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 those of 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 and thus slows exponential growth. When conversions fall 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 model 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 population is N = U + A + B, and the total number in the church is C = A + B.
Parameters
The model has only five parameters.
| PARAMETER | DESCRIPTION |
|---|---|
| Reproduction Potential Rp | This is the number of unbelievers converted and made into enthusiasts by one existing enthusiast, given that the entire population are unbelievers. It measures how much an enthusiast can “reproduce” themselves from the pool of unbelievers. |
| Duration of Enthusiastic Phase τa | The average time an enthusiast is active in conversion before they become an inactive believer. |
| Fraction of Converts Enthusiast g | The fraction of new converts who become enthusiasts. The remaining people become inactive believers immediately after conversion. |
| Leaving Rate α | The total rate at which inactive believers leave the church. |
| Initial Fraction of Church Enthusiast | The fraction of the church that are enthusiasts at the start of the model. |
Results
There are implications for the church’s long-term growth and decline. As with the basic limited enthusiasm model, there is a revival growth threshold beyond which the church will see rapid growth. However, there is also an extinction threshold of unity below 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 that is driven by a subpopulation.
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.
