## The Logistic Model

Birth rates and death rates are not constant but vary with population size: birth rates decline and death rates rise as the population grows. The logistic (loh-JIS-tik) model of population growth builds on the exponential model but accounts for the influence of limiting factors. The logistic model includes a new term, carrying capacity (symbolized by K), the number of individuals the environment can support over a long period of time.

A graph of logistic growth looks like a stretched-out letter S. Examine Figure 19-9. When the population size is small, birth rates are high and death rates are low, and the population grows at very near the exponential rate. But as the population size approaches the carrying capacity, the population growth rate slows because of the falling birth rate and the increasing death rate. When a population size is at its carrying capacity, the birth rate equals the death rate and growth stops. This pattern of growth is known as logistic growth.

The logistic model, like the exponential model, contains some assumptions. One such assumption is that the carrying capacity is constant and does not fluctuate with environmental changes. In reality, carrying capacity does fluctuate. It is greater when prey is abundant, for instance, and smaller when prey is scarce. The logistic and exponential models are not universal representations of real populations, but they are an important tool that scientists use to explain population growth and regulation.

figure 19-9

This graph of logistic population growth is typical of populations in new environments. In the first phase, the population shows rapid, nearly exponential growth. In the second phase, the growth rate slows until the carrying capacity, K is reached. In the third phase, the population has become stable, neither increasing nor decreasing in size. Real populations may fit this pattern for some period of time but rarely remain stable.

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