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

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Activity 5C: Simulating carrying capacity and limits to growth

In this activity the aim is to explore a population growth model which includes an internal mechanism for achieving homeostasis.

In Activity 5B you explored a system dynamics model with two feedback loops: the ‘birth loop’ and the ‘death loop’. These loops contained fixed birth and death rates as the simulation evolved. To develop a more realistic simulation of population dynamics, you will now explore a simulation in which the death rate changes according to circumstance during the actual simulation. We can introduce the change to the model with the following rationale: without the possibility of emigration, as the population grows, population density increases. This speeds up the depletion of resources, spread of diseases and death from conflict (here I am looking at a case where there is no technological innovation or change in human behaviour that would reduce resource exploitation, combat disease or eliminate conflict). Thus, the higher the population, the greater the death rate. Not only are a greater number of individuals dying, but a greater proportion of individuals are dying compared to the total population number as resources become more limiting, diseases spread faster, and conflicts escalate.

Activity 3

Open the model Activity_5C.nlogo – you will see that there is an additional feedback loop between the ‘Population’ stock and the ‘Death_Percentage’ variable (Figure 5.12).

Figure 5.12
Figure 5.12 System dynamics diagram of population stock, birth and death rates and associated birth percentage variable and death percentage variable with additional feedback loop.

To include this additional feedback effect into the mathematical side of the model, I have replaced the constant within the ‘Death_Percentage’ Variable with the following simple expression:

Population / 1000000000

thus, as the population nears 1 billion individuals, the death rate will reach 100 per cent.

With the birth percentage rate still at 0.027, what do you think will happen now? Try and work through the implications of this modified feedback loop and then carry out the simulation (remember to press ‘setup’ then ‘go’, and to stop the simulation press ‘go’ again). It might help to sketch an additional graph here, where you plot the death rate against population number. You can then use this new graph to estimate population change over time.

Answer

The figure below shows an initial exponential growth in human population as the positive feedback loop within the birth rate dominates. However, after about 400 years within the simulation, population growth begins to dampen as a result of the surge in the death rate. A steady state of about 27 million people is reached once we pass 800 years of simulation. At this stage, the death and birth rates are equal. The overall shape of the graph is therefore S shaped (Figure 5.13).

Figure 5.13
Figure 5.13 Graph showing the results of the Activity_5C.nlogo simulation

You may have been able to predict that the additional ‘death loop’ stabilises population growth. This new addition to the death rate calculations is therefore a much better representation of the homeostatic effect of negative feedback: the greater the population, the greater the death percentage; and the lower the population, the lower the death percentage. This significantly reinforces the negative feedback effect of the death rate. Stability is reached when the population number is such that the death rate equals the birth rate.

This new model formulation describes the concept of ‘carrying capacity’. Here, the population reaches a maximum sustainable number which we have hypothesised is controlled by the availability of natural resources, the spread of disease and the level of conflict. The new ‘Death_Percentage’ formula is a surrogate for limiting population according to an ‘ideal’ density where natural resources are not over depleted, diseases do not turn into pandemics, and conflicts do not turn into genocide.

In this model, the death rate rapidly increases to avoid numbers exceeding the carrying capacity. The idea of carrying capacity is intimately linked with the concept of ‘limiting resources’, which affect the population directly (i.e. people starve to death during famine) or indirectly (i.e. limited shelter or food etc., increases in the spread of diseases). Every species depends on a number of resources, such as food and shelter, which, when limited, weaken individuals so that both birth and death rates are affected. For example, there might not be enough nutrition to maintain the same levels of fertility, or poor quality shelter may allow diseases to spread more rapidly. The Rwandan 1994 genocide, where an estimated 800,000 people were killed in six weeks, has been partly attributed to pressures resulting from a rapid decrease in natural resource availability in Africa’s most densely populated country. For an account of the environmental factors contributing towards the Rwandan genocide, read Jared Diamond’s Collapse: how societies choose to fail or survive (2005).

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