Monday, 29 May 2017

Church Growth and the Perils of Second Order Feedback

Feedback is the process where an action has an effect, which in turn changes the original action. Either it magnifies it ­– reinforcing feedback; or it corrects it – balancing feedback. In the last blog I showed how such feedback loops affect the growth and decline of the church, where changes in the size of people groups within the church feed back on its growth and decline processes [1].

For example, consider the enthusiasts – those responsible for recruiting new people to church. The more enthusiasts, the more converts recruited, thus even more enthusiasts – reinforcing feedback with accelerating growth. This is an example of first order feedback – that is, it only involves one stock – the enthusiasts in this case. A stock is an accumulation – taking time to increase or deplete. Populations are stocks, and first order feedback means they directly change themselves: short-term changes, those up to a couple of generations. Church growth and decline is mainly governed by such first order feedback [2].

Second Order Feedback

Second order feedback involves two stocks. A change in a population stock, like church, affects another stock, before it comes back to change the original population through accumulation or depletion. The “peril” of this second order effect is that the feedback process has inertia, meaning that there is a delay in the feedback process, either magnifying, or regulating, the population changes. This feedback introduces long-term changes that are very hard to control [3]. The following example from church dynamics will illustrate this.


When a new church or movement starts, it is normally quite small, free of centralised control, and spiritually lively. They are often in the hands of people where growth and spiritual purity are to the fore, and they grow rapidly, generating many people who operate in the same unfettered manner.

However, as church gets larger, there is a growing need for organisational structures to regulate church life, train ministers, construct and maintain buildings and finance salaries. These are some of the traits of institutionalism. The bigger the church, the more institutionalised it becomes. It leads to an institutionalised mindset, where maintenance and acceptance by society become a higher priority than growth and spiritual issues.

Institutionalism is a stock, figure 1. It takes time to accumulate, and is hard to remove once it is there. Institutional inertia is well known. Second order feedback, balancing loop B3, happens because institutionalism undermines the conversion process, the first order reinforcing loop R1. The institutional shift moves the bulk of the church from mission to maintenance, thus conversions, loop R1, fall to a value under the church leaving rate and deaths, loop B2. Thus church declines. But the second order effect means that by the time the decline becomes really noticeable, the church is not able to act fast enough to remove the institutionalism to increase conversion, thus decline continues [4].

Figure 1: Institutional Model of Church Growth

Overshoot and Decline

Figure 2 shows the effects of this second order institutional loop on church numbers. There is a long period of rapid growth, helped by the delays in the loop as institutionalism takes so long to build. This is the good side of second order feedback – it takes ages to have an impact. The bad side – the peril of the second order feedback – is that by the time institutionalism has turned growth to decline it has become too large to deal with. There is some natural depletion of institutionalism, (loop B4, figure 1), but it requires deliberate action by the church to dismantle it. Unfortunately, institutionalism is also a mindset, and that action is too little, too late, and the church heads for extinction, figure 2.  

Figure 2: Overshoot and Decline Behaviour of Church Growth and Institutionalism

The result is the institutional lifecycle of figure 2. This is the current state of most of the older denominations in the UK and other Western countries. The revivals of the 18th- 19th centuries have transitioned to organisations with much wider concerns than just saving souls. The timescale of this second order effect, about 300 years in this case, is much longer than that of the revival growth dynamics.

Thus the current decline in church is not primarily due to events happening now, or from the immediate past, but events of a hundred or more years ago being naturally worked out.  I have blogged before how the UK church reached its peak around the 1870s, about 140 years after the commencement of revival [5]. Institutionalism was too high, stifling revival, spiritual life and doctrinal orthodoxy. Decline followed, and 140 years further on again, the churches are in the same declining phase, but much nearer the end.

A Way Forward?

