Table 1 Descriptions of terms.

Ecosystem stability and function

We use a range of complementary metrics in a broad sense⁵⁰ to assess stability in terms of speciesʼ ability to persist with limited popuation variabity and function in terms of stocks and flows of biomass. We apply the metrics both to the whole ecosystem (all species in the network) and to individual guilds in the network at near steady-state dynamics of our simulations.

Biomass variability (CVs)

Coefficient of Variation (CV = standard deviation/mean) of species’ or guilds’ biomass evaluated over the last 1000 timesteps of the simulations when their dynamics are approximately at steady state. Increased temporal stability corresponds to decreased biomass variability.

Consumption rate

Total amounts of biomass extracted by consumers per unit time.

Diversity

Number of species within a network. Networks of initial diversity S are subjected to dynamical simulations, during which species may go extinct or persist. The resulting number of species is the final diversity, or simply diversity.

Guilds

Guilds are groups of species with similar types of consumer−resource interactions including: plants without pollinators; plants with pollinators; floral rewards of plants with pollinators (when relevant to analyze their biomass and flows separately from vegetation, Fig. 1); herbivores: species that only eat plant vegetation in the original niche-model food web (Fig. 2a); omnivores: species that eat vegetation and animals in the original niche-model food web (Fig. 2a); carnivores: species that eat only animals; added (+) herbivores/pollinators: herbivores or herbivorous pollinators added by the RO or RP treatments that consume rewards (multiplex treatments) and/or vegetation (FW treatments; Fig. 2, yellow-green nodes); added (+) omnivores/pollinators: omnivores or omnivorous pollinators added by the RP treatment that consume rewards (multiplex treatment), animals, and/or vegetation (Fig. 2e, orange nodes). Added animals include both +herbivores/pollinators and +omnivores/pollinators.

Local stability

The tendency of abundances of species within a system to return to their equilibrium after a very small perturbation⁴.

Persistence

Fraction of species that survive to the end of simulations (=initial diversity/final diversity).

Productivity

Total rates of biomass increase due to plant growth and food assimilated by animals minus loss due to animals’ metabolic maintenance costs and plants’ costs of producing rewards.

Steady state dynamics

Formally, dynamics in which all species have constant abundance (dB_i/dt = dR_i/dt = 0 for all i). At the end of 5000 timesteps, our systems approximate steady-state dynamics (Fig. 3) as quantified by very small variability in total ecosystem biomass over the last 1000 timesteps of the simulations (CVs < 0.0001).

Multiplex networks

Ecological networks that include more than one type of species interaction. Here, we focus on multiplex networks that combine food webs (including carnivorous and herbivorous feeding interactions, Fig. 2a) and pollination networks (including feeding interactions and reproductive services, Fig. 2b).

Rewards Only (RO) treatment

Network construction treatment in which pollinators can access floral rewards of plants w/ pollinators as their only resource (Fig. 2d). These networks are subjected to multiplex dynamics, which include pollination in addition to traditional food web dynamics.

Rewards Plus (RP) treatment

Network construction treatment in which pollinators can access floral rewards plus plant vegetation and/or animal biomass resources (Fig. 2e). These networks are subjected to multiplex dynamics, which include pollination in addition to traditional food web dynamics.

Pollination link or mutualistic interaction

A pollination link or mutualistic interaction between pollinator i and plant w/ pollinator j describes both the consumption of j’s floral rewards by i and the reproductive services provisioning to the vegetative growth rate of j by i (Fig. 1). In the FW treatments, pollination links are switched to links in which i consumes the vegetative biomass of j, i.e. to herbivory links.

Food Web (FW) treatments

Ecological networks with the structure similar to multiplex networks, where all pollination links are switched to herbivory links (also corresponding to zero rewards productivity). In the Rewards Only Food Web (RO FW) construction, animal i is strictly an herbivore (Fig. 2d), while in the Rewards Plus Food Web (RP FW) construction, animal i could be an omnivore or herbivore (Fig. 2e). These networks are subjected to traditional food web dynamics.

Rewards productivity (β)

Parameter in our multiplex model specifying the rate of rewards biomass produced by plants w/ pollinators per unit of their vegetative biomass. “Low” (β = 0.2) and “High” (β = 1.0) are arbitrary values chosen to illustrate two behaviors of the multiplex model compared to traditional food web dynamics. “None” corresponds to Food Web treatments. We interpret rewards productivity as a proxy for the intensity of pollination interactions. See Supplementary Fig. 2 for persistence results across a range of β values.

Feedback control

To test whether transient mutualistic feedbacks isolated from rewards availability lead to the differences between our multiplex and Food Web treatments, we initialized simulations forced with rewards availability from multiplex simulations but with feedbacks (dashed and purple arrows in Fig. 1) turned off. We then observed potential changes in steady-state ecosystem stability and function.