INTRODUCTION

Pollination services are a well-studied example of mutualistic interactions between species and considered essential for regulating ecosystems (IPBES 2016). These interactions can be exclusive between a single plant and pollinator species (e.g. Kritsky, 1991; Pellmyr and Huth, 1994). More often, however, pollination networks consist of hundreds of interlinked species (Primack, 1983; Herrera, 1988), with plants that have dozens of different pollinators (Ollerton, 1996; Waser et al., 1996).

A key aspect in understanding pollination networks is that their structure strongly affects the fitness of the species in the community (Muchhala et al., 2010). While plants and pollinators in generalised networks benefit from a variety of pollen-distributing animals and reward-providing plants, respectively, benefits provided by each partner partly depend on the number of other species that the partner is linked to (Simmons et al., 2019). Generalist pollinators that are shared by many different plant species would be less favoured because they have a more undirected pollen transport (Bell et al., 2005; Wheelwright & Orian 1982), resulting in pollen loss and stigma clogging (Murcia and Feinsinger, 1996; Brown and Mitchell, 2001). The same can be true for the pollinator side: a specialist plant produces food rewards such as nectar exclusively for one pollinator species while sharing a plant decreases the amount of rewards that is available to each pollinating species (Kunin and Iwasa, 1996). Ideally, a species is thus a generalist, linked to many specialised species, but there is no network that satisfies this condition for all parties.

The effect of pollination network structure on the fitness of species suggests that the network structure itself is under the influence of adaptive evolution. Here arises the question about how evolutionary mechanisms can produce networks in the wide range of the specialist–generalist spectrum, when plant and pollinator species often have different interests. The situation is further complicated because plants evolve to attract new pollinators over evolutionary time while pollinators can adapt foraging behaviour within days (Burd, 1995; Tong et al., 2019). Previous modelling studies have linked specialised pollination networks to scenarios in which the density of a certain species is high and constant over time (Waser et al., 1996; Sargent and Otto, 2006). However, these assumptions might not always be met (Carvalheiro et al., 2008) and, even more disputably, these studies only look at the plant or only at the pollinator side and often miss the critical factor played by other species in the community.

Whether plants and pollinators specialise on each other or interact with a range of different species can be regarded as a strategy to increase their fitness. Here, the average fitness of a plant for example, not only depends on its own strategy (or the predominant strategy in its species) but also on the strategies of all the other plant species in the community. Such a community could consist of two plant species, P₁ and P₂, which are pollinated by different pollinator species. At some point, plant species P₁ could evolve to attract a specific pollinator A that is also visiting plant P₂. This change in P₁’s strategy is affecting the fitness of the pollinator A because it will get more food rewards. It will also affect the fitness of P₂ plants because the pollinator will visit more infrequently and waste some of P₂’s pollen on flowers of P₁.

This situation resembles a typical problem in game theory, the mathematical study of decision-making among interacting parties. In a game, the different parties or players can choose between strategies to maximise their payoff. By deriving certain payoff equilibria (Nash, 1950; Selten, 1965), game theory can, for example, explain why individuals or groups might not cooperate with each other, even if it is in their best interests to do so (Hardin, 1968). Game theoretical models have been used to understand ecological and evolutionary processes (Maynard-Smith and Price, 1973) and can offer tools to analyse the evolution of stable network structures (Skyrms and Pemantle, 2000). There is, however, no study that analyses the evolution of entire pollinator networks using a game theoretical approach.

Following previous work on coevolutionarily stable strategies and networks (Matsuda and Namba, 1991; Roughgarden 1996), we developed a new game theoretical model to explore the evolution of stable network structures under different scenarios of pollinator availability. The model uses the different plant and pollinator species of a community as players in a game, and takes into account the different time scales of plant and pollinator adaptation. We analyse small network communities with up to 10 plant and 10 pollinator species.

