Exponential Astonishment

1. Exponential Astonishment 8

2. Unit 8C Real Population Growth

3. Example The average annual growth rate for world population since 1650 has been about 0.7%. However, the annual rate has varied significantly. It peaked at about 2.1% during the 1960s and is currently (as of 2013) about 1.1%. Find the approximate doubling time for each of these growth rates. Use each doubling time to predict world population in 2050, using a 2013 population of 7.1 billion.

4. Example (cont) Solution Using the approximate doubling time formula (Unit 8B), we find the doubling times for the three rates: For 0.7%: For 2.1%: For 1.1%:

5. Example (cont) To predict world population in 2050, we use the formula new value = initial value × We set the initial value to 7.1 billion and note that 2050 is t = 37 years after 2013: 0.7%: 2050 population = 2.1%: 2050 population = 1.1%: 2050 population =

6. Definitions Overall Growth Rate The world population growth rate is the difference between the birth rate and the death rate: growth rate = birth rate – death rate For any particular species in a given environment, the carrying capacity is the maximum sustainable population—that is, the largest population the environment can support for extended periods of time.

7. Example In 1950, the world birth rate was 37 births per 1000 people and the world death rate was 19 deaths per 1000 people. By 1975, the birth rate had fallen to 28 births per 1000 people and the death rate to 11 deaths per 1000 people. Contrast the overall growth rates in 1950 and 1975.

8. Example (cont) Solution In 1950, the overall growth rate was In 1975, the overall growth rate was The growth barely changed because death rates fell almost as much.

9. Logistic Growth Rate When the population is small relative to the carrying capacity, logistic growth is exponential with a fractional growth rate close to the base growth rate r . As the population approaches the carrying capacity, the logistic growth rate approaches zero. The fractional logistic growth rate at any particular time depends on the population at that time, the carrying capacity, and the base growth rate r:

10. Exponential versus Logistic Growth In the exponential case, the growth rate stays equal at all times. In the logistic case, the growth rate starts out equal and as time progress the growth rate becomes smaller, until it finally reaches zero as the population levels out at the carrying capacity.

11. Example The global population growth rate has been slowing since around 1960, when the growth rate was about 2.1% and the population was about 3 billion. Assume that these growth rate and population values represent one point in time on a logistic growth curve with a carrying capacity of 12 billion. Does this model successfully predict the 2013 growth rate of 1.1% given the 2013 population of 7.1 billion? Explain.

12. Example (cont) Solution We begin by using the 1960 data to find the base growth rate r in a logistic model that uses the 1960 data values. You should confirm that solving the logistic growth rate formula for r gives

13. Example (cont) Substituting the 1960 growth rate of 2.1% = 0.021, population of 3 billion, and a carrying capacity of 12 billion, we find We now use this value of the base growth rate r to predict the growth rate for the 2013 population of about 7.1 billion:

14. Example (cont) This logistic model successfully predicts the 2013 growth rate of 1.1%. It is reasonable to say that human population has been growing logistically since about 1960. If population growth continues to follow this logistic pattern, then the growth rate will continue to decline and the population will gradually level out at about 12 billion.

15. Example (cont) However, human population has not been following logistic growth over longer periods. The base growth rate we found for our logistic model, r = 0.028, implies that the actual population growth rate should have started long ago at 2.8% and gradually declined to 2.1% by 1960. The 1960 growth rate was an all-time peak. While we see evidence that population has followed a logistic trend since about 1960, it has not followed this trend over longer periods. Therefore, it is still too soon to conclude that the logistic trend will continue in the future.

16. Overshoot and Collapse The phenomenon where a real population increases beyond the carrying capacity in a relatively short period of time is overshoot. If the overshoot is substantial, a rapid and severe decrease in the population can occur—a phenomenon known as collapse.

17. What is the Carrying Capacity? Any estimate of carrying capacity is subject to greatuncertainty, for at least four important reasons: • The carrying capacity depends on consumption of resources such as energy. However, different countries consume at different rates. • The carrying capacity depends on assumptions about the environmental impact of the average person. A larger average impact on the environment means a lower carrying capacity. • The carrying capacity can change with both human technology and the environment. • The earth is such a complex system that precisely predicting the carrying capacity may be impossible.

18. Difficulties of Population Prediction The historical population of Egypt SOURCE: T.H. Hollingsworth, Historical Demography (Ithaca, NY: Cornell University Press, 1969).

19. Mathematical Models This example offers an important lesson about mathematical models. They are useful for gaining insight into the processes being modeled. However, mathematical models can be used to predict future changes only when the processes are relatively simple. For example, it is easy to use mathematical modeling to predict the path of a spaceship because the law of gravity is relatively simple. But the growth of human population is such a complex phenomenon that we have little hope of ever being able to predict it reliably.