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.2022 Jun;19(191):20220182.
doi: 10.1098/rsif.2022.0182. Epub 2022 Jun 1.

Rate-induced collapse in evolutionary systems

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Rate-induced collapse in evolutionary systems

Constantin W Arnscheidt et al. J R Soc Interface.2022 Jun.

Abstract

Recent work has highlighted the possibility of 'rate-induced tipping', in which a system undergoes an abrupt transition when a perturbation exceeds a critical rate of change. Here, we argue that this is widely applicable to evolutionary systems: collapse, or extinction, may occur when external changes occur too fast for evolutionary adaptation to keep up. To bridge existing theoretical frameworks, we develop a minimal evolutionary-ecological model showing that rate-induced extinction and the established notion of 'evolutionary rescue' are fundamentally two sides of the same coin: the failure of one implies the other, and vice versa. We compare the minimal model's behaviour with that of a more complex model in which the large-scale dynamics emerge from the interactions of many individual agents; in both cases, there is a well-defined threshold rate to induce extinction, and a consistent scaling law for that rate as a function of timescale. Due to the fundamental nature of the underlying mechanism, we suggest that a vast range of evolutionary systems should in principle be susceptible to rate-induced collapse. This would include ecosystems on all scales as well as human societies; further research is warranted.

Keywords: collapse; evolutionary rescue; extinction; nonlinear dynamics; rate-induced tipping points.

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Figures

Figure 1.
Figure 1.
Dynamics of the model (note units ofx andn are arbitrary). The black dot and black line atn = 0 denote stable fixed points (equilibria), while the white dot denotes an unstable saddle point. The timescale separation means that the system equilibrates rapidly towards the dn/dt = 0 nullcline (shown by black arrows), and only then adjusts slowly towards the stable fixed point (if the population is not already extinct).
Figure 2.
Figure 2.
Perturbations to the system can lead to extinction. We assume that the system was initially at the stable high-n equilibrium, and the optimal traitx* is instantaneously increased. The system will then rapidly adjust towards the neighbourhood of dn/dt = 0 (the critical manifold). Ifx is not too far fromx* (green trajectory), the system comes close to the extant critical manifold (grey), and evolutionary rescue occurs: the system recovers back to the high-n stable state. Ifx is far enough fromx*, the trajectory passes below the local minimum (fold) in the extant critical manifold, and extinction occurs instead. Separating these two cases is a set of ‘canard trajectories’ that stay close to the unstable branch of the extant critical manifold, initially heading towards the low-n saddle point in figure 1; these can go towards either extinction or recovery, as shown.
Figure 3.
Figure 3.
Evolutionary rescue and rate-induced extinction. We change the optimal traitx* linearly over two slightly different time periods. When we do it more slowly, the system is able to recover (evolutionary rescue) (a). When the system is perturbed more quickly, the system collapses (rate-induced extinction) (b). (c) We also visualize this in the space ofn,x,x*: here, the extant critical manifold becomes a surface (black line = fixed point), and the distinction between extinction and recovery is given approximately by whether or not the system passes the fold (dashed black line) in the extant critical manifold.
Figure 4.
Figure 4.
The scaling of the critical rate with time, in the minimal differential-equation model. It scales withτ−1 below the evolutionary timescaleτev, and is constant on longer timescales. Far enough below evolutionary timescales, adaptation is negligible, and so there is an effective critical amount of change inx* for which extinction occurs.
Figure 5.
Figure 5.
Demonstration of the basic features of the many-agent model. For every typex that exists at timet, a point is plotted; its colour shows how many individuals of that type currently exist. The model is seeded with 1000 agents withx = 100 at timet = 0, and over evolutionary time intervals (tens of thousands of time steps), evolutionary branching occurs. We eventually reach a quasi-evolutionary stable strategy (qESS).
Figure 6.
Figure 6.
Rate-induced extinction and evolutionary rescue in the many-agent model. In both cases, the community is subjected to a linear ramp in the parameterβ (mortality cost of investing in public goods). If this is done more slowly (a), some branches can persist (evolutionary rescue). If it is done more quickly (b), the entire community goes extinct (rate-induced extinction).
Figure 7.
Figure 7.
Critical rate of change for extinction for the many-agent model. The probability of extinction is estimated throughout rate–timescale space, using results from around 20 000 simulations. Below the evolutionary timescaleτev, we recover the now familiarτ−1 scaling, with a discrete boundary. Aboveτev, we see a flattening slope indicating a constant critical rate, but there is also no longer such a clear boundary between low and high probabilities of extinction. Compare to figure 4.
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