MATERIAL AND METHODS

The theoretical framework allows analysing the evolution of a community consisting of fast adapting animal species with slowly evolving plant species. A key aspect of the model is that it involves two time scales.

Interactions between plants and animals happen over ecological time. Here, the following events happen in order. First, plants offer nectar or other food rewards to attract pollinators. Each plant species can either be a generalist that does not distinguish between the animal species or it can specialise on a certain number of animal species by reducing food rewards for the others. Second, after the plants made their food offers, animals decide which plants to visit. We assume that decision-making follows optimal foraging. To optimise their food reward, each animal can choose a generalist strategy and visit all plant species with equal probability, or a specialist strategy and visit only one or a certain number of plant species. Third, pollination happens as a byproduct of the animal visits. We assume that each animal individual randomly visits the plant individuals it is linked to. Pollen uptake happens at a first and pollen delivery at a second plant individual.

Plant evolution happens over evolutionary time. We assume that each plant species is a unit of selection and that selection acts on the degree of specialisation towards the different animal species. Fitness is measured by the percentage of conspecific pollen transport. Though we are aware that many other factors affect plant fitness, we argue that conspecific pollen transport is most relevant for our study because it is critically affected by the degree of specialisation in the community. It should be further noted that while the plants can influence the pollinator’s strategy of choice, the pollinators ultimately determine the network structure as their visits are the observable interactions that form the links in the pollination network.

In the theoretical analysis, we search for interaction networks that are stable against both evolutionary changes of plants and behavioural changes of animals. For this, we introduce the term evolutionary stable network structure (ESNS). A formal definition of ESNS and details of the model are given below.

Game theoretical model

In general, a game consists of (1) a set of players, (2) a strategy place consisting of all possible strategies the players can adopt and (3) one or several pay-off matrices that describe pay-off for the players for all situations that are possible in the game. In our model, the players are the N plant and M animal species in the community (we always refer to the species and not plant or pollinators individuals if not stated otherwise).

The plants strategy space consists of all N × M matrices S whose elements s_ij can take either of two values, H or L (1  H > L  0). If plant species i specialises on animal species j, then animals of species j get a high amount s_ij = H of the food reward, whereas all other animal species get a low amount L of the reward. If plant species i is a generalist it attracts all pollinators equally with a high reward H. The strategy space of animals consists of all N × M matrices K whose elements k_ij can take on values 0 and 1. The elements are defined as k_ij = 1 if animal species j is visiting and pollinating plant species i, and k_ij = 0 otherwise. Plant and pollinator strategies in the 2 × 2 game are displayed in Fig. 1.

The payoff of each plant and pollinator species depends on the strategy composition in the community. We therefore consider a fixed set of plant and pollinator strategies (S, K) and calculate the payoff of each species as a function of K and S. We note that matrix K determines the animal visits and therefore the structure of the interaction network.

To first calculate the payoff for the plants, we assume that each plant and pollinator species has a constant population size with p_i individuals for plant species i and a_j individuals for animal species j. We further assume that each plant individual produces a certain amount of pollen (r), which is removed by its visitors with equal probabilities. However, because of limited uptake by each animal individual (u), some pollen may be left on the plants. We assume that pollinators do not differ in pollination service, that is, they all lose or consume pollen at the same rate.

The payoff for each plant species (Q_i) is then calculated as the average amount of its pollen that is transported conspecifically during pollination. The total number of animal individuals visiting plant species i for pollen uptake computes to . Here, is the probability of an animal individual of species j to visit plant species i among all the plant species it is connected to. The sum is then taken over all animal species pollinating plant species i and weighted by the animal population sizes a_j.

The V_i animal visitors remove either all or only a part of the pollen of plant species i. Note that the total pollen production by this plant species is rp_i. As each animal individual has a limited amount of pollen uptake u, the maximal possible pollen uptake from plant species i is uV_i. Accordingly, there is a pollinator shortage for uV_i < rp_i and a pollinator oversupply for uV_i > rp_i. These two cases must be distinguished to calculate the pollination efficiency Q_i.

