Tipping points in complex ecological systems

Abstract

Tipping points are one of the hot topics in modern physics of complex systems. But what is a tipping point? A generic definition declares it as ``a state of the system where a small change in its parameters can lead to a significant change in its properties''. Additional ingredients that often enter the definition of tipping process are the abruptness of the resulting change and its irreversibility, i.e. it is impossible to recover the initial state if one reverses the protocol of change of the parameters. However, there exists a number of different mathematical structures that can show this behavior, the one that was originally suggested as a tipping point (nowadays usually referred to as bifurcation induced tipping) is just one of many. Different preconditions and/or different level of details included into the model, reflecting also different environmental forcing, can lead to a variety of tipping mechanisms. Furthermore, in a spatially extended system and/or a system with multiple scales, different parts can react to a change in environmental conditions differently or at a different time, interacting with each other to create a tipping cascade. In this paper, using ecosystems as a paradigm of complex nonlinear open systems, we provide a critical overview of the progress made in tipping point science over the last 15 years. We highlight the main findings, identify gaps in our knowledge, and outline a roadmap for further progress.

Figures (6)

• Figure 1: Tipping point in a prototypical one-dimensional system of the form $\dot{x}=f(x,\nu)=-\mathrm{d}V(x,\nu)/\mathrm{d}x$ with a few steady states. (a) A stable steady state of the system (shown by asterisks / the upper branch of the curve) can merge with an unstable state (shown by open circles / lower branch of the curve) and disappear when a system's parameter $\nu$ crosses its critical value, say $\nu=h$ (shown by vertical red line). The system is therefore in a 'safe' state for any $\nu<h$ (as is visualized by the green line) but will experience a 'free fall' transition to another state for $\nu>h$. (b) The disappearance of the steady state in terms of $V(x,\nu)$: a sufficiently (overcritical) increase in the bifurcation parameter $\nu$ will eliminate the local minimum.
• Figure 2: For the caption, see next page.
• Figure 3: a) Idealised multiscale quasi-potential $\Phi$ defining the landscape of a metastable stochastic dynamical system as a function of the variable $X$. b) Corresponding idealised structure of hysteresis diagrams due to multistability obtained by changing the value of the parameter $P$. Reproduced from margazoglou2021dynamical; see also Lohmannetal2024 for an example of multiscale quasi-potential in an oceanographical context.
• Figure 4: Cascading tipping on trophic networks. (a) An example of heterogeneous foodweb where tipping (species extinctions) at nodes 1-5 leads to tipping in other parts of the web. Adapted frompalmer2023. (b) Effect of tipping cascade in an ancient foodwebRoopnarine2006 resulting in a highly nonlinear, threshold-type response of the ecosystem to the initial tipping in parts of the web.
• Figure 5: Different types of cascading tipping. (A) Dynamical pattern of cascading tipping in a system of coupled bi-stable subsystems $X_i(t)$ on a network. Initially, all subsystems stay in the lower (non-tipped) state; then further tipping of each subsystem occurs as a separate event. For the same observed pattern shown in (A), the mechanism of cascading can be different (quasi-cascading; multi-phase cascading; slow domino effect, noise-induced cascading without domino effect, etc), and this depends on the nature of the variation of the control parameter. (B) Joint cascading tipping (joint tipping/fast domino effect): regime shifts of $X_i$ occur simultaneously. (C)-(D) Scenarios of variation of the control parameter, resulting in cascading in shown (A) and (C), respectively: a gradual increase (solid line); stochastic variation (dashed line).