Fig 1.
Multistability in the plant-pollinator network.
(A) Raw network structure of an empirical mutualistic network from actual observation data [40]. (B) The proportion of different stable states (y-axis) with random removal of a certain percentage of pollinators (x-axis). Total 1000 sub-networks are simulated in each column except for the first (0% represents the original network). (C) A typical plant-pollinator network after removing 80% of pollinators, in which tristability occurs. (D) The abundance of 17 plant species and 13 pollinator species (indicated by each column) in low, intermediate and high stable states using default parameters. The default values are set as follows: ,
,
, κ = 1.07, γ0 = 1, δ = 0.5, h = 0.2, μ(P) = μ(A) = 0.001. (E) Illustration of seven possible stable states, including monostable state, bistability and tristability. H: high state, HI: high-intermediate bistability, HL: high-low bistability, HIL: high-intermediate-low tristability, IL: intermediate-low bistability, L: low state, T: tetrastability.
Fig 2.
Multistable landscape and transition path reveal the role of intermediate state.
(A) The phase diagram under parameter variation (κ=0.5∼1.5, γ0=0.75∼1.25). The same notations apply as in Fig 1E. I: intermediate state. (B) Landscape and transition paths after projection onto new coordinates in the tristable system (κ = 1.1, γ0 = 1). The white lines indicate the collapse process while the magenta lines represent the recovery process. The solid and dashed lines correspond to direct and indirect transition paths, respectively. (C) Multidimensional transition paths between different attractors after normalization, where two left pictures are indirect paths (through the intermediate state) from high to low state (top) and low to high state (bottom); conversely, the two on the right are direct paths. Each row represents one of 30 species in the network and the upper 17 rows are plants while the lower 13 rows are pollinators.
Fig 3.
Landscape changes with parameters κ and γ0.
All pictures share the coordinates of κ=0.99 and γ0=0.94, which are uniquely labelled for simplicity. For each landscape, small and large values of plant PC1 / pollinator PC1 correspond to low state (lower left corner) and high state (upper right corner) respectively, and moderate value is related to the intermediate state. Blue color indicates the basin of attraction corresponding lower potential energy or higher probability, while yellow color indicates high potential energy or lower probability. The transition paths between different stable states are also displayed, as in Fig 2B.
Fig 4.
Consistency among barrier height, transition action and MFPT.
(A-C) A line graph of RBH between stable-state pairs with varying κ (A: intermediate and high state; B: low and high state; C: low and intermediate state). We also label the turning point (purple point), where two stable states have the same potential energy value (RBH = 0). (D-F) The difference for the minimum actions between the forward and backward transitions (ΔS) vary with κ (D: intermediate and high state; E: low and high state; F: low and intermediate state). (G) Approximate linear relationships between RBH and log(MFPT) (blue line), as well as between ΔS and log(MFPT) (green line).
Fig 5.
Barrier height serves as a new EWS to predict collapse.
(A and B) The process of system collapse (A) and recovery (B) is simulated from the Langevin equation. We present two-dimensional landscape, in which the same coordinates are used to ensure comparability. The left arrow indicates the proportion of each state (H: blue, I: green, L: gold). The transitions between attractors are marked by thick arrows. (C) Calculated RBH between low and intermediate states with increasing κ. We regard the point as an EWS when the BDS statistic for the sequence up to this point is significant (blue: p<0.05, gold: p<0.01, red: p<0.001). (D) For complete collapse to the low state, the RBH can serve as the earliest warning signal compared with other metrics based on time series, such as AR(1), variance (Var), coefficient of variation (CV) and fano factor (fano). The color in RBH has the same meaning as (C), indicating the predicted critical κ value based on different p-values.
Fig 6.
Global sensitivity analysis for parameters identifies the key factors.
The parameters are perturbed 10% from the default value. The left panel corresponds to the transition between intermediate and low states, with parameter perturbation in the direction of system collapse, and the right panel represents the opposite perturbations. α*, β*, κ* and denote perturbations of the only surviving pollinator in the intermediate state on the intrinsic growth rate, intraspecific competitions, average decay rate and per capita mutualistic strength respectively. Since rising 10% of γ0 makes the system exhibit high state only, the action is 0 for intermediate to high state transition, but infinity conversely. The corresponding relative changes are -1 and infinity (denoted as 2.5).
Fig 7.
Changes in ecosystem state from ongoing environmental degradation illustrated by potential energy landscape.
The global decay in pollinators caused by the damage results in landscape topography changes, and further leads to transitions between stable states. The ecosystem state is represented by the value of plant PC1, and the true potential energy is obtained from actual simulations. The phase diagram versus κ is also shown.