2.2.1. Purpose of the model.

This rule- and individual-based model SOLBEE simulates spatial pollen foraging behaviour of central place foragers and is designed to be applicable to multiple solitary bee species foraging individually for pollen (Fig 1a). One simulation measures the mean performance of a group of individual bees sharing the same focal traits, in terms of fitness (potential number of brood cells based on collected pollen) and pollination activity (flowers visited, percentage visited foraging habitat, mean distance flown from the nest). The aims of the model are: 1) understanding how solitary bees interact with landscape structure by comparing their performance in different landscapes, 2) understanding how solitary bees with different traits interact with landscape structure by comparing the performance of groups (differing in focal traits) and 3) to give insight into the solitary bee's needs (concerning resource distribution) within a landscape and the consequences for pollination from the perspective of the bee. To reveal performance differences among groups caused by fragmentation only, the maximum expected number of brood cells per bee (per capita resource availability) is set equally. Bee densities (number of individuals) of each type are therefore scaled with body length and the amount of pollen in the landscape.

2.2.2. Entities, state variables and scales.

The model comprises a spatially explicit grid-based landscape with a given resource distribution and a community of pollen-collecting bees. A list of all parameters (italic in the main text) and their values is given in Table 1.

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Table 1. List of model parameters with definitions, units and simulated values.

https://doi.org/10.1371/journal.pone.0188269.t001

A model landscape requires at least habitat units that can be used for nesting, foraging, both or none (matrix), and for our specific questions with a pre-defined landscape fragmentation. We use the midpoint displacement algorithm [46] to stochastically (landscape stochastic factor) generate a coarse grid (detail set by a grain size of 50 m, defined in landscape element size) with a given proportion of vegetation units suitable for foraging (foraging habitat availability) and a given landscape fragmentation that separates vegetation units from non-vegetation units (Fig 1d panel D and E, 20 by 20 grid cells). We rescale the landscape to a finer resolution (grain size of 5m, defined in landscape detail) to define small homogeneous habitat units that can be visited by foraging bees and be used as nest habitat. This final grid has a spatial extent of 1 km (landscape size) subdivided into 200 by 200 grid cells.

The vegetation for each habitat unit is simplified with a "single generic