Abstract
Shifting ecosystem disturbance patterns due to climate change (for example, storms, droughts and wildfires) or direct human interference (for example, harvests and nutrient loading) highlight the importance of quantifying and strengthening the resilience of desired ecological regimes. Although existing metrics capture resilience to isolated shocks, gradual parameter changes, and continual noise, quantifying resilience to repeated, discrete disturbance events requires different analytical tools. Here, we introduce a mathematical flow–kick framework that uses dynamical systems tools to quantify resilience to disturbances explicitly in terms of their magnitude and frequency. We identify a boundary between disturbance regimes that cause either escape from, or stabilization within, a basin of attraction. We use the boundary to define resilience metrics tailored to repeated, discrete perturbations. The flow–kick model suggests that the distance-to-threshold resilience metric overestimates resilience in the context of repeated perturbations. It also reveals counterintuitive triggers for regime shifts. These include increasing the periods between disturbance events in proportion to increases to disturbance magnitude, and—in systems with multiple dynamic variables—increasing time periods between disturbances of constant magnitude. Sustainability depends on the resilience of natural, social and engineered systems. This theoretical study quantifies resilience to repeated disturbances, synthesizing understanding of how the sizes of shocks, or ‘kicks’, and recovery, or ‘flows’, contribute to maintaining systems in desirable states.
Highlights
Climate change and other human impacts are altering disturbance patterns in Earth’s systems
Using an example from fisheries, we show the shortcomings of resilience metrics based in state space for detecting this type of resilience, introduce the flow-kick model of disturbance, and propose new resilience metrics based in what we term disturbance space
In the context of a lake eutrophication model, we describe connections between this new framework and existing resilience metrics, and generalize the approach to include stochastic kicks and recovery times
Summary
Climate change and other human impacts are altering disturbance patterns in Earth’s systems. The match stems from the fact that as the flow time and the kick decrease toward zero in a fixed ratio (an average nutrient addition rate), the flow-kick system limits to a continuous system with P input continuously augmented by this average rate (Supplementary Information Section 3) These echoes of existing resilience metrics in the resilience boundary R reiterate that distance-to-bifurcation gives a good measure of resilience to nearly continuous disturbance (small kicks and short flow times, such as those in box D3 in Figure 4b), while the distance-to-threshold indicator approximates resilience to very rare disturbances (large flow times). The phenomenon in which increasing recovery time between disturbances triggers escape from a basin (Figure 5d) does not occur in models with one state variable, but this climate example alerts us to the possibility of similar behavior in ecological and other systems with multiple state variables
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