I am studying some biology system and arrives at this simplified dynamics:
\begin{align} x_1' &= a_1 + a_2x_2 - a_1x_2 - a_4x_1 - a_5\frac{1+a_6x_3}{1+a_7x_3}x_1\\ x_2' &= a_5\frac{1+a_6x_3}{1+a_7x_3}x_1 - a_2x_2\\ x_3' &= a_8x_1 - a_9x_3 \end{align} Where as the context and notation dictate: all coefficients $a_i$ are strictly positive. I also showed all $x_i$ to be strictly positive and bounded.
I have obtained locally-asymptotically stability for the unique positive fixed point. I also tried various lyapunov functions, but without much luck. I would really appreciate any idea to approach this problem -please do not provide me with full solution, perhaps a form of Lyapunov function, a direct method, etc that I should try.