## Introduction

Mutualistic networks of plants and their pollinators are key promoters of terrestrial biodiversity1 and crucial for our society’s food security2,3. Unfortunately, the introduction of alien species into native ecosystems, together with climate change, widespread application of pesticides, habitat loss, and degradation severely threatens the integrity of these systems and their critical ecosystem services2,4. Despite the importance of plant–pollinator networks, we still lack predictive understanding of the factors and mechanisms determining their invasibility5,6 and the subsequent effects on native species7,8. Fortunately, such understanding of invasions in complex food webs has recently increased9,10. Here, we build on those efforts by further developing consumer–resource theory to elucidate the determinants of invasion success and the impacts of alien pollinators in complex plant–pollinator networks.

Several challenges inhibit a better understanding of the processes an properties of the plant–pollinator systems that determine the success and subsequent impacts of alien invaders on native species. For example, the high mobility of pollinators limits the duration of population experiments11,12 which obscures critically important longer-term impacts of alien species on native communities13. Still, researchers have empirically evaluated impacts of one-third of the 80 bee species introduced as pollinators to date12. Although most of that evidence is inconclusive12 and methodologically suspect11,13, many studies of the honeybee, Apis mellifera, illuminate several potential impacts that alien pollinators may exert on native communities. One such impact is increased competition for floral resources inflicted on native pollinators14,15,16,17,18 though little or no effect on native populations through shared resources is also frequently observed13,19,20,21. Similarly, several studies show that introduced honeybees reduce the reproductive success of native plants22,23,24,25, while others demonstrate that honeybees effectively pollinate native plants26,27,28,29.

Despite the inconclusive and seemingly contradictory results, invasion studies of food webs10 suggest that the potential effects of alien pollinators on ecological networks may be successfully predicted based on the characteristics of the alien species and its host community. For example, previous theory13,30 predicts that extraction of substantial amounts of shared limiting resources by aliens may increase the partitioning or decrease the abundance of resources and extirpate native populations. Alternatively, if the resources extracted by the invader are minor in quantity or otherwise not limiting or not shared with natives, native pollinators may be unaffected13,30. Regarding plants, theory predicts30 that alien pollinators may affect native plants negatively or positively depending on whether the aliens act as major pollinators, secondary pollinators, or floral parasites of the plants. Although these predictions constitute important advances, they insufficiently consider the complex networks of plant–pollinator interactions that determine the dynamics of those systems31,32,33,34,35. More explicit and quantitative network theory suggests that impacts of pollinators depend on floral resources being shared by pollinator species and also on the effectiveness of plant reproductive services provided by pollinator species33,34.

To explore these issues here, we use a dynamic consumer–resource approach that incorporates adaptive foraging of pollinators to mechanistically model pollinators’ consumption of floral rewards and reproductive services to plant species33,34. We simulate species introductions into models of plant–pollinator networks and compare the same networks before and after alien species introductions9,10. This comparison provides a more complete understanding of the invasion process and its effects on the native system. Specifically, we address (1) how traits of alien pollinator species determine invasion success, (2) how network structure affects community resistance to species introductions, and (3) how network structure and traits of alien pollinator species interact to determine impacts on native species. We find that species traits of alien pollinators alone predict their introduction success into plant–pollinator networks, while information on the network structure is needed to predict the impact of invading pollinators on native species. Our results mimic several disparate observations conducted in the field and potentially elucidate the mechanisms responsible for their variability.

## Results

### Overview

We use several common terms to designate the final density of introduced aliens (Methods). “Successful” aliens maintain their density above the pollinators’ extinction threshold (10−3) through to the end of the simulations. “Unsuccessful” aliens venture below that threshold and are removed from the network. “Naturalized” aliens maintain their density in-between the extinction threshold (1.0 × 10−3) and their initial density (1.5 × 10−3). If aliens increase their density to above 0.5 (i.e., more than 333 times its initial density), they are considered “invaders”. No aliens have densities between 0.5 and their initial densities at the end of our simulations. Overall, we find that the abundance of aliens increases with their number of interactions for efficient but not for average foragers (Supplementary Fig. 1).

### Explanatory mechanisms

Invaders either weakly increased or decreased the total density of native plants depending on the balance between the pollination services the plants directly receive from the invaders and the services the plants stop receiving from native pollinators lost due to the invasion. This balance becomes negative when many of the native pollinators contributing to plant reproduction go extinct, which happens at high invaders’ density and in networks with many specialist pollinators. At low-to-moderate invader density within networks of few specialist pollinators, invaders weakly increase native plant density by increasing the reproduction of native plants without outcompeting native pollinators that also contribute to the plant reproduction.

## Discussion

Our consumer–resource approach to complex plant–pollinator networks provides a first step to understand and predict invasion success and impacts on native species of alien pollinators based on the aliens’ traits, the network structure of plant–pollinator communities, and the adaptive behavior of alien and native pollinators. This approach allowed us to explicitly study mechanisms behind the invasion process of alien pollinators. Studying such mechanisms in the field is very challenging due to the high mobility of pollinator species11,12.

