I am studying a biological system (HIV) and arrived at this simplified dynamical system:

\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 truly appreciate any idea to approach this problem, perhaps a form of Lyapunov function, a direct method, etc that I should try.

Additionally, I have done extensive numerical simulations, which suggests the positive steady state is globally asymptotically stable.

Edit: for those interested in the original system - especially to back up the various claims that I made in my question. **Please note that the parameters do not match to the system in the question**. It's just a convenient way to write the parameters.

\begin{align} x_1' &= a_1x_2 -a_2x_1\\ x_2' &= -(f(y_2) + a_1) x_2 + a_3x_3 + a_2x_1\\ x_3' &= f(y_2)x_2 - a_3x_3\\ y_1' &= b_1x_2 - (b_2+b_3)y_1\\ y_2' &= b_2y_1 - b_4y_2\\ f(y_2) &= \frac{c_1}{c_2} \frac{1 + c_2(y_2/c_3)^n}{1+(y_2/c_3)^n}. \end{align} The only conditions on the parameters are that they are positive and $c_2>1$. Note that the first 3 equations are conservative, hence it can be reduced. And at any time, $x_1 + x_2 + x_3 = 1$. This system is part of a larger system, but is decoupled from the rest.

generalpositive coefficients $a_j$, so, unless you tell us a bit more about your coefficients, it is hard to give you a good advice. Why don't you post the problem exactly as you have it? $\endgroup$ – fedja Apr 28 '19 at 23:28