conditions for asymptotic comparison to hold

I have the following simple dynamical system: \begin{align} x_1' &= a - f(x_2)x_1\\ x_2' &= bx_1 - cx_2, \end{align} where all parameters and initial conditions are positive. $$f(x_2)$$ is a positive and increasing function with respect to $$x_2$$. Suppose I want to study the asymptotic behavior of this system and decide to do the following.

First, I note that the more $$x_1$$ I have, the higher the production rate for $$x_2$$ will be. Secondly, I note that the larger $$x_2$$, the higher $$f(x_2)$$ would be. Thus I choose to replace $$f(x_2)$$ by a function $$g(x_1)$$ such that:

1. $$g(x_1) > 0$$ and $$\frac{dg}{x_1} > 0$$.

2. $$g(x_1)$$ gives the same fixed points for $$x_1$$ (perhaps through something like a quasi-steady-state-approximation for $$x_2$$).

Together, I obtain: $$$$x_1' = a - g(x_1)x_1.$$$$

Due to the construction, the asymptotic behavior of $$x_1$$ in this equation should be the same as that of $$x_1$$ in the original equation. I tried this out with $$f$$ and $$g$$ being simple hill equation and it works.

This is just a toy example. My question is: for higher dimension and more complicated functional responses, if I only care about asymptotic behavior, when can something similar be carried out? I would appreciate any references on this topic.

• I'm just curious: have you proved anything rigorously even for this simple 2D system and if so, how? I completely agree that it is the simplest case to consider before trying to attack the general problem but can you really do it in full generality? – fedja May 1 at 3:46
• @fedja I only proved in the case $f$ and $g$ take the form of hill equation (unique steady state). The method that I used is Bendixon-Dulac criterion. And the approximation for $x_2$ is $x_2 \approx b/c x_1$. You are right about the problem with the generality, which is what I am asking in this post? I am curious if there is existing work on this topic. – Paichu May 1 at 4:14
• @fedja Yes, for the system as given by the OP the global asymptotic stability of a unique equilibrium in the non-negative orthant can be rigorously proved, see my comment on MSE. – user539887 May 1 at 9:13