We are given a continuous dynamics $x(t) \in \mathbb{R}^n_{> 0}$ that follows $\frac{dx}{dt} = g(x)$, where $g : \mathbb{R}^n_{> 0} \rightarrow \mathbb{R}^n$ is a smooth continuous function. We know that from any starting point $x(0) \in \mathbb{R}^n_{> 0}$ the dynamics always stays in $\mathbb{R}^n_{> 0}$ and converges to the unique point $x^*$. ($g$ is well-behaved and it pushes us either parallel to or away from the boundary $x_i = 0, \infty$ for all $i$ and eventually leads us to $x^*$.)
Now, we are given a rate multiplier $r \in \mathbb{R}^n_{> 0}$. Does the dynamics $\frac{dx}{dt} = r g(x) = (r_i g_i(x))_{i \in [n]}$ also converge to $x^*$ from any starting point? Intuitively, seems to be true as we are multiplying the direction of movement along each coordinate by a scaling factor.