Emergence of cascading dynamics in interacting tipping elements of ecology and climate — TEI-tagged view

Ann Kristin KlosepersName, Volker KarlepersName, Ricarda WinkelmannpersName, Jonathan F. DongespersName (2020date). “Emergence of cascading dynamics in interacting tipping elements of ecology and climate.” Royal Society Open Sciencetitle·j.

Object of study — the qualitative long-term behaviour of interacting cusp-like tipping elementsterm·dom in ecology and climate

Research question — how interactions between tipping elementsterm·dom shape their tipping behaviour and when tipping cascadesterm·dom emerge

Contribution — tipping rules (facilitated, impeded, back-tipping) and stability mapsterm·mth for two and three coupled tipping elementsterm·dom

Full text — tagged

Emergence of cascading dynamics in interacting tipping elementsterm·dom of ecology and climate Abstract In ecology, climate and other fields, (sub)systems have been identified that can transition into a qualitatively different state when a critical threshold or tipping pointterm·dom in a driving process is crossed. An understanding of those tipping elementsterm·dom is of great interest given the increasing influence of humans on the biophysical Earth system. Complex interactions exist between tipping elementsterm·dom, e.g. physical mechanisms connect subsystems of the climate system. Based on earlier work on such coupled nonlinear systems, we systematically assessed the qualitative long-term behaviour of interacting tipping elementsterm·dom. We developed an understanding of the consequences of interactions on the tipping behaviour allowing for tipping cascadesterm·dom to emerge under certain conditions. The (narrative) application of these qualitative results to real-world examples of interacting tipping elementsterm·dom indicates that tipping cascadesterm·dom with profound consequences may occur: the interacting Greenland ice sheetterm·dom and thermohaline ocean circulation might tip before the tipping pointsterm·dom of the isolated subsystems are crossed. The eutrophication of the first lake in a lake chainterm·key might propagate through the following lakes without a crossing of their individual critical nutrient input levels. The possibility of emerging cascading tipping dynamics calls for the development of a unified theory of interacting tipping elementsterm·dom and the quantitative analysis of interacting real-world tipping elementsterm·dom.

Keywords: tipping pointterm·dom, critical threshold, hysteresis, tipping cascadeterm·dom, Earth system, eutrophication Introduction Many natural systems exhibit nonlinear dynamics and can undergo a transition into a qualitatively different state when a critical threshold is crossed. Those systems are called tipping elementsterm·dom and the corresponding threshold in terms of a critical parameter is the tipping pointterm·dom of the system. A precise mathematical definition is given in [ 1 ]. Examples for tipping elementsterm·dom can be found in ecology as a specific type of regime shifts [ 2 , 3 ] such as the transition of a shallow lake from a clear to a turbid state [ 4 – 9 ]. Furthermore, subsystems of the Earth system [ 1 , 10 , 11 ] such as the thermohaline circulation [ 12 – 18 ] or the Greenland ice sheetterm·dom [ 19 , 20 ] have been identified as tipping elementsterm·dom.

The term tipping pointterm·dom among other roots originated from describing the changing prevalence of ethnically diverse population in a US community [ 21 – 25 ] and has been applied to natural systems more recently. However, the idea that systems may show such nonlinear behaviour has already been developed within the frameworks of dynamical systems and catastrophe theory [ 26 – 30 ]. The latter theory received increasing attention and has been applied to several real-world systems in the period after its introduction [ 31 ]. Its extensive use has been criticized [ 32 – 35 ] so that it became a mathematical theory without much recent influence [ 36 ]. Mostly independently of the results given by catastrophe theory, critical transitions, tipping pointsterm·dom and regime shifts have been analysed in ecology [ 2 , 9 , 37 – 39 ] using the concepts of multistability and resilience [ 40 , 41 ]. Some first attempts to define a climatic tipping elementterm·dom relating to abrupt climate shifts can be found in [ 42 – 44 ].

Different types of tipping pointsterm·dom are discussed in the literature [ 9 , 23 , 37 – 39 , 45 ]. First, a qualitative change of the system’s state when a continuously changing control parameter crosses a threshold is called bifurcation-induced tippingterm·dom [ 23 , 46 – 48 ]. Noise can induce a transition into an alternative stable state without a change of the system’s control parameter [ 23 , 38 , 49 ]. Furthermore, rate-induced tippingterm·dom describes the shift to a qualitatively different state when the rate of change of a control parameter crosses a critical threshold [ 23 , 45 , 49 , 50 ].

It is known that bifurcation-induced tippingterm·dom, even though often mentioned, is not the only possible type of tipping [ 24 , 45 , 49 , 51 ]. Nevertheless, the response of many natural systems to a control parameter can be described in terms of a double fold bifurcation [ 48 , 52 , 53 ].

Real-world tipping elementsterm·dom are not independent from each other [ 52 ], but there may exist complex interactions between them. Potential interactions through various physical mechanisms were revealed for tipping elementsterm·dom in the climate system [ 54 ]. As an example, meltwater influx into the North Atlantic as a result of a tipping of the Greenland ice sheetterm·dom could weaken the Atlantic meridional overturning circulationterm·dom [ 55 ]. Lake chainsterm·key or rivers can be seen as an ecological example for coupled tipping elementsterm·dom. Each lake or river section in the chain can undergo a transition from a clear to a turbid state in response to nutrient input [ 5 , 6 , 8 ]. The single lakes can in reality be connected through small rivers or streams and can therefore not be considered independently [ 56 – 58 ].

