Understanding the environment: Complexity and chaos
Understanding the environment: Complexity and chaos

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Activity 5D: Simulating overshoot, collapse and regeneration

In this activity the aim is to understand the effects of time lags, overshoot and collapse.

In the model used in Activity 5C the slightly more sophisticated ‘death loop’ results in a steady-state population. The assumption in the model is that the death rate increases rapidly as the population nears its maximum carrying capacity. This maintains the population below the carrying capacity threshold. However, this is not the typical behaviour of populations – it is often the case that the overall growth rate takes a population over the carrying capacity. This is because there is a time lag between the carrying capacity threshold being crossed and negative feedback kicking in. This occurs when, for example, populations use up resources faster than the resources can regenerate themselves – the population can still grow by drawing down on the resource, but once this resource is depleted and no alternative is available, collapse ensues.

Activity 5

Let’s explore the concept of overshoot and collapse by extending the population growth model.

Open the model Activity_5D.nlogo. You will notice that this new model has an additional stock called ‘Biocapacity’ (Figure 5.14). ‘Biocapacity’ is the area of biologically productive land and water required to provide the resources we use and to absorb our waste. The World Wildlife Fund for Nature’s (WWF) 2006 Living Planet Report estimated that in 2003 we had just over 11 billion global hectares (gh) of biologically productive area ( Living Planet Report, 2006). You may remember that you came across the term ‘global hectares’ in Section 3 as this was the unit used in measuring your personal ecological footprint. Here is where things get really exciting, in that the rather static ecological footprint calculations take on a new meaning through dynamic simulations. This biocapacity will represent the key limiting factor for the population in this new model. ‘Biocapacity’ is renewed by a flow of new resources into it, which I have called the ‘Regeneration’ rate and a flow out of it, which I have called the ‘Consumption’ rate.

Figure 5.14
Figure 5.14 System dynamics diagram of the NetLogo model, Activity_5D.nlogo, showing the interrelationship between population growth and biocapacity availability

The ‘Consumption’ rate has to be divided by the number of individuals in the population, so a link has been inserted from the ‘Population’ stock to the ‘Consumption’ rate. Also, a new variable called ‘Consumption_per_Individual’ has been added and assigned the value of ‘1.8’ (every individual in the population requires the equivalent of 1.8 gh of biologically productive area per person per year, the value considered to be sustainable at current population levels as estimated by WWF for 2003). This new variable has been linked to the ‘Consumption’ rate. The equation in the ‘Consumption’ rate is as follows:

Population * Consumption_per_Individual

The above expression calculates how fast the ‘Biocapacity’ stock is drawn down. But how do limitations in biocapacity affect the population birth and death rate?

To address this a new ‘Biocapacity_Multiplier’ variable has been added which links the ‘Population’ and ‘Biocapacity’ stocks. The following is a simple equation that characterises the impact of population, biocapacity and consumption on birth and death rates:

Biocapacity / (Consumption_per_Individual * Population)

This basically calculates the amount of biocapacity there is per individual: high levels of biocapacity, and low levels of consumption and population, will result in a high ‘Biocapacity_Multiplier’ value. This ‘Biocapacity_Multiplier’ is simply introduced to the birth rate as a multiplier:

Population * Birth_Percentage * Biocapacity_Multiplier

Thus, the higher the biocapacity there is per individual, the higher the birth rate.

The death rate is affected in the opposite direction:

Population * Death_Percentage * (1 / Biocapacity_Multiplier)

The higher biocapacity there is per individual, the lower the death rate.

How is the ‘Biocapacity’ stock replenished? A ‘Regeneration_Rate’ variable is introduced which simply multiplies the amount of biocapacity remaining every year. However, this is limited by a negative feedback loop (the ‘Limit’ and ‘Comparator’ variables), which make sure that biocapacity does not exceed the physical limits determined by the size of the Earth system:

Regeneration_Rate * Biocapacity * Comparator

That is it! But with just 2 stocks, 4 flows, 7 variables and 16 links, the model has significantly exceeded Miller’s magical number seven, plus or minus two.

Trying to explore the dynamics through mental modelling would be challenging to most people – and this is one of the simplest population models that I have managed to develop that still reflects the most important basic dynamics. If you can figure out an even simpler model which produces similar behaviours please let me know!

  • Can you predict what will happen with the current ‘sustainable’ biocapacity consumption per individual?
  • What do you think will happen if you decrease or increase biocapacity consumption per individual?

Try and sketch a series of graphs, including population change over time and the availability of biocapacity over time.

Once you have completed your sketches, carry out the 100 year simulation and try changing the ‘Consumption_per_Individual’ variable (I would recommend that you try slowing down the simulation speed so that you don’t miss out on the dynamics). While you are experimenting with the model you may get a Runtime Error message. This is because either the Earth’s biocapacity and/or the human population has crashed to zero i.e. you have just simulated the sixth mass extinction! Just click ‘Dismiss’ to continue.


The simulation plot with a ‘Consumption_per_Individual’ value of 1.8 should show you the following results (Figure 5.15).

Figure 5.15
Figure 5.15 Simulation plot of the Activity_5D.nlogo model

This clearly demonstrates that, at least initially, stability is not reached, either in the population size or biocapacity. This is because there is a lag between a change in biocapacity and a change in population number so that periodically the population is in ‘overshoot’, i.e. biocapacity is being consumed at a faster rate than it can renew itself. As a result when the biocapacity drops below a certain level the population crashes. Lower population levels then allow biocapacity to rapidly build up again only for the growing population to precipitate a subsequent crash. This cycle is repeated over and over again until, eventually, a stable state is reached.

A seminal 1995 paper by Odum et al. (Odum et al., 1995) presented a new paradigm proposing that many stable systems actually demonstrate ‘repeating oscillations that are often poised on the edge of chaos’. In other words, these systems are able to maintain stability by operating regular pulsing checks and balances. This makes intuitive sense since nature already manifests regular pulses of energy, matter and information, for example night/day, high/low tide, summer/winter, dry/wet season, drought/flood cycles, ice age/interglacial, etc. It is no surprise that living systems have evolved cybernetic controls involving a range of feedback relationships to make the most of these overlapping biophysical oscillations. Yet, as stated earlier, these systems tread a fine line between stability and chaos, with seemingly small perturbations sometimes resulting in drastic reorganisation of system structures and processes. For example, your experimentations with variable changes may have detected a threshold where a small increase in biocapacity consumption per individual results in the sudden collapse of both the ‘Biocapacity’ and ‘Population’ stocks.

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