The climate system as well as ecosystems might undergo relatively sudden qualitative changes in the dynamics when environmental parameters or external forcings vary due to anthropogenic influences. The study of these qualitative changes, called tipping phenomena, requires the development of new methodological approaches that allow phenomena observed in nature to be modeled, analyzed and predicted, especially concerning the climate crisis and its consequences. Here we briefly review the mechanisms of classical tipping phenomena and investigate rate-dependent tipping phenomena which occur in

The climate system consists of many interacting components

The study of the impact of nonlinearities in the geosciences has a long history concerning investigations of (i) chaotic dynamics leading to obstructions to predictability

In this paper, we review the classification of tipping phenomena, explain their mechanisms, and give examples of their occurrence in climate science and ecology (Sect.

In many cases, tipping phenomena require the simultaneous existence of several different stable states of a system under the same given environmental conditions. Bi- and multistability can best be illustrated by a stability landscape represented as a potential (Fig.

Here, stability of a state means linear stability with respect to small perturbations given by the eigenvalues of the corresponding Jacobian matrix for, e.g., steady states

In this setup, critical transitions are associated with a relatively sudden qualitative change of the dynamics in which the system moves from one stable state to another; i.e., the system tips by getting from one valley into the other via different mechanisms. In general, those mechanisms are related to certain disturbances, kicking the system out of the position in the valley such that the other valley can be reached. However, it is essential to note that a specific tipping phenomenon, namely rate-induced tipping, does not necessarily require the existence of bi- or multistability. Instead, in those critical transitions, it is sufficient that the system trajectory moves into a part of the state space with different properties. Therefore tipping, in general, cannot always be identified with the well-known classical bifurcations but can, particularly in rate-induced tipping, only be explained as bifurcations in non-autonomous systems.

Often the picture of the stability landscape mentioned above is translated into a specific bifurcation diagram exhibiting hysteresis, showing the two stable states and the unstable one separating those two depending on the intrinsic parameters of the system or the external forcing (Fig.

Relationship between the stability landscape and the control parameters/the forcing of the system. Cyan arrows indicate the changes in control parameters or forcings; the light-green and light-brown arrows refer to changes in the state variables.

Next, we illustrate the different tipping mechanisms in Fig.

Environmental changes, e.g., increasing habitat destruction, can affect the growth of plants. If those environmental changes are very slow, then the ecosystem state would slowly “move” along the solid red line

Illustration of the four different tipping mechanisms:

Let us discuss some examples of bifurcation-induced tipping in the climate system and ecology. Over the past decade, a number of tipping elements, i.e., climate phenomena, have been identified as candidates expected to tip in the further course of climate change

The possible melting of the Arctic Sea ice is also discussed in terms of bistability comprising two stable states, where in one of which the Arctic Sea ice disappears to a large extent in summer and only shows ice cover in winter

Examples of alternative states in ecosystems have been discussed in the literature (

It is important to note that bifurcation-induced tipping is not restricted to the saddle-node bifurcation shown, but many other bifurcations, such as Hopf bifurcations, torus bifurcations and homoclinic bifurcations, can be related to tipping phenomena

This tipping process is caused by fluctuations (Fig.

Such noise-induced tipping is hypothesized to be responsible for the regime shift observed in the dominance of two species – a brittle star and a burrowing mud shrimp species – living in the sediment of the North Sea. This change in dominance took place at the end of the 1990s without any significant changes in environmental conditions and, hence, cannot be attributed to bifurcation-induced tipping but rather to a change in fluctuations in the water movement

Noise-induced transitions have also been shown to be a crucial mechanism of tipping in the climate system, as climate change involves not only shifting mean values such as global temperature associated with global warming

While noise-induced tipping causes the system to tip through a whole sequence of small disturbances, shock-induced tipping is caused by a single large disturbance (Fig.