When one church lifecycle is ending, there is more space for another to start. The Methodist movement came out of the decline of a previous church lifecycle that had started at the reformation and, with much political turmoil, had run its course by the end of the 1600s. Methodism started as a renewal movement in the Church of England in the 1730s, then eventually separated in the late 1700s, allowing both to flourish, along with the other denominations also caught up with the revival. Separation allowed Methodism to free itself of the stock of institutionalism of the established church, especially the parish system, thus breaking the effect of the second order feedback loop – at least for the next 100 years before it developed its own institutionalism.

Perhaps now is the time for those who have been part of the evangelical and Charismatic renewal of the last 60 years to separate from the declining denominations. Just tinkering with effects of second order feedback will not turn decline into growth, instead denominations will continue to head to extinction.  The more radical approach of separation is needed.

Such separation is not a recipe for division. The division is already there as is seen in the fights between the biblically orthodox and liberal wings of the older denominations. A separation now would allow both camps to concentrate on what they see as their missions, rather than battles for control of organisations that have run their course. Recently, Free Church minister David Robertson has suggested this strategy for the Church of Scotland, but it really applies to all existing institutional churches – those that have been around for 150 years or more [6].

Some of the older denominations will no doubt wither away. Others may indeed save themselves from that fate, once the internal dynamics have changed. Hopefully, in the new separated churches, it would allow the revival work that has been simmering away since the start of Pentecostalism to flourish and result in many conversions. This is a controversial solution to church decline, but the history of the church is full of such separation. This year is the 500th anniversary of Luther and his objections to the Catholic Church that led to a major parting of the ways [7]. The time may well be right for another parting of the ways in the Christian denominations.


[1] Feedback and Church Growth, Church Growth Modelling Blog

[2] The Limited Enthusiasm Model is largely a model with first order loops

In a population, birth and death precoesses are first order, as are some capacity issues

[3] Overshoot and Collapse is archetypal behaviour caused by a second order loop. See the interaction of a deer population with its environment

[4] The Institutional Model of church growth and decline is described more fully elsewhere:

and applied to the GB Methodist Church:

[5] The connection between revival and church growth and decline is illustrated from the history of the Welsh Presbyterian Church:

[6] Lessons from the Disruption – How a Church Split Can be Positive! The Blog of David Robertson, 19/5/17,

[7] The Quincentenary of the Reformation, Lutheran Council of Great Britain

Sunday, 14 May 2017

Feedback and Church Growth

Feedback in a system that changes over time is the mechanism where an action results in an effect, which then in turn influences the original action. The action literally “feeds back” on itself.

Reinforcing Feedback

For example, if a population grows through births, the more in the population, the more people are born, thus even more are added to the population and it grows faster. This is a reinforcing feedback loop; the result is exponential growth, that is growth that speeds up [1].

Sometimes church leaders think that churches grow the same way. The larger the church becomes, the more people are added to the church, thus the church grows even faster, figure 1. People may be added through births and through conversion. The rectangle “Church” is called a stock. It is an accumulation of people. The pipe “add to church” is called a flow and represents the addition of people in a fixed period of time. Figure 1 is an example of a system dynamics model.

Fig 1. Reinforcing feedback (R) applied to church growth

It is true that the early phase of a church’s growth is often exponential. However, with a bit more thought, not everyone in a church is engaged in the process of adding people to the church. Some are inactive in church altogether. Others may be active in church life but may have never brought anyone new to church. Others may have invited new people in the past but are no longer doing so. For many of the churches I have studied I have estimated that at any one time less than 5% of church members are active in adding people to their church.


As an alternative to the hypothesis in figure 1, I have proposed an alternative feedback hypothesis where only a subset of the church actively adds people to the church [2]. I call these people enthusiasts, after the nickname given to the Methodists in the 18th century – people who were very active in evangelism. Figure 2 expresses this hypothesis as a reinforcing feedback loop

Fig 2. Reinforcing feedback (R1) and enthusiasts

Those from outside the church who are made enthusiasts are also added to the church, and make more enthusiasts – the feedback effect. In addition, enthusiasts convert others who, although added to the church, are not enthusiasts themselves, converting no one. This latter mechanism has no feedback loop. The feedback of figure 2 is weaker than that of figure 1, however it is the mechanism seen in revival, evangelistic campaigns and courses such as Alpha, and can result in considerable exponential growth in the church.