For a pollinator shortage, it holds that

(1)

The sum on the right-hand side of eqn (1) is the total number of animals that visit plant species i twice in a row, once for pollen uptake and once for pollen delivery. Multiplication by u yields the total amount of pollen that is delivered conspecifically, and further division by rp_i gives the pollination efficiency Q_i.

For pollinator oversupply, the pollination efficiency computes to

(2)

The sum on the right-hand side is the same as in eqn (1). Differences occur with respect to the total amount of conspecifically delivered pollen, which is calculated by multiplying the sum in eqn (2) by rp_i/uV_i. The factor rp_i/uV_i is interpreted as the average amount of pollen taken up by the animal visitors under pollination oversupply. As above, multiplication by u/rp_i yields the pollination efficiency Q_i.

Assuming that one pollinator can remove one pollen package per visit, r = u = 1, we combine both eqns (1) and (2) into a single equation for Q_i:

(3)

Note that the one plant–two pollinator model by Waser et al. (1996), which uses more general terms for visiting frequency and pollination efficiency, can be transformed to describe the plant strategies in a two plant–two pollinator game by rewriting N_j = a_j, V_j = , .

The payoff for the pollinators is measured in food reward. The reward R_j of animal species j is defined as

(4)

The first factor gives the reward for pollinator individuals. If there are more plant than pollinator individuals, each pollinator individual gets a full reward, . If there are more visiting pollinators than plant individuals, each pollinator individual only gets a fraction of n_i. This amount is then multiplied by the pollinator specific reward access s_ij. The last factor , gives the fraction of pollinator species j that actually visit plant species i. This means that the number of plant individuals a pollinator individual can visit stays constant, regardless how many different plants it is linked to.

It can happen that the reward for a pollinator species is the same, whether it visits a certain plant species or not. In this case, we assume that the pollinator always visits the plant to not spend time discriminating between flowers. We thus introduce a small cost ε to the pollinator payoff in case it chooses specialism.

Game theoretical analysis

First, we analyse games with small species numbers, two and four plant and animal species, respectively, by analytically calculating payoffs for plants and pollinators as a function of animal species abundance (a = a₁ = a₂). The ESNS were then found by drawing a decision tree, as shown in Fig. 2, using the method of backward induction as suggested by Osborn and Rubinstein (1994). Backward induction first determines the decisions of the players that decide last – here the pollinators – and then the ones that decide first – the plants. Most stable states can be found like this, for special cases see Fig. S3.

The plant pollinator game for larger species numbers was analysed numerically using a discrete time model implemented in Python 2.7. Simulations start with an initial community in which plant and animal species are randomly linked, and high or low reward access, s_ij, is also distributed randomly. In each time step, there is an evolutionary change in one plant species i that is chosen randomly, and the reward access to a random pollinator species j, is changed from high to low or from low to high. All pollinator species can then adapt to this evolutionary change: they can establish new links with any unconnected plant species or break up existing links with any connected plant species, until they cannot increase their reward any more. After the time step, it is checked whether the evolutionary change in the plant could prevail. Its pollination efficiency Q_i is re-evaluated and the change is accepted if it increased and rejected if it decreased Q_i. If pollination efficiency stayed the same, the change is accepted with a chance of 50%. This procedure is repeated 5,000 times to reach equilibrium.

It should be noted that equilibria reached with the simulations are only stable against stepwise evolutionary changes. Both the simulation and the analytical approach give the same results for smaller networks, but the simulations might only reach local optima for larger networks and are thus repeated several times. All parameters are kept constant during the simulations with r = 1 and u = 1, H = 100% and L = 50% for analysis and simulations.

