The number of yeast cells in a laboratory culture increases...

The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function $$ n=f(t)=\frac{a}{1+b e^{-\theta \pi}} $$ where $t$ is measured in hours. At time $t=0$ the population is 20 cells and is increasing at a rate of 12 cells/hour. Find the values of $a$ and $b$. According to this model, what happens to the yeast population in the long run?

The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function
$$
n=f(t)=\frac{a}{1+b e^{-\theta \pi}}
$$
where $t$ is measured in hours. At time $t=0$ the population is 20 cells and is increasing at a rate of 12 cells/hour. Find the values of $a$ and $b$. According to this model, what happens to the yeast population in the long run?

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Key Concepts

Logistic Growth Model

This model is a form of the logistic growth equation, often written as N(t) = L/(1 + Ae^{-kt}). It describes a situation in which populations grow rapidly at first but then level off as resources become limiting, approaching a maximum carrying capacity. The parameter representing the carrying capacity indicates the maximum population size that the environment supports, while other parameters account for the initial population and the rate of growth.

Initial Conditions and Parameter Estimation

In applied models, known values such as the initial population size and its initial rate of change are used to calculate unknown parameters. By applying these boundary and rate conditions to the logistic equation, one can solve for the parameters that define the specific growth curve. This process involves substituting the initial values into the equation and its derivative and solving the resulting equations.

Long-run Behavior (Asymptotic Analysis)

The asymptotic behavior of the logistic growth model is crucial to understanding the model’s predictive capacity. As time increases indefinitely, the exponential term decays to zero, and the population approaches the carrying capacity. Thus, in the long run, the population stabilizes at the environment’s maximum sustainable level, without unbounded growth.

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