Balancing Feedback

Balancing feedback is the process where the effect of an action attempts to restrain rather than reinforce the action.  For example when a population is declining through deaths, the more deaths, the less people in the population, thus deaths are reduced. The result is the exponential decay of the population, decline that slows down [3].

Limited Enthusiasm

There is good evidence that enthusiasts do not remain so indefinitely. Some run out of people to invite to church, others get taken up with other aspects of church life and forget evangelism. John Wesley complained that his converts went on to become much better people as a result of the Holy Spirit in their lives, so good they became prosperous, and lost their zeal for religion! This is often called Wesley’s law of the decline of pure religion [4], and can be expressed as balancing feedback:

Fig 3. Balancing feedback (B2) and enthusiasts

Limited Population Size

Of course converting people and making them enthusiasts does not occur in a population of an unlimited supply of people. Populations are finite and as people are made enthusiasts, this pool of unbelievers declines, and it becomes harder to make enthusiasts. This is balancing feedback on unbelievers:

Fig 4. Balancing feedback (B3) and unbelievers

Because unbelievers become enthusiasts, then the effect of the balancing feedback on unbelievers, slowing its decline, is mirrored in its equal and opposite effect on enthusiasts, slowing their growth. Feedback loops exert forces on the population, speeding them up or slowing them down, and this mirroring effect is the equivalent of Newton’s third law of motion. Some readers may remember that from high school physics.

Balance of Forces

It follows that the stock of enthusiasts is subject to three forces from the three feedback loops. The action of the enthusiasts themselves, R1, accelerates their growth; the decline of unbelievers, B3, slows the enthusiasts’ growth, turning growth to accelerating decline; the loss of enthusiasts, B2, eventually slowing their decline to zero, figure 5. Growth in enthusiast numbers eventually turns to decline because the effect of the unbelievers reduces the production of enthusiasts to a level below their losses.

Fig 5. The effects of feedback on the growth and decline of enthusiasts (percentage of population)

The effect of enthusiasts’ activity is that the total church, enthusiasts and inactive Christians, follows S-shaped growth, figure 6.  The final level of the church falls short of the total population. The enthusiasts have burnt out before they reach all people, just under 40% of the population in this example, the result of the three competing feedback loops.

Fig 6. Church growth resulting from feedback (percentage of population)

Other Feedback Loops

Over time people leave the church and people die, both balancing feedback. However new are born into the population, replenishing the pool of potential converts, a reinforcing loop. Thus enthusiasts never quite go to zero and a stable balance of church numbers is possible, despite losses and deaths. Nevertheless if conversions are not sufficient, both enthusiasts and church can head for extinction, the situation currently faced by many UK denominations.

Second Order Feedback

This “limited enthusiasm” model described here works well for a couple of generations, but over longer periods the effectiveness of churches in conversion changes for other reasons, usually resulting from “second order” loops.

All the feedback described above is “first order”, that is, only one stock is involved in the loop, see figures 1-4.  First order feedback means that its effect on increasing or decreasing a population is immediate, and thus relatively easy to follow. Not so for second order feedback, which involves two stocks. Its effects are often delayed and counter intuitive.

To give an example of  second order feedback, consider the case of loops R1, figure 2, and B3, figure 4, acting together, figure 7. Although B3 is first order on unbelievers as only one stock is involved, figure 4, the combination is second order on enthusiasts, figure 7. As enthusiasts increase, more people are taken from unbelievers, thus unbelievers falls, thus less are made enthusiasts and their numbers eventually slow – a balancing effect with two stocks. The result, with B2 (figure 3) added, is growth of enthusiasts changing to decline, figure 5, a type of behaviour that cannot occur in a stock that only has first order feedback.

Fig 7. Combining feedback loops B3 and R1 is a “second order” effect.

