As a start, let us take the common metaphor that “we're all in the same boat” literally and represent the state of the Earth system with all its natural and socio-economic parts at each point in time by a single small boat floating or being rowed somewhere on a rather complex system of waters such as in Fig. .

The boat can only be on water, not on land, and will generally float along with the stream that represents the inherent dynamics of the Earth system over hundreds and thousands of years (the “default trajectory”), but it may also be rowed in more or less different directions depending on how strong the current of the stream is, and this possibility of rowing represents humankind's agency in deliberately influencing the Earth system's course to some extent by some or other form of what we will call “management” below. Let us assume that the main qualitative distinction with regard to where humanity wants their boat to be is represented by a division of the whole region into a desirable, “sunny” region on the left and an undesirable, “dark” region on the right, both containing several parts of the waters that may be connected in any imaginable ways, and with the natural water flow possibly drawing the boat back and forth between these two regions. The sunny region is meant to consist of all those possible states of the natural and socio-economic parts of the Earth system in which some generally agreed environmental and living standards are met, such as those defined by the human rights charter or the sustainable development goals (global goals) recently adopted by the United Nations. An alternative definition of the sunny region has been put forward in the planetary boundary framework , where states lying within the corridor of Earth system variability during the Holocene that human societies are adapted to are considered as desirable.

We will show in this article that in such a setting, no matter how the waters look exactly, the general situation is in a certain sense always equivalent to the situation depicted in Fig. . There will in general be a certain sunny water region where one does not need to row at all in order to stay in the sun forever but can simply lean back and let the boat float around inside that region. In the picture, this region is the top-left tranquil tarn, but in general this region may also consist of several disconnected parts which we will call the shelters to emphasize their desirable and safe nature. Indeed, we will argue below that these shelters may be the most natural candidates for being called a “safe and just operating space for humanity”, only that we may not yet be in them. In the Earth system, there may be several such shelters, one of which might correspond to resilient states of the world where humanity lives reconnected to the biosphere and no active intervention or constant large-scale management is needed.

Connected to the shelter(s), there will in general also be other parts of the sunny region where it would not be safe to just lean back since the flow would then draw the boat into the dark after some time, but from where the shelters can still be reached by some suitable rowing, as show to the left of the “danger” sign in the image. For their “almost-safe” character, we will call such regions glades. If the glade is for some reason more desirable or offers more flexibility in terms of where one may row, one may face a dilemma when in a glade, i.e. a qualitative decision problem, namely whether to prefer staying in the safety of the shelter or in the more desirable but unsafe glade.

The shelters may also be reached by rowing from some places within the dark region (e.g. to the right of the “danger” sign) or through such a dark region from some other sunny places (such as those above the “keep out” sign). Among these latter sunny places from where the shelters can be reached only through the dark, there will generally be some places where one may alternatively stay forever in the sun by continuous rowing instead of passing through the dark and leaning back eventually. Such special places as the one above the “keep out” sign will be called lakes here, and they are characterized by a moderate current towards a dark place that one can row against and by the decision dilemma that results from the question of whether one should indeed do so or rather row to a shelter through the dark.

All these regions together will be called the upstream region for reasons that should become clear soon. In any system's state space, the upstream consists of all states from which the shelters can be reached by management, and it is partitioned into one or several shelters, glades, dark upstream parts, lakes, and some remaining sunny upstream parts where it is not possible to stay in the sun forever. In Fig. , the upstream ends where the rapids left of the “keep out” sign begin since there the stream becomes so strong that it becomes impossible to row against it in order to eventually reach a shelter. Once the boat has left the upstream via such a rapid, there is no hope of leaning back eventually and staying in the sun, and for this reason the borders of the upstream may be called the “no-regrets planetary boundaries”, forming a middle level of a hierarchy of planetary boundaries we will suggest in Sect. .

