The concepts of tipping points, regime shifts, and multiple stable points are potentially powerful organizing principle for studying the dynamics of nonlinear systems. Social-ecological systems certainly have many important characteristics, including the presence of both fast and slow variables, which suggest that using ideas based on tipping points may be useful in informing management to avoid undesirable outcomes. However, the underlying framework of tipping points is really based on ideas from one dimensional systems and the use of tipping points as a framework depends on other assumptions as well. The question of how best to use ideas form tipping points in complex systems remains.
I will review underlying principles of the dynamics of highly nonlinear systems, emphasizing both the potential difficulties that arise in the application of simple ideas to complex systems and the advantage of recognizing the possibility of relatively simple classes of possible sudden change in complex systems. I will draw upon examples of different social-ecological systems and purely ecological systems that can exhibit sudden changes in dynamics in response to slow external changes to show that great care needs to be used in applying ideas from simple systems to single realizations of the behavior of complex systems. Examples will include coral-algal-grazer systems, invasive species, and simple food webs. Classes of systems where simple ideas can and cannot be used will be described. I will emphasize the potential importance of explicitly recognizing time scales in the response of complex systems to slow changes in external conditions and that if both state variables and external conditions can be changed rapidly enough that undesirable changes can be reversed.