I have a nonlinear system of equations which I'm struggling to solve and actually do not have much hope to find an explicit solution. I do not need to solve it; but I want to prove a property of the solution of this system, namely 'monotonicity' of the combination of solution components depending on the parameter. The system is: $$ \begin{align} x_1&=(\hat{d}_1+q_1)(q_1+q_2+pq_1)\\ x_2&=(\hat{d}_2+q_2)(q_1+q_2)\\ q_1&=\bar{d}_1-\Bigl( k_1 \sqrt{d^2_1+x_1} + \frac{x_1} {x_1+x_2}k_0\sqrt{d^2+(x_1+x_2)}+\frac{x_1}{x_1+x_2}k_0d\Bigl)\\ q_2&=\bar{d}_2-\Bigl( k_2 \sqrt{d^2_2+x_2} + \frac{x_2}{x_1+x_2}k_0\sqrt{d^2+(x_1+x_2)}+\frac{x_2}{x_1+x_2}k_0d\Bigl) \end{align} $$ where $(x_1, x_2, q_1, q_2)$ are unknowns, $p$ is a parameter and all others are constants:
$k_0, \ k_1, \ k_2, \ d_1, \ d_2$ are unrelated and $$ \begin{align} d&=d_1+d_2\\ \hat{d_i}&=2d_i\\ \bar{d_i}&=\bigl(2/(k_0+k_i-1)+2k_0+k_i\bigl)d_i \end{align} $$
The statement (which is a hypothesis but I believe it should be true) I'm trying to prove is: $$ \frac{x_1}{k_1+k_0-1}+\frac{x_2}{k_2+k_0-1} \quad \text{ increases as } p \text{ increases}$$
Any ideas would be very helpful. Many thanks in advance!