Further down the stream there will typically be places where it is still possible to stay in the sun forever, only that one has to row over and over again to do so, such as in the slow-moving side branch below the “keep out” sign in the picture. Such regions, called backwaters here, are similar to lakes, only without the option of rowing to a shelter, so that the lake dilemma does not occur since the only chance one has is to row against the slow current to stay in the backwater. While the upstream was defined by being able to reach a shelter, the downstream is now defined as all places from where a backwater but not a shelter can be reached, including the backwaters, some dark parts such as the slow-moving dark part just right of the backwater in the picture, and maybe some remaining sunny downstream parts from where one may reach a backwater only through the dark. An example of a backwater could be a “machine world” where humanity can fully control nature to its very minute detail. While they can stay within the sunny region for infinite time through this management, there is no way of reaching a shelter anymore because the ecosystem has been changed irreversibly.

The waterfall in Fig.  indicates that besides the upstream and downstream regions, where it is possible to stay in the sun eventually, there will in general be further, less hopeful places the system may be in, from where one cannot avoid entering the dark over and over again. In some of those, one can at least make sure that one also spends some time in the sun over and over again, as depicted by the kayak in the picture. Since this is typically connected to some form of cyclic motion, we will call such regions eddies. In some eddies, failing to row correctly may push the boat into an even less desirable region, called an abyss, from where one can no longer avoid ending up in the dark forever eventually, as in the ring-shaped abyss shown inside the eddy in the figure. Finally, the dark region from where there is no escape, depicted in the centre of the abyss, will be called a trench.

This completes our main partitioning of the Earth system's or any other manageable system's state space into qualitatively different regions: upstream and downstream, defined by being able to reach shelters or backwaters; abysses, defined by not being able to avoid ending up in a trench; and eddies in between, defined by being at least able to switch between sun and dark forever. Figure  summarizes all these regions in the form of a decision tree, where one can identify the region the system is in by answering a small number of questions. That our partitioning is indeed complete and can be given a suitable and unambiguous mathematical form for all kinds of systems is shown in the next section.

While in Fig. , each of the introduced set of system states is just one topologically connected region, in general most of these sets are composed of several disjoint regions, so there may be several shelters, glades, lakes, etc. On a finer level, these may be analysed further by looking at which parts may be reached from which other parts, and this leads to a finer, hierarchical partition into ports, rapids, harbours, docks, etc. and to several new types of dilemmas, as shown in Fig. .

All of the five types of dilemmas listed in Table  can easily occur in the collective “management” or governance of the Earth system by humanity. A glade dilemma may occur if adaptation is seen as preferable to mitigation for welfare reasons but turns out to be a riskier option due to a higher uncertainty of the corresponding climate impacts. A lake dilemma can arise if a great transformation of the global energy system towards a carbon-free economy would temporarily lead to welfare losses in poorer countries. A port dilemma may come from the option of increasing welfare by extending industrial agriculture causing biodiversity loss (decreasing flexibility) due to the related large-scale land-use change. A harbour dilemma could occur in the future when colonization of other planets (increasing flexibility) becomes feasible but extremely costly. Finally, a dock dilemma arises whenever a very promising new technology with some unknown risks and side effects (such as genetically engineered food production) could be introduced on a planetary scale.

The main idea of the coarsest of our classifications of states is to first identify (i) a safe region where management is unnecessary, called the shelters S, and (ii) a less safe but larger manageable region M where one can permanently avoid the dark at least by management. Then we classify all states with regard to whether and how X+, S, and M can be stably reached from the current state by management. For each state, we ask the following questions. (iii) Can S be stably reached, and if so, can the dark be avoided on the way? (iv) If not, can M be stably reached? (v) If not, can we stably reach X+ over and over again, or at least once again? We will see that these criteria lead to a partition of state space into a “cascade” consisting of five main regions: upstream U, downstream D, eddies E, abysses Υ, and trenches Θ. Each of these will then be split up further into sets such as glades G, lakes L, and backwaters W by asking further qualitative questions. In choosing these figurative terms, we try to avoid a too technically sounding language and rather extend the useful and common metaphor of “flows” and “basins” in a natural way without trying to match their common-language meanings too accurately.

To acknowledge the fact that all real-world dynamics and management will be subject to at least infinitesimal noise and errors, we base the formal definition of these state space regions on certain notions of invariant open kernel, sustainability, and stable reachability, whose symbolic mathematical definitions and algebraic properties are detailed in Appendix .

