. Energy flows into x, initially increasing its kinetic energy – x accelerates. But B removes more and more energy, causing x to lose kinetic energy, slowing it down until kinetic energy becomes constant.

Figure 6 shows the energy balance on x. The energy injected from y (red dashes) is positive. The energy removed by B is negative (black dots). Their sum is equal to the change in kinetic energy (blue solid).

change in kinetic energy of x = work done by y on x + work done by B on X

Energy balance with external force and one balancing loop
Figure 6: Energy balance for the model in Figure 4. Change in kinetic energy, work done by y on x and work done by x on itself via B

The energy injected into x by y is partly absorbed into the kinetic energy of x, making it change faster. The remainder is removed from the system by B. For example, at time = 10, the work done by y on x is 0.370, and the change in kinetic energy is 0.2. The remainder is the work done by B, -0.170. Eventually, the kinetic energy stops changing, and x continues to change uniformly (Figure 5).

Summary

One stock can inject or remove energy from another stock. First-order balancing loops remove energy from a stock-flow system. It can be shown that first-order balancing loops inject energy, second-order balancing loops exchange energy, and, under some conditions, may be conserved. In all other loops, energy is injected into a system regardless of their order, whether they are reinforcing or balancing. A first-order balancing loop is the only structure that removes energy from stocks. They are essential for stability to be possible.