Solution $x:\mathbb R_+\to [0,1]$ of the ODE $\dot x = \tfrac 1 2 x^2 - (1-x) (1-e^{-at})$ is a concave function of $g(t)=1-e^{-at}$.

Question

Is any solution $x_t:\mathbb R_+\to [0,1]$ of the ODE $\dot x = \tfrac 1 2 x^2 - (1-x) (1-e^{-at})$ is a concave function of $g_t=1-e^{-at}$ for any $a>0$?

Note: A solution has to converge to the critical point $x_* = \sqrt{3}-1$, otherwise it would explode (and leave the interval $[0,1]$, so it won't be a solution). There is in fact unique such solution.

This is a simplification of the question: Convexity / concavity of $(g_t,x_t)$, where $x_t$ solves the ODE $\dot x_t=\tfrac 1 2 x_t^2 - (1-x_t) g_t$., which is a simplification of my original problem: System of quadratic autonomous ODEs - convexity of the solution curve

A slightly more general question would be:

Is any solution $x:\mathbb R_+\to [0,1]$ of the ODE $\dot x = \tfrac 1 2 x^2 - b(1-x) (1-e^{-at})$ a concave function of $g_t=1-e^{-at}$ for any $a,b>0$?

I will appreciate a solution or a reference to any similar problems.