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 $$\mathrm{n}=\mathrm{f}(\mathrm{t})=\frac{\mathrm{a}}{1+\mathrm{b} \mathrm{e}^{-0.71}}$$ 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
$$\mathrm{n}=\mathrm{f}(\mathrm{t})=\frac{\mathrm{a}}{1+\mathrm{b} \mathrm{e}^{-0.71}}$$
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?

Key Concepts

Rate of Change

The rate of change in a population model indicates how quickly the population increases or decreases at a given time. It is typically described by a derivative with respect to time and is fundamental in understanding the dynamics of population growth, especially when used alongside initial conditions to set up and solve the model.

Long-term Behavior

Long-term behavior in a logistic model concerns what happens to the population as time becomes very large. Typically, the model predicts that the population will approach the carrying capacity, reflecting the idea that resource limitations eventually cap the growth of the population, leading it to stabilize.

Parameter Estimation

Parameter estimation is the process of using known data—like initial population size and growth rate—to solve for the constants within a model. In the context of logistic growth, the parameters often include the carrying capacity and a constant related to the growth rate, which are determined by substituting the initial conditions into the model equation.

Logistic Growth Model

The logistic growth model describes how populations grow in environments with limited resources. It is characterized by an initial phase of exponential growth that gradually slows as the population nears a maximum limit, known as the carrying capacity. This model is commonly used in biology to understand how populations reach equilibrium over time.

Initial Conditions

Initial conditions refer to the specific values provided at the start of a process, such as the initial population size and the initial rate of growth. These values are essential for calibrating mathematical models, as they allow for the determination of unknown parameters in the equations governing the system.

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