What the book covers

1. Folds and hysteresis. The two saddle-node bifurcations and the hysteresis loop of a grazing (harvesting) model; reversible, hysteretic and irreversible lakes in a phosphorus model; the evidence from lakes and coral reefs, and the dispute about reefs.
2. Critical slowing down. The recovery rate goes to zero as a square root near a fold; derivation and simulation check of autocorrelation and variance in the Ornstein–Uhlenbeck process; evidence from palaeoclimate and experimental systems.
3. The limits of early warning. ROC evaluation of Kendall’s $\tau$ indicators; noise-induced flips, flickering and the “prosecutor’s fallacy”; performance in long-term ecological data.
4. Engineering and ecological resilience. Pimm’s “speed of return” versus Holling’s “how much disturbance can be absorbed”; how flattening small fluctuations or removing reserves can shrink a basin of attraction.
5. Cascading failure. Motter and Lai’s capacity–load model and the spare capacity $\alpha$; the discontinuous collapse of interdependent networks and the model assumptions behind it.
6. Tight coupling and modularity. Perrow’s theory of normal accidents and the theory of high-reliability organisations; modularity as a firebreak, and its cost.

Chapter 7 reads this research beside “Treading on frost, the hard ice is coming” in the Book of Changes, “Danger comes to one who rests secure in his position” and “To know the incipient is divine” in the “Great Commentary”, chapter 64 of the Laozi and the “Illustrations of Laozi” chapter of the Han Feizi. Chapter 8 offers a “four-step resilience and early-warning checklist” and the protocol of a pre-registered randomised controlled trial.

The book is related to two other 5D Library volumes. Chapters 3 and 4 of Volume 13, The Classical and Statistical Mechanics of Action, have already discussed Kramers escape rates in a double well and hysteresis in the cusp model; Chapter 6 of Volume 20, Credit Cycles, Self-Organised Criticality and the Statistical Physics of Market Phase Transitions, has already demonstrated critical slowing down and a Kendall’s-$\tau$ warning rule on autoregressive series and introduced Boettiger and Hastings’s critique. This book does not repeat those programs; it returns to the original ecological literature and moves on to ROC evaluation, the distinction between two kinds of resilience, cascading failure and modularity. For the general principles of ROC curves, see Chapter 4 of Volume 26, Probabilistic Judgement, Brier-Score Calibration and the Statistics of Collective Intelligence.

Claims corrected on checking

Several claims in the author’s original framework needed correction or qualification once they were checked. First, the grazing equation $\mathrm{d}x/\mathrm{d}t = rx(1-x/K) – c\,x^2/(x^2+a^2)$ is the standard form of this class of models; the graphical analysis of multiple stable states in grazing systems by growth and consumption curves, sigmoid ones among them, goes back to Noy-Meir’s 1975 paper, and May’s 1977 review extended it to grazing, insect pests and host–parasite systems. In the model the collapsed state is a low but non-zero level of vegetation; with the illustrative parameters, pressure must fall from about 2.60 to below about 1.79 for recovery, which is hysteresis, not irreversibility in the strict sense; strict irreversibility occurs only for some parameters of the lake phosphorus model. Second, the evidence for multiple stable states is good for lakes; whether “phase shifts” on coral reefs represent alternative stable states is seriously disputed (only about 4 % of 1,851 reefs were dominated by macroalgae). Third, variance and autocorrelation are meaningful only if they rise together: rising variance alone may simply mean that the noise has grown. Fourth, Boettiger and Hastings’s 2012 “prosecutor’s fallacy” paper in Proceedings of the Royal Society B is about selection bias and contains no ROC analysis; the ROC evaluation comes from their paper of the same year in the Journal of the Royal Society Interface, and the ROC program in this book is an illustrative simulation in that spirit. Fifth, the name “engineering resilience” comes from Holling’s 1996 essay; Pimm in 1984 spoke only of “resilience” (the speed of return). Sixth, “flattening fluctuations and draining redundancy shrink the basin of attraction” is an argument Holling and colleagues drew from case studies; this book presents it as a conclusion that holds under stated model conditions plus a generalisation from cases, not as a general law established by controlled tests. Seventh, Motter and Lai did not give a universal critical value $\alpha_c$; in their scale-free networks, attacking the highest-load node reduced the largest connected component by more than 20 % even at $\alpha = 1$; the $\alpha_c$ in this book (about 1.4) is an operational definition that applies only to one illustrative network. Buldyrev and colleagues’ $p_c = 2.4554/\langle k\rangle$ rests on assumptions such as random wiring and complete one-to-one dependence; when dependence is weakened, the collapse becomes continuous. Eighth, on the classical texts: both passages from the “Great Commentary” begin “The Master said”, that is, they are sayings of Confucius quoted in that commentary; “a dyke a thousand zhang long is breached through the hole of a mole cricket or an ant” comes from the “Illustrations of Laozi” in the Han Feizi, where the received text reads “a thousand zhang”, while “a dyke a thousand li long” is a later popular form; and the dates of the persons in the story of Bian Que and Duke Huan of Cai cannot be reconciled, so it is best read as a parable; when the Records of the Grand Historian tells a similar story, the ruler is Marquis Huan of Qi. Each chapter says so where the point arises, and the Editor’s Notes published with the book collect them.

How to read the boxes

As in other 5D Library books, different kinds of statement are marked separately.

About this edition

This is version 1.0.0, the first complete edition: a preface and eight chapters, in Chinese and English. The book contains 15 programs and 4 vector figures. The text, boxes and code were drafted with an AI writing tool from the author’s title, framework and notes; the author is responsible for reviewing and revising them and for checking every source. Corrections and missing references are welcome through the 5D Library website.