RESULTS

In the plant–pollinator game with two plant and two animal species, the network could take three possible stable states: both pollinators were specialist (S_A, S_A), both were generalist (G_A, G_A) or one pollinator specialised and the other visited both plants (S_A, G_A) (Fig. 1). We found that two of these three network structures can be evolutionarily stable, but that stability depended on the abundance of pollinators in the community (Fig. 3). The analysis revealed that the evolutionary stable network structures (ESNS) formed three different zones: a zone of generalism at low pollinator abundance (pollinator shortage), a zone where multiple ESNS coexist at intermediate pollinator abundance and another zone of generalism for high pollinator abundance (pollinator oversupply).

Variation in the ESNS resulted from the interplay between pollinators and plants. Pollinators are assumed to optimise foraging. They only have a limited time to collect food rewards and thus try to avoid plants that yield little reward. Plants, however, experience two costs that affect their fitness. These costs are incomplete pollen uptake, which occurs when pollinators visit on too few occasions to remove all pollen, and ineffective heterospecific pollen transport, which occurs when different plant species share pollinators.

When pollinators are overly abundant (zone (iii) in Fig. 3), all pollens are removed in the community and plants do not suffer from incomplete pollen uptake. Therefore, it would be beneficial for the plants to segregate pollinators to avoid heterospecific pollen transport. However, the pollinators are experiencing a ‘flower shortage’ in this case. They thus visit as many plant species as possible to optimise their foraging. This results in a generalised network because it is the pollinators in our model that determine the final network structure.

When pollinator abundance is low on the other hand (zone (i) in Fig. 3), plants suffer from incomplete pollen uptake. In this case, each plant species can increase the amount of dispersed pollen by attracting more animal pollinator species with a high reward offer. The pollinators follow these offers because they optimise foraging and the resulting ESNS are characterised by a high number of links (network G in Fig. 1). Interestingly, the fitness for both plant species would be highest if both specialised on a separate pollinator, as the pollinators would then segregate too, but such a specialised network structure is not evolutionarily stable (Fig. 4). Thus, when pollinators are scarce, plants suffer not only from incomplete pollen uptake but also the inefficient use of pollinators due to a ‘pollination dilemma’.

It is only when plant and pollinator abundance are approximately balanced and neither plants nor pollinators experience a strong shortage that the network can evolve to more specialised states (zone (ii) in Fig. 3). For such intermediate pollinator abundance, there are two stable states: one where pollinators are efficiently used in a community (S in Fig. 3) and one ‘dilemma state’ (G in Fig. 3). Here, the community’s initial network structure determines the course of evolution. If the network starts in the ‘dilemma state’, it will stay there. All other starting conditions lead to an efficient community (Fig. S3).

It should be noted that the ratio of pollinator-specific reward L and H can have a large effect on the evolved network structure, especially in zone (iii) at pollinator oversupply. Keeping H at 100% and varying L simulates the effectiveness with which plants can exclude pollinators from their rewards. If L is very low and the plants can thus effectively exclude unwanted pollinators, the network shifts towards specialism which is more beneficial for the plants. If L is rather high and pollinators can still extract large amounts of reward from specialised plants, the network shifts more towards generalism. See Figure S8 for details.

The analysis of the plant–pollinator game for four plant and four animal species found six ESNS in total (Fig. 5). All these ESNS are regular, as links are equally distributed among plant and animal species, except for I₃. In I₃, all plant species have three links, but the animal species have either two or four links. This results in a slightly higher average pollination efficiency compared to the regular I₄, in which all plant and pollinator species have three links. The reward for the pollinators in I₃ is the same whether they have two or four links.

As above, there are three qualitatively different zones: (1) inefficient networks with high degree of generalism evolve for low animal abundance due to a pollination dilemma, (2) multiple stable network structures ranging from specialism to generalism occur for intermediate animal abundance and (3) inefficient networks with some degree of generalism evolve due to resource competition among pollinators under animal oversupply. Stable ranges of the intermediate ESNS in zone (ii) all overlap but differ slightly in length.