Calculating these smallest disturbances in ecological networks also provides valuable information about which system parts are most vulnerable to extreme disturbances. In this way, it can be shown that in networked ecosystems of plants and their pollinators, particularly those species have the highest extinction risk that are specialists or species that are part of a tree-like structure in the graph of the network with only a very loose connection to the core of the species' network

This tipping mechanism describes a system's response to an environmental change associated with a particular trend. It differs from those discussed so far by three essential points: (1) in this mechanism, the relationship between the timescales of the physical, chemical and/or biological processes in the system under consideration, i.e., the intrinsic timescales and timescale or rate of the trend of environmental changes, plays a decisive role. (2) This mechanism does not necessarily require the existence of alternative stable states. (3) The critical threshold value is not determined by a specific environmental parameter itself but by the

While all aforementioned tipping mechanisms are related to the coexistence of alternative states, rate-induced tipping can also occur when there is only one stable state present and the system is characterized by different timescales (slow–fast system). The dynamics of such systems can be described by so-called critical manifolds in the case of a perfect timescale separation or slow manifolds, when the timescale separation is finite. In the case of a complex structure of the critical manifolds, for instance when these critical manifolds have stable and unstable parts which meet in a fold, a rate-induced crossing of this fold can make the trajectory visit very different parts of the state space far away from the original stable state and perhaps even dangerous for the system (for a more mathematical description including the conditions under which this transition occurs, see

Again, let us look at an ecosystem as an example: if, for example, environmental changes occur very slowly, as in bifurcation-induced tipping, the species in the ecosystem have enough time to adapt to the changed environment. Conversely, if, e.g., climate change happens too fast, species adaptation fails, and, as a result, ecosystems can collapse. For example, this mechanism can be demonstrated in predator–prey systems, where the prey's habitat is destroyed by climate change or anthropogenic influences like land use change. It is possible to determine a critical rate of environmental changes beyond which the ecosystem collapses

In the previous analysis, the considered systems were spatially homogeneous and, therefore, usually modeled by ordinary differential equations (ODEs) or time-discrete systems (maps). However, complex systems in space are often characterized by the fact that they can spontaneously form spatially inhomogeneous patterns resulting from, e.g., a Turing bifurcation

In the course of climate change it is becoming more and more important to find appropriate methods to predict tipping points and to identify early warning signals. One method that has been developed in the physics and chemistry literature is critical slowing down (CSD) of the restoring forces when a bifurcation-induced transition is approached

To analyze the role of timescales as well as the role of saddles (invariant sets of saddle type possessing stable and unstable manifolds), we will employ different simple models from population dynamics. We start with a one-dimensional model of the growth of a population influenced by an Allee effect. The Allee effect describes the ecological fact that certain populations need a minimal critical population density to grow

Let us now assume that changes in the environment lead to changes in the critical population density

We apply a linear drift of parameter

Varying the critical population density means moving the two quasi-stationary states

We find that for this given rate of change of environmental conditions already, quite a large number of trajectories' initial conditions tip to extinction. Increasing the rate of change

Time evolution of trajectories of system Eq. (

The mechanism, i.e., how trajectories in this one-dimensional case tip, is observable in Fig.

To gain more insights into the interplay between the rate of moving basin boundaries due to environmental change and the intrinsic dynamics' timescale, we analyze a higher-dimensional problem in which smooth basin boundaries can be considered hypersurfaces partitioning the state space into regions of different qualitative behavior. The basin boundaries correspond in the frozen-in case to the stable manifolds of a saddle point. For the sake of simplicity, we would like to analyze two coupled bistable systems, which can be coupled in two different ways, unidirectional and bidirectional. In the context of the simple population dynamical model analyzed above, it could be interpreted as two habitats (patches) bearing the same species which can move or migrate between the habitats. An ecologically relevant bidirectional coupling would be migration based on the population differences between the habitats, i.e., a diffusive coupling.