The invariant open kernel of a set A ⊆ X, denoted Aι∘, is the largest open subset of A that contains the default trajectories of all its own points. The shelters are the invariant open kernel of the sunny region, S=X+ι∘. S contains all sunny states whose default trajectories stay in the sunny region X+ forever without any management even when infinitesimal (or small enough) perturbations occur. In other words, when inside S, one will “stably” stay in X+ by default.

We call an open set A sustainable (in the basic sense of the word, simply meaning that it can be sustained) iff it contains an admissible trajectory for each of its points. The sustainable kernel of a set A ⊆ X, denoted AS, is the largest sustainable open subset of A. We call the sustainable kernel of the sunny region the manageable region: M=X+S⊇S. In other words, when inside M, one can stably stay in X+ by management.

In Appendix , we introduce a suitable notion of stable reachability to overcome two problems with the classical notion of (plain) reachability known from control theory. For now, let us assume we know what we mean when saying that a state y or a set Y ⊆ X is stably reachable from some state x through some set A ⊆ X, denoted x ⇝A y or x ⇝A Y. Using this notion of stable reachability for the choice A = X (other choices of A will be used in the next section), we can now define the upstream U as the set of states from where the shelters S can be stably reached at all. Likewise, the downstream D consists of all states from which the manageable region M but not the shelters can be stably reached: U=⇝XS⊇S,D=⇝XM-⇝XS=⇝XM-U⊇M-U.

On the other, dark end of what we will call the main cascade, we first define the trenches Θ as that region in the dark from which one cannot stably reach the sunny region even once, Θ=X-⇝XX+ (this concept approximately corresponds to the “catastrophe domains” of ).

Now we turn to the region from where one cannot avoid ending up in the trenches. We define the abysses Υ as the closure of this region, minus the trenches: Υ=x∈X|∀μ∈Mx∃t⩾0:μ(t)∈Θ‾-Θ. The closure is taken since even an infinitesimally small perturbation from a point in this closure can make the trenches unavoidable.

Finally, the eddies E are the remainder of X, i.e. the part from where the manageable region cannot be stably reached but the trenches can be avoided: E=X-U-D-Υ-Θ=X-⇝XM∩(X-(Υ+Θ)). Thus, when in the eddies, even though one can reach the sunny part over and over again, one cannot stay there forever but has to visit the dark repeatedly.

A connected component of Θ, Υ, or E will be called an individual trench, abyss, or eddy, and the latter two typically have sunny and dark parts.

The system C = {U, D, E, Υ, Θ} is a partition of X which we call the main cascade because of the following mutual reachability restrictions: ¬(Θ⇝Υ),¬(Υ⇝E),¬(E⇝D),¬(D⇝U). In other words, one might at best be able to go in the “downstream” direction by default or by management, from upstream to downstream to the eddies to the abysses to the trenches, but not in the other, “upstream” direction (see also Fig. ).

Some of the states in the manageable region M may be in U = (⇝XS) but not in (⇝X+S). This motivates the definition of two subsets of M via the relation of sunny stable reachability, ⇝X+, namely (i) the glades G, from where the shelters can be stably reached through the sun, and (ii) the lakes L, from where the shelters can be stably reached only through the dark: G=⇝X+S-S,L=M∩U-⇝X+S=M∩U-S-G. Glades and lakes are two particularly interesting types of regions since in both one has a qualitative decision problem. The glade dilemma occurs if a glade is for some reason more desirable than its shelter, since then one has to decide whether to stay in the more desirable but unsafe glade or row to the less desirable but safe shelter. The lake dilemma exists in every lake: shall one stay in the sun by rowing over and over again, but risking to float into the dark if the paddle breaks, or shall one move into a shelter, accepting a temporary passage through the dark, to be able to recline in safety eventually? In other words, the lake dilemma is a choice between uninterrupted desirability and eventual safety. Below we will encounter more qualitative dilemmas of this and other types.