The tipping probability of a certain tipping elementterm·dom might be influenced by the behaviour of other interacting tipping elementsterm·dom [ 54 , 59 ]. As a consequence, crossing of a critical threshold of a first tipping elementterm·dom could trigger, as a domino effect, a critical transition in a coupled tipping elementterm·dom or even tipping cascadesterm·dom [ 37 , 59 – 64 ]. In the following, we use the term tipping cascadeterm·dom for a critical transition triggered by a preceding tipping event of an influencing system. The heterogeneity of the subsystems as well as the coupling strengthterm·key may be important factors that influence the overall system behaviour [ 65 ] and should be considered in the analysis of coupled tipping elementsterm·dom. In the case of interacting climate tipping elementsterm·dom, a tipping cascadeterm·dom may impose a considerable risk on human societies [ 66 ].

Different attempts to analyse the influence of coupling between different tipping elementsterm·dom on their tipping behaviour have been followed. The development of critical transitions in lake chainsterm·key was studied using established models of lake eutrophication [ 58 , 67 , 68 ]. In analogy to wave propagation in discrete media [ 69 – 72 ], the spread of local disturbances in spatially extended, bistable ecosystems was analysed for explicit ecological examples [ 73 ] and more theoretically [ 74 ]. In addition, cascades may occur on networks [ 75 – 78 ] and networks of networks [ 79 – 83 ]. [ 52 , 84 , 85 ] analysed the system behaviour of special cases of coupled cusp catastrophesterm·key. The possible appearance of tipping cascadesterm·dom in coupled bifurcational systems and, in particular, in the climate system was supported by results from coupling conceptual models of the Atlantic meridional overturning circulationterm·dom and El Niño-Southern Oscillation [ 86 ].

Consequences of interactions between tipping elementsterm·dom, their nonlinear dynamics as well as the possible development of tipping cascadesterm·dom in systems of interacting tipping elementsterm·dom have not been assessed systematically so far. Here, we make an advance in the theory of interacting tipping elementsterm·dom focusing on bifurcation-induced tippingterm·dom in the form of cusp catastrophesterm·key. The special cases presented in [ 52 , 85 ] were extended by looking at uni- and bi-directional coupling and tipping chains consisting of two and three elements. We explored the tipping behaviour of the interacting system extensively under the influence of different coupling types (positive and negative interactions) and the coupling strengthterm·key and particularly focused on identifying conditions that favour tipping cascadesterm·dom. In addition, we applied our theoretical results to real-world systems to reveal mathematically possible tipping cascadesterm·dom in ecological systems such as lake chainsterm·key and in the climate system.

Model We use a conceptual model of tipping elementsterm·dom in order to investigate the qualitative long-term behaviour of coupled subsystems each of which exhibits critical transitions. In particular, we consider a continuous dynamical system x i ˙ ( t ) = f i ( x 1 , … , x n ) in n -dimensions, where each component x i ( t ) ∈ R corresponds to a generic tipping elementterm·dom X i .

The dynamics of the tipping elementsterm·dom is modelled with the topological normal form of the cusp bifurcation [ 52 , 85 ], i.e. the most generic polynomial system exhibiting this type of tipping behaviour ( figure 1 ). The long-term behaviour of many real-world systems in terms of the system’s state, such as the strength of the thermohaline circulation [ 12 , 15 , 17 , 18 ], the ice thickness of the Greenland ice sheetterm·dom [ 19 ] and the algae density in shallow lakes [ 6 , 8 ], is qualitatively represented by a slice of the cusp catastropheterm·key, a double fold bifurcation, showing bistability, hysteresis properties and transitions to a qualitatively different state when a critical threshold is crossed [ 87 ]. In contrast to other possible bifurcations such as the transcritical, pitchfork or Hopf bifurcations, the double fold bifurcation as a ‘dangerous’ bifurcation [ 46 ] captures the catastrophic nature of tipping which is of major interest here. A tipping elementterm·dom X i is then represented by where a i , b i > 0 and f i 0 corresponds to the uncoupled case.

Schematic coupled tipping elementsterm·dom. Two subsystems X i and X j are coupled via the coupling functions C i and C j with coupling strengthsterm·var d ji and d ij . The dynamics of both subsystems is given by the normal form of the cusp catastophe. Shown are the equilibria x i ∗ , x j ∗ depending on the tipping parameter c i , c j .

The parameter a i corresponds to the distance between the upper and lower layers of stable equilibria of the cusp and the parameter b i controls the strength of the nonlinearity in the system. Both parameters are fixed throughout the analysis and chosen to assure bistability of the subsystems for a certain range of the parameter c i allowing for tipping behaviour. The control parameter c i is then associated with a tipping parameter the size of which determines whether the system is in this bistable range or not. By leaving the bistability range, a critical transition from one stable state to another may arise ( figure 1 ): For − c i crit < c i < c i crit , equation ( 2.1 ) has one negative stable equilibrium x i − ∗ < 0 and a positive stable equilibrium x i + ∗ > 0 as alternative stable states. We call x i − ∗ < 0 and x i + ∗ > 0 the normal and the alternative state, respectively. Increasing the control parameter c i such that the threshold c i crit is crossed, the normal state x i − ∗ disappears and only the alternative state x i + ∗ exists. If the system resided in the normal state x i − ∗ , it transitions to the alternative state x i + ∗ for c i > c i crit . Analogously, for c i < − c i crit , only the normal state x i − ∗ exists and for lowering the control parameter c i below − c i crit the system falls from the disappearing alternative state x i + ∗ to the normal state x i − ∗ ( figure 1 ). The transition of the uncoupled tipping elementsterm·dom through a changing control parameter c i at the critical manifold [ 88 ] given by the roots of the polynomial can be quantified: depending on the sign of the discriminant D i 0 = ( b i c i / 2 ) 2 − b i ( a i / 3 ) 3 there are either one ( D i 0 > 0 ) or two ( D i 0 ≤ 0 ) stable equilibria determined by f i 0 ( x i ∗ ) = 0 .