Jay Forrester, the founder of system dynamics, suggested that our “life and mental processes have been conditioned almost exclusively by first order negative feedback loops” [5].  By contrast second order feedback takes us by surprise and we tend not to respond to it effectively. This is so true for church growth and decline – but that is the next blog!


[1] Reinforcing feedback is also called positive feedback.

Martin, LA. 1997. An Introduction to Feedback, Road Maps, MIT System Dynamics in Education Project, D4691.

Also on the Creative Learning Exchange, Road Maps

[2] Hayward J. 1999. Mathematical Modeling of Church Growth, Journal of Mathematical Sociology. 23(4), 255-292.

Hayward J. 2005. A General Model of Church Growth and Decline. Journal of Mathematical Sociology, 29(3), 177-207.

[3] Balancing feedback is also called negative feedback. See [1] above.

[4] Kelley D. (1986). Why Conservative Churches are Growing: A Study in the Sociology of Religion. Mercer University Press.

Wesley's Law of the Decay of Pure Religion

[5] Forrester, JW. 1969. Urban Dynamics,  p.109. MIT Press.

Monday, 30 January 2017

A Prophecy of Church Decline

At the beginning of the New Year, David Robertson, a well known Free Church of Scotland minister and blogger, gave ten prophecies for 2017 [1]. One of these particularly caught my eye:

The Church of Scotland and the Church of England in the UK will continue to decline …

One of the reasons he gives to support his prophecy is the turning away of these churches from the gospel. In fact, their decline for this year is virtually guaranteed because their senior age profile gives them a high death rate that births into the church cannot overcome. To stop declining both churches would need a massive number of conversions, far higher than anything seen in generations. So, sadly, I think David Robertson will certainly be able to tick this prophecy off!

These two denominations are not alone; the pattern of long-term decline is typical of nearly all pre-1900 denominations. To show this, let me present some data for eight UK denominations and examine their growth and decline patterns.

Institutional Nature of Decline

One cause of changes in church membership is the general growth of the population through births, deaths, and migration. To see genuine growth, rather than look at the actual numbers in a church, I will examine the church’s percentage of the total population of the country. If the percentage is positive, the church was gaining members from society; that is enough conversions were taking place to more than counterbalance its inability to keep all its children. If the percentage is negative, the church was failing to keep pace with any population growth, effectively losing its “market share”. This method works as it is safe to assume none of these churches had a per capita birth rate greater than that of society.

Figure 1 shows membership percentages for four of the larger denominations [2,3], three established and one free (Welsh Presbyterian) [4].  The Church of Scotland and Church in Wales show moderate percentage increases followed by long declines as they fail to keep pace with population growth. The Church of England would probably show the same pattern, but I suspect the membership figures in the 1800s are inflated due to inaccurate measures of electoral roll. However, its recent figures are more realistic, thus its percentage decline since the 1940s is genuine [5].

Figure 1: Membership of Church of England, Church in Wales, Church of Scotland, & Presbyterian Church of Wales, as percentages of the population of their countries, where data is known.
By contrast, the Welsh Presbyterians had a much higher percentage membership from the 1700s to the 1870s, due to very high conversion rates resulting from revival sustained over a number of generations [6]. Nevertheless, it has declined ever since and is now only a marginal part of Welsh life.  
All four churches show the pattern of the institutional lifecycle. A rise to prominence, caused by the freedom to channel spiritual life into evangelism, followed by a long decline, as the trappings of institutional maintenance and theological revisions distract churches from a passion for conversions. The Welsh Presbyterian church peaked earlier, perhaps because it was the one not established. Thus, it is further advanced in the lifecycle, explaining why is it now a smaller percentage than the Church in Wales.  The Church of Scotland has the fastest decline, probably because it has started with a much larger percentage of the host population.

Figure 2 shows four smaller denominations. The Methodist Church has the same institutional lifecycle as the Welsh Presbyterian, which is to be expected as both were at the forefront of the 18th century revival in England and Wales [7]. However, the United Reformed Church (URC) is similar to the institutional churches in figure 1. It was formed from the English Presbyterian and Congregational churches, which, like the established churches, predate the 18th century revivals.