While {S, G, L} is a partition of M ∩ U, the downstream D may also contain a manageable part, the backwaters W. This is the region where one may stay in the sun forever by rowing over and over again, but where one may not stably reach the shelters at all, not even through the dark: W=M∩D=M-U. This completes the manageable partition M=S+G+L+W. Also, both U and D may contain points outside M, which we call the dark upstream/downstream, U-=U∩X-,D-=D∩X-, and the remaining sunny upstream/downstream, U(+)=U∩X+-M,D(+)=D∩X+-M, leading to the upstream and downstream partitions U=S+G+L+U(+)+U-,D=W+D(+)+D-. Finally, one can divide the eddies and abysses into sunny and dark parts: E±=E∩X±,Υ±=Υ∩X±.

All the sets introduced so far are summarized in Fig.  in the form of a decision tree that allows for a fast classification of individual states.

In addition to the glade and lake dilemmas introduced above, there exist at least three further types of qualitative decision problems, all related to the question of which parts or subregions of the above introduced regions may be stably reached from which other parts, and whether corresponding transition pathways exist that do not leave the shelters or at least the sunny region, or only through the dark. In order to study these questions, we introduce three additional, successively finer partitions derived from the reachability relations ⇝X (stable reachability) and ⇝X+ (stable reachability through the sun) that we used already above, and from the even more restrictive relation ⇝S (stable reachability through the shelters).

To summarize, we have now a hierarchy of ever-finer partitions of the system's state space at our hands. We began with the main cascade C = {U, D, E, Υ, Θ}, its refinement into the partition {S, G, L, U(+), U-, W, D(+), D-, E+, E-, Υ+, Υ-, Θ} (see Fig. ), and the further refinement by topological connectedness into individual shelters, glades, lakes, backwaters, eddies, abysses, and trenches. These partitions represent the qualitative differences in stable reachability of the shelters or the manageable set, thus allowing for a first classification of states with regard to the possibilities of sustainable management, and may reveal decision problems of the type of glade or lake dilemma which will occur in many of the examples below, where one has to choose between higher safety and higher desirability or flexibility or between uninterrupted desirability and eventual safety.

A different refinement of C into the ports-and-rapids network is still based on stable reachability alone but contains other details suitable for the identification and discussion of possible port dilemmas that involve a choice between higher desirability and higher flexibility. Inside the desirable region X+, this partition can be refined into the harbours-and-channels network suitable for the discussion of harbour dilemmas that involve a choice between uninterrupted desirability and eventually higher desirability or flexibility, and further into the docks-and-fairways network suitable for the discussion of dock dilemmas that involve a choice between uninterrupted safety and eventually higher desirability or flexibility (Table ).

These three networks may also be interpreted as a three-level “network of networks” with nodes representing state space regions of different quality and size. A network-theoretic analysis of it using methods such as the node-weighted measures of may especially be interesting in the context of varying system parameters and bifurcations such as those in Fig. , but this is beyond the scope of this article.

Our first example is from natural Earth system modelling and illustrates which of the above-introduced regions occur most often for systems that possess only a single, globally stable, and desirable attractor.

proposed a conceptual model of the global carbon cycle capturing its main features while keeping the model sufficiently low-dimensional to be able to discuss the planetary boundaries concept with it. We use their model for pre-industrial times, which has three dynamical variables cm, ct and ca = 1 - cm - ct representing the maritime, terrestrial, and atmospheric shares of the fixed global carbon stock. The dynamics are of the form c˙m=amca-βcm,c˙t=fca,ct-αct, where am and β are diffusion parameters, f is a function representing photosynthesis and respiration, and α governs the human offtake rate from the terrestrial carbon stock. See for details and parameter values.

Since the parameter α can be considered the natural human management option for this system, we assume the default flow has a value of α = α+ = 0.5, while management can reduce it by half to α = α- = 0.25, which results in the trajectories shown in Fig. . Both have a unique stable fixed point in the interior of the state space which is globally attractive for all states with ct > 0.

In order to roughly represent the planetary boundaries relating to climate change, biosphere integrity, and ocean acidification , we require a “sunny” state to have sufficiently low atmospheric carbon, at least a minimum value of terrestrial carbon, and not too large maritime carbon, leading to a dark region of the shape shown in Fig.  in grey. If, as shown, the unmanaged fixed point is sunny, one obtains a purely upstream situation with a shelter surrounding the fixed point, a glade, and a remaining sunny upstream U(+) as shown in the figure. For our (quite arbitrarily) chosen parameter values, a trajectory starting in the sunny upstream is likely to first cross the climate boundary and then the biosphere boundary before getting back into the sunny region, whereas it seems quite unlikely to cross the acidification boundary.