For given a i and b i and setting D i 0 = 0 , the critical value for the control parameter c i crit ( a i , b i ) = ± 2 1 b i ( a i 3 ) 3 can be calculated, where the transition into a regime with only one equilibrium takes place. We call c i crit ( a i , b i ) the intrinsic tipping pointsterm·dom for an uncoupled tipping elementterm·dom as given by equation ( 2.1 ).

In the following, we couple the subsystems with each other using a coupling function C i ∈ R ( figure 1 ). Subsystem X i then becomes with a i , b i > 0.

For simplicity, we choose a linear coupling [ 52 , 85 ]. The linear coupling function for subsystem X i then reads as where a coupling d ji ≠ 0 indicates an influence of another subsystem X j to subsystem X i . Even though equations ( 2.2 ) and ( 2.3 ) provide the simplest equations to describe the threshold behaviour of n coupled tipping elementsterm·dom, they can be used for understanding the qualitative features of all systems with the same critical behaviour. Using the concept of topological equivalence [ 88 ], the critical behaviour of a class of more complicated real-world systems can be mapped to the system above. Table 1 provides an overview of special cases of coupling between interacting cusp catastrophesterm·key investigated in the literature [ 52 , 84 , 85 ]. In addition, more theoretical analyses on bifurcations of coupled cell systems (with symmetry properties) have been conducted (e.g. [ 89 ]).

Overview of linearly coupled tipping elementsterm·dom studied in the literature.

For n = 2, the corresponding equations read with a i , b i > 0. With d 21 = 0 and d 12 ≠ 0, we recover a generic master–slave configuration. The stable equilibria can be determined analogously to the uncoupled case with f i ( x 1 ∗ , x 2 ∗ , … , x n ∗ ) = 0 ∀ i . The discriminant for the second tipping elementterm·dom X 2 becomes D 2 = ( b 2 ( c 2 + d 12 x 1 *)/2) 2 − b 2 ( a 2 /3) 3 .

Note that D 2 is a function of the control parameter c 2 , the coupling strengthterm·key d 2 and the equilibrium x 1 *. The number of stable equilibria of subsystem X 2 depends on the sign of the discriminant. For D 2 ≤ 0, we find two stable equilibria and for D 2 > 0, we find one stable equilibrium. The threshold of the control parameter c 2 at which the number of solutions changes is obtained by solving D 2 = 0 and is given by where as the effective tipping pointterm·dom of the coupled tipping elementterm·dom X 2 .

In the following section, we will elaborate on how one can infer the qualitative behaviour of the coupled system using this expression.

Results Different types of tipping behaviour of a coupled system can be derived for the governing system of equations ( 2.4 ). For simplicity, let us consider the case a i = 1, b i = 1 (arbitrary b i can be achieved by rescaling x i ) for i = 1, 2 here and thereafter and d 21 = 0, i.e. unidirectional coupling. Subsystem X 2 leaves the bistable range for following expression ( 2.5 ) for the effective tipping pointterm·dom of the coupled tipping elementterm·dom X 2 in the previous § 2 , possibly giving rise to a critical transition to the alternative state x 2 + ∗ > 0 .

Based on this simple system, rules on the spread of tipping processes in the considered system of coupled tipping elementsterm·dom are formulated in the following. These tipping rules depend on the type of coupling, i.e. whether the subsystems are positively or negatively coupled, as well as on the relation between the control parameters c 1 (determining the possible stable states of subsystem X 1 ) and c 2 and the absolute value of the coupling strengthterm·key d 12 .

Let d 12 > 0, i.e. subsystem X 2 is positively coupled to subsystem X 1 . Then: — Facilitated tippingterm·dom ( figure 2 , upper panel): Assume that subsystem X 1 is in its alternative state x 1 + ∗ . Note that this assumption can be fulfilled if either subsystem X 1 transitions to the alternative state by crossing its intrinsic tipping pointterm·dom c 1 crit with an increase of the control parameter c 1 or if subsystem X 1 simply occupies the alternative state (which is in general possible for c 1 > − c 1 crit ). Then subsystem X 2 is pushed towards its tipping pointterm·dom in our model and can undergo a critical transition to its alternative state x 2 + ∗ for c 2 ≥ c 2 crit − d 12 x 1 ∗ . The effective tipping pointterm·dom of subsystem X 2 is lower than its intrinsic tipping pointterm·dom c 2 crit . The higher the coupling strengthterm·key, the lower the necessary critical value of the control parameter c 2 for which subsystem X 2 can tip.

— Impeded tippingterm·dom ( figure 2 , lower panel): If subsystem X 1 is in its normal state x 1 − ∗ , subsystem X 2 is pulled away from its tipping pointterm·dom in our model and can undergo a critical transition for c 2 ≥ c 2 crit + d 12 | x 1 ∗ | . The effective tipping pointterm·dom of subsystem X 2 is higher than its intrinsic tipping pointterm·dom c 2 crit . The higher the coupling strengthterm·key, the higher the necessary critical value of the control parameter c 2 for which subsystem X 2 can tip.

— Back-tipping : Assume that subsystem X 1 is in its normal state x 1 − ∗ . If subsystem X 2 occupies the alternative state x 2 + ∗ , subsystem X 2 can tip back to the normal state x 2 − ∗ for c 2 < − c 2 crit + d 12 | x 1 ∗ | ( figure 3 , upper panel). This behaviour especially occurs for a high coupling strengthterm·key d 12 and small values of the control parameter c 2 . However, subsystem X 2 is staying in the alternative state x 2 + ∗ if − c 2 crit + d 12 | x 1 ∗ | < c 2 . Here, subsystem X 2 is pushed into the bistable range of the system ( figure 3 , lower panel). This behaviour especially occurs for a high coupling strengthterm·key d 12 and high values of the control parameter c 2 .

