Box 1. Critical transitions in the fold catastrophe model

From the following article:

Early-warning signals for critical transitions

Marten Scheffer, Jordi Bascompte, William A. Brock, Victor Brovkin, Stephen R. Carpenter, Vasilis Dakos, Hermann Held, Egbert H. van Nes, Max Rietkerk & George Sugihara

Nature 461, 53-59(3 September 2009)

doi:10.1038/nature08227

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The equilibrium state of a system can respond in different ways to changes in conditions such as exploitation pressure or temperature rise (Box 1 Figure a, b, c). If the equilibrium curve is folded backwards (Box 1 Figure c, d), three equilibria can exist for a given condition. The grey dotted arrows in the plots indicate the direction in which the system moves if it is not in equilibrium (that is, not on the curve). It can be seen from these arrows that all curves represent stable equilibria, except for the dashed middle section in Box 1 Figure c, d. If the system is driven slightly away from this part of the curve, it will move further away instead of returning. Hence, equilibria on this part of the curve are unstable and represent the border between the basins of attraction of the two alternative stable states on the upper and lower branches. If the system is very close to a fold bifurcation point (for example point F1 or point F2), a tiny change in the condition may cause a large shift in the lower branch (Box 1 Figure c). Also, close to such a bifurcation a small perturbation can drive the system across the boundary between the attraction basins (Box 1 Figure d). Thus, those bifurcation points are tipping points at which a tiny perturbation can produce a large transition. Small perturbations can also cause large changes in the absence of true bifurcations, provided that the system is very sensitive in a certain range of conditions (Box 1 Figure b). Finally, a shift in system state may simply be caused by a sudden large external force (Box 1 Figure a). Early-warning signals tend to arise as systems approach a bifurcation point such as in Box 1 Figure c, d, and also if systems approach a non-catastrophic threshold such as the one shown in Box 1 Figure b.

Early-warning signals for critical transitions 

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