relative volume of a blow-up is given by $$\e_i = \frac{8}{3}e^{2\m\f}z_i^2=\frac{2}{3}|{\bf c}|^2\frac{\z_i^2}{\z^2}
        \left(\frac{z_{0i}}{d_i}e^{-y}+\frac{z_{1i}}{d_i}e^y\right)^2\; .$$ We recall that our solution can only be trusted as long as $\e_i\ll 1$. Since, from Eq. , the pre-factor on the RHS of the above equation is of order one (for at least one $i$) we have to require the term in bracket be smaller than one. This leads to two cases, namely

-   $\e_i\ll 1$ if $|z_{0i}/d_i|\ll 1$ and $\ln |z_{0i}/d_i|\ll y\ll -1$. In this time range the breathing mode evolves as $\f\simeq -\m y +{\rm const}$.

-   $\e_i\ll 1$ if $|z_{1i}/d_i|\ll 1$ and $1\ll y\ll\ln |d_i/z_{1i}|$. In this time range the breathing mode evolves as $\f\simeq \m y +{\rm const}$.

In both cases the breathing mode and, hence, the orbifold volume $\cV_0$, can increase or decrease depending on the choice of the sign $\m$. However, independent of this choice, the relative blow-up volume $\e_i$ will always leave the allowed range $\e_i\ll
1$ when one of the limits of the given time ranges is approached. This means that after a finite proper time, both in the past and in the future, at least one of the blow-ups will take up a significant portion of 