The number Y of yeast organisms in a culture is given by the...

The number $Y$ of yeast organisms in a culture is given by the model $$Y=\frac{663}{1+72 e^{-0.547 t}}$$ where $t$ represents the time (in hours). (a) Use a graphing utility to graph the model. (b) Use the model to predict the populations for the 19th hour and the 30 th hour. (c) According to this model, what is the limiting value of the population? (d) Why do you think this population of yeast follows a logistic growth model instead of an exponential growth model?

The number $Y$ of yeast organisms in a culture is given by the model $$Y=\frac{663}{1+72 e^{-0.547 t}}$$ where $t$ represents the time (in hours).
(a) Use a graphing utility to graph the model.
(b) Use the model to predict the populations for the 19th hour and the 30 th hour.
(c) According to this model, what is the limiting value of the population?
(d) Why do you think this population of yeast follows a logistic growth model instead of an exponential growth model?

Key Concepts

Comparison with Exponential Growth

Unlike exponential growth models, which assume unlimited resources and predict continuous, unchecked growth, logistic growth models account for environmental constraints. This results in a deceleration of growth as the population nears the carrying capacity. The logistic model thereby provides a more realistic depiction of biological populations where factors such as resource depletion and competition exist.

Limiting Value (Carrying Capacity)

The limiting value in a logistic model, often referred to as the carrying capacity, represents the maximum population size that the environment can sustainably support. As time approaches infinity, the population predicted by the logistic equation will level off at this value, reflecting the impact of limited resources on population growth.

Population Prediction Using Models

Using a mathematical model to predict population involves substituting specific time values into the model equation and computing the resulting population size. This process illustrates how the model can be used to forecast future states based on current trends and parameters, providing a practical application of the theoretical framework.

Logistic Growth Model

A logistic growth model is a mathematical representation of population growth wherein the rate of reproduction is proportional to both the existing population and the amount of available resources, leading to a slowdown in growth as the population approaches a maximum limit, known as the carrying capacity. This model is characterized by an S-shaped curve and is widely used in biological contexts where resource limitations affect population dynamics.

Graphing Logistic Functions

Graphing a logistic function involves plotting the S-shaped curve that starts with a slow growth phase, then a rapid growth phase, and finally a leveling off as it approaches the carrying capacity. A graphing utility can be used to visualize this behavior, helping to understand how the model behaves over time and identify key features such as inflection points and asymptotes.

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