Biologists have created a second model to account for the behavior of populations under finite resources and have called it logistic growth. If the number of individuals in a population undergoing logistic growth is plotted as a function of time, the curve resembles a flattened S shape. In other words, the curve is initially flat, but then curves upward at a progressively faster rate, much like exponential growth. At some point (called the inflection point), however, the curve begins to turn to the right and flatten out. Ultimately, the curve becomes horizontal, indicating a constant population over time.
An important aspect of pure logistic growth is that the population approaches, but does not exceed, a certain level. That level is called the carrying capacity, and is represented by the symbol K in most mathematical treatments of logistic growth. The carrying capacity is the maximum number of individuals that the environment can support, based on the space, food, and other resources available. When the number of individuals is much fewer than the carrying capacity, the population grows rapidly, much as in exponential growth. As the number increases, however, the rate of population growth becomes much less than the exponential rate. When the number approaches the carrying capacity, new population growth virtually ceases. If the population were to increase above the carrying capacity for some reason, there would be a net loss of organisms from the population.
There are few studies that have documented logistic growth in nature. It would be necessary to
The study of population growth must consider how often an animal reproduces and how many offspring it produces at one time. Some animals, such as the elephant, have only one offspring with a long gestation period, while others, such as the fruit fly, lay thousands of eggs at a time. Significantly, elephants are an endangered species while fruit flies are not. (Corbis)
watch a species in a habitat from the time of its first introduction until its population stabilized. Such studies are necessarily of a very long duration and thus are not normally conducted. Logistic growth has been found in a number of experimental studies, however, particularly on small organisms, including protozoans, fruit flies, and beetles.
An important aspect of logistic growth is that, as the population increases, the birthrate decreases and the mortality rate increases. Such ef-
fects may be attributable to reduced space within which the organism can operate, to less food and other resources, to physiological and behavioral stress caused by crowding, and to increased incidence of disease. Those factors are commonly designated as being density-dependent. They are considered much different from the density-independent factors that typically arise from environmental catastrophes such as flooding, drought, fire, or extreme temperatures. For many years, biologists argued about the relative importance of density-dependent versus density-independent factors in controlling population size. It is now recognized that some species are controlled by density-independent factors, whereas others are controlled by density-dependent factors.
Classically, when a species undergoes logistic growth, the population is ultimately supposed to stabilize at the carrying capacity. Most studies that track populations over the course of time, however, find that numbers actually fluctuate. How can such variability be reconciled with the logistic model? On the one hand, the fluctuations may be caused by density-independent factors, and the logistic equation therefore does not apply. On the other hand, the population may be under density-dependent control, and the logistic model can still hold despite the fluctuations. One explanation for the fluctuations could be that the carrying capacity itself changes over time. For example, a sudden increase in the amount of food available would increase the carrying capacity and allow the population to grow. A second explanation relates to the presence of time lags; that is, a population might not respond immediately to a given resource level. For example, two animals in a rapidly expanding population might mate when the number of individuals is less than the carrying capacity. The progeny, however, might be born sev-
eral weeks or months later, into a situation in which the population has exceeded the carrying capacity. Thus, there would have to be a decline, leading to the fluctuation.
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