You are attempting to model the growth of bacteria and decrease in nutrient over time in a well-stirred flask of volume, \(V\). You want to chart the total number of cells per unit volume (cell concentration), \(N\), and the concentration of nutrient, \(c_{n}\). An initial inoculate of cells, \(N_{o}\) is charged to the flask. The initial nutrient concentration in the flask is \(c_{n o}\). You may assume the reproductive frequency of the bacteria, \(k^{\prime \prime}=k_{o}^{\prime \prime} c_{n}\) is directly proportional to the concentration of nutrient in the flask. The reproductive rate, \(r_{p}=k^{\prime \prime} N\) is a product of the reproductive frequency and the number of cells per unit volume. For every new cell that is formed, \(\alpha\) units of nutrient are consumed \(-r_{n}=\alpha r_{p}\) where \(r_{n}\) is the rate of nutrient consumption.

a. Using this information, derive the two differential equations describing the cells concentration and the concentration of nutrient.

b. Show that these two equations can be collapsed into a single equation for the cell concentration:

\[\frac{d N}{d t}=k_{o}^{\prime \prime} N\left[c_{n o}-\alpha\left(N-N_{o}\right)\right]\]

c. Solve the set of equations to determine the cell concentration and nutrient concentration as a function of time.

d. What about cell death?