Figure 2: Membership of the Methodist Church of Great Britain, the Pentecostal & New Churches, Free Church of Scotland, & United Reformed Church, as percentages of the population of their countries, where data is known.
To make sense of these patterns I would suggest the pre-revival churches, established and URC constituents, were declining before the 18th century revival at the end of a previous lifecycle. The revival was driven by new movements, Methodist and Welsh Presbyterian [8], which subsequently became churches and later institutionalised. Some congregations of the pre-revival churches became part of this revival, but many didn’t, hence the slower 19th century growth rate.  These churches were already theologically mixed and not predominantly evangelical, so a partial take-up of revival is not surprising.

By the end of the 19th century the Methodists and Welsh Presbyterians had become institutionalised, revival was no longer welcome, and they entered the decline phase of the lifecycle. The pre-revival churches just moved back on to the decline phase of their lifecycle that they had been on over a century earlier. Data from all these denominations have been fitted to the limited enthusiasm model of church growth and all are well under the extinction threshold, except the Church of England who are marginal For some that extinction is predicted before the end the middle of the century [5]. Lifecycles can come to an end.

The Contrast to Decline

There are still two churches I have not mentioned. I have added together the Pentecostal churches with the “New” churches. The former came from early 20th century revival, whereas the latter came from the 1960-1990 charismatic renewal, thus share similar traits. These churches are at the start of the lifecycle, with accelerated growth coming from the late 1990s due to strategic church planting and immigration enhancing the existing revival. Of course, it is not enough to make up for the losses in the other churches, but if growth is sustained it may do in the future.

I have also added in the Free Church of Scotland, as it was a comment by a Free Church minister that inspired this blog. Although it looked as if it were on the decline phase of the lifecycle it has recently flattened out and seen some small growth.

Cause of Decline

If, as David Robertson suggests, church decline is caused by “turning away from the gospel”[1], then there should be some correlation between the decline and theological liberalism.  Figure 3 shows the average membership change over the last 10 years; the evangelical churches are growing, the theologically mixed are declining. Of the declining ones, only the Church of England shows a significantly smaller decline rate. It could be argued that it has a greater fraction of theologically conservative congregations than the others; hence its decline is softened. Figure 3 suggests that theological liberalism, one of the causes of institutionalism, is driving church decline, a conclusion that has been reached by others before me [9].
Figure 3: Annual membership changes, averaged over 2005-2015, or closely similar period where data is uncertain.

Seriousness of Decline

The critical nature of church decline can be seen by noting that the rate of decline is increasing for most denominations. Figures 4 and 5 shows the annual percentage change in membership, averaged over ten years, for two denominations, compared with population change. This accelerated decline is due to aging in churches where there are few conversions to bring the average age down. This later stage of the institutional lifecycle contrasts markedly with the earlier revival phase, figure 4, where the Methodists managed growth well in excess of population growth for over a hundred years.

Figure 4: Annual change in Methodist Church membership, smoothed over a 10 year period, compared with the annual smoothed population change. See note [11] for explanation of periodicity in the numbers from 1900.

Figure 5: Annual change in Church of Scotland membership, smoothed over a 10 year period, compared with the annual smoothed population change.
Church leaders often express some mild concern over falling numbers [10]. A quick glance at either figure 4 or 5 should provoke the utmost alarm in those leaders and a determination to radically change direction. The trend is so clearly getting worse that it is almost unbelievable that this is not top of the agenda of every church denominational leadership meeting, rather than the endless debates on the church’s response to social trends.

Reversing Decline

It is the Pentecostal church grouping that is seeing clear growth. The lesson for the other denominations from this is that if you want to see growth, embrace the charismatic revival with its Bible centred evangelical doctrines! However, it may not be as simple as that.