Our results mimic many of the disparate observations conducted in the field while potentially elucidating the mechanisms that may be responsible for variability (and apparent contradictions) in the empirical results. In particular, our results closely mimic the empirical findings of the honeybee’s invasion process and impacts on natives, as well as the displacement of the Patagonian bumblebee by European bumblebees36 (Bombus terrestris and Bombus ruderatus). Like the honeybee and European bumblebees, the invaders in our simulations were highly efficient foragers. Our simulations support the empirical findings of (1) European bumblebees outcompeting their native congeners with very similar niches for floral resources36, (2) honeybees negatively impacting native pollinators through increased competition for floral resources14,15,16,17,18, (3) little or no effect of invasive pollinators on native pollinators13,19,20,21, and (4) hump-shaped effects of the abundance of alien pollinators on native plants37. We find that the network structure of the host community mediates the presence and strength of impacts on native pollinator by an invader like the honeybee. For example, we expect weaker (or no) effects of honeybees on native pollinators in networks where the most-generalist plant is weakly connected and where a low fraction of native plant species is visited by the honeybee. Similarly, our simulations support empirical studies showing that introduced honeybees reduce the reproductive success of native plants22,23,24,25 and also other studies demonstrating that honeybees effectively pollinate native plants26,27,28. Our results suggest a resolution to this apparent contradiction by demonstrating that invasive pollinators can either increase or decrease the reproduction of native plants depending on the invaders’ density and the fraction of native pollinators specialize on only one plant. Additionally, our work supports the previously described hump-shaped effects of the abundance of alien pollinators on native plants37 and suggests that this pattern can be effectively explained with consumer–resource mechanisms operating within a complex network. While floral damage may also reduce floral rewards37, our results suggest that such damage is not necessary. Finally, our results do not support previous theoretical work, suggesting that pollinator species indirectly benefit each other by sharing mutualistic partners32 (i.e., plants). Our results rather support the absence of such positive effects in empirical records12. Such indirect positive effects among pollinator species do not occur in our model because the depletion of floral rewards strongly decreases native pollinator density even when aliens increase the density of plants pollinated by those natives. Still, pollinator density only increases plant density until processes in the plant life cycle other than pollination such as seedling recruitment or adult survival limit plant density.

Our study may help to productively focus empirical research on a few of many different factors influencing invasion success and impacts of alien pollinators. These few factors are aliens’ foraging efficiency, diet overlap with native pollinators, fraction of plant species visited by the alien pollinator, connectivity of the most-generalist plant in the host community, and fraction of native pollinators with only one interaction. Such focus informs trait-based prediction in ecology38 by suggesting that explanations based on species’ traits alone may be limited. In particular, our results show that traits of alien species (foraging efficiency) predict the invasion success of alien pollinators, while traits of networks are needed to predict the impact of invading pollinators on native species. Research that continues this trend of incorporating biological processes such as consumer–resource interactions, reproductive services, and species traits into the study of ecological networks may further help scientists discover other informative, surprising, and profoundly counterintuitive behaviors of complex ecological networks.

## Methods

### Network structures

Following previous dynamical studies of ecological network dynamics34,39, we distinguish two fundamental components: the structure of the networks and the dynamics occurring in those networks. The structure of a network broadly describes which links are present or absent between all plant and pollinator species in a system irrespective of the strength of the link. The dynamics occurring within plant–pollinator networks consist of changes in the abundance of the interacting species and/or the strength of the interactions, that is, changes in the values of the nodes and/or links, respectively. The network structures we used are those of 1200 networks previously generated34 using Thèbault and Fontaine’s stochastic algorithm39. This algorithm randomly and independently assigns each plant i and pollinator (animal) j the respective interaction probabilities of P Pi and P Ai drawn from a power-law distribution of degree −2. This creates relatively few generalist species that interact with many other species and many specialist species that interact with few other species as is typically seen in empirical networks31. Then, with a probability “pnest”, each species’ interacting partners are sequentially chosen from all potential partners with a probability $$\frac{{P_{P_i}}}{{\mathop {\sum}\limits_{k = 1}^{S^P} {P_{P_k}} }}$$$$(or\frac{{P_{A_j}}}{{\mathop {\sum}\limits_{l = 1}^{S^A} {P_{A_l}} }})$$. With a probability of 1-pnest, partners are chosen with a probability 1/SP (or 1/SA), where SP (or SA) indicate the number of plant (or animal) species. When “pnest” is high, the algorithm generates more-nested networks.

Our generated networks exhibit empirically observed patterns of species richness (S) inversely varying with connectance (C = L/(SA * SP) where L = number of links). This pattern was followed by stochastically generating three sets of 400 networks with each set broadly centered at three combinations of S and C: S = 40 and C = 0.25, S = 90 and C = 0.15, and S = 200 and C = 0.06. Half of the networks within each set were significantly nested. Nestedness of these networks (NODFst40) varies from −0.33 to 2.3 which includes the empirically observed range of nestedness (−0.37 to 1.3, ref. 34). The ratio of the number of animal to plant species (SA/SP) also matched the empirically observed mean41 of ~2.5.

