You know from Equation 54.4 that a population will grow, over some time period, as long as the per capita growth rate, r, is greater than zero (that is, when b > d). During this time period, the population will add a number of individuals that is r times its initial size. As the number of individuals in a population (N₀) increases, the number of new individuals added per unit of time accelerates—that is, rN₀ increases—even though the per capita growth rate (r) remains constant (Figure 54.6). This multiplicative pattern is known as exponential growth. It is also clear from Figure 54.6 that the value of r can make a big difference in the rate of growth of the population—a population with r = 1.00 will have a much faster rate of growth than a population with r = 0.25, even though both populations will display an exponential growth pattern.

Question

The blue curve (r = 1) has a faster growth rate, and thus a steeper slope, than the red curve (r = 0.25).

Equation 54.4 gives us the exponential growth of a population, but it does not allow us to predict the size of a population at a later time. Using calculus, we can integrate Equation 54.4 and get

N_t = N₀e^rt                                                                (54.5)

where N_t = the population size at time t, N₀ = the population size at time 0, e = a constant, the base of the natural log (e = 2.718), r = per capita growth rate, and t = the time interval between time 0 and time t. Knowing the starting population size and the per capita growth rate, we can use Equation 54.5 to forecast the population size at some later time, assuming exponential growth. This is similar to the equation banks use to calculate compound interest on a loan.

One application of projecting population sizes into the future comes in the form of predicting human population growth. The human population increased relatively slowly until 1825, when it was 1 billion, and now stands at 7.4 billion (Figure 54.7). United Nations data show that world population growth rates have fallen from a high of 2.30 percent (note that the percent population growth rate is r × 100) in the early 1960s to 1.18 percent today (Investigating Life: Will the Global Human Population Growth Rate Decline?). Projections indicate that human population growth rates will continue to decline to 0.50 percent in 2050 and to 0.10 percent in 2100, leading to a “best estimate” population size of 9.1 billion by 2050 and 11 billion by 2100. These projections in human population size beg the question: will world resources be able to sustain such large populations into the future? Is 11 billion above the maximum number of humans that can be supported sustainably on Earth? We will consider these questions at the end of the chapter.

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Some populations may grow at rates close to their maximum, depending on the circumstances. During the 39 years following the ban on commercial whaling, North Pacific humpback whale populations grew exponentially at a rate of 5–7 percent per year. Since the whales were released from hunting, they have had ample habitat, abundant food, and no predators, so there appears to be little limiting their population growth at this point. However, changes in ocean conditions and thus food resources, entanglement in fishing gear, ship strikes, whale-watching harassment, and harvesting for scientific purposes all pose potential threats to humpback whale population recovery. In addition, there is some limit to the population sizes whales can attain, given intraspecific competition for food resources and breeding areas.

We will turn next to what happens to population growth when limiting resources affect the number of births and deaths within a population.