A pretty standard biological reaction is that of an enzyme on it's target which might have the following differential equation to describe it's evolution:
\begin{align} \dot T &= D - \gamma T \\ \dot M &= D - \gamma M - \frac{kTM}{K + M } \end{align}
In this reaction species $T$ is being produced and enzymatically degrades $M$. This kind of equation produces "pulse" behavoir, and there is a distinct maximum in $M$ that I would like to analyze - what does it depend on?
Unfortunately this kind off differential equation seems pretty hard to solve. We can solve for $T = D/\gamma (1 - e^{-\gamma t}) $, but the $M$ equation is difficult. If you solve for $\dot M = 0$, you get
$$ \gamma M^2 + ( \gamma K_M - D + k \frac{D}{\gamma}(1- e^{-\gamma t})) M - D K = 0 $$
This is an equation that relates $M$ to $t$ at the max, but I need another constraint to solve for $M$ and $t$. Normally I guess this would come from solving the differential equation, but I'm not sure a closed form exists in this case. I'm just trying to get a closed form for the maximum $M$, is this possible without solving the equation?
Edit: I guess you shouldn't be able to do this without taking into account the initial conditions, maybe this isn't possible.