I am trying to solve the following system (derived from a Michaelis-Menten kinetics model for an enzymatic chemical reaction):
$\dot{y}_a = r_p x_a - \lambda_p y_a$
$\dot{x}_b = \frac{\alpha_0 + \alpha_1 (\frac{y_a}{K})^n}{ 1 + (\frac{y_a}{K})^n} - \lambda_m x_b$
Ideally, for all $n\in\mathbb Z$, but I would already be quite happy with $n \in \{-2, -1, 1, 2\}$
Currently, I use Fourier series expansions of $x_a$, $y_a$ and $x_b$ to rewrite the system and estimate the values I need...
I am wondering if there might be a closed-form solution to this system?
I think $y_a$ should be rewritable as an exponential function of $x_a$, but reinjecting this in the second equation got me nowhere (even when taking the $\log$... which straightens the fraction, but makes a mess of the rest).
I'd be really grateful for any pointer toward a closed-form solution (or indication that there is none)...