Pentecostal growth has not come out of the blue, but from a sustained commitment to its theological stance over generations. Revival is often a long haul. Likewise, the other churches’ decline has come from generations of theological and institutional lethargy. It cannot be turned around overnight. But individual congregations could be turned into growth if the central denomination allows those who wish to go down a Bible-based and revival route, to do so unhindered by denominational pressures. Unhindered doctrinally, financially and in terms of ministerial training and appointment.

Perhaps even small lessons can be learned. Why has the Free Church of Scotland stopped declining? It is not dramatic, but is better than decline! I could point to similar modest success in the independent evangelical churches in England. What have they discovered that the declining denominations have failed to grasp? If nothing else the leaders of the denominations facing future extinction owe it to their members to learn from denominations and congregations that have bucked the trend.

References and Notes

[1] Ten Prophecies re the Church in 2017, David Robertson, The Wee Flea, 11/1/17

[2] Currie, R., Gilbert, A. D., & Horsley, L. S. (1977). Churches and Churchgoers: Patterns of Church Growth in the British Isles since 1700. Oxford University Press.

Brierley P. (2014). UK Church Statistics 2010-2020, and previous editions. Brierley Consultancy. Also Religious Trends Vol 1-7, Christian Research.

Statistics for Mission. Church of England

Statistics for Mission, various editions, Methodist Church.

Church of Scotland Blue Books.

Attendance figures are not known over a time range of centuries, even obtaining membership data is challenging. Attendance tends to be higher than membership in growing churches, and lower in declining ones.  This is due to delays in joining and leaving churches, and due to evangelistically active churches have a large fringe of uncommitted people.

[3] Percentages are calculated in terms of the relevant host population, such as England, Wales, Scotland, Great Britain (GB), the sum of the previous three.

[4] Though technically disestablished, the Church in Wales occupies a similar position in law as the Church of England, sharing rules on marriages and church schools, thus having points of contact with the state.

[5] Blog: Anglican Church Decline in the West – The Data

Blog: Anglican Church Decline in the West – Possible Reasons

[6] Blog: Church Decline Caused by Lack of Conversions

Blog: Church Decline cause by Lack of Revivals

Blog: Why Revivals Stopped in the UK

[7] Not the sudden drop in combined Methodist membership in the middle of the 19th century. Although the data includes all the constituent Methodist churches before and after splits and mergers, it is likely that for a couple years a number of members failed to be counted. The temporary drop is a recording issue not a genuine drop in real members. The same effect can be seen in figure 4.

[8] The Welsh Presbyterians were Methodists, but of the Calvinist variety. The name change to Presbyterian came at the beginning of the 20th century.

[9] The correlation between theological stance and growth patterns has been discussed many times in the literature, much as a response to the seminal work by Dean Kelley:
Dean Kelley, Why Conservative Churches are Growing: A Study in the Sociology of Religion. Mercer University Press, revised 1986, originally 1972.

[10] The following is typical of church leaders avoiding the issue. The soon-to-be-retiring Archbishop of Wales in an interview with the BBC admitted attendances at church had not done "terribly well" but said he has seen church communities become "more engaged" with society than ever before.  Given there has been sixty years of decline in the Church in Wales, which has accelerated in the last few years, then his comments are something of an understatement.

Archbishop of Wales Dr Barry Morgan on his retirement, BBC News, 27/1/17.

[11] Someone asked me the excellent question as to why there is periodicity in the time series for the Methodist membership change data figure 4 ( and to a lesser extent the Church of Scotland figure 5)

a. Unlike the previous graphs these concern the change of membership. Rates of change of a quantity always exaggerate effects which may be present in the original quantity itself because a rate is attempting to be instantaneous. In this case the time period was 10 years, not an instant, but it still exaggerates any underlying periodicity. This periodicity may be in the original membership figures - but that is an accumulation and may not be as noticeable. Also I gave no graph of the membership figures, only the percentage of the population, figure 2, so the periodicity could be lost in population changes.