In this example, all non-upstream regions are empty, and so is the lake region; hence, no lake dilemma occurs. On the other hand, if one considers a higher ct to be preferable, we get an example of the glade dilemma since the managed fixed point in the less safe glade has higher ct than the unmanaged fixed point in the safer shelter. Note that this is neither a port, harbour, or dock dilemma since both points are in the same port and harbour and only the unmanaged one is in a dock.

If, instead, we had chosen the minimum value for ct to be larger than the unmanaged equilibrium value, the shelter would be empty and the whole situation would change from upstream-only to either a downstream-only or an abyss-and-trench situation. This type of topological bifurcation will be studied in Sect. . In the next example, we will see a lake dilemma instead of a glade dilemma.

The second example, from ecology, demonstrates how the lake dilemma may occur in a multistable system with a sunny and a dark attractor.

In this fictitious example, two plant types (1 and 2) compete for some fixed patch of land, modify the soil, and are harvested. Their growth follows logistic-type dynamics, with land cover proportions x1,2 ∈ [0, 1] following the equations x˙1=x1K1x1,2-x1-h1x1,x˙2=rx2(K2(x1,2)-x2)-h2x2. In this, r > 1 is a constant productivity quotient, h1,2 are the harvest rates, and the two dynamic capacities K1(x1,2) = x1(1 - x2) ⩽ 1 and K2(x1,2) = x2(1 - x1) ⩽ 1 represent the fact that each type modifies the soil quickly to its own benefit but to the other type's disadvantage (see Supplement 1 for a discussion of the model design based on , , , , , and .

For our illustration, we assume that, on the default trajectories, both harvest rates h1,2 equal some rather high value h+, leading to low equilibrium harvests. We assume management can repeatedly choose between this default and two types of alternative trajectories. Type 1 has a lower value for both harvest rates, h1,2 = h- < h+, representing management by restricting harvests politically in order to yield higher long-term harvests, but without aiming to change the plant mix, as depicted in Fig.  (left panel). Type 2 management option has harvest rates h2 = 0 and h1 = 2h+, representing management by temporarily protecting type 2 in order to change the plant mix to the higher productivity plant; we assume that this moratorium results in more intense harvesting of type 1, as depicted in Fig.  (right panel). We assume that both options exist simultaneously at all times (the separate plots of Fig.  are only for better discernibility of the trajectories). We set the desirable region to where x1 + x2 > ℓ for some ℓ > 0 in order to ensure some minimum harvests.

For the choice r = 2, h+ = 0.2, h- = 0.1, ℓ = 0.65 of the figure, the desirable high-productivity stable fixed point of the default dynamics at ≈ (0, 0.79) is in the sunny region and is thus contained in a shelter S. The latter is delimited by the default trajectory that meets the boundary to the undesired region tangentially. S can be stably reached from all states with x2 > 0, and hence the upstream is U = {(x1, x2)|x2 > 0}. The border of the glade G next to S can be found by backtracking the “widest” admissible trajectory that meets the boundary to the undesired region tangentially; this turns out to be a type 2 management trajectory as seen in Fig.  (right panel). This shows how the boundaries of regions may often be found by identifying tangential or otherwise significant points and backtracking the default and alternative trajectories leading to them.

The lower-productivity stable fixed point of the default dynamics (with h1,2 = h+) at ≈ (0.52, 0) is undesired for this choice of X+. From it one cannot only navigate to S but can also (and faster) get to the higher productivity stable fixed point of the first type of managed dynamics with h1,2 = h-, at ≈ (0, 0.79), and stay there as long as management holds. Hence the region around (0, 0.79) is part of the manageable region M. The exact boundary of this region (which soon turns out to be a lake, L) is the “widest” admissible trajectory that meets the boundary to the undesired region tangentially; in this case, this trajectory turns out to be a type 1 management trajectory as seen in Fig.  (left panel). To get from this type 1-dominated region to the type 2-dominated shelter S via the other management option of protecting type 2, one has to cross the undesired middle region in which both types coexist at a low level due to soil conditions that are suboptimal for both types. Hence the region around (0, 0.79) is a lake. The associated lake dilemma is similar to a glade dilemma in that staying in a lake is unsafe as in a glade, but it differs in the reason why one may want to stay there: while staying in a glade may be attractive simply because the glade may be more desirable than the shelter in some quantitative sense, staying in a lake may seem attractive since that avoids having to pass through the dark to reach safety.

