We examine a simple model problem designed to elucidate how easily a hole in a diffusion flame can close and how easily an isolated region of burning in a mixing region (a flame isola) can grow. With the thickness of the mixing layer/diffusion flame defining the characteristic length, we find that a hole of diameter Ο· 1 closes for all but a tiny interval of Damko Β¨hler numbers above the one-dimensional quenching value; a hole of diameter Ο· 3 closes for Damko Β¨hler numbers that exceed the quenching value by more than 30%-40%, depending on the Lewis number; and a hole of infinite diameter closes for Damko Β¨hler numbers that exceed the quenching value by more than Ο³70%. Even larger Damko Β¨hler numbers are required for isolas to grow, and we calculate these values for different radii. We investigate the speeds with which hole edges or isola edges advance or retreat and provide evidence that for a shrinking hole or isola, a meaningful speed can be defined that depends only on the combustion parameters and the instantaneous hole/isola radius. Fig. 1. Typical multivalued responses in combustion systems, showing possible end-point states for edge flames.