If a church has more people dying and leaving than joining, then it will decline. If, over time, the rates remain the same, then the overall rate of decline is a constant percentage (per capita) rate, leading to a negative exponential curve. This result assumes that the age distributions of joiners and leavers are similar to those of the remaining population, thereby keeping the population’s average age constant.

However, if there is a large discrepancy between the joining and loss rates, especially if the losses are due to death, then population ageing becomes a significant factor. The smaller the church becomes, the more pronounced the effect of population ageing, which increases the death rate, reduces the birth rate, and thus accelerates the overall decline. Church decline then ceases to be negative exponential and approaches a straight line. Therefore, care is needed when modelling churches near the end of their lives, as ageing tends to straighten the decline, bringing extinction earlier than models based on constant percentages would predict.

The following analysis compares different models of deaths and ageing in a population to illustrate these effects.

Birth-Death Model of Church

Consider a simple model of births and deaths applied to the church. Additions are through the reinforcing feedback loop R: more people, more births, and thus more people added. This assumes all those born to church members join the church. Subtractions are through the balancing loop B: more people, more deaths, thus more people removed (if the population is increasing).

The model also assumes no one from outside the church joins, and all in the church die rather than leave. This is the simple birth-death model, shown in Figure 1. The stock Church represents the number of people in the church at any given time. Mathematically, it reduces to the differential equation dx/dt = (b-d)x where x is the number in the church, represented by the stock in Figure 1, and b, d the birth and death rates, represented by the converters in the figure.

Figure 1: Birth and Death Model of Church.

Let the birth and death rates be those for the UK for 2013, b = 0.01226, d = 0.00933 (Index Mundi). The solution for the next 100 years is given in Figure 2. It is a growing exponential, x = exp( (b-d) t), where t is time.

Figure 2: Birth and Death Model, Result for a Growing Church.

The exponential curve appears slight because the birth and death rates are close. If the birth rate was less than the death rate, the result would be exponential decline.

Smoothed Death Model of Church

Assume now the church has no births. That is, all those born to church members no longer join the church. The model can be simplified to the pure draining process in Figure 3, with the solution determined by the balancing loop B and the per capita death rate. Such a process is called first-order smoothing (Sterman 2000) as it treats the attributes of all people in the church as equal, including their age.

Figure 3: Death Model of Church using First Order Smoothing.

The solution of this model is a negative exponential decline, determined by the death rate, as shown in Figure 4:

Figure 4: Death Model with First Order Smoothing, Result for a Declining Church.

The result is clearly wrong, as it predicts the church will still have a healthy number of people a hundred years later, in 2113! The mistake lies in the first-order smoothing because, as the church declines, the model has assumed the age distribution, e.g. the average age of the congregation, remains the same. Clearly, it will not. Instead, without any additions of younger people, the church will get older and cease to exist when the last person dies, probably before the 100 years are up.

Cohort Death Model of Church

For a true model of the effect of deaths on the church, the stock church needs to track the ages of its members. This is done by splitting the stock into cohorts, Figure 5:

Figure 5: Death Model of Church using Cohorts.

This type of stock is called a conveyor and works like a conveyor belt. The people are assumed uniformly distributed in age. That is, they have been in stock for different lengths of time and thus are placed uniformly along the conveyor belt. Death is determined by how long a person stays in the stock. In this case, 85 years is taken as a reasonable upper limit to lifespan. The resulting decline is now a straight line, as the church not only declines but also ages (Figure 6).

Figure 6: Death Model with Cohorts, Result for a Declining Church.

The first-order smoothing process has now been replaced by a pure delay process, sometimes called a pipeline delay (Sterman 2000). Now the church ceases after 85 years, the lifetime of the population. Because the population were spread uniformly throughout the stock, there was at least one person who had just been born at the back of the conveyor. They are the last one to die, as it is assumed all people live the full lifespan. Crucially, the decline is not negative exponential but straight, which suggests that a church dominated by the death process will more closely follow a straight-line decline.

In such a church, the life expectancy of the remaining people is going down. That is, the percentage/per capita death rate is rising, as the church declines. For example, if the church consists of all people in their eighties and the human lifespan is 85, then the church ceases in 5 years, however large it is. The per capita death rate in the first year is 20%, in the second year 25%, and so on until the final year when it is 100%.

If randomness is introduced, because some people do live longer than the lifespan, then the church may last a little longer, but now the results can only be given in probabilities.

Death Model Represented by Feedback Loops

Alternatively, it is possible to examine the situation using the first-order smoothing in Figure 3 if the death rate is allowed to rise as the population declines. There are now two loops, the balancing loop B for deaths, and the reinforcing loop R2 for ageing, Figure 7:

Figure 7: Death Model of Church using First-Order Smoothing and Ageing Loop.

