exupero's blog

# A sigmoidal population

Here's a graph of the worldwide membership of an evangelical Christian denomination since 1900:

To check if this growth is exponential, we can plot the y-axis logarithmically and see if the result is linear:

It looks mostly linear, except for maybe the beginning and the end. Here's an exponential fit with R2 = 0.98.

It's close, but it looks like it could be closer. Possibly the data was more exponential before 2000 than since 2000, so here's an exponential fit that ignores more recent data (R2 = 0.9988 before 2005):

The data since 1990 looks quite linear (R2 = 0.9967):

That explains why the logarithmic plot droops at the end: growth switched over from exponential to linear.

Rather than splicing two models together, we can use a trend function that naturally exhibits a transition from exponential to linear behavior: the logistic function (R2 = 0.9988 for all years):

What troubles evangelical Christians about logistic growth is that after becoming linear it flattens:

In this particular model, membership tops out between 31 and 32 million, suggesting the denomination is already at 70% of its maximum membership, and will be at 95% within 30 years.

That's not to say conversion efforts will fail to bring in new members. This particular denomination has a well-known revolving door problem, such that even when adding a million new converts a year, half a million existing members may be subtracted off the rolls. By 2050, when net growth looks like it will have mostly stalled, there may still be a million new members joining, but there may also be a near-equal number of members exiting.

Sigmoidal population growth like this usually indicates an environment with a carrying capacity. In natural settings, an ecosystem doesn't have an indefinite amount of food or space to support an ever-growing population, so at some point growth flips over into competition for resources. The population hits a ceiling. That seems like an unusual characteristic to observe in a religion, which is fundamentally an idea. Do resource limitations cap the adoption of an idea? Probably it depends. Some ideas, like gravity, spread over time via existing resource networks (such as schools) or through word of mouth and common sense. Other ideas, such as religions, require new resource networks (churches and private schools) and have to be maintained more deliberately through weekly sermons and devotional material. Those resource dependencies may give an idea a carrying capacity. As the population grows, competition for space, time, financial resources, and personal influence can begin to drive adherents away.

Whether this particular denomination's growth is truly sigmoidal or whether the linear phase simply represents a pause in exponential growth that could resume with a cultural or political shift, only time will tell.