Populations of many simpler organisms, like bacteria or amoebas,  grow by doubling each generation, under good conditions. The numbers of cells or individuals ( y ) depend on the number of generations ( x ), according to the equation y = 2^x. It is useful to draw a graph of this, with x from 0 to 10, to show how a very small initial population can produce over 1 000 descendants (2¹ =2; 2¹⁰ = 1 024). Of course, the time scale varies greatly with animal species, because generation time varies. The number of offspring per generation may be much more than 2 in some species. For example, a pair of fruitflies can produce 200 little flies in 2 weeks; thus populations of Drosophila do not really double; they explode.

As Darwin (1859) pointed out, Linnaeus had calculated that an annual plant that produces only two seeds that develop into seedlings each year can have over one million offspring after 20 years. (2²⁰ = 1 048 576).         

The doubling principle works in reverse when considering ancestors : 2¹ parents, 2²  grandparents, 2³  great-grandparents etc. With an average generation time of 30 years, about 12 generations have passed since the 17^th century (1640 + 360 = 2000). Thus each person now living has 2¹² ancestors from that era. Estimates of world population (Table 7-2) make it clear that the number of people living then was far too small to allow each modern individual to have an unique set of 4 096 ancestors. Therefore, it is highly probable that any two people from the same part of the world, say Norway or Mongolia, have one or more ancestors in common.

A good graph has scales on both the vertical (y) and horizontal (x) axes, shows all the points, and is clearly labelled. The log plot is linear, becausea logarithm is an exponent. Remember the firm rule that: when quantities with the same base are multiplied, the exponents are added.

The number of living individuals is affected by survivorship, that is, the overlapping of generations. Future population sizes can be estimated by subtracting death rate from birth rate. For example, in Florida in 1999, the birth rate was 13.0 live births per 1000 population, and the death rate was 10.8 per 1000. If the total population was 15 982 378, then the net increase would be (13.0 - 10.8)/ 1000 = + 0.0022 per year.

This predicts an increase of:

0.0022 X 15 982 378 = 35 161 people added by the following year.

Population growth rates can be expressed as the ratio of present census to an earlier one. For example, the population of Sarasota County, FL was 277 776 in 1990, and 325 957 in 2000. The growth rate was:  325 957 / 277 776 = 1.17348871; 1.17348871 – 1 = +17.35% in 10 years = 1.735%/yr.

In the same years, the population of Newark, NJ changed from 275 221 to 273 546. The rate was negative:

0.993913981 -1 = -0.006086019 in 10 years = -0.0609%/yr.

Perhaps 1 675 people moved from Newark to Sarasota in that decade. (If so, they made a good decision.) These population changes may be compared to the increase in the entire USA. The first census was done in 1790. and it was repeated every ten years since then (Table 7-1).

This result suggests that Sarasota population grew more rapidly than the average for the whole country.

These graphs look much like the graphs of y = 2^x (Exercise 7-1), and suggest doubling, or some other exponential process, like money at compound interest. Thanks to calculus, it is possible to derive an equation, similar to the compound-interest equation, which allows calculation of annual growth rates, and projections of future populations.  

The basic equation is: dN/dt = rN, where N is number, t is time, and r is a rate constant. As Banks (1998) wrote, “the rate at which a particular quantity grows is directly proportional to the amount of the quantity present at any moment.”

 The problem is to obtain r from counts taken at two times. In ecology, r is a natural rate of increase, a characteristic of a species when its population growth is unrestrained.  To obtain r from census data, the basic equation must be transformed into one that allows calculation. By a theorem of calculus, the process of integration yields :

N_t/N_o = e^rt

N_o is the initial number; N_t is the number at time t;  e is a constant, the base of the system of natural logarithms is symbolized ln rather than log. The logarithmic form of the equation above is:          

r = (ln N_t - ln N₀) / ∆t

Remember that log refers to logarithms of  the base 10. A LOGARITHM IS AN EXPONENT.  e is an irrational number, approximately 2.718. Your scientific calculator will give you a more precise value of e. Because the base is smaller, the ln of a number is about 2.3 times the log of the number.

Now, let us find r for a particular population. In 1995, the population of Pakistan was 131.5 million, and in the year 2000, it was 144.6 million.

r = (ln 144.6 - ln 131.5)/ 5 y = (4.97397131 – 4.879006852)/ 5

= 0.094964458/5 = 0.01899/y = 1.899 % per year

Note the similarity to a calculation of rate of appreciation of property (Chapter 6). The main difference is the use of  base e, rather than base 10 logarithm. The compound-interest equation seems a variant of this more fundamental equation.

This result shows a much more rapid population growth in the early history of the country. Was it due to immigration, or a very high birth rate, or both?

In ecology, r is often a maximum rate, observed when population growth is unrestrained by environmental factors. For humans, this is about 3.0% per year.

Future populations can be predicted by using the most recent r value, according to a different form of the same equation:

ln P_t = ln P₀ + r∆t

To predict the population of Pakistan in 2005:

ln P_t = ln 144.6 + 0.01884 X 5 y = 5.06817131

Use the e^x function to find P_t= 158.9 million.

This predicts a year 2020 population very much greater than the year 2000, and  raises the question of saturation level or carrying capacity of the earth. One view is that the more people the better; advances in agriculture and engineering can always supply more food and water. The only real problem is distribution. Another view is that a finite earth cannot support an unlimited number of people, and the earth is already dangerously overpopulated. Thus, ways to limit population growth must be found. Which is correct?