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Growth limited by interpersonal conflict?

In the previous post I showed a population that seems to be turning over from exponential growth to stagnation. Such leveling off usually suggests competition for resources that limits the environment's carrying capacity, however human populations have some other dynamics that can come into play. For example, humans can begin worrying about population size before resource limitations become a problem, and they can influence growth with state policies and programs.

Here, when looking at the declining growth of a worldwide evangelical denomination, something else may be going on. In The Tipping Point, Malcolm Gladwell popularized the idea that human social groups are limited to about 150 people because the complexity of the human brain can only track around 1502 relationships. In groups larger than 150 people, populations form implicit sub-groups, which create subtle divisions that can lead to factions and schisms.

Thus, in human populations, some of the limited resources that can influence population size include attention and relational psychology. To explore this idea, here's an iterative model of population that factors in the size of the population, its size squared, and its size cubed:

Fitting this function to membership data produces this curve (R2 = 0.9989):

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The approximate coefficients that produce this curve are

These values suggest that as the membership grows, it grows faster, and as the number of relationship grows, membership grows faster, but as the number of trios grows, total membership actually grows more slowly. a indicates a base growth rate of almost 5%, but to understand b and c we also need to think in terms of per capita, for which it helps to rewrite the model as

In this case, bPn is 0.057, suggesting that about 6% of potential relationships in the denomination bring in a new convert each year. cPn2, on the other hand, is -0.102, indicating that more than 10% of potential three-person relationships in the denomination actually drive out a member each year.

It's hard to know how much credit to give this model. Three coefficients is only one step away from von Neumann's elephant, so it could be argued that this is simply overfitting and we shouldn't read too much meaning into these coefficients. On the other hand, as Nassim Taleb observes in The Bed of Procrustes, a "politician doesn't try to convince the other side, only the audience." The cubic term may represent an audience effect. It's generally believed that people leave a church less over theology than how they're treated, but the above model may indicate that people leave the church more because of how other people are treated. Thus, the Pn3 term doesn't reflect the number of trios as much as the number of relationships to other relationships.

Again, time may tell. Unlike the logistic model, this model doesn't predict a slow stagnation, but an incredibly rapid one, peaking short of 23 million members within the next decade:

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If the low growth of the last two years continues, this may be an insightful model. If growth recovers and the last two years were only a brief interruption (completely plausible), then we'll have to wait and see whether logistic growth plays out. Either way, if you have suggestions for further analysis, please email me.