Continuously replenished reaction
The first step is to set the discriminant equal to zero, and thereby solve for the critical value of $r_5$. When I did this, I got $$r_5a_0=\frac{r_2c_1}{1+2\sqrt{\frac{r_2r_3}{r_1r_4}}}$$.  The next step is to substitute this into the equation for m, and find the two  corresponding equal values of m.