The limited enthusiasm model can be extended for long-term church growth analysis by adding reversion. This model is easier to handle than the demographics model as it has fewer parameters. Nevertheless, it can model long-term dynamics provided births and deaths are similar. As such, the model is helpful for analysing denominational growth and decline where the only data are membership or attendance figures.

The results of this model are included with the demographics model.

### System Dynamics Model

The limited enthusiasm model has two types of Christians: Enthusiasts – those active in conversion; and inactive believers – who play no part in conversion. In this simplified model, only inactive believers leave the church. They immediately become open to re-conversion, figure 1.

Growth is driven by the reinforcing loop R1, where enthusiasts reproduce themselves through conversion. The feedback: more enthusiasts, more conversions, more enthusiasts, gives exponential growth. Growth is opposed by B1, which reduces conversions, thus slowing the exponential growth. When conversions have been reduced below the number who lose enthusiasm, B2, the number of enthusiasts starts to decline. Therefore church growth slows and eventually halts.

B3 controls the unbelievers who leave the church and are open to rejoining. Thus once revival growth is over, the church may decline rather than plateau. There is a reinforcing loop from recycling R2, which may help the church to survive. R2 is the combination of the three loops B1-B3, which create long cycles of growth and decline.

### Equations

The system dynamics reduces to three differential equations:

U is the number of unbelievers; A is the number of enthusiasts; and B is the number of inactive believers. The total populations N = U + A + B and the church numbers are C = A + B.

### Parameters

The model has only five parameters.

### Results

There are implications for the long-term growth and decline of the church. Like the basic limited enthusiasm model, there is a revival growth threshold over which the church will see rapid growth. However, there is also an extinction threshold, equal to unity, under which the church will decline to zero members or attenders.

Figure 2 shows the increase in the two categories of believers when the reproduction potential is above the revival threshold. The number of enthusiasts rises rapidly in the first 7 years. This is the revival period. The rise in inactive believers occurs later, leading to a peak in church numbers after 10 years, figure 3. Subsequently, church numbers oscillate, tending to equilibrium. This equilibrium is less than the first peak in church numbers. Overshoot is a feature of growth where only a sub-population drives the growth.

Reducing the reproduction potential reduces the size of the revival, figure 4. Also, growth takes longer. If the reproduction potential falls below 1, the church heads for extinction.