Multistability is a common phenomenon which naturally occurs in complex networks. often one of the coexisting stable states can be identified as being the desired one for a particular application. We present here a global approach to identify the minimal perturbation which will instantaneously kick the system out of the basin of attraction of its desired state and hence induce a critical or fatal transition we call shock-tipping. the corresponding Minimal Fatal Shock is a vector whose length can be used as a global stability measure and whose direction in state space allows us to draw conclusions on weaknesses of the network corresponding to critical network motifs. We demonstrate this approach in plant-pollinator networks and the power grid of Great Britain. in both system classes, tree-like substructures appear to be the most vulnerable with respect to the minimal shock perturbation. Many processes in nature can be well described in terms of nonlinear dynamical systems possessing multiple stable states for constant external conditions 1,2. As long as perturbations are small, it is sufficient to consider the linearized problem around an attracting state to evaluate its 'local' stability. However, in real systems, the most threatening perturbations are usually not small but large, even constituting extreme events like storms, earthquakes or financial crises 3-5. Obviously, large perturbations in multistable systems call for a 'global' stability paradigm. Adequate frameworks have been introduced in the ecological literature-by Holling 6,7-as well as in the engineering literature-by Soliman and Thompson 8 (see 9 for a recent review)-who both argued that the local approach to stability needs to be accompanied by non-local measures capturing the characteristics of a state's basin of attraction. Recently, this global approach has been picked up in the field of complex networks in which multistability and large disturbances occur naturally as well 10-14. Suitable aspects of a basin of attraction are its size, shape and depth, which until now have all been used-separately 15-19 or in combination 20-22-to measure the stability of multistable systems. The suitability of a certain stability criterion for a specific problem also depends on the shape or distribution of the perturbations that occur. If perturbations are best described as noise, a suitable approach is to analyze the most likely escape path from the basin of attraction 23-25. This path is determined by the position of the saddle point on the basin boundary possessing the lowest barrier to escape 26. A very different situation occurs if perturbations are singular, large and abrupt. The response of nonlinear systems to such perturbations has been considered in terms of linear response theory as well as transient dynamics in climate science 27 , fluid dynamics 28,29 and energy networks 18,30,31. In networks, the most frequently applied characteristic to measure a system's stability against large abrupt perturbations is the relative basin size which is usually esti...

The transformation of ecosystems proceeds at unprecedented rates. Recent studies suggest that high rates of environmental change can cause rate-induced tipping. In ecological models, the associated rate-induced critical transition manifests during transient dynamics in which populations drop to dangerously low densities. In this work, we study how indirect evolutionary rescue—due to the rapid evolution of a predator’s trait—can save a prey population from the rate-induced collapse. Therefore, we explicitly include the time-dependent dynamics of environmental change and evolutionary adaptation in an eco-evolutionary system. We then examine how fast the evolutionary adaptation needs to be to counteract the response to environmental degradation and express this relationship by means of a critical rate. Based on this critical rate, we conclude that indirect evolutionary rescue is more probable if the predator population possesses a high genetic variation and, simultaneously, the environmental change is slow. Hence, our results strongly emphasize that the maintenance of biodiversity requires a deceleration of the anthropogenic degradation of natural habitats.

Multistability is a common phenomenon which naturally occurs in complex networks. If coexisting attractors are numerous and their basins of attraction are complexly interwoven, the long-term response to a perturbation can be highly uncertain. We examine the uncertainty in the outcome of perturbations to the synchronous state in a Kuramoto-like representation of the British power grid. Based on local basin landscapes which correspond to single-node perturbations, we demonstrate that the uncertainty shows strong spatial variability. While perturbations at many nodes only allow for a few outcomes, other local landscapes show extreme complexity with more than a hundred basins. Particularly complex domains in the latter can be related to unstable invariant chaotic sets of saddle type. Most importantly, we show that the characteristic dynamics on these chaotic saddles can be associated with certain topological structures of the network. We find that one particular tree-like substructure allows for the chaotic response to perturbations at nodes in the north of Great Britain. The interplay with other peripheral motifs increases the uncertainty in the system response even further.

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Today, the transformation of ecosystems proceeds at unprecedented rates. Recent studies suggest that high rates of environmental change can cause rate-induced tipping. In ecological models, the associated rate-induced critical transition manifests during transient dynamics in which populations drop to dangerously low densities. In this work, we study how indirect evolutionary rescue - due to the rapid evolution of a predator’s trait - can save a prey population from the rate-induced collapse. Therefore, we explicitly include the time-dependent dynamics of environmental change and evolutionary adaptation in an eco-evolutionary system. We then examine how fast the evolutionary adaptation needs to be to counteract the response to environmental degradation and express this relationship by means of a critical rate. Based on this critical rate, we conclude that indirect evolutionary rescue is more probable if the predator population possesses a high genetic variation and, simultaneously, the environmental change is slow. Hence, our results strongly emphasize that the maintenance of biodiversity requires a deceleration of the anthropogenic degradation of natural habitats.