Results of the Limited Enthusiasm Model with Demographics

The model has four population variables: Unbelievers, who are open to conversion; Hardened Unbelievers; enthusiasts, believers who drive the growth; and inactive believers, who are inactive in recruitment, though may be active in other areas of church life. Principles can be established for the dynamics of long-term church growth:

  1. Equilibrium in the proportions in each population category.
  2. The threshold of revival growth.
  3. The extinction threshold.
  4. Delayed growth.
  5. Recurrent growth.

The term “revival growth” means a rapid exponential growth in the numbers in church and the number of enthusiasts. It is similar to the epidemic phase of the spread of a disease.


After a sufficiently long time, the numbers in the church will stabilise to an equilibrium value, be it zero or non-zero, see figure 1. If the church is part of a growing population, then the proportion of the church in society will stabilise to equilibrium.

Figure 1

The value of this equilibrium and the length of time taken to stabilise depend on the parameters in the system. Generally, the reproduction potential has the biggest effect on the ultimate size of the church. The loss rates have a more moderate effect. Indeed, as long as the reproduction rate is high and new people are being converted, then the loss of children only has a minor effect on the ultimate size of the church as they converted later in life. However, for most church situations, the growth and loss rates are more finely balanced. In this case, the loss of children has a very significant effect on the future size of the church.

Threshold of Revival Growth

As in the case of short-term growth, there is a threshold determined by the loss parameters, the duration of the enthusiastic period, and the proportion of society not in the church. If the reproduction potential exceeds this threshold, the number of enthusiasts increases and the growth of the church becomes rapid, figure 2. The number of enthusiasts starts declining once the threshold is exceeded. However, unlike in short-term growth, the number of enthusiasts never completely dies away as the pool of unbelievers is replenished by new people being born, and people who have left the church. The number of enthusiasts eventually settles to some equilibrium value, like that of the church’s numbers.

Figure 2

Note that the revival growth threshold (curve 1 in the above graph) rises as the church rises as there are fewer unbelievers to convert. Eventually, the revival growth threshold will equal the reproduction potential. This is equilibrium. It is harder to get revival growth in a church that is a large proportion of society compared with one that is small in comparison to society.

If the losses from the church are higher, the revival growth threshold will be correspondingly higher. It is harder to get revival-type growth if the losses from the church are high.

Extinction Threshold

A second threshold, also determined by the loss rates and the duration of the enthusiastic period, determines whether a church survives. If the reproduction potential is under this extinction threshold, the enthusiasts are not reproducing themselves fast enough to survive. As such, the church ends up declining to extinction with a speed proportional to its losses. The extinction threshold is the dotted line in figure 2. This is discussed further on the long-term decline page.

Delayed Growth

When the church is small and declining, an increase in reproduction potential can bring it back to growth and survival. However, it can take some time before growth in the number of enthusiasts (figure 3) can lead to growth in the church. In this case, an enthusiastic period = 5 years, it takes nearly 25 years for the rise in enthusiasts to turn the church around.

Figure 3

Recurring Growth

An intense period of growth can be followed by a significant decline if the growth has exceeded the equilibrium value. Growth has occurred so fast that there are insufficient unbelievers left to sustain the number of enthusiasts. Thus, the church starts to decline. That decline will fall below the equilibrium value leading to further growth again, although not so intense this time (figure 4). The result is revival-type growth recurring at intervals with decreasing intensity.

Figure 4

The period of recurrence and the intensity of subsequent periods of growth depends on the various parameter values. Note that this mechanism alone cannot account for the recurrence of revivals in a country like Wales as the later revivals (1859 and 1904) were the most intense. Also, in the case of Wales, there was no period of decline between revivals only slower periods of growth.