Complex systems can be conceptualized and studied as networks of nodes with varying connectivity between nodes. In well-connected systems, local disturbance of individual nodes can be countered by input from neighbouring nodes, buffering the system against local change. Thus, pronounced change in a well-connected system may not occur until the system hits a threshold or ‘tipping point’ that drives a shift to an alternative, system-wide state. In contrast, poorly connected systems are more prone to gradual node-by-node change. We use forward-in-time simulations of multi-locus evolution to test these general predictions concerning complex systems. We do so in the context of local adaptation in patchy environments comprised of many demes (i.e., nodes) of two habitat types. We vary connectivity by manipulating migration rate and the spatial clustering of habitat types. We find gradual and ‘deme-by-deme’ dynamics of local adaptation when connectivity is low. The dynamics transition towards more sudden, system-wide shifts as population connectivity is increased (i.e., many demes adapt more suddenly and simultaneously). Our results support a trade-off between local and system-wide resilience, and we discuss their implications for the conservation of species living in patchy and fragmented habitats.