Fig 1.
Dynamics of pollinator communities under external stressors.
(A) Causal dynamics of the adaptive foraging model. Plants and pollinators boost each other’s abundance through mutualistic interactions; pollinators receive resources from plants and plants are pollinated by pollinators. Interactions are specified by a bipartite plant-pollinator nested network. The strength of the interactions changes over time through adaptive foraging. Pollinators adapt their investment in connected plant species based on the supply-demand ratio of plant resources. Besides, species experience intraguild competition and an intrinsic growth rate. The system is subjected to a driver of decline (dA)—an external stressor that negatively influences the intrinsic growth rate of all pollinator species. (B) The mutualistic network has a nested structure with a few highly connected species (generalists) and many sparsely connected species (specialists). The edge colors show the weights of the adaptive foraging matrix α, denoting the investment of each pollinator species in plant species. (C) Hysteretic impact of driver of decline dA (without adaptive foraging). At a critical value of the driver of decline, for any increase of the driver of decline, the pollinator community undergoes bifurcation-induced tipping. At this tipping point, the pollinator community collapses—all pollinator species go extinct (very low abundance). By lowering the driver of decline, the system has another bifurcation-induced tipping point, where, for any decrease of the driver of decline, it recovers at the point of recovery where the first species are reintroduced.
Fig 2.
Species persistence collapses for high rates of environmental change within environmental ranges that are otherwise presumed safe.
(A) Three scenarios are represented for communities without adaptive foraging, perturbed to start with a low initial species abundance (out of equilibrium). (Inset) Stress in communities increases over time as the driver of decline increases at different rates, λ, up to a maximum value, . The black line represents the point of collapse above which a fixed value of the driver of decline leads to the collapse of all communities and below which some communities are sustained. The maximum value of the driver of decline in each simulation is denoted by the fraction θ of the point of collapse
. (A) Dotted orange line represents an increase in the driver of decline up to 90% of the critical value,
, dot-dashed green line an increase up to 50%, and dashed pink an increase up to 20%. In this panel, species persistence is calculated as the fraction of pollinator species alive relative to the number of species alive at the lowest rate of change measured (λmin = 10−4). (B) The persistence of species decreases as a function of the maximum value of the driver of decline, represented as a fraction θ of the point of collapse, for a fast rate of change (λ = 1). Communities without adaptive foraging see a critical transition in species persistence when the driver of decline increases to a value close to, but lower than, the point of collapse
at a fast enough rate. In this panel, species persistence is calculated as the fraction of pollinator species alive relative to the number of species alive at θ = 0 (no external stressor). Initial species abundance Sinit = 0.1 for all simulations. The results are averaged over 100 feasible networks, for which all 15 plant and 35 pollinator species survive under no stress, with the bands representing the first to third quartile ranges. Other parameters in Table A in S1 Text.
Fig 3.
Pollinator communities with adaptive foraging still collapse at high rates of change but less abruptly in the extent of environmental change.
Figure equivalent to Fig 2, but considering adaptation and resource congestion. Adaptive communities respond to an increasing driver of decline by reweighing their connections. (A) Rate-induced transitions are still present, with some communities exhibiting rate-dependent tipping at 50% of the point of collapse. Non-monotonicity is within the error range, thus, non-significant for the number of simulations. (B) Overall, pollinator persistence is more sensitive to rates of change in a larger domain of changes in the driver of decline, θ, than for communities without adaptive foraging. Some particular networks see an increase in persistence, especially for small changes and low rates of change, leading to distinct relative persistence levels above 1. For all simulations, initial species abundance of Sinit = 0.1. Adaptation strength of ν = 0.7 and resource congestion q = 0.2. The results are averaged over 100 feasible networks, for which all 15 plant and 35 pollinator species survive under no stress, with the bands representing the first to third quartile ranges. Other parameters in Table A in S1 Text.
Fig 4.
Higher rates of change in the driver of decline lead to extinction at lower values of the driver of decline only for perturbed communities.
The figure shows the value of the driver of decline dA at which all pollinators go extinct, , as a function of the rate of change λ of the driver of decline. For a low initial pollinator abundance (left panels), after a critical value of the rate of change λ,
has a nonlinear response. This effect disappears for higher resource congestion q, while it increases with stronger adaptation. For a high initial pollinator abundance,
increases monotonically with the rate of change λ. The results are averaged over 100 feasible networks per case (lines), shown with the standard deviation across networks (band). ν = 0.7 for the case with adaptive foraging. The initial abundance for all species is Sinit = 0.1 for the low initial abundance condition and Sinit = 2 for the high initial abundance condition. Other parameters in Table A in S1 Text.
Fig 5.
The coevolution of foraging effort and species abundances under increasing driver of decline dA.
The top row shows the evolution of pollinator and plant abundances under linearly increasing driver of decline dA with rate λ = 0.05, starting from a low abundance condition of Sinit = 0.1, with adaptation strength of ν = 0.7 and resource congestion q = 0.2. Each line style and color combination represents a single pollinator species in all graphs (except the plant abundance graph, top right). For example, there is one pollinator species with degree 3. Since the degree is 3, there are three solid blue lines (one for each connection to a plant species). Another example is the two pollinator species with degree 4, thus showing eight lines (four solid lines for one species and four dashed lines for another species). Since the values of the foraging effort α for each individual species add up to 1, the evolution of the foraging effort α is not shown for pollinators with degree one since they have a constant α = 1 to their single connected plant species. Pollinators with high degree rapidly become the most abundant. Furthermore, the foraging effort α drastically changes—especially around 10 time units when most species reach their peak abundance. The two pollinator species with degree 9 survive the longest and also have one plant species in which they invest most of their foraging effort after 10 time units. Other parameters in Table A in S1 Text.
Fig 6.
Influence of adaptive foraging and resource congestion on the bifurcation diagram.
Adaptation changes the bifurcation diagram. The collapse is less abrupt for pollinator communities with adaptive foraging. Resource competition decreases the size of the bistable area if there is no adaptive foraging. S1 Text Figs B-E show the full bifurcation diagrams of all individual pollinator species for different settings of ν and q. The results are averaged over 100 networks per parameter setting, with the error bands showing the standard deviation. Parameters in Table A in S1 Text.
Fig 7.
Adaptability and resource congestion affect hysteretic patterns and the viability of plant-pollinator networks.
For adaptive pollinators, intermediate levels of resource congestion increase the overall persistence of ecological networks. The point of collapse and recovery of pollinator species increases as a function of resource congestion q without (A) and with adaptive foraging (B). For low resource congestion, the system possesses bistable states which disappear after a critical value of the resource congestion. Resource congestion also affects the feasibility of the networks—networks for which all 15 plant and 35 pollinator species survive under no stress. An intermediate level of resource congestion is required for the adaptive model to produce feasible networks. The orange arrows indicate the resource congestion strength q chosen for the simulation of Figs 2 and 3. These values were chosen such that the systems possess bistable states—as observed in the non-overlapping points of collapse and recovery—and have a high fraction of feasibility. For low resource congestion, adaptability increases the range of drivers of decline at which pollinator communities do not collapse, increasing resilience in the Holling sense [41]. (A) ν = 1 and (B) ν = 0.7. The results are averaged over 100 networks per value of resource congestion q with the error bars showing the standard deviation. Other parameters in Table A in S1 Text.