6.1 Population growth

The greatest shortcoming of the human race is our inability to understand the exponential function.

Exponential growth

Charles Darwin, in his theory of evolution by natural selection, was greatly influenced by the English clergyman Thomas Malthus. Malthus published a book in 1798 stating that populations with unlimited natural resources grow very rapidly. This accelerating pattern of increasing population size is called exponential growth .

The best example of exponential growth is seen in bacteria. Bacteria can undergo cell division about every hour. If 1000 bacteria are placed in a large flask with an unlimited supply of nutrients (so the nutrients will not become depleted), after an hour, there is one round of division, resulting in 2000 organisms—an increase of 1000. In another hour, each of the 2000 organisms will double, producing 4000, an increase of 2000 organisms. After the third hour, there should be 8000 bacteria in the flask, an increase of 4000 organisms. The important concept of exponential growth is that population growth (G) —the number of organisms added in each reproductive generation—is accelerating; that is, it is increasing at a greater and greater rate. After 1 day and 24 of these cycles, the population would have increased from 1000 to more than 16 billion. When the population size, N , is plotted over time, a J-shaped growth curve is produced ( [link] ).

The bacteria example is not representative of the real world where resources are limited. Furthermore, some bacteria will die during the experiment and thus not reproduce, lowering the growth rate. Therefore, when calculating the growth of a population, the number of deaths ( D (number organisms that die during a particular time interval) is subtracted from the number of births ( B ) (number organisms that are born during that interval). This is shown in the following formula:

Now let's examine how the average number of births and deaths relate to population growth. The average birth and death rates based on the number of individuals in a population is a per capita bases. So, the per capita birth rate ( b ) is the number of births during a time interval divided by the number of individuals in the population at that time, and the per capita death rate ( d ) is the number of deaths during a time interval divided by the number of individuals in the population at that time. See the equations below for a simple formula.

Now returning to the simple population growth equation above, we can convert this simple model to one in which births and deaths are expressed on a per capita basis for a time interval. Thus, B (births) = bN (the per capita birth rate “ b ” multiplied by the number of individuals “ N ”) and D (deaths) = dN (the per capita death rate “ d ” multiplied by the number of individuals “ N ”). When substituting bN for B and dN for D in simple growth equation, we can examine the population growth rate based on per capita birth and death rates as seen the equation below.