Consider the equation $\dot{x} = r+x^2$. When $0 < r \ll 1$, this system experiences a bottleneck effect. Then the time $T$ spent in this bottleneck can be approximated by:
$$T_{bn} = \int_{-\infty}^{\infty} \frac{dx}{r+x^2}$$
Now consider a two-dimensional system by:
$$\dot{x} = x(3-x-2y)\qquad \dot{y} = y(2-x-y)$$
It's easy to show that $(1,1)$ is a saddle point and $(3,0)$ and $(0,2)$ are stable fixed points. Thus $(1,1)$ should experience a bottleneck effect as well. Is there a way to expand on the one-dimensional case to approximate the time spent in the bottleneck of the two-dimensional case? Then in theory, one could build this up to any higher dimension.