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Why is the logistic function used to model bacterial populations and crop yields?

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Line graph showing world population growth from 1700 to 2000 with a steep rise after 1950, followed by three projection scenarios for 2000 to 2100: high, medium, and low, reaching between 7 and 14 billion. — population Graph of the world's estimated human population from 1700 until 2000, with population projections extending until 2100.

The logistic function is used to model bacterial populations and crop yields because it accurately reflects real-world constraints on growth, which are often limited by available resources.1

Unlike models of unrestricted exponential growth, the logistic function accounts for the fact that growth rates eventually stabilize as a system reaches its maximum capacity. In biology, this maximum value, denoted as L, is known as the carrying capacity of the environment.1

The model, developed by Pierre-François Verhulst, assumes that the rate of reproduction is proportional to both the existing population size and the availability of resources. This results in a characteristic S-shaped, or sigmoid, curve that tracks three distinct stages of development:1
* An initial period of accelerating growth.
* A midpoint of inflection where growth begins to slow.
* A final stage where growth levels off as the population or yield approaches its saturation point.1

By incorporating these limiting factors, the logistic function provides a more realistic representation of how bacteria and agricultural crops behave as they interact with their environment.1