This form of the lake dilemma can also occur in other multistable systems when one of the attractors is in the dark but sufficiently close to the sunny region so that constant management can sustain the system in a sunny place near that attractor, and when other management options may push the system towards another, sunny attractor after crossing the dark.

Note that, in this example, the lake dilemma falls together with a port dilemma since after leaving the lake for the shelter, one cannot return. If we choose a slightly larger sunny region by lowering ℓ to ℓ = 0.45, the unmanaged fixed point with y = 0 gets into X+ and the former lake around it now becomes a second shelter, which might be called a shelter–lake transition. But from this shelter the other, more desirable shelter can still only be reached through the dark. Since the two shelters correspond to two harbours in the reachability network, this means the former lake dilemma has been converted into a harbour dilemma.

The example also shows that the more management options exist, the less trivial it is to find the boundaries between regions even in two-dimensional systems. For higher dimensions, one will usually have to rely on specialized numerical algorithms such as the viability kernel algorithm of from viability theory.

Our third example concerns a purely socio-economic part of the Earth system that bears some similarity to the preceding example but features regions from both ends of the main cascade: upstream and abyss/trench, without having the intermediate regions of downstream and eddies.

Instead of plants, in this example a certain produced good (e.g. electric energy) comes in two types which are economically perfectly substitutable but whose production processes use two different technologies – one “dirty” and one “clean” (e.g. conventional and renewable energy). The production costs C1 and C2 are convex functions of production output per time yi and decrease over time via learning-by-doing dynamics that are similar to Wright's law : Ciyi=γiyi1+σi/1+σixiαi. In this, xi is cumulative past production (with x˙i = yi), γi are cost factors, σi > 0 are convexity parameters, and αi > 0 are learning exponents. We assume that demand D depends linearly on price, D(p) = D0 - δp, δ > 0; that demand equals production, D=y1+y2 (“market clearance”); and that price equals marginal costs, p = ∂C/∂yi = γiyiσi/xiαi, due to perfect competition among producers. One can then uniquely solve for the produced amounts yi, getting some formula yi = fi(x1, x2). This results in a two-dimensional dynamical system with state variables x1, x2 and equations x˙i=fix1,x2.

Let us put D0 = 1, δ = 1, σi ≡ 1/5, αi ≡ 1/2, and assume that the default dynamics have γi ≡ 1, so that the long-term default behaviour is p(t) → 0, D(t) → 1. If the dirty technology (1) is the traditional one, so that x1(0) > x2(0), we have x1(t) → ∞, x2(t) → x^2 < ∞, y1(t) → 1, and y2(t) → 0, i.e. usage of the clean technology (2) will die out. If instead x1(0) < x2(0), technology 1 will die out. Hence the system is bistable as in the plant example, but with attractors at infinity. To depict the diverging behaviour, we used the transformation zi = xi/(0.3 + xi) in Fig. .

The main dynamical difference to the plant example is, however, not the diverging behaviour, but has to do with the choice of management options. While in the plant example, the choice of management options led to an upstream-only situation in which the more desirable fixed point could be reached from everywhere, in this example we will get regions from which the desirable fixed point cannot be reached and which are thus non-upstream. We consider the management option of lowering γ2 to a value of, say, 1/2 by subsidising the clean technology to induce a technological change . This leads to the alternative dynamics depicted in Fig. , showing that for some initial states with x1>x2 one can now get x2(t) → ∞ and y1(t) → 0. The goal of keeping the usage of the dirty technology below some limit, y1 < ℓ < 1, corresponds to a desirable region in terms of x1, x2, whose border can be computed as x2 = x1(1/ℓ - 1 - 1/ℓ4/5x1)2/5 (see Fig. ). That goal is automatically fulfilled in the top-left shelter region, can also be sustained by management (subsidies) in the glade region below it, and can at least be reached eventually from the remaining sunny upstream U(+) below the glade and from the dark upstream U-, which is delimited by the management trajectory that meets the upper right corner.

