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logistic curve
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Side-by-side line graphs compare exponential and logistic population growth. The exponential growth graph shows a curve rising steeply upward, while the logistic growth graph shows an S-shaped curve leveling off at the carrying capacity of the environment.
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Side-by-side line graphs compare exponential and logistic population growth. The exponential growth graph shows a curve rising steeply upward, while the logistic growth graph shows an S-shaped curve leveling off at the carrying capacity of the environment.
carrying capacity and exponential versus logistic population growth In an ideal environment (one that has no limiting factors) populations grow at an exponential rate. The growth curve of these populations is smooth and becomes increasingly steep over time (left). However, for all populations, exponential growth is curtailed by factors such as limitations in food, competition for other resources, or disease. As competition increases and resources become increasingly scarce, populations reach the carrying capacity (K) of their environment, causing their growth rate to slow nearly to zero. This produces an S-shaped curve of population growth known as the logistic curve (right).

The logistic curve, also known as the sigmoid function, is a mathematical equation that describes growth rates of quantities over time. It is graphed as an S-shaped curve. The equation for the logistic function is (P(t) = \frac{L}{1 + e^{-r(t-t_0)}}), where (P(t)) is the quantity changing with time (t), (r) is the growth rate, (t_0) is the midpoint, (e) is the base of the exponential function, and (L) is the maximum value or carrying capacity.12

This function is used in various fields, including biology, sociology, agriculture, and economics, to model phenomena like population growth, product sales, and crop yields. The curve shows an initial growth stage, a midpoint where growth rate changes, and a final stage where growth levels off towards the maximum value. The standard logistic function, (f(t) = 1/(1 + e^{-t})), is often referred to as the sigmoid function.1345

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