b. I used a 10 year average in figures 4 and 5 - but I did it by taking the gradient between the beginning and end point of the 10 year period as I was only interested in comparing it with population change. With there be about 20 years between the end of world war 1 and the start of world war 2 there is a danger my smoothing/averaging period has exaggerated the effects of the war and counted them twice at the beginning point and the end point.

c. Looking at the actual annual membership change, figure 6, it can still be seen there is some periodicity in the data - though not as pronounced now. But there is a remakable smoothness in the changes

Figure 6

d. To double check I tried a variety of smoothing periods and this time averaged all the data points in the period, not just the end points. Figure 7 shows once such graph with a 7 year period, deliberately chosen to be out of phase with the 20 year inter-war period. 

Figure 7

The periodicity is still there, so short of doing a statistical test, we can assume it is real. 

The first negative low point is 1914-18 - the effects of world war 1, the inevitable losses and records not fully kept up to date. The rise to 1926 is post war membership readjustment, with the real state of affairs following up to 1939. The same happens again with a low point early 1940s  and it peaking again 1954 as records are updated in the post war period. Remember these are not the membership figures but change in membership.

The next trough is 1978, a longer period than before and a peak in the early 1980s. This is likely an effect of the baby boom now becoming adults and entering membership and thus temporarily slowing the decline. It has only been down since.

Thus the best explanation of the periodicity is change is demographics, due to the two world wars - a 25 year effect, and the effect of world war 2 on birth rates, a roughly 30 year effect.

e. There is a peak in 1906, 20 years before the periods looked at above; and another peak at 1882, 24 years before that. These times may be coincidence. The 1880s saw membership fall behind population as the period of revivals came to an end. Early 1900s saw rapidly falling birth rates - but this could be a generational effect from the 1880s drop in conversion.

Sudden changes in populations, where the whole population, or a church sub population, can often see a knock on demographic effect 20-30 years later, so these effects are not surprising.

Wednesday, 30 November 2016

Infectious Church Growth

An “Agent-Based” View

The central hypothesis I use to model church growth is that religious belief spreads like an infectious disease. This principle is built in to the limited enthusiasm model of church growth. The church contains enthusiasts, who pass their faith on to unbelievers, who convert to the faith. Some of those new converts also become enthusiasts for their newfound faith. Eventually enthusiasm wanes, the convert runs out of people to positively influence; they cease to be enthusiasts, thus becoming inactive believers.

The essence of this model is expressed in the stocks and flows of system dynamics, figure 1, and it works well for periods of revival, and, with a few additions, sustained periods of growth or decline [1,2].

Figure 1: Outline of Limited Enthusiasm Model in System Dynamics

This type of model is called macroscopic, because people of the same type are treated as a single unit. The stock, “Enthusiasts”, stands for the total number of enthusiasts at any one time. What this type of model does not do is give a picture of events at the individual level. For this we need a microscopic model; one example of which is called an agent-based model.

Agent-Based Model

In an agent-based model each person is modelled individually – they are called agents. Every agent is capable of being in more than one state, and the hypotheses of the model determine how an agent changes their state. There are also hypotheses that describe how agents relate to each other – their network [3].

For the limited enthusiasm model of church growth, the agents are the people who can have one of three states, depending on whether they are: unbelievers, enthusiasts, or inactive believers. The simplest form of network is to use a rectangular grid so that each person, a mini square, relates to 8 neighbours. In figure 2, the green squares are unbelievers (U), the red squares are enthusiasts (E), and the blue squares are inactive believers (B).

Figure 2: Outline of Limited Enthusiasm Model in an Agent-Based Model 

The unbeliever at the centre of figure 2 can be potentially influenced by any of the 8 surrounding agents. Conversion may occur in a given time period if there is at least one enthusiast connected to the unbeliever. The more enthusiasts in that network, the more likely the conversion. Thus in figure 2, there is a 2 in 8 (= 25%) chance of the central unbeliever having a contact that may lead to conversion. The green turns red. That conversion is still not inevitable, but the longer an unbeliever has an enthusiast in their network, the more likely a conversion becomes.