Schematic of the tipping rules of facilitated (upper panel) and impeded tippingterm·dom (lower panel) for d 12 > 0. The blue dot in the left bifurcation diagram represents a possible state of the master system X 1 . The master system X 1 influences the slave system X 2 via a linear coupling with a coupling strengthterm·key d 12 > 0 and results in the shift of the uncoupled slave system’s state (indicated by a blue dot in the right bifurcation diagram) along the blue line.

Schematic of the tipping rule of back-tipping for d 12 > 0. The blue dot in the left bifurcation diagram represents a possible state of the master system X 1 . The master system X 1 influences the slave system X 2 via a linear coupling with a coupling strengthterm·key d 12 > 0 and results in the shift of the uncoupled slave system’s state (indicated by a blue dot in the right bifurcation diagram) along the blue line.

Let d 12 < 0, i.e. subsystem X 2 is negatively coupled to subsystem X 1 . Then: — Impeded tippingterm·dom : Assume that subsystem X 1 is in its alternative state x 1 + ∗ . Note that this assumption can be fulfilled if either subsystem X 1 transitions to the alternative state by crossing its intrinsic tipping pointterm·dom c 1 crit with an increase of the control parameter c 1 or if subsystem X 1 simply occupies the alternative state (which is in general possible for c 1 > − c 1 crit ). Subsystem X 2 is pulled away from its tipping pointterm·dom in our model and can undergo a critical transition for c 2 ≥ c 2 crit + | d 12 ∥ x 1 ∗ | . The effective tipping pointterm·dom of subsystem X 2 is higher than its intrinsic tipping pointterm·dom c 2 crit . The higher the coupling strengthterm·key, the higher the necessary critical value of the control parameter c 2 for which subsystem X 2 can tip.

— Facilitated tippingterm·dom : If subsystem X 1 is in its normal state x 1 − ∗ , subsystem X 2 is pushed towards its tipping pointterm·dom in our model and can undergo a critical transition to its alternative state x 2 + ∗ for c 2 ≥ c 2 crit − | d 12 ∥ x 1 ∗ | . The effective tipping pointterm·dom of subsystem X 2 is lower than its intrinsic tipping pointterm·dom c 2 crit . The higher the coupling strengthterm·key, the lower the necessary critical value of the control parameter c 2 for which subsystem X 2 can tip.

— Back-tipping : Assume that subsystem X 1 is in its normal state x 1 − ∗ . If subsystem X 2 occupies the alternative state x 2 + ∗ , subsystem X 2 can tip back to the normal state x 2 − ∗ if c 2 < − c 2 crit − | d 12 ∥ x 1 ∗ | . This behaviour especially occurs for a high coupling strengthterm·key d 12 and a low value of the control parameter c 2 . However, subsystem X 2 stays in the alternative state if c 2 ≥ − c 2 crit − | d 12 ∥ x 1 ∗ | . Here, subsystem X 2 is pulled to the bistable area of the system. This behaviour especially occurs for a high coupling strengthterm·key | d 12 | and high values of the control parameter c 2 .

These tipping rules based on the analytic solution of two unidirectionally coupled tipping elementsterm·dom give an impression of the interplay between certain system parameters and their influence on the tipping process. Going beyond this, using numerical calculations of fixed points and their stability (via the eigenvalues of the system’s Jacobian at the respective fixed point), the overall qualitative long-term behaviour of up to three interacting tipping elementsterm·dom with uni- and bi-directional coupling of varying sign has been assessed.

A stability mapterm·mth displays the number of stable equilibria of the system under consideration in the control parameter space for fixed coupling strengthsterm·var. Multiple stability mapsterm·mth have been calculated for various combinations of the coupling strengthsterm·var. The stability mapsterm·mth have been arranged in the form of a matrix, where one matrix element corresponds to one stability mapterm·mth. For illustrative purposes, we refer to the example given in figure 4 for two interacting tipping elementsterm·dom.

Number of stable fixed points of the system consisting of two bidirectionally coupled tipping elementsterm·dom depending on the control parameters c 1 and c 2 and the coupling strengthsterm·var d 21 ≤ 0 and d 12 ≥ 0 in a matrix of stability mapsterm·mth. A stability mapterm·mth shows the number of stable fixed points in the ( c 1 , c 2 )–space for a specific coupling strengthterm·key, where a certain number of stable fixed points is associated with a specific colour. Note that different areas in the control parameter space with the same colour have the same number of stable fixed point but they do not necessarily have the same phase portrait. The dashed lines represent the intrinsic tipping pointterm·dom of the respective subsystem. The position of a stability mapterm·mth in the matrix is determined by the coupling strengthterm·key. In the blue–grey region for high coupling strengthsterm·var with opposite sign but same magnitude indicating the absence of stable equilibria a stable limit cycle (Kadyrov oscillations in [ 85 ]) can be observed.

The system loses or gains stable fixed points through the variation of one or various control parameters c i for fixed coupling strengthsterm·var, which is associated with switches between the areas of different number of stable equilibria in the control parameter space by crossing the boundaries between the coloured areas. In addition, the phase space portrait may change.

Depending on the changes in the phase space (in terms of the stable fixed points and the flow) and the occupied state of the system, different types of system behaviour emerge. Combining the stability mapsterm·mth with phase space portraits (example given in figure 5 for illustrative purposes), the different areas in the stability mapsterm·mth can be characterized in terms of the emerging system behaviour, and possible critical transitions can be identified. Depending on which state the system was in, critical transitions can occur or not. For example, if the system resided in an equilibrium which lost stability and disappeared through the variation of one (or multiple) control parameters, the flow in the phase space suggests the state to which the system may transition.

