Populations can grow rapidly because they increase by multiplication

Equations 11.1 and 11.3 show that populations increase by multiplication, not addition: at each point in time, the population changes in size according to the multiplier λ or r.

Consider our own population. In this chapter's Case Study, we stated that the current annual growth rate of the human population was 1.0%, which is an r = 0.01. If we set the year 2022 as time t = 0, we have N₀ = 7.9 billion, the size of the human population in 2022. Plugging these values of r and N₀ into Equation 11.4, we calculate that the population size 1 year later should be N₁ = 7.9 ? e⁰’⁰¹, which equals 7.98 billion people. Thus, in 2022, the human population was increasing by 80 million people per year (7.98 billion - 7.90 billion = 0.08 billion = 80 million). Since populations grow by multiplication, if r remained constant at 0.01 for an extended period of time, the yearly increments to the human population would become astronomical. For example, after 225 years, there would be 75 billion people, and our population would be increasing in size by almost a billion people each year.

Turning from humans to other species, what do field studies reveal about the growth rates of their populations? Some species, such as the woodland herb Asarum canadense (wild ginger), have maximum observed values of λ that are close to 1 (λ = 1.01 in young forests, λ = 1.1 in mature forests) (FIGURE 11.6). Similar values were observed for a population of 25 reindeer introduced to Saint Paul Island off the coast of Alaska in 1911.

FIGURE 11.6 Some Populations Have Low Growth RatesThegrowthratesofa population of wild ginger (Asarum canadense) in a young forest vary from year to year. The maximum growth rate in this forest is 1.01. However, growth rates are often less than 1.0, suggesting that the population will decline in size unless conditions improve. (Data from H.

Damman and M. L. Cain. 1998. JEcol 86: 13-26.) View larger image

Considerably higher annual growth rates have been observed for populations of many species, including western gray kangaroos (λ = 1.9), field voles (λ = 24), and rice weevils (λ = 10¹⁷), which are insect pests of rice and other grains. Some bacteria, such as the mammalian gut inhabitant Escherichia coli, can double in number every 30 minutes, resulting in the unimaginably high annual growth rate of λ = 10⁵’²⁷⁴.

Recall that when λ > 1 (or r > 0) for an extended period, populations increase exponentially in size, forming a J-shaped curve like that in Figure 11.4A. In natural populations, λ > 1 (or r > 0) when key factors in the environment are favorable for growth, survival, and reproduction. But can such favorable conditions last for long?