But from below the latter trajectory, the shelter cannot be reached. In other words, when in U-, one has to act fast in order not to lose the option of reaching S. From the dark part denoted Θ, not even the sunny region is reached, and hence that region is a trench, while the sunny part to its left is the abyss leading to that trench. There are no intermediate regions (downstream or eddies) between upstream and abyss in this example.

Our fourth example models the coevolution (in the sense of joint time evolution) of a natural Earth system component coupled with a socio-economic Earth system component and shows how different parameters may qualitatively move the resulting state space topology through the whole main cascade, from an upstream-only situation via downstream-only and eddies-only to an abyss-and-trench situation.

The model was used in to explain the rise and fall of the native civilization on Rapa Nui (Easter Island) before western contact, but it may also be interpreted as a conceptual model of global population–vegetation interactions. It is derived from simple economic principles and leads to a modified Lotka–Volterra model with a finite resource. The human population x is preying on the island's forest stock y, which itself follows logistic growth dynamics: x˙=δx+ϕγxy,y˙=ry(1-y/κ)-γxy for some parameters γ, δ, κ, ϕ, and r representing growth and harvest rates and the stock's capacity.

We assume management will either reduce the default harvest rate γ0 to some smaller value γ1 < γ0 to avoid over-exploitation of the resource or increase it to a larger value γ1 > γ0 to avoid famine. Our choice of the sunny region relies on two principles. The absolute population should not drop below a threshold xmin and the relative decline in population under the default dynamics, -x˙/x, should not exceed a value of ℓ. Hence X+ = {x > xmin and y > ymin = max(0, -(ℓ + δ)/ϕγ0)}.

The resulting state space partition is depicted in Fig.  for ϕ = 4, r = 0.04 and different choices of γ0, γ1, δ, κ, xmin, ymin. One either gets an upstream-only situation, a downstream-only one, an eddy-only one, or an abyss-and-trench situation, depending on whether the unmanaged and managed fixed points belong to the desired or undesired region. In Appendix , these kinds of transitions are more formally interpreted as bifurcations.

An interesting case occurs when the whole state space is a single eddy as in Fig.  (bottom-left panel): one can then repeatedly visit the sunny region by suitably switching between a low default harvest rate and a managed higher harvest rate, but one cannot avoid getting back into the undesired region of a low or fast declining population. An “optimal” management strategy would then lead to slowly but strongly oscillating behaviour.

While in the above examples typically only some of the possible regions were non-empty for each parameter combination, the following example from classical mechanics displays a rich diversity of state space regions that coexist at a single choice of parameter values. Despite extremely simple dynamics, it features both a glade and a lake dilemma, an eddy, and a trench at the same time.

In the model, people sit in a fun ride resembling a gravity pendulum with angle θ and angular velocity ω and default dynamics given by θ˙=ω,ω˙=-sin⁡θ. An optional additional clockwise acceleration of the pendulum of magnitude a > 0 (“management”) leads to alternative admissible trajectories on which for some time interval(s) one has ω˙ = -sin⁡θ - a. The sunny region is where |ω| < ℓ, for some ℓ > 0 representing a safety speed limit above which people might get sick.

The unique shelter S is delimited by the default trajectory leading through the points θ = 2kπ, ω = ±ℓ that surrounds the stable resting state of θ = ω = 0 (see Fig. ). If a state lies on a default trajectory that has ω > 0 (anticlockwise pendulum motion) at least some of the time, then there is an admissible trajectory from it leading into the shelter, generated by the management strategy of “braking” whenever ω > 0. Hence the upstream U equals the region strictly above the default trajectory with ω < 0 that connects the unstable saddle point at θ = (2k + 1)π, ω = 0 (pendulum balancing upright) with itself.

