Adapting to a changing environment: non-obvious thresholds in multi-scale systems

Abstract

Many natural and technological systems fail to adapt to changing external conditions and move to a different state if the conditions vary too fast. Such ‘non-adiabatic’ processes are ubiquitous, but little understood. We identify these processes with a new nonlinear phenomenon—an intricate threshold where a forced system fails to adiabatically follow a changing stable state. In systems with multiple time scales, we derive existence conditions that show such thresholds to be generic, but non-obvious, meaning they cannot be captured by traditional stability theory. Rather, the phenomenon can be analysed using concepts from modern singular perturbation theory: folded singularities and canard trajectories, including composite canards. Thus, non-obvious thresholds should explain the failure to adapt to a changing environment in a wide range of multi-scale systems including: tipping points in the climate system, regime shifts in ecosystems, excitability in nerve cells, adaptation failure in regulatory genes and adiabatic switching in technology.