Continuity Test of Causal Measures

As the effect of a causal link (or loop) on a stock gets smaller, then the measure used for that causal link should tend to zero.

This test ensures continuity between the model with casual link or loop and the model without.

This test is illustrated with the three loop limits-to-growth model, figure 1. The stock x is subject to a growth mechanism, loop R0. The effects of this growth are limited by capacity M, loop B1. The third loop is a drain from x, loop B0, which causes the stock to fall short of capacity

Figure 1

The loop gain of B0 is b. Figure 2 shows the stock behaviour as b is reduced from 0.05 to zero. The trajectory of x gets closer to that of the system with loop B0 removed, that is b = 0.

Figure 2

In this model, the impact of B0 is just the loop gain b. Thus, loop impact satisfies this continuity test in this instance. The percentage of the total impact of the three loops can also satisfies the continuity test. Figure 3 shows the percentage impact of B0 tending to zero as loop gain tends to zero:

Figure 3

The way impact is constructed it will satisfy the continuity test in any circumstances where weakening a loop of link. For example, in the model of figure 1, letting M tend to infinity would produced behaviour that tends to the linear model, with just R0 and B0. Correspondingly, the impact of R0, I(R0) = a(1-x/M) -> a. Likewise, the impact of B1 = –ax/M -> 0. The analytical expressions of loop impact in this model are derived in Hayward & Roach 2017 and in:

Loop Impact in the Three-loop Limits-to-Growth Model