Tipping points in complex ecological systems

Tipping points are one of the hot topics in modern physics of complex systems. But what is a tipping point? A generic definition declares it as ``a state of the system where a small change in its parameters can lead to a significant change in its properties''. Additional ingredients that often enter the definition of tipping process are the abruptness of the resulting change and its irreversibility, i.e. it is impossible to recover the initial state if one reverses the protocol of change of the parameters. However, there exists a number of different mathematical structures that can show this behavior, the one that was originally suggested as a tipping point (nowadays usually referred to as bifurcation induced tipping) is just one of many. Different preconditions and/or different level of details included into the model, reflecting also different environmental forcing, can lead to a variety of tipping mechanisms. Furthermore, in a spatially extended system and/or a system with multiple scales, different parts can react to a change in environmental conditions differently or at a different time, interacting with each other to create a tipping cascade. In this paper, using ecosystems as a paradigm of complex nonlinear open systems, we provide a critical overview of the progress made in tipping point science over the last 15 years. We highlight the main findings, identify gaps in our knowledge, and outline a roadmap for further progress.