Yeast Growth The amount Y of yeast in a culture is given by...

Yeast Growth The amount $Y$ of yeast in a culture is given by the model $$Y=\frac{663}{1+72 e^{-0.547 t^{\prime}}}, \quad 0 \leq t \leq 18$$ 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 19 th hour and the 30 th hour. (c) According to this model, what is the limiting value of the population? (d) Why do you think the population of yeast follows a logistic growth model instead of an exponential growth model?

Yeast Growth The amount $Y$ of yeast in a culture is given by the model
$$Y=\frac{663}{1+72 e^{-0.547 t^{\prime}}}, \quad 0 \leq t \leq 18$$
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 19 th hour and the 30 th hour.
(c) According to this model, what is the limiting value of the population?
(d) Why do you think the population of yeast follows a logistic growth model instead of an exponential growth model?

Key Concepts

Logistic Growth Model

The logistic growth model is a mathematical representation of population expansion that starts with an exponential increase when resources are abundant, but then slows as the population approaches a maximum, stabilizing near a carrying capacity. The model is characterized by an S-shaped curve, reflecting the shift from rapid growth to a plateau as environmental limitations come into play.

Carrying Capacity

The carrying capacity is the upper limit of a population that an environment can sustain indefinitely. It represents the maximum population size that can be supported by available resources, and in logistic models, it is approached asymptotically as time goes on, indicating a stabilization of growth.

Graphing Utility

Graphing utilities, such as graphing calculators or software, are essential for visualizing dynamic models like logistic growth functions. They help in plotting the behavior of the model over time, which aids in understanding key features such as the inflection point and the asymptote corresponding to the carrying capacity.

Exponential vs Logistic Growth

Exponential growth describes an unlimited increase where the rate of growth is proportional to the current size, leading to unbounded outcomes. In contrast, logistic growth incorporates the concept of resource limitations by introducing a carrying capacity, making it a more realistic model for biological populations and other systems where growth is naturally constrained.

Population Prediction Using Mathematical Models

Using a mathematical model to predict future population values involves substituting specific time values into the equation to compute the expected outcome. This process allows researchers and practitioners to forecast population dynamics and assess the impact of growth limitations over time.

Transcript

Please give Ace some feedback