Number of stable fixed points and phase space portraits in a master–slave system with a low positive coupling strengthterm·key d 12 = 0.2 > 0 depending on the control parameters c 1 and c 2 . The dashed lines represent the intrinsic tipping pointterm·dom of the respective subsystem. The phase space portraits allow to derive the possible critical transitions in the master–slave system. Within the phase space portraits stable fixed points are shown in orange, while unstable fixed points are shown in red. The background colour indicates the normalized speed v = x ˙ 1 2 + x ˙ 2 2 / v max going from close to zero (purple) to fast (yellow–green).

In the following, results for selected examples of coupled tipping elementsterm·dom are shown. First, a simple master–slave system with a unidirectional coupling is presented in example 3.1. In example 3.2, the previous system is extended by an additional negative coupling resulting in a bidirectionally coupled system of two tipping elementsterm·dom. Finally, the propagation in a unidirectionally coupled system consisting of three tipping elementsterm·dom is described in example 3.3. We only analyse the system behaviour for c i ≥ 0.

Master–slave system for d 12 > 0, e.g. a pair of lakes The behaviour of a master–slave system ( n = 2) given by equations ( 2.2 ) and ( 2.3 ) with i = 1, 2 and positive coupling d 12 > 0 and d 21 = 0, which was used for the derivation of the tipping rules, is described in more detail in the following. This type of coupling can be seen as an example for a pair of interacting lakes. Each lake may undergo a transition from a clear to a turbid state when some critical magnitude of the nutrient input as the tipping parameter is exceeded [ 6 , 8 ]. It is assumed that the lakes are connected [ 57 , 58 , 67 ] through a unidirectional water stream. Critical transitions can be derived using the numerically calculated phase portrait in combination with the stability mapterm·mth for a fixed coupling strengthterm·key d 12 > 0 ( figure 5 ). Note that figure 5 is a zoom into figure 4 at the lower right. The system has four stable equilibria for small values of the control parameters c 1 and c 2 which are separated by four saddles and an unstable node in the centre of the phase space.

With increasing control parameter c 1 , a critical transition of subsystem X 1 occurs in our model when its intrinsic tipping pointterm·dom c 1 crit is crossed ( figure 5 , moving from lower left to lower right along the green arrow), given that the system occupied one of the stable equilibria which lose stability for c 1 > c 1 crit .

With increasing control parameter c 2 , a critical transition of subsystem X 2 occurs in our model even if c 2 < c 2 crit , given that subsystem X 1 is in the alternative state ( figure 5 , moving upwards from the lower left along the yellow arrow). The coupled subsystem X 2 tips at an effective tipping pointterm·dom lower than its intrinsic tipping pointterm·dom c 2 crit .

For an increase of the control parameter c 1 above the intrinsic tipping pointterm·dom c 1 crit and a slight increase of the control parameter c 2 , a tipping cascadeterm·dom, starting from the normal states of X 1 and X 2 , with a critical transition in subsystem X 1 and a following transition in subsystem X 2 arises in our model ( figure 5 , moving from the lower left to the right column along the pink arrow). Note that the cascade occurs before the intrinsic tipping pointterm·dom of subsystem X 2 is crossed.

There is a change in the system behaviour for an increasing coupling strengthterm·key (electronic supplementary material, figure S1). The previously described area with only one stable fixed point of the two subsystems in the alternative state for c 1 > c 1 crit exists for extremely low values of c 2 << c 2 crit . Therefore, a tipping cascadeterm·dom can occur for even lower values of the control parameter c 2 << c 2 crit than for a system with lower coupling strengthterm·key. For c 1 < c 1 crit and low values of the control parameter c 2 , subsystem X 2 either transitions to the alternative state for c 2 < c 2 crit , given that subsystem X 1 is in the alternative state, or subsystem X 2 tips back from the alternative state to the normal state, given that subsystem X 1 has not tipped. For c 1 < c 1 crit and an increased control parameter c 2 , the critical transition of subsystem X 2 to the alternative state, given that subsystem X 1 occupies the alternative state, is the only transition that can be observed.

Bidirectional interaction of two tipping elementsterm·dom, e.g. Greenland ice sheetterm·dom and Atlantic meridional overturning circulationterm·dom Consider a system consisting of two ( n = 2) tipping elementsterm·dom given by equations ( 2.2 ) and ( 2.3 ) with i = 1, 2 and a bidirectional coupling where d 21 < 0 and d 12 > 0. This type of coupling can for instance be found in the interaction of the Greenland ice sheetterm·dom (GISterm·dom) and the Atlantic meridional overturning circulationterm·dom (AMOCterm·dom), whose long-term behaviour may be represented by a double fold as suggested by (simple) models [ 14 , 17 , 90 ]: increased meltwater influx into the North Atlantic due to tipping of the GISterm·dom could lead to a weakening or even shutdown (tipping) of the AMOCterm·dom [ 55 ], i.e. introducing a positive coupling. At the same time, a slowdown of the AMOCterm·dom leads to a relative cooling around Greenland and hence corresponds to a negative coupling [ 54 ]. The system is analysed for a low and a high coupling strengthterm·key (where d 21 and d 12 have opposite signs but the same magnitude) due to a substantial change of the qualitative behaviour towards higher coupling strengthterm·key. There currently is not sufficient knowledge on the strength of the interaction between the GISterm·dom and the AMOCterm·dom so that neither a low nor a high coupling strengthterm·key can be excluded for certain. The number of stable equilibria and possible critical transitions for different parameter settings are given in figure 6 as zoom into figure 4 for low coupling strengthsterm·var (lower right). A starting point of the analysis of the system behaviour for the low coupling strengthterm·key is the area of four stable equilibria for low values of the control parameters c 1 and c 2 in figure 6 .