The process of an enthusiast becoming inactive does not depend on neighbours. Instead, after a fixed period of time, there is a chance the enthusiast will cease to have influence. Thus the red turns blue.

The two preceding hypotheses are called transition rules. They are the algorithms that drive the model. The transitions green to red to blue are the individual level equivalent of the stock/flow diagram of figure 1.

A Simulation.

For a simulation I will use the agent-based simulation software NetLogo [4]. Let us start with a church of enthusiasts in one block, figure 3, plot 1. The world is the entire green square of 121 by 121 cells, 14641 agents.  Some of the initial enthusiasts have no contact with unbelievers, but the ones on the edge do. Thus as time progresses the church grows, plot 2, leaving a church of mainly inactive believers in its wake. These are inactive in conversion, though they may be active in other aspects of the church.

Figure 3: Four Snapshots From Agent-Based Version of the Limited Enthusiasm Model. Green Unbelievers; Red – Enthusiasts; Blue – Inactive Believers

As time runs on further, the church grows, figure 3, plots 3 and 4, though there is always a chance that it could stall as it runs out of enthusiasts. Many of its boundaries with society have no enthusiasts – thus no conversions. Also there are unconverted people who can no longer be reached, the green agents surrounded by blue ones.

The church eventually stops growing leaving many people unconverted, figure 4. A fundamental result of the spread of an infection is that it burns out before everyone gets the disease.  Likewise the limited enthusiasm model predicts that revivals burn out before everyone is converted. Figures 3 and 4 show this principle at the microscopic level.

Figure 4: Final State of a Simulation of the Limited Enthusiasm Model. No Enthusiasts Remain and Church Stops Growing.

The total number of people in the church (red and blue) can be plotted over time, figure 5; along with the total number of enthusiasts (red), figure 6. The growth patterns of the agent-based model are very similar to the system dynamics model, figure 7, but the former has more randomness due to its microscopic nature [5, 6]. To fully replicate the smooth system dynamics predictions, the agent-based model would need to be run many times and the results averaged [7].

Figure 5: Growth of Church Over Time in Agent-Based Limited Enthusiasm Model

Figure 6: Number of Enthusiasts Over Time in Agent-Based Limited Enthusiasm Model

Figure 7: Results of System Dynamics Limited Enthusiasm Model


Agent-based models give a very visual view of how the church grows and running a simulation can really bring a model to life. Try the online version of the agent-based model to get feel for its behaviour:

(Instructions under Model Info)

The real drawback with agent-based models is that it is very hard to describe the model. Unlike system dynamics, the agent-based methodology has no intuitive and visual representation of the model structure and its hypotheses. Also you do need to run an agent-based model many times to see a clear conclusion. A system dynamics model can achieve this in one run, and quickly connect behaviour with model structure. Try the system dynamics version of the limited enthusiasm model yourself and see the contrast:


Hopefully this quick introduction to agent-based modelling has given further insight into the nature of church growth, and the mind of the mathematical modeller!

John Hayward
Church Growth Modelling


[1] The limited enthusiasm model of church growth is explained in:

[2] I have other models of church growth, whose main aim is to explain why the effectiveness of believers in conversion may vary over time. Issues include limited resources, the generation of spiritual life and the accumulation of institutional baggage.

[3] For introductions to agent-based modelling see:

[4] NetLogo is freely available from

[5] The argument runs that the behaviour of an individual cannot be exactly predicted. Prediction only becomes possible when the behaviour of large numbers of individuals are combined and the unpredictability is smoothed out. Getting the nature and extent of randomness in individual behaviour is far from straightforward.

[6] The system dynamics simulation was performed in Stella Architect available from isee Systems

[7] Averaging does not give a complete replica of the system dynamics results. This is partly due to the unrealistic network used. Some people have many contacts; some have few. Scale free and small world networks give better results. Further replication would require an improved model of how long an agent remains infectious, i.e. an enthusiast.