Biology Differential Equations/Dimensional Analysis

[ceejay2000]

http://img209.imageshack.us/img209/7596/mbiolq1.jpg


Hi PF, any help with parts a, b and c would be most appreciated. I haven't had a go at these simply because I don't know where to start. There are parts d) etc. onwards that I have done already; it seems to be the simple stuff I struggle with!
The question comes from a past paper for an exam I am revising for and the exam is on Wednesday so I'm desperate! Ha ha, thanks in advance!

 

[Curious3141]

a) A chemostat is a tank with liquid culture medium kept at a fixed volume. The chemostat receives a constant influx of nutrients at a fixed flow rate, and medium is siphoned off at the same flow rate (in order to maintain constant volume). A fixed inoculum of bacteria is introduced at the start of the process, and only natural multiplication increases their numbers.

Given this info, can you decipher the coupled differential equations? Think of what processes cause the bacterial concentration to respectively increase and decrease. You may assume that death of bacteria is not a factor here (as the tank is kept adequately supplied by nutrients at all times). Now think of what processes cause the nutrient concentration to respectively increase and decrease.

b) The dimensional analysis is easy. The LHS of both equations has the dimensions of concentration/time. Every individual term on the RHS (separated by addition or subtraction) has the same dimension. Multiplying and dividing dimensions works just like in algebra. Can you now work out the dimensions of those constants? Which one (only one) is dimensionless?

c) This is a bit more tricky. Start by substituting $x = kb, y = pn$ and $\tau = qt$ where $k, p$ and $q$ are constants into the reduced set of differential equations (in $x$ and $y$). Work through the calculus (you'll need to use Chain Rule to handle the time parameter $\tau$) and get the equations into a form comparable with the original set (in $b$ and $n$). Then compare coefficients to deduce the values of $k, p$ and $q$.

You should find $\mathbb{D} = \frac{\phi{r}}{\gamma^2}$. Use this to check your final answer.

 

[ceejay2000]

That was really helpful; thanks for your time!