<p>The p53 tumor suppressor responds to DNA damage by inducing transcriptional programming that can lead to cell growth arrest or apoptosis. However, the molecular threshold mechanisms for choosing arrest or apotosis are still not well understood biologically and mathematically. In this paper, we propose a threshold mechanism underlying DNA damage repair, characterized by simultaneous isolas, with isolated closed curves of equilibria or limit cycles, in a p53-Mdm2 network model of DNA damage, inspired by dynamical systems theory. To the best of our knowledge, this is the first study to identify coexistence of isolas of equilibria and limit cycles in a p53-Mdm2 model, corresponding to abrupt transitions between non-oscillatory and oscillatory DNA damage repair mechanisms. We also locate the isola and bidirectional isola regions, and identify a Zero-Hopf (ZH) bifurcation point that signals the emergence of semi-isolas and unstable isolas, indicating transitions between distinct non-oscillatory repair states. We provide numerical simulations to illustrate our theoretical results, and discuss biological implications of the isola mechanism for DNA damage repair. Our findings provide novel insights into the dynamics underlying DNA damage repair.</p>