The balancing loop B, representing deaths, exerts an opposing force on the stock Church, leading to a negative exponential curve, with the church approaching equilibrium at zero. This loop is like a brake, slowing the decline. The reinforcing loop R2, representing ageing, exerts a positive force on Church, leading to a positive exponential curve. This loop accelerates the decline. If there are no other additions or subtractions from Church then the two forces of acceleration and braking are in balance, and there is no net force. Thus, the decline is a straight line, figure 8, line 1. Essentially, the effects of the two loops cancel out.

Figure 8: Smoothed Death Model with Ageing, Result for a Declining Church.

Note the per capita death rate, line 2, is increasing as the population declines, and thus ages. Towards the end of the church’s lifetime, the death rate increases dramatically to 100% as the last people die. Notice that the lifespan of the people in the church declines as no one is joining, which would have lowered the average age (Figure 9). If people did join and an equal number left, then the age difference between joiners and leavers would affect the age distribution in the church, causing it to decline more slowly, though still a straight line. If the joining and leaving are unequal, the decline will exhibit some curvature, but not as much as in Figure 4 above.

Figure 9: Smoothed Death Model with Aging, Lifespan.

Thus, care needs to be taken with models of churches in the last stages of decline, as ageing will tend to straighten the decline and bring extinction faster than predicted by models based on percentage, per capita rates, that is exponential/logarithmic models.

Discrete Ageing Chain

An ageing chain is a model of a population split into cohorts to record the number of people in each age group as the population changes. Thus, a model of the population with deaths only, no births, additions or subtractions, could be modelled with, for example, three age groups, Figure 10:

Figure 10: Discrete Ageing Chain Model of a Population with only Deaths.

Note that the widths of the first and last age groups are 20 years, with the middle group double-width at 40 years. It is assumed that only the older age categories have deaths. Thus, for the first 20 years, only the youngest age category (plot 1) declines as they age and join the middle group. In the other two categories, the inflows match the outflows, whether age progression or death.

Figure 11: Result of Ageing Chain Model.

From 20 years onwards, the middle age category (plot 2) starts to decline as there are no more young people to progress. The oldest age group does not begin to decline until the middle category is depleted. It has been assumed all age groups start uniform. If this were a church, it would be getting progressively older. The population declines as a straight line, as in Figure 6.

This ageing chain is the type of model used in demographics and can be easily simulated with a spreadsheet. It models time in discrete steps according to the required degree of age resolution. Often they are annual or quarterly. Thus it is called a discrete ageing chain.

Continuous Ageing Chain

Most social models are constructed using continuous time. This is because the processes involved in interactions among different population groups are mixed across the entire population, leading to very short time scales over which changes occur. Although this is not true for species subject to seasonal effects, it is nevertheless a reasonable approximation. However, for human populations, the time step is usually a small fraction of a day, thereby requiring a continuous-time model. The types of nonlinear processes in these models behave very differently in continuous time than in discrete time. The latter may induce oscillations and chaos that are not present in the continuous-time case or in the real world.

Thus, it would be wrong to use discrete models where processes involving contact within populations are required. As a consequence, ageing chains are often modelled continuously with first-order smoothing, Figure 12:

Figure 12: Continuous Ageing Chain Model of a Population with only Deaths.

Comparing the results with the discrete case, Figure 11, it is seen that errors start to creep into the solution as the model smooths what should be strict age progression, Figure 13.

Figure 13: Result of Continuous Ageing Chain Model.

Notice the middle age group declines immediately as more are leaving the category than entering it. Effectively, they are being sent into the next cohort too early because first-order smoothing treats all individuals in the stock the same at all times. Note also that there are people still in the young category even at 50 years, long after they should have aged, as the first-order smoothing keeps the ageing progression rate the same even though the average age in the young category is advancing. This effect is referred to as cohort blending. The population declines exponentially, and wrongly, as in Figure 4.

Of course, a population with only deaths is an extreme case. This is very rare in human populations and only seen in subgroups such as organisations and churches near the end of their lifespan. In nearly all cases, when births and other additions and subtractions are included, the continuous and discrete models of ageing give very similar results as cohort blending averages out. The age distribution is now changing very slowly and does not unduly affect the progression rates.

The two cases which require discrete ageing are: (1) where age-related deaths dominate, as above; (2) where there is sudden migration in a specific age group. Mixing discrete ageing with continuous processes is both analytically and computationally difficult, and compromises are usually necessary. However, the issue has been solved in system dynamics by Eberlein & Thompson (2013) where they took the integration step as a very narrow cohort age group. People are working on other solutions to the problem of cohort blending.