Number of stable fixed points and phase space portraits for two bidirectionally coupled tipping elementsterm·dom with d 21 = −0.2 < 0 and d 12 = 0.2 > 0 and low coupling strengthsterm·var where | d 21 | = | d 12 | depending on the control parameters c 1 and c 2 . The dashed lines represent the intrinsic tipping pointterm·dom of the respective subsystem. The phase space portraits allow to derive the possible critical transitions in the master–slave system. Within the phase portraits stable fixed points are shown in orange, while unstable fixed points are shown in red. The background colour indicates the normalized speed v = x ˙ 1 2 + x ˙ 2 2 / v max going from close to zero (purple) to fast (yellow–green).

With increasing control parameter c 1 , a critical transition of the GISterm·dom as subsystem X 1 is possible for c 1 < c 1 crit in our model, given that the AMOCterm·dom as subsystem X 2 is in the normal state ( figure 6 , moving from lower left to the right along the yellow arrow). The GISterm·dom might tip at an effective tipping pointterm·dom which is lower than the intrinsic tipping pointterm·dom of the isolated subsystem. A critical transition of the GISterm·dom for the AMOCterm·dom being in the alternative state is possible with a further increase of the control parameter c 1 for c 1 ≫ c 1 crit ( figure 6 , moving from lower left to the right along the second yellow arrow).

With increasing control parameter c 2 , a critical transition of the AMOCterm·dom as subsystem X 2 to a state with weakened strength (i.e. the alternative state) is possible in our model for c 2 < c 2 crit , given that the GISterm·dom as subsystem X 1 has already tipped ( figure 6 , moving from lower left upwards along the yellow arrow). The AMOCterm·dom might tip at an effective tipping pointterm·dom which is lower than the intrinsic tipping pointterm·dom of the isolated subsystem. Given that the GISterm·dom is in its normal state, a critical transition of the AMOCterm·dom is possible with a further increase of the control parameter c 2 for c 2 ≫ c 2 crit at an effective tipping pointterm·dom higher than its intrinsic tipping pointterm·dom ( figure 6 , moving from lower left upwards along the second yellow arrow). As a result of the model formulation, the GISterm·dom would still pull the thermohaline circulation (THC) away from its tipping pointterm·dom even though it has already started to melt but has not tipped to x 1 + ∗ (i.e. 0 > x 1 ∗ > − 1 ). It would be possible to adjust the coupling function or the dynamics of each tipping elementterm·dom (e.g. [ 76 ]) so that already a slight change of the GISterm·dom state towards the alternative state without a complete critical transition to full loss of the ice sheet would push the AMOCterm·dom towards its own tipping pointterm·dom.

For a slight increase of both control parameters c 1 and c 2 , a critical transition of the GISterm·dom as well as the AMOCterm·dom to the alternative state is possible in our model for c 1 < c 1 crit and c 2 < c 2 crit before their respective intrinsic tipping pointsterm·dom are crossed ( figure 6 , moving from lower left along the pink arrow). In contrast to the previous example 3.1, the additional negative coupling results in the tipping of both interacting subsystems at an effective tipping pointterm·dom below their intrinsic tipping pointsterm·dom. In a master–slave system (example 3.1), the master system X 1 needs to tip through an increase of its control parameter above its intrinsic tipping pointterm·dom c 1 > c 1 crit to trigger a critical transition in the slave system X 2 at an effective tipping pointterm·dom c 2 < c 2 crit .

With increasing coupling strengthsterm·var d 21 and d 12 the system behaviour changes. The system has one unstable fixed point and Kadyrov oscillations [ 85 ] occur for a wide range of the control parameters in the considered part of the ( c 1 , c 2 )–parameter space (upper left of figure 4 ).

Master–slave–slave system, e.g. propagation of critical transitions in lake chainsterm·key Consider a system consisting of three ( n = 3) unidirectionally coupled tipping elementsterm·dom given by equations ( 2.2 ) and ( 2.3 ) with i = 1, 2, 3; d 12 , d 23 > 0 and d 21 , d 32 , d 31 , d 13 = 0. This type of coupling corresponds to the behaviour of a lake chainterm·key subject to an external input of nutrients as a control parameter (as e.g. in [ 57 ]). As in example 3.1, it is assumed that the lakes are connected through a unidirectional water stream [ 57 , 58 , 67 , 91 ]. The behaviour of subsystem X 1 corresponds to the behaviour of an uncoupled tipping elementterm·dom. Therefore, the eutrophication of the first lake in the lake chainterm·key, i.e. the tipping of subsystem X 1 , is possible with an increase of its control parameter c 1 > c 1 crit . For c 1 > c 1 crit only stable fixed points with subsystem X 1 in the alternative state exist. Additionally increasing the control parameters c 2 , c 3 or both results in the loss of further stable fixed points and allows for critical transitions in the slave systems X 2 and X 3 ( figure 7 ). In the following, the system behaviour with c 1 > c 1 crit for low coupling strengthsterm·var d 12 , d 23 > 0 is analysed (see figure 7 , lower left and electronic supplementary material, figure S3 for a zoom-in).

