Calculating Steady States: Exploring the "Two Cases

[mt91]

I've got this question here, I know to calculate steady states you set dn/dT and dc/dT to 0 and then solve. However can anyone help me understand what it means by the "two cases" and how you go about this?

 

[HOI]

Yes, the "steady" in "steady state" means that it does not change. "Steady State" solutions for n and c are constant solutions so the derivatives are 0. The equations become:
$\frac{n}{n+1}- \beta nc=0$ and
$\alpha- \mu c= 0$.

Since the second equation involves only c, no n, I would solve it first:
$c= \frac{\alpha}{\mu}$.

Now put that into the first equation:
$\frac{n}{n+1}- \frac{\alpha \beta}{\mu}n= 0$
$\frac{n}{n+1}= \frac{\alpha \beta}{\mu}n$

$n= \frac{\alpha\beta}{\mu}n(n+ 1)$

Obviously n= 0 is a solution to that so one steady state solution is $c= \frac{\alpha}{\mu}$, $n= 0$.

If n is not 0 we can divide by it to get $1= \frac{\alpha\beta}{\mu}(n+ 1)$.
Then $n+ 1= \frac{\mu}{\alpha\beta}$,
$n= \frac{\mu}{\alpha\beta}- 1= \frac{\mu- \alpha\beta}{\alpha\beta}$.

So if n is not 0 then $c= \frac{\alpha}{\mu}$, $n= \frac{\mu}{\alpha\beta}- 1$ is a steady state solution.

But notice the "$\frac{\mu}{\alpha\beta}- 1$". If $\frac{\mu}{\alpha\beta}= 1$ then we are back to the "n= 0" solution, above. If $\frac{\mu}{\alpha\beta}> 1$ the steady state solution for n is positive, if $\frac{\mu}{\alpha\beta}< 1$ the steady state solution for n is negative.

Was this the entire question or was there some application in which n being positive or negative would be important?