Cascade tipping, cascade hopping and indirect tipping in ecological systems

Tipping occurs when an abrupt transition is observed in a system's long-term behavior. This study demonstrates tipping in ecological systems using a unidirectional, three-species coupled system, where only the first subsystem experiences a ramp-like parameter drift. Using a model of bacteria, phytoplankton, and fish populations, we explore how environmental changes, coupling strength, and nonlinearity drive sudden transitions. This study identifies three crucial tipping phenomena based on the coupling strength: tipping cascade, cascade hopping, and indirect tipping. While prior research has focused on tipping cascade and cascade hopping, this work introduces indirect tipping as a critical phenomenon, where the tertiary subsystem tips even when both the primary and secondary subsystems do not tip. The results emphasize the vulnerability of coupled systems to minor changes and highlight the necessity of closely observing systems that appear untipped in coupled systems.