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Sustainability Matters 12: Population Growth and Resource Availability


Population Growth and Resource Availability

Suppose we put a single bacterium in a bottle at 11 AM.   I will tell you this kind of bacteria divides once per minute. I will also tell you that precisely at noon, the bacteria have filled the bottle. At this point, they have no more room to grow, so all the bacteria die. Here is a question: at what time was the bottle HALF full?

You have all the information you need to answer the question: you don’t need to know how big the bottle is or how many bacteria there are. It’s not a trick question, it’s a thinking game.”

Al Bartlett
University of Colorado

Answer: The bottle would be half full at 11:59AM!


Ecologists have observed two types of growth in natural populations: arithmetic growth and geometric (exponential) growth.  The growth described in the above question is descriptive of geometric growth.

How does Population fit into Arithmetic Growth?

In the case of arithmetic growth, a population grows based on linear pattern 1,2,3,4,5… or perhaps 2,4,6,8,10…. Arithmetic growth, when plotted on a graph, appears as a straight line.  Each succeeding unit of time across the X-axis (the horizontal arm of the graph) increases by the same amount along the Y-axis (the vertical arm of the graph).

Figure 4-1 Four examples of Arithmetic Growth

In the case of this graph:   

  • The blue line shows an increase of .5 units per period.
  • The red an increase of 1 unit per period
  • The green an increase of 1.5 units per period.
  • The purple an increase of 2 units per period.

Most mammal populations grow arithmetically, with some limitations. These populations grow gradually until they experience pressures such as disease, famine, drought, loss of habitat which limit their growth. When that occurs, the population diminishes. When there is less pressure, the population rises again. Based on all the factors that contribute to the growth and limitation of population size, there is a theoretical carrying capacity for any population. The carrying capacity means the maximum number of individuals in a population that can be supported by the local environment on an ongoing, sustainable basis.

In ecological terms, the carrying capacity of an ecosystem is the size of the population that can be supported indefinitely upon the available resources and services of that ecosystem. ‘

Living within the limits of an ecosystem depends on three factors:

  • the number of resources available in the ecosystem,
  • the size of the population
  • the number of resources each individual is consuming.

In reality, the numbers in a population tend to fluctuate around this carrying capacity. When the numbers rise past this point, some limiting factor such as a lack of food or disease causes the number of individuals in a population to decline.

As a result of these oscillations, arithmetic growth curves in ecological systems look more like an S on its side.

Figure3-2 Arithmetic Growth – S Curve pattern

Geometric or Exponential Growth

The other way that populations grow is geometrically, or exponentially.  (These terms can be used interchangeably.)  “A quantity grows exponentially when its increase is proportional to what is already there.” In this type of growth pattern, the quantity will double over time, and will show a pattern similar to the one in Figure 3-3 below. Each interval represents a doubling.

There is a relationship between the percentage rate of growth and the length of time it takes for a quantity to double. The higher the percentage rate of growth, the faster the rate of growth. 

The economic equivalent of exponential growth is Compound Interest.

Compound interest is the eighth wonder of the world. He who understands it, earns it … he who doesn’t … pays it.” Albert Einstein

Think of it like you would money in the bank 

The Rule of 70

By dividing the percent growth rate (or interest rate) by 70 yields the approximate doubling time.

– the greater the interest rate, the faster your money doubles.   If you have $1,000 in the bank at 7%, you will have $2000 dollars in ten years (70/7 = 10).  Similarly, $1,000 at 14% would become $2000 in five years (70/14 = 5). 

Now think about the same growth pattern for a population 

A population growing at an exponential rate, the demand on  a valuable resource to that population and the resulting pollution all grow in similar fashion to your savings account or the virus we’re watching devastate us. 

Here are the numbers from the first ten minutes of the ‘Thinking game’ at the beginning of this chapter. 


Number of Bacteria





















Figure 3-3 below represents the first ten minutes. Look carefully at the shape of the line. It is sometimes referred to as a J-curve or a hockey stick, and you are going to see this pattern repeatedly.   

Figure 3-3 Exponential -Geometric Growth. Hockey stick curve

In nature, bacteria and rodents and as we’re seeing throughout the world, viral infections are examples of J-shaped growth curves. What is typical of populations that grow this way is that when they overshoot their carrying capacity, the population crashes.

Figure 2-4 Arithmetic and Geometric growth patterns in nature

  • Series 1 = Arithmetic Growth
  • Series 2 = Geometric Growth followed by crash

No matter how a population grows, the ecosystem in which it exists has a carrying capacity. In the case of arithmetic growth, when the population exceeds the carrying capacity (for example, when the population grows larger than the amount of food the ecosystem can produce for it), the population is reduced through starvation. The population subsequently drops below the carrying capacity of the ecosystem. When this limitation is no longer a factor, the population begins to grow again until another limiting factor is encountered.

This is represented by the S-curve growth pattern we saw in Figure 3-2 above.

In the case of geometric growth, the population quickly grows past the carrying capacity and keeps growing until it runs out of a key resource (a limiting factor) and crashes. After the crash, the population, if not extinct, will resume growing in the same pattern as shown in Figure 3-4.

Both types of population growth patterns are represented in Figure 3-5, which shows the numbers of Snowshoe Hares and Lynx which were reported as trapped by the Hudson Bay Company over the course of 20 years. What you see is the Snowshoe Hare population increasing exponentially, which provides food for the interdependent Lynx population.  

Figure 3-5 J and S shaped curves
(colored horizontal lines represent theoretical carrying capacities)

As the Lynx population grows arithmetically, its increased numbers put pressure on the Snowshoe Hare population which drops as a result. Less available food causes the Lynx population to drop, and as the pressure on the Snowshoe Hare population diminishes, its population begins to grow again.

Figure 3-6 Lynx preying on Snowshoe Hare
(this image kindly provided by Tom and Pat Leeson, who retain the copyright)

Regardless of its growth pattern, when a population exceeds its carrying capacity, some natural factor such as lack of food, water or disease will limit the population. In the case of geometric growth, the population adjustment is a crash. Assuming the diminished species survives, the factors that encourage and limit growth rebalance themselves and the pattern repeats itself.   

Human Population Growth and Energy Consumption

Human population grew arithmetically for the first 99% of human existence.  For more than a million years, we existed as hunter-gatherers. At a point in the 18th century, the human population began growing in an exponential fashion. The following two charts demonstrate that growth.






Human population growth over 10,000 years                                    Human population growth since 1750
Developing vs Industrialized countries 

Figure 3-7: Human Population Growth

The chart on the left represents the last 12,000 years, and though not to scale, gives a sense of the abruptness of the unprecedented worldwide growth of the human population. The data in the right chart is the calculation of world population growth since 1750 and projected out to the year 2050.

Figure 3-8 below shows the correlation between human population growth and energy consumption for the past one million years. Notice that the sharp rise in the energy consumption as humans learned how to make fossil fuels work for them. This was a time when fossil fuels became cheaper and more plentiful than energy gained through animal or human labor (including slavery).  

Was it a historical coincidence that we, the US banned slavery at the time fossil fuels became commercially available?

Figure 3-8: Historic Global Energy Consumption.
Data sources: US Census Bureau, HYDE database, EIA, Kremer (1993)

Clearly, there is a relationship between energy consumption and human population growth. As humans invented new ways of using energy, that energy has helped humans to push back the limiting factors which once restricted the size of our population.  

Next week we’ll discuss limits to growth concerning food and population!

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