Number of stable fixed points of the system consisting of three unidirectionally coupled tipping elementsterm·dom for c 1 = 0.4 > c 1 crit depending on the control parameters c 2 and c 3 and the coupling strengthsterm·var d 12 ≥ 0 and d 23 ≥ 0 in a matrix of stability mapsterm·mth. A stability mapterm·mth shows the number of stable fixed points in the ( c 2 , c 3 )–space for a specific coupling strengthterm·key, where a certain number of stable fixed points is associated with a specific colour. Note that different areas in the control parameter space with the same colour have the same number of stable fixed points but they do not necessarily have the same phase portrait. The dashed lines represent the intrinsic tipping pointterm·dom of the respective subsystem. The position of a stability mapterm·mth in the matrix is determined by the coupling strengthterm·key.

With an increasing control parameter c 2 or c 3 , a critical transition in the corresponding subsystems is possible in our model for c 2 < c 2 crit (electronic supplementary material, figure S4) or c 3 < c 3 crit (electronic supplementary material, figure S5), given that the preceding subsystem occupies the alternative state or undergoes a transition into the alternative state through a continuously changing control parameter. Lake X 2 and X 3 in the lake chainterm·key can, therefore, become eutrophic before the intrinsic critical level of nutrient input of an isolated lake is crossed, given that the preceding lake has already become eutrophic.

For a simultaneous, slight increase of the control parameters c 2 and c 3 of both subsystems X 2 and X 3 , a critical transition in both subsystems X 2 and X 3 is possible in our model for c 2 < c 2 crit and c 3 < c 3 crit . As a result, a tipping cascadeterm·dom can be observed, given subsystem X 1 has already tipped or tips with c 1 > c 1 crit (pink trajectory in figure 8 ). Consequently, after the eutrophication of the first lake, a critical transition to the turbid state of a lake can spread in the lake chainterm·key even if the intrinsic critical value of nutrient input known from an isolated lake is not crossed.

Tipping cascadeterm·dom in a system of three unidirectionally coupled tipping elementsterm·dom for an increase of the control parameters c 2 << c 2 crit and c 3 << c 3 crit and c 1 > c 1 crit (as indicated by the pink arrow in the stability mapterm·mth, upper left panel). The central cube shows the flow in the ( x 2 , x 3 )-space as part of the three-dimensional phase space and the remaining stable fixed point (in orange) for c 1 = 0.4 > c 1 crit , c 2 = c 3 = 0.2 with d 12 = 0.2 > 0 and d 23 = 0.2 > 0. The tipping cascadeterm·dom is highlighted by the pink trajectory. Two-dimensional plots arranged around the central cube show the flow in the ( x 2 , x 1 )- and ( x 3 , x 1 )-space corresponding to the lateral surfaces of the cube.

Discussions and conclusion The qualitative long-term behaviour of interacting, cusp-like tipping elementsterm·dom has been assessed in a simple analytic and an extensive numerical analysis of a conceptual model. Depending on the type of coupling and the coupling strengthterm·key, qualitatively different behaviours of the systems of interacting tipping elementsterm·dom were observed. In particular, tipping cascadesterm·dom where a critical transition in one subsystem triggers the tipping of a coupled subsystem may occur under certain conditions.

Simple analytic calculations resulted in the formulation of tipping rules for the spread of tipping processes in systems of interacting tipping elementsterm·dom: a shift in the threshold value of the control parameter, at which an interacting tipping elementterm·dom undergoes a transition into a qualitatively different state, can occur. We call the threshold value of the isolated subsystem the intrinsic tipping pointterm·dom of the tipping elementterm·dom. If an interaction with another tipping elementterm·dom exists, the tipping process takes place when the so-called effective tipping pointterm·dom is crossed. Depending on the coupling direction, the effective tipping pointterm·dom can occur at either lower (facilitated tippingterm·dom) or higher (impeded tippingterm·dom) values of the control parameter than the intrinsic tipping pointterm·dom.

We have generalized and extended existing studies of special cases of coupled cusp-like tipping elementsterm·dom [ 52 , 85 ] through an extensive numerical analysis of two and three interacting tipping elementsterm·dom with one- or bidirectional coupling of varying direction. The behaviour of the special cases including a window with both subsystems of a simple master–slave system [ 52 ] in the alternative state, a tipping cascadeterm·dom in a positively coupled master–slave–slave system [ 52 ] and the Kadyrov oscillator [ 85 ] for two bidirectionally coupled tipping elementsterm·dom for a high coupling strengthterm·key of same magnitude but with opposite signs is consistent with the system behaviour observed in our analysis.

In addition, our extensive analysis allowed to identify types of coupling that favour critical tipping scenarios. Conditions in terms of coupling strengthterm·key and control parameters of the subsystem under which the tipping scenarios occur have been determined. Cascades of tipping processes that occur before the crossing of intrinsic tipping pointsterm·dom, i.e. where the effective tipping pointterm·dom lies at lower values than the intrinsic tipping pointterm·dom of the uncoupled tipping elementterm·dom, are of special interest. In a simple master–slave system with positive coupling, a critical transition in the master system due to a crossing of its intrinsic tipping pointterm·dom triggers a critical transition of the slave system at an effective tipping pointterm·dom lower than its intrinsic tipping pointterm·dom. By contrast, a negative coupling would prevent a facilitated tippingterm·dom of the slave system in the case of a master system being in the alternative state. In a system of two tipping elementsterm·dom with bidirectional coupling, a tipping cascadeterm·dom is favoured if one of the coupling terms is negative. In a master–slave–slave system with d 12 > 0 and d 23 > 0, the initial tipping of the master system can trigger cascading tipping processes in the following subsystems before the intrinsic threshold of the corresponding control parameter is crossed. Such a tipping cascadeterm·dom before the crossing of the corresponding intrinsic thresholds cannot be observed after the introduction of a negative coupling (results not shown here). Tipping processes are suppressed instead in this case and do not spread into all subsystems.