Just left of the shelter is the unique glade G. Depending on the parameter values, the stable fixed point of the managed dynamics (hanging pendulum inclined by constant acceleration) may either belong to the shelter or to the glade. In the latter case (Fig. ), we have a glade dilemma since the inclined position is preferred to the resting position by the riders but is unsafe since if the engine breaks, people will get sick.

An even more exciting position is close to the upright balancing saddle point, at θ slightly larger than (2k + 1)π and ω ≪ 1, where there is an admissible trajectory that stays close to there (by braking repeatedly for short intervals while staying almost upright), so that this point is in the manageable region M. This is a typical example of how a region close to a saddle point of the default dynamics may become manageable due to an alternative feasible trajectory that has a slightly shifted saddle point, so that in the diamond-shaped region between the two saddle points, one can concatenate unmanaged and managed trajectories into periodic orbits.

However, for choices such as a = 0.6 and ℓ = 0.5 (Fig. ), there is no admissible trajectory leading from the exciting region with θ ≈ (2 k + 1)π, ω ≈ 0 into the shelter without entering the region with |ω| > ℓ. In that case the diamond-shaped region is a lake and we have a lake dilemma.

Finally, the region below and including the default trajectory that touches the line ω = -ℓ from below is the trenches since one cannot brake in that direction, and the region between the trench and the upstream is the eddies. Downstream and abysses are empty in this example.

This final example system is designed to illustrate the relationship of reachability and bifurcations of a dynamical system that can be managed through a parameter and shows bifurcations of the type typically associated with tipping elements of the Earth system .

It has a two-dimensional state space X = {(r, y)}, where the “fast” variable y ∈ R has default dynamics y˙=h(y|r)=-4+r23y3+2r2-14+r2y+er-10, which cannot be managed directly, and r ∈ R is a “slow” variable with (approximately) no default dynamics (r˙ = 0) which, however, can be changed by management up to a velocity at most 100 and with arbitrarily large acceleration, leading to admissible trajectories with r˙ ∈ [-100, 100] and y˙ = h(y|r). We assume that values of y ⩽ -1/3 are undesirable.

If r is instead interpreted as a parameter of the one-dimensional system y˙ = h(y|r), the set X can be interpreted as its bifurcation space in which one can plot a bifurcation diagram consisting of the loci of stable (solid lines) and unstable (dotted lines) fixed points, as shown in Fig. . As one can see, there are three saddle-node bifurcations at r1 ≈ -2.2, r2 ≈ 1.735, and r3 ≈ 4.9 with monostable parameter regimes r1 < r < r2 and r > r3, and bistable parameter regimes r < r1 and r2 < r < r3. Individual and paired saddle-node bifurcations (which often result from fold bifurcations) occur frequently in bistable Earth system components such as the hysteretic thermohaline circulation , monsoonal soil–vegetation feedbacks , or other tipping elements . Hysteresis also occurs on other spatial and temporal scales, e.g. in local hydrology and in long-term glacial climate dynamics .

The main part of the resulting network of ports and rapids of our example system is depicted in Fig. . On its coarsest level, there are two ports, each containing one of the two connected loci of stable/unstable fixed points, and a rapid in between through which one can pass from the left to the right port but not back. If the right port seems more attractive, e.g. because it allows a higher value of y, we have a port dilemma since by leaving the left port for the right one, we lose flexibility in terms of reachable regions.

The right port contains two harbours, similarly connected by a narrow “internal” channel, as well as another “exit” channel leading from the right harbour to the dark region. Note that on the leftward-pointing dashed management trajectory in the middle of the bifurcation diagram, there is a leftmost point from where one can still “turn around” and reach (if only unstably) the right part without entering the dark region; this point is a corner of the right harbour (but not belonging to it, for stability reasons), and below it is a channel leading to another harbour in the bottom left. Again, if the right harbour seems more attractive, we have a dilemma, this time a harbour dilemma, since in order to reach the right harbour from the left one, we have to pass through the dark.

Finally, the right harbour contains two docks again connected by a fairway, plus some more fairways. Again, we get a dilemma if the top-right dock is more attractive than the top-left one: the dock dilemma is that, in order to reach the top-right dock from the top-left one, one has to pass through the unsafe middle region and risk ending up in the dark if management breaks down.