These zones were also found for larger networks with 10 plant species and 10 pollinator species (Fig. 6). Regular networks were checked for stability before simulations were used, to determine evolving network structures starting from randomly connected networks. The simulation results show the same basic trend with the three zones. Some evolved networks, however, do not fall in line with the shown ESNS. This is partly because we only analysed regular networks for stability. Other unregular ESNS exist, as found in the four plant and four animal species system (I₃ in Fig. 5). It is also possible that attracting clusters of unstable structures exist, for example, with plants having a mix of two, three and four links for pollinator oversupply. Networks that take any of the structures within such a cluster could then only evolve to other structures within the cluster and get stuck. Finally, a small number of networks did not reach their stable states despite the high number of simulation steps due to the randomised evolutionary process.

DISCUSSION

In this study, we present a new game theoretical approach to model complex coevolutionary processes and apply it to pollination networks. The evolution of plants to attract or distract pollinators, combined with the pollinators’ adaptive choices of which plants to visit, not only alters the network structure but also modifies the pollination efficiency and, in turn, drives further plant adaptation. The direction and endpoint of this complex coevolutionary process are, according to the model, strongly dependent on the availability of pollination services. More specifically, the specialised interactions only occur when the demand and supply of pollination services are balanced. Communities that experience a pollinator shortage or oversupply will evolve to more closely connected, generalist networks.

From the plants’ perspective, the most efficient network would be a highly specialised network, in which plant species do not share pollinators and can avoid pollen loss and stigma clogging (Mitchell et al., 2009). However, the model predicts that such an efficient network is hard to emerge, due to the game theoretical nature of pollination network evolution. Either pollinator shortage or oversupply pushes the system towards generalism, even though it is less efficient in terms of conspecific pollen transfer. Under pollination shortage, a one-to-one network cannot be maintained, as the plants evolve to attract pollinators that are already utilised by other plants. In communities with pollinator oversupply, the pollinators experience a shortage of plants and rewards and thus drive the network more towards generalism (as observed by Fontaine et al., 2008).

An important insight of our analysis is that the situation for pollinator shortage resembles the famous prisoner’s dilemma (Axelrod and Hamilton, 1981). The prisoner’s dilemma is a two-player game, in which both players have maximal payoff if both cooperate. However, cooperation is not stable because each player can increase its payoff by not cooperating. Once either player becomes non-cooperative, so does the other, ending up in the situation where both players have lower payoff than in the cooperative situation. Our results suggest that plants face a similar dilemma if there is pollinator shortage in the community (Fig. 3 and Fig. 4). We therefore call it ‘pollination dilemma’. Remarkably, the ‘specialisation’ strategy appears to be cooperative in our model, and the ‘generalism’ strategy appears to be non-cooperative. The finding of the pollination dilemma adds a new flavour to community ecology and highlights the potential of game theory for the analysis of network evolution.

The model further predicts that pollination networks may have multiple stable states (Fig. 5 and Fig. 6), often overlooked as specialism–generalism studies tend to focus on two plant, two pollinator systems only (e.g. Waser et al., 1996; Sargent and Otto, 2006; Muchhala et al., 2010; Song and Feldman, 2014). Our results show that when demand and supply of pollination services are approximately balanced, multiple stable states can exist and which state is reached depends on the initial state. Yet, once a stable network structure is reached, it tends to keep this structure despite changes in pollinator abundance, due to their self-sustaining nature. It is only when a tipping point is reached that the system flips between different states. This innate tendency to keep the current state is called hysteresis. It can lead to unexpected regime shifts at the tipping points (Scheffer et al., 2001; Andersen et al., 2009) and could have implications for conservation efforts. The globally observed pollinator loss (Hallmann et al., 2017, Sánchez-Bayo & Wickhuys 2019) might not change network structures for a long time, until plants suddenly have to attract every available pollinator, and drift into a pollination dilemma.

