INTRODUCTION

No real population can maintain exponential growth for very long. The rate of increase of a population that is experiencing such rapid growth will eventually slow because environmental limitations cause birth rates to drop and death rates to rise. In fact, over long periods of time, the size of most populations fluctuates around a relatively constant number. Another way to say this is that, dN/dT (i.e., the rate of change in the size of the population) averages zero or some value very close to zero.

One way to picture the limits imposed by the environment is to assume that an environment can support no more than a certain number of individuals of any particular species. This number, called the environmental carrying capacity (K), is determined by the availability of resourses, as well as disease, predators, and perhaps social interactions.

Recall the equation for exponential growth:

    (1)

How might this equation be modified to include the concept of carrying capacity? The simplest such derived equation is that for logistic growth:

    (2)

The biological assumption in this equation is that each additional individual depresses population growth equally as it competes with others in the population for available resources.

What do you think the size of a population following the logistic growth equation looks like over time? To find out, select Graph. Click/Tap on New Plot to see a plot for given values of r, N₀ (initial population size), and K. To see the value of N for a given time, click/tap at the desired location. To clear, click/tap again.

The dashed horizontal line is the carrying capacity. The solid line is the logistic growth plot. The exponential growth curve (same values of N₀ and r) is shown using "+" symbols.

The limitations imposed by the carrying capacity mean that, rather than being exponential, population growth follows a curve that flattens out as the population approaches the carrying capacity. If the initial population (N₀) is less than the carrying capacity, the curve has a characteristic S-shape. If N₀ is greater than K, then the population will decrease until N = K. From our equation of logistic growth

    (2)

notice that when N = K, (K - N) = 0, and thus dN/dT = 0.

Also notice that when K is much larger then N, (K - N) / K approaches 1, and equation (2) reduces to the equation for exponential growth:

    (1)

This perhaps fits with your initutive notion that when a population is just starting to fill up its environment its growth is nearly exponential. However, there is a serious flaw at extremely low values of N. The equation suggests that when N is low (e.g., when the population is on the verge of extinction), the growth rate is at its highest. Can you think of reasons why this is not true for most real population? [Answer]