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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: \begin{equation} x_1' = a - g(x_1)x_1. \end{equation}

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.

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  • $\begingroup$ 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? $\endgroup$ – fedja May 1 at 3:46
  • $\begingroup$ @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. $\endgroup$ – Paichu May 1 at 4:14
  • $\begingroup$ @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. $\endgroup$ – user539887 May 1 at 9:13

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