Scratching out some thoughts here to address an issue that continues to bug me with the way the literature discusses these two very different issues.

The ecological literature on early warning signals and related subjects continually seems to confuse the concept of state and regime.

Let’s get this straight.

Regime \(=\) Attractor

State \(\neq\) Attractor

Nonlinear state change \(\neq\) Regime shift

Evidence that a regime shift occurred, and when the shift occurred (e.g. change-point analysis) is not presented here. The theory of early warning signals rests on the notion of a change in the stability of a state (a bifurcation that impacts the stability of an attractor, in the dynamical systems sense, causing a transition to an alternative attractor). This process is distinct from one which may change the state-space coordinates of an attractor without impacting it’s stability (for instance, in a simple continuous-time logistic model the location of the attractor is determined by the carrying capacity parameter, K, while the stability of the attractor is determined by the growth-rate, r. If K depends on some manipulated parameter in a nonlinear way, than changes to this parameter will change the location of the attractor in a non-linear way while having no impact on stability and exhibit no classical warning signals (that is, the eigenvalue of the system does not approach zero, the mechanism posited for the increases in autocorrelation, coefficient of variation, etc). Consequently, it is difficult to distinguish between “early warning up to a year in advance of the regime shift” and “evidence that the system is being perturbed” by the periodic pulse introductions of a large predator.

Both the theory of early warning signals and the theory of sudden or catastrophic regime shifts are based in dynamical systems theory of attractors and bifurcations.

We can change state by moving a single attractor around. This doesn’t imply any change in stability of the attractor. Because everyone likes drawing ‘ball-and-cup’ diagrams[^1] in these discussions, this is the ball-in-cup version of a state shift:

This is the ball-and-cup diagram of the “regime shift” (change of attractors, in this case due to a bifurcation).

The point of this is not to introduce a semantic battle. If we want to decide “state” and “regime” should be used interchangeably that’s fine. Only we should not discuss a theory of “state-shifts” when we are considering shifts between alternate attractors and loss of stability.

A catastrophic regime shift is characterized by the loss of stability

[^1] Motivated by potential-energy plots – the negative integral of the force over state (e.g. position). I find this use somewhat peculiar, since while in physics we work with the potential energy directly all the time, in ecological dynamics we almost exclusively work with the growth rate equations, e.g. \(\tfrac{dN}{dt} = N \cdot f(N, t)\), which is rather the “equation-of-motion” or force-representation, rather than it’s negative integral.