To be more general, we choose to consider not only an ecological example but a general bistable model of the following form:

Since our focus is on the role of timescales, we have introduced an additional parameter,

As the first coupling scheme, we consider a unidirectional coupling corresponding to a drive–response configuration:

Rescaling the time in terms of

System 2 appears as a driver for system 1. We analyze the dynamics in the most intuitive way and use the concept of nullclines, which are given by the algebraic equations

Basins of attraction, attractors and nullclines for the autonomous system shown in Eq. (

To investigate the impact of a time-varying environment, we change the coupling parameter

The most interesting dynamics happen with a parameter drift along which the number of attractors changes and saddle-node bifurcations lead to new attractors. This setup is suitable to elucidate the relative size of the basins of attraction and tipping probabilities for different initial conditions depending on the rate of change in the driver strength

As mentioned above, classical bifurcations occur along the course of parameter variation, giving rise to new invariant sets (in this case, steady states) and new basins of attractions, including their boundaries that lead to a new “partitioning” of the state space with tremendous consequences for single trajectories. To illustrate this, we investigate the following scenario, which is inspired by the scenarios of parameter drift used by

As in the previous subsection, we vary the rates of environmental change by varying the time interval

Time evolution of the non-autonomous system shown in Eq. (

To understand the tipping in more detail, we have plotted some particular trajectories and observed that some trajectories tip while others do not. While tipping trajectories change their course in state space when the bifurcation occurs to reach the newly emerging stable states, the tracking (non-tipping) trajectories follow their path largely undisturbed. In Fig.

Looking at the basins of attraction as a whole, we note by comparing the right panel of Fig.

To study that further, we now look at the variation of the relative size of the basins of attraction for different rates of environmental change (Fig.

The relative size of the non-autonomous basins of attraction

To get deeper insights into the mechanism of basin boundary crossing of a particular trajectory, we analyze for

Tipping of trajectories starting next to the non-autonomous basin boundary in the points indicated by small filled black circles;

So far, we have only varied the rate of environmental change but left the intrinsic timescales of the two systems equal to make the systems identical. However, different intrinsic timescales contribute also to a change in the dynamics. This approach is illustrated in Fig.

Time evolution of the non-autonomous system shown in Eq. (

Overall, we can conclude that bifurcations which change the topological structure of the state space have a tremendous impact on the evolution of trajectories. This impact depends crucially on the relation between the intrinsic dissipative timescale and the timescale of environmental change. It turned out, that following the trajectories, which can be considered observables, does not necessarily detect those transitions. There is a detection limit beyond which bifurcations which happen in state space are not noticed by the observables. As a consequence, transitions due to other tipping mechanisms like noise-induced tipping can happen without any warning.

The second type of coupling we consider is a mutual coupling of the two systems having different timescales and different strengths of impact on each other. This results in the following system of differential equations:

Let us consider the symmetrical case in which we assume the same coupling

Time evolution of the non-autonomous system shown in Eq. (

When we continuously vary the rate of environmental change to identify the rate-dependent masking effect, we find that the relative size of the non-autonomous basins of attraction and the tipping probabilities vary non-monotonously in the case of different timescales

The relative size of the non-autonomous basins of attraction

Finally, we address the third coupling scheme, diffusive coupling, often used when coupling systems bidirectionally with the same coupling strength

For ecological systems, this would be the appropriate coupling when considering two populations in two different habitats coupled through species migration. The same coupling would be used for coupled chemical systems, where diffusion is assumed to be the most important spatial transport process. Following the same protocol of numerical simulations with the same parameter values, we observe the same qualitative behavior as for the other coupling schemes with one important difference. The masking effect occurs for much larger

The relative size of the non-autonomous basins of attraction

We aimed to evaluate the consequences of a time-dependent variation of parameters or external forcing following a prescribed trend in a multistable system. In contrast to many other studies, our focus was not on the stable long-term behavior, i.e., the attractors, but on the unstable sets of saddle type since their stable manifolds make up the basin boundaries. Specifically, we were interested in how the relative size of the basins of attraction varies in a non-autonomous system and how the “movement” of the corresponding basin boundaries influences the trajectories in state space. As already known from earlier works on rate-induced tipping, the time-dependent forcing implies that attractors like stationary points become quasi-stationary and “move” through the state space according to the trend

