Consider a church of initially 68 people with a total leaving rate of 6% per year, including deaths. Church members are assumed to be very effective at building its reputation, the resource, which enables growth at 19 recruits per year per resource/reputation unit. However the church starts with lowish reputation, at 25% of its maximum, due to the church not having had a large impact on society. Reputation is easily depletable at 50% per year, i.e. popularity is lost easily if society is not reminded of the church. This reflects a new paradigm church rather than one with generations of history. It is built at a rate of 0.01 per person per year assuming there is no resistance to building from the resource itself.
Such a highly effective church grows over 80 years to get near its equilibrium value of 254. This is an upper limit that becomes increasingly harder to approach due to the difficulty in generating the reputation/ resource, figure 1. I.e. the church is about as popular as it is possible to get.
Figure 1: Church Growth Through Resource. 1 = Church, 2 = Equilibrium, 3 = Dominance of Feedback Loops
Church Growth Limited by Reputation
Results of the Self-Enhancing Resource Model
A church which relies on a resource such as popularity or reputation alone to generate its growth will always be limited as that resource becomes harder to produce. Additionally, if for other reasons the church or resource falls below a critical size, then the church heads for extinction as it is unable to regenerate its reputation fast enough.
Initially the growth of the church accelerates as the recruitment reinforcing loop has the strongest impact on the church. By time 35 this impact has fallen so that the leaving loop dominates, figure 1. Church growth now slows. The slowing period is much longer than the acceleration period and occurs when the church is 135 less than half way to its final equilibrium value. When growth is through reputation the pattern is often not symmetric, with most of the church growth coming while it is slowing down.
The resource follows a similar pattern, figure 2, and approaches its equilibrium at 80% of is possible maximum faster than the church approaches its limit.
Figure 2: Resource Generation by Church. 1 = Resource, 2 = Equilibrium, 3 = Dominance of Feedback Loops
For the first 22 time units growth in the resource accelerates as resource generation with the self-enhancing "accelerator" has the highest combined impact. Though it slows down before the church does at time 25, the self-enhancing nature of popularity keeps the resource of reputation accelerating for longer than with most other types of resources. Compare with figures 1 and 2 of the bounded resource model where the resource slows down long before church growth slows, 10 and 20 time units respectively.
At time 22 the combination of two balancing feedback loops, the "brake" and depletion, dominates and the increase in reputation starts slowing down, figure 2. The "brake" is the increasing difficulty of generating reputation as it gets large. It can be shown that this increasing difficulty is the cause of the church reaching a limit, not that actual bound on the resource. At 80% reputation has stopped well short of this bound.
Let the church described above start with 58 people, and all other parameters the same. The leaving and recruitment forces are just in balance as can be seen in figure 3 where the balance is initially changing between the two forces. The church stays almost level with very little growth. Eventually the church tips into growth after time 50. The church numbers, at 58 in this case, is the critical mass that determines survival for the given parameters. Any lower, e.g. 57, the church heads for extinction. Thus when a resource is self-enhancing, such as reputation, popularity, or legitimacy, then there is a critical mass of numbers, and reputation, it must achieve to grow. Thus if a church tries to grow on popularity alone it must start at a sufficient size and reputation in order to grow. This could be achieved if it is a church plant of a nationally or internationally known denomination or megachurch. The new church can use the existing church's reputation.
Figure 3: Resource Generation by Church. 1 = Resource, 2 = Equilibrium, 3 = Dominance of Feedback Loops
Figure 4 shows the reputation resource, which is just being held at a critical value as depletion and generation forces fight for control. Mathematically this value is just above an unstable equilibrium point and thus the generation wins. If either church or reputation were lower it would be under this critical value and the church would be unable to get reputation to rise.
Figure 4: Resource Generation by Church. 1 = Resource, 2 = Equilibrium, 3 = Dominance of Feedback Loops
The critical mass effect can be seen in the reputation as well. Initial church is set at 68 people. Curve 1 in figure 5 is where reputation starts at 25%, above the critical mass and the church grows, curve 1 figure 6. Curve 2 is where reputation starts at 15%, just above the critical reputation value. Reputation grows but struggles to rise until around time 30. Likewise the church stays flat until time 30 (curve 2, figure 6); it is just on the margins of heading for extinction.
Figure 5: Resource Generation by Church. 1 = Resource, 2 = Equilibrium, 3 = Dominance of Feedback Loops
Dropping the initial resource to 13%, curve 3, shows that by time 15 the church reputation moves from growth to decline, figure 5, and the church heads for extinction, figure 6. For church growth to occur through reputation alone both church numbers, and reputation, must be above critical values. Thus a new church/small church cannot expect to survive or grow on popularity alone - there must be other growth mechanisms.
The existence of a critical mass when growth is largely through reputation means that a church that suffers a loss of reputation is vulnerable to decline and maybe ceasing to exist. Curve 1 in figures 7 & 8 shows a church above the critical threshold that grows through reputation. The same church is examined again with a drop in reputation at time 40, perhaps due to a scandal, or a change in attitude or law in society over a specific church teaching. At time 50 the church recovers, perhaps the scandal or conflict is forgotten or resolved. Curve 2 has a drop in reputation to a value just above its threshold, curve 3 reputation drops slightly more. In the case of curve 2 the church is able to recover and start growing again, by time 60. However in curve 3 the reputation has fallen too far. Although reputation grows again at time 50, figure 8, the church is declining too fast for sufficient reputation to be maintained and both eventually decline to extinction, figure 7. Thus a church that grows on reputation alone is vulnerable to irreversible decline if it suffers damage to its reputation.
Figure 7: Church suffers a drop in reputation at time 40 for 10 years
curve 2 and curve 3, with no such drop, curve 1 for comparison.
Figure 8: Church suffers a drop in reputation at time 40 for 10 years
curve 2 and curve 3, with no such drop, curve 1 for comparison.