Figure 4.

Map of responses to perturbations (Box 3). Some food chains display bottom-up patterns where variations in abundance are correlated across all trophic levels (Frederiksen et al. 2006). Other chains display top-down trophic cascades, with anti-correlations between adjacent levels (Tronstad et al. 2010). These correlations can be measured in directional trends, resulting from press perturbations such as nutrient enrichment, or in undirected population fluctuations. (a) Long-term response to a press perturbation. The matrix V_ij measures the relative biomass change ΔB_i/B_i in response to a relative change of growth rate Δg_j/B_j. (b) Covariance of fluctuating time series. Applying a stochastic perturbation on one level k at a time gives a covariance matrix $C_{ij}^{(k)}$, different for each perturbed level k. The diagonal element $C_{ii}^{(k)}$ is the squared coefficient of variation (CV) of level i. The CV of the perturbed level is highlighted for comparison. In (a) and (b), colours (and arrows on the side diagrams) represent the sign and strength of the response, rescaled here by the largest coefficient of each matrix. We see that the main diagonal axis of the map, λ = ma×εa (the feedback of a level on itself through its predators), determines the nature of the response. Matrices V and C^(k) are both invariant along the other diagonal κ = εa/ma, which controls the chain’s top-heaviness in Fig. 3. This invariance does not hold for absolute stability metrics, see Fig. S1. In the bottom-up region (λ≪1), the perturbed species is the one that responds most strongly, and perturbations only propagate upward, as illustrated by C⁽¹⁾ and C⁽⁴⁾ where we affect either the basal or top level. In the top-down region (λ≫1), a trophic cascade pattern (anti-correlated levels) is seen in V and C^(k). We also find an alternating pattern in the CV, with the top level being least variable and its prey being most variable, no matter which level is perturbed (Shanafelt & Loreau 2018).