We have demonstrated the mechanism of basin boundary crossing by employing a system from population dynamics possessing an Allee effect. In this model, the basin boundary crossing occurs for varying the critical population density corresponding to the “moving” saddle point making up the basin boundary. In the case of crossing, the trajectory tips when it meets the moving saddle point, i.e., the moving basin boundary. Then we addressed the question of what happens in higher dimensions when the basin boundaries are not just saddle points but hypersurfaces in state space that are moving and/or even changing their shape. In addition, basins of attraction can even appear and disappear in bifurcations, e.g., in a saddle-node bifurcation. The classical computation of the relative size of a basin of attraction applies only to the frozen-in case with fixed parameters. To extend this approach to non-autonomous systems, we have called a non-autonomous basin of attraction the union of all those initial conditions which converge to a particular quasi-stationary state including the parameter drift. As an example, we analyzed two coupled, rather general bistable systems with different coupling schemes, which have been studied in a similar form already in the context of tipping cascades. In climate science, the most interesting coupling is drive–response coupling, which is particularly used to investigate tipping cascades in two or a few coupled natural systems

To study the impact of moving non-autonomous basin boundaries in detail, we have focused on a simple system with smooth basin boundaries in the whole parameter range, not fractal ones. The most straightforward situation in which basin boundaries are important is the saddle-node bifurcation of steady states in which a new stable steady state occurs together with a saddle point, whose stable manifolds make up the basin boundary for the newly appearing steady state. Apart from details that are related to the different coupling schemes, the main findings are qualitatively the same for all of them. In our setup, two different timescales are involved, the intrinsic dissipative timescale of the dynamics of each subsystem and the timescale of the environmental change, which in our case was influencing only the coupling strength. We found that the relative size and the shape of the non-autonomous basins of attraction depend strongly on the rate of environmental change of parameters or external forcing. As a consequence, initial conditions that would converge to one attractor in the frozen-in case tip into another basin of attraction during the parameter drift. This leads to a partial tipping of trajectories. In addition, we showed that for each finally tipping initial condition, there exists a critical rate of environmental change for which this tipping occurs for the first time. Hence this tipping process fulfills the requirements of rate-induced tipping as defined in

The tremendous consequences of that masking effect become more evident when we discuss it from the point of view that anthropogenic changes in parameters/forcing are accelerating corresponding to an increasing rate, which means lowering the time interval

We have further unraveled the mechanism of how the trajectory tips from one basin of attraction to the other by crossing the basin boundary. This rate-induced basin crossing happens at the saddle point; either the trajectory “meets” the saddle point directly, or first it approaches the neighborhood of its moving stable manifold and travels along it until the saddle point is reached for the crossing.

An analysis of the relative size of the basins of attraction and the tipping probabilities in a highly multistable system with fractal basin boundaries has been provided by

This paper contains numerical simulations obtained with freely available numerical integrators for ordinary differential equations (Runge–Kutta integrator Dormand–Prince 5(4)), available at

All data in this paper are produced numerically; equations and parameters are given in the text.

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This article is part of the special issue “Centennial issue on nonlinear geophysics: accomplishments of the past, challenges of the future”. It is not associated with a conference.

Ulrike Feudel would like to thank Ann Kristin Klose and Johannes Lohmann for inspiring discussions, Marie Arnold for some preliminary simulations and Everton Medeiros for a careful reading of the manuscript. Moreover, Ulrike Feudel thanks one of the anonymous reviewers for their manifold suggestions to improve the figures. Ulrike Feudel acknowledges support from the European Union's Horizon 2020 Research and Innovation program under the Marie Skłodowska-Curie Action Innovative Training Networks grant agreement no. 956170 (CriticalEarth).

This research has been supported by the Deutsche Forschungsgemeinschaft (grant no. 454054251) and the Horizon Europe Marie Skłodowska-Curie Actions (grant no. 956170).

This paper was edited by Reik Donner and reviewed by two anonymous referees.