Collapses and regime changes are pervasive in complex systems (such as marine ecosystems) governed by multiple stressors. The demise of Atlantic cod (Gadus morhua) stocks constitutes a text book example of the consequences of overexploiting marine living resources, yet the drivers of these nearly synchronous collapses are still debated. Moreover, it is still unclear why rebuilding of collapsed fish stocks such as cod is often slow or absent. Here, we apply the stochastic cusp model, based on catastrophe theory, and show that collapse and recovery of cod stocks are potentially driven by the specific interaction between exploitation pressure and environmental drivers. Our statistical modelling study demonstrates that for most of the cod stocks, ocean warming could induce a nonlinear discontinuous relationship between fishing pressure and stock size, which would explain hysteresis in their response to reduced exploitation pressure. Our study suggests further that a continuing increase in ocean temperatures will likely limit productivity and hence future fishing opportunities for most cod stocks of the Atlantic Ocean. Moreover, our study contributes to the ongoing discussion on the importance of climate and fishing effects on commercially exploited fish stocks, highlighting the importance of considering discontinuous dynamics in holistic ecosystem-based management approaches, particularly under climate change.