S1 Metrics and Components of Resilience

Supplementary Material

Towards a comparable quantification of resilience
Ingrisch, Johannes; Bahn, Michael

Institute of Ecology, University of Innsbruck, Sternwartestr. 15, 6020 Innsbruck, Austria
Corresponding author: Bahn, Michael ()

S1 Metrics and components of resilience

Figure S1: (A) Hypothetical response of system state S to a disturbance (orange box). The impact at the end of the disturbance is the difference of S and the baseline St0 (=pre-disturbance state). The rate of recovery R after the disturbance determines the recovery time rt, i.e. the time after the disturbance needed by the system to return to its baseline. The perturbation P is the difference between system state and baseline integrated over the recovery time. (B) The response trajectory, normalized to its baseline. The baseline-normalized impact denotes the fraction by which system state is changed at the end of the disturbance. The perturbation during recovery Pbasecan be measured as the area between the trajectory and the normalized baseline. The baseline-normalized rate of recovery Rbase denotes the fraction of the baseline by which the system recovers per unit of time. (C) The impact-normalization relates the system state to the impact . Consequently, the impact-normalized rate of recovery denotes the fraction of impact recovered per unit of time.

Table S1 Definition of abbreviations

Absolute Response

For definitions and explanations of abbreviations also see Figure S1A and Table S1.

The impact of the disturbance on the ecosystem statecan be calculated as

(1)

Where St0 is the baseline (=pre-disturbance) state of the system and Sti the system state at the end of the disturbance (time ti).

The post-disturbance change of the system state at time tx is defined as

(2)

The rate of recoveryR describes the post-disturbance change of the system per unit of time

(3)

The recovery time rt is the time it takes for a system to recover to the baseline state after the disturbance

(4)

The recovery time depends on the impact of the disturbance and the recovery rate R

(5)

The absolute perturbation P (grey triangle Figure S1A) depends on the impact and the recovery time rt

(6)

Baseline-normalization

The response of the system state S can be normalized by its baseline St0 (Figure S1B).

(7)

The baseline-normalized impact normalizes the impact by the baseline:

(8)

Likewise, the post-disturbance change of system state can be normalized by the baseline, i.e. the system sate of the system prior to disturbance

(9)

We define the rate of baseline-normalized recovery Rbase as

(10)

The recovery time rt can be calculated as

(11)

The rate of baseline-normalized recovery denotes the fraction of the baseline that is recovered per unit of time.

The perturbation from the baseline Pbase (grey triangle in Fig S1(B)) is defined as

(12)

Impact-normalization

The post-disturbance change of system state can be normalized by the impact of the disturbance (Figure S1C)

(13)

The rate of impact-normalized recovery is defined as

(14)

At the time tx it can be calculated as:

(15)

The recovery time can be calculated as:

(16)

Thus, the impact-normalized rate of recovery is the reciprocal of the recovery time (Fig S2). The rate of impact-normalized recovery corresponds to the fraction of impact recovered per unit of time.

Figure S2: The relationship of rate of impact-normalized recovery and recovery time.

The impact-normalized perturbation (grey triangle, Figure S1C) can be calculated as

(17)

S1 Alternative representation of metrics in the proposed bivariate scheme

Figure S3: Representation of resilience composed of baseline-normalized impact Impactbase and recovery rate Rbase for (A) hypothetical response trajectories from Figure 2A and (B) response trajectories from Figure I.Background colors represents the amount of baseline-normalized perturbation.Dashed arrows point to the rate of impact-normalized recovery, which is represented as third axis. Response trajectories with similar Rimpact, and thus recovery time, align along identical arrows (e.g. Case 4, Reference, Case 9 in (A); c and d in (B)).