### Network dynamics

We simulated the dynamics within our 1200 networks using Valdovinos et al.’s consumer–resource model34 of population dynamics with adaptive foraging. This model describes the population dynamics of each plant and animal species, the dynamics of the total floral rewards of each plant species, and the adaptive dynamics of the per-capita foraging preferences of each pollinator species for each plant species. The model parameters are described below and in Supplementary Table 1 along with their units. The model calculates the change of the density (p i ) of plant individuals, each with a single flower, of species i over time as

$$\frac{{{\rm{d}}p_i}}{{{\rm{d}}t}} = \gamma _i\mathop {\sum}\limits_{j \in A} {e_{ij}\sigma _{ij}V_{ij}} - \mu _i^Pp_i$$
(1)

where the first and second terms on the right represent population gains and losses, respectively. The realized fraction of seeds that recruit to adults is γ i

$$\gamma _i = g_i\left( {1 - \mathop {\sum}\limits_{l \ne i \in P} {u_lp_l} - w_ip_i} \right)$$
(2)

where g i is the maximum fraction of seeds that can recruit to adulthood. We subject g i to both interspecific (u l ) and intraspecific (w i ) competition with u l  < w i . e ij in Eq.  (1) is the constant expected number of seeds produced by a pollination event. We address the impacts of pollinator sharing on plant fitness by calculating σ ij , the fraction of visits of animal j to plant i that successfully pollinate plant i

$$\sigma _{ij} = \frac{{\varepsilon _iV_{ij}}}{{\mathop {\sum}\limits_{k \in P_j} {\varepsilon _kV_{kj}} }}$$
(3)

where ε i is the pollen production of plant i and V ij is the frequency of visits by animal species j to plant species i

$$V_{ij} = \alpha _{ij}\tau _{ij}a_jp_i$$
(4)

where V ij  = 0 if plant i and animal j do not interact. The dimensionless function discussed further below, 0 ≤ α ij  ≤ 1, is the foraging preference of pollinator j on plant i. τ ij is the pollinator’s visitation efficiency on plant i, which corrects for units and is fixed at 1 in this study. μ i P in Eq. (1) is the constant density-independent per-capita mortality rate of plant i.

The change of the density of pollinator individuals (a j ) of species j over time is

$$\frac{{{\rm{d}}a_j}}{{{\rm{d}}t}} = \mathop {\sum}\limits_{i \in P} {c_{ij}} V_{ij}b_{ij}\frac{{R_i}}{{p_i}} - \mu _j^Aa_j$$
(5)

where c ij represents the constant per-capita conversion efficiency of pollinator j converting plant i’s floral resources into j’s births. b ij is the constant efficiency of pollinator j extracting plant i’s floral resources (R i ) whose change over time is

$$\frac{{{\rm{d}}R_i}}{{{\rm{d}}t}} = \beta _ip_i - \varphi _iR_i - \mathop {\sum}\limits_{j \in A_i} {V_{ij}b_{ij}\frac{{R_i}}{{p_i}}}$$
(6)

where β i is plant i’s per-capita resource production rate and ϕ i is a constant self-limitation parameter. μ j A in Eq. (5) is pollinator j’s constant density-independent per-capita mortality rates.

Adaptation of pollinator j’s foraging preference on plant i (α ij in Eq. (4)) is

$$\frac{{{\rm{d}}\alpha _{ij}}}{{{\rm{d}}t}} = G_j\alpha _{ij}\left( {c_{ij}\tau _{ij}b_{ij}R_i - \mathop {\sum}\limits_{k \in P_j} {\alpha _{kj}c_{kj}\tau _{kj}} b_{kj}R_k} \right)$$
(7)

where G j is the basal adaptation rate of foraging preference and Σα ij  = 1 for all plants that each pollinator j visits. Pollinator j allocates more foraging effort to plant i whenever such reallocation enhances j’s food intake.

The parameters for plants, including competition coefficients, floral reward productivity, and mortality stochastically vary among plant species within 10% of constant values. The non-structural parameters for constraining pollinator dynamics, including visitation efficiency and mortality do not vary among native pollinator species. This parameter choice allowed us to disentangle the effect of network structure from the population dynamics of native pollinators, which were more sensitive to invasions than plants. We ran our model for each of our 1200 networks for 10,000 time steps and then measured several topological properties and dynamic variables described below as response variables. At t= 10,000, we stopped the simulations to introduce an alien species and then ran the model for another 10,000 time steps, after which we remeasured the response variables. Most networks achieve stable equilibrium at around 3000 time steps. Running the model for longer ensures that transient dynamics minimally affect the differences between the network dynamics before and after introductions. Sensitivity analyses of the dynamic model have been performed in previous studies34,39, and the main results presented here (i.e., foraging efficiency of alien pollinators is sufficient to predict invasion success, while information on the network structure is also required to predict the invaders’ impact on natives) are qualitatively robust to variation in parameter values.