An interesting question would be if the predicted relationship between network specialisation and pollinator abundance is also observable in nature. On the one hand, there is good empirical evidence that the degree of specialisation varies geographically in pollination networks. Comparative studies revealed that specialisation increases from tropical to temperate latitudes (Schleuning et al., 2012; Sakai et al., 2016), and that generalism is more prevalent on islands than on mainlands (Barrett et al., 1996; Kaiser-Bunbury et al., 2010). On the other hand, there is some empirical evidence suggesting significant variations in the availability of pollination services and its temporal changes among regions. Vamosi et al. (2006) showed that regions with a high species richness, which are more prevalent in the tropics (Pianka, 1966), are associated with pollen limitations. Comparative studies on fruit set and species diversity suggest a more pollination-limited plant reproduction in the tropics and on islands (Larson and Barrett, 2000; Tremblay et al., 2005). In addition, the balance of demand and supply of pollinator services may drastically change in certain habitats such as deserts due to asynchronous responses of plants and pollinators to fluctuating environments (Fleming et al., 2001; Wright et al., 2015). It is an important subject of future study to integrate these known geographical variations using the framework presented by this study.

Our theoretical framework is based on some simplifying assumptions. Here, we discuss possible extensions that can be made in future studies. A factor that is likely to affect the pollination network structure, but not dealt with here, is the variation of population sizes among plants and pollinator species. Since a plant species with a higher density in an area, or at a certain season, receives consecutive visits of generalist pollinators more frequently than one with lower density does, it is more likely to stay a generalist. Therefore, variation in population density will cause variation in the degree of specialisation among plants (Sargent and Otto, 2006). We started to look at networks with varying population sizes and food rewards, which resulted in the evolution of more irregular structures, while the general pattern of specialisation depending on pollinator abundance stayed the same (see S3 in Supporting Information). The absence of more irregular structures might also explain the lack of nestedness in our resulting networks, even though nestedness is a common feature of pollination networks (Bascompte et al., 2003).

The role of pollinator behavioural characteristics could be another interesting factor. Some pollinators species show individual preferences for certain flower types that they visit consecutively, and thus have a higher conspecific pollen transfer rate, the so-called flower constancy (Waser, 1989). On the other hand, many pollinators including the honeybee have a lower rate of pollen transfer as they collect pollen as a food source (Brodschneider and Crailsheim, 2010). These behavioural characteristics, together with plant mechanisms which improve pollen carryover (Morris et al., 1995), should affect the results. Moreover, pollinator supply and network structure could dynamically affect each other: A pollinator shortage could raise pollinator and reduce plant populations over time (and the reverse for a pollinator oversupply), pushing the system towards the central zone of multistability. The presented model will be a useful tool to study such feedback mechanisms, whether for certain species or for the whole community. Other kinds of ecological networks, such as seed-dispersal or host–microbiome networks also show mutualistic interaction in which the two sides could co-evolve on different time scales (Herrera, 1985; Lewin-Epstein and Hadany, 2020). Whether the presented game theoretical approach is applicable to these networks will depend on the particular interactions involved.

Finally, his study provides implications for the management of pollination services. Most theoretical and empirical studies of human effects on pollination networks have focused on changes in the number of members of the networks, that is, extinctions and invasions of plants and/or pollinators (e.g. Memmott et al., 2004; Kaiser-Bunbury et al., 2010; Rohr et al., 2014). In contrast, the presented results highlight the importance of quantitative changes in plants and pollinators on network structures. While changes in the population size of pollinators may be less apparent compared with changes in whole species numbers, we may need to pay more attention to these quantitative changes and how they impact pollination network architecture and plant and pollinator communities.

AUTHORSHIP

SS initiated the study and conceived the project idea; AT developed the game theoretical framework and introduced the terms, evolutionary stable network structure and pollination dilemma; SM developed the detailed model with SS and AT and conducted the numerical and analytical analysis with SS, MK and AT; SM, SS and AT wrote the manuscript with contributions from MK; all authors were involved in interpreting the results, and contributed to the final draft of the paper.