Applying the qualitative system behaviour to selected interacting real-world tipping elementsterm·dom revealed possible tipping scenarios, which are relevant for the future development of the Earth system, and in addition, due to the consequences of tipping, such as sea-level rise [ 92 , 93 ], for the economy, infrastructure and society more broadly. In particular, the analysis of the qualitative long-term system behaviour of two bidirectionally coupled tipping elementsterm·dom with opposite sign but same magnitude suggests that the Greenland ice sheetterm·dom and the AMOCterm·dom might tip before their intrinsic tipping pointsterm·dom are reached. In other words, the meltdown of the Greenland ice sheetterm·dom and the slowdown of the AMOCterm·dom might begin before the intrinsic threshold ranges identified for isolated tipping elementsterm·dom [ 94 ] is crossed. The possible existence of such tipping cascadesterm·dom increases the risks that anthropogenic climate change poses to human societies, since the intrinsic threshold ranges of some climatic tipping elementsterm·dom including the Greenland ice sheetterm·dom are assumed to lie even within the 1.5–2°C target range of the Paris agreement [ 94 ].

When it comes to the application of tipping behaviour to real-world systems, it should be noted that tipping elementsterm·dom were described in an idealized way using the normal form of the cusp catastropheterm·key onto which, by the concept of topological equivalence [ 88 ], the critical behaviour of a class of real-world systems can be mapped. The proposed model of interacting tipping elementsterm·dom, therefore, shows a hypothetical, but mathematically possible system behaviour. It was motivated by its catastrophic features [ 46 , 87 ] in contrast to other bifurcational systems allowing the transition into a qualitatively different state by the variation of a bifurcation parameter and the appearance of the double fold bifurcation in many real world systems [ 6 , 8 , 12 , 15 , 17 – 19 , 95 ]. However, processes which are not taken into account in the conceptual representation of tipping elementsterm·dom, but are present in the real world, might influence the system and its tipping behaviour. In addition to a direct coupling of tipping elementsterm·dom in the climate system [ 54 ], an indirect, ‘diffusive’ interaction through e.g. the global mean temperature [ 1 ] could be considered as a potential coupling mechanism. Furthermore, chains and pairs of tipping elementsterm·dom have been analysed isolated from the larger network of interacting climatic tipping elementsterm·dom [ 54 ], i.e. possible interactions with other climatic tipping elementsterm·dom have been neglected. Here, we focused on bifurcation-induced tippingterm·dom assuming that the control parameter varies sufficiently slowly for the system to keep track with the stable states. It should be noted that a change of the control parameter with a high rate is likely, given the increasing influence of humans on the Earth system, possibly giving rise to rate-induced tippingterm·dom [ 45 , 50 ].

The qualitative and theoretically possible system behaviours studied here and their application to real-world systems, therefore, introduces further research questions regarding tipping elementsterm·dom and their interactions in ecology, climate science and other fields. The conceptual approach should be extended to networks of tipping elementsterm·dom as already suggested in [ 84 ] and motivated in [ 54 ]. Networks of interacting tipping elementsterm·dom can be analysed using methods of statistical mechanics [ 96 ]. Critical transitions may spread across a whole network of tipping elementsterm·dom depending on the clustering and the spatial organization of the network [ 76 – 78 ]. Taking the important interactions of climatic tipping elementsterm·dom into account in a network approach, realistic complex models must be used for quantitatively approximating the effective tipping pointterm·dom. In addition, interacting tipping elementsterm·dom with heterogeneous intrinsic threshold and varying internal time scales should be considered [ 65 ] as, for example, the critical nutrient input of lakes varies with their depth [ 7 ]. Finally, generic early warning signals for tipping cascadesterm·dom comparable to already existing indicators for critical transitions of isolated tipping elementsterm·dom [ 97 , 98 ] are desirable to forecast cascading tipping events and counteract undesired consequences of tipping (cascades). A first step towards early warning indicators of tipping cascadesterm·dom has been presented only recently [ 86 ]. However, up to now, it remains an open question whether early warning signals for tipping cascadesterm·dom based on critical slowing down [ 97 , 98 ] exist.

Supplementary Material Acknowledgements This work has been performed in the context of the copan collaboration and the FutureLab on Earth Resilience in the Anthropocene at the Potsdam Institute for Climate Impact Research. Furthermore, we acknowledge discussions with and helpful comments by N. Wunderling, J. Heitzig and M. Wiedermann.

Data accessibility Data and relevant code for this research work are stored in GitHub: https://github.com/pik-copan/pytippinginteractions and have been archived within the Zenodo repository: https://doi.org/10.5281/zenodo.3768371 .

Authors' contributions A.K.K. and V.K. performed mathematical and numerical analysis. A.K.K. led the writing of the manuscript and created figures and tables. J.F.D. and R.W. conceived of the study, designed the study and coordinated the study. All authors helped in drafting the manuscript and critically revised the text. All authors gave final approval for publication and agree to be held accountable for the work performed therein.

Competing interests We declare we have no competing interest.

Funding V.K. thanks the German National Academic Foundation (Studienstiftung des deutschen Volkes) for financial support. J.F.D. is grateful for financial support by the Stordalen Foundation via the Planetary Boundary Research Network (PB.net), the Earth League's EarthDoc program and the European Research Council Advanced Grant project ERA (Earth Resilience in the Anthropocene). We are thankful for support by the Leibniz Association (project DominoES).

References Associated Data Supplementary Materials Data Availability Statement Data and relevant code for this research work are stored in GitHub: https://github.com/pik-copan/pytippinginteractions and have been archived within the Zenodo repository: https://doi.org/10.5281/zenodo.3768371 .