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The aim of this post is to highlight the 2009 book Critical Transitions in Nature and Society by Prof. Scheffer, which is perhaps more timely than ever and I strongly recommend this book to essentially anyone.
Far too often we (maybe not you) assume that systems are simply reversible, e.g. based on overly simplified models (think of a model corresponding to a flow).
The crux of this book is that for many ecosystems such a reversal might be very hard or even practically impossible.
Evidently, this observation has many serious ramifications with respect to understanding and tackling climate change.
Mathematically, this can be understood via bifurcation theory, in particular, via folds and cusps, introducing a form of hysteresis.
The hysteresis is important here, as this means that simply restoring some parameters of an ecosystem does not directly imply that the state of the ecosystem will return to what these parameters were previously corresponding to.
The book makes a good point that catastrophe- and chaos theory (old friends of bifurcation theory) themselves went through a critical transition, that is, populair media (and some authors) blew-up those theories.
To that end, see the following link for a crystal clear introduction to catastrophe theory by Zeeman (including some controversy),
as well as this talk by Ralph Abraham on the history (locally) of chaos theory.
One might argue that some populair fields these days are also trapped in a similar situation.
Scheffer stays away from drawing overly strong conclusions and throughout the book one can find many discussions on how and if to relate these models to reality.
Overall, Scheffer promotes system theoretic thinking. For example, at first sight we might be inclined to link big (stochastic) events, e.g. a meteor hitting the earth, as the cause of some disruptive event, while actually something else might been slowly degrading, e.g. some basin of attraction shrunk, and this stochastic event was just the final push. This makes the book very timely, climate change is slowly changing ecosystems around us, we are just waiting for a detrimental push. To give another example, in Chapter 5 we find an interesting discussion on the relation between biodiversity and stability. Scheffer highlights two points: (i) robustness (there is plently back-up) and (ii) complementation (if many can perform a task, some of them will be very good at it). Overall, the discussions on ‘‘why we have so many animals’’ are very interesting.
One of the main concepts in the book is that of resilience, which is (re)-defined several times (as part of the storyline in the book), in particular, one page 103 we find it being defined as ‘‘the capacity of a system to absorb disturbance and reorganize while undergoing change so as to still retain essentially the same function, structure, identity, and feedbacks.’’ Qualitatively, this is precisely a combination of having an attractor of some system being structurally stable. However, the goal here is to quantify this structural stability to some extent. Indeed, one enters bifurcation theory, or if you like, topological dynamical systems theory.
Throughout, Scheffer provides incredibly many examples of great inspiration to anyone in dynamical system and control theory, e.g. the basin of attraction, multistability and time-separations are recurring themes. My favourite simple example (of positive feedback) being the ice-Albedo feedback, (too) simply put, less ice means less reflection of light and hence more heat absorption, resulting in even less ice (as such, the positive feedback). More details can be found in Chapter 8. Another detailed series of examples is contained in Chapter 7 on shallow lakes. This chapter explains through a qualitative lens why restoring lakes is inherently difficult. Again, (too) simply put, as with ice, if plants disappear in lakes, turbidity is promoted, making it more difficult for plants to return (they need sunlight and thus prefer clear water). For this lake model, one can find some equations in Appendix 12, which is essentially a Lotka-Volterra model.
As such, some readers might be unsatisfied when it comes to the mathematical content*. Moreover, at times Scheffer himself describes bifurcations as exotic dynamical systems jargon and throughout the book one might get the feeling that the dynamical systems community has no clue how to handle time-varying models. Luckily, the latter is not completely true. Perhaps the crux here is that we can use more cross-contamination between scientific communities. Especially recognizing these bifurcations (critical transitions), not just in models, but in reality is still a big and pressing open problem.
*A fantastic and easy-going short book on catastrophe theory is written, of course, by Arnold.
The aim of this book was to have a widely accesible mathematical reference.
The introduction to this book reads as:
‘‘This booklet explains what catastrophe theory is about and
why it arouses such controversy. It also contains
non-controversial results from the mathematical theories of
singularities and bifurcation.’’
whereas Arnold ends with:
‘‘A qualitative peculiarity of Thom's papers on catastrophe
theory is their original style: as one who presenses the
direction of future investigations, Thom not only does not have
proofs available, but does not even dispose of the precise
formulations of his results. Zeeman, an ardent admirer of this
style, observes that the meaning of Thom's words becomes
clear only after inserting 99 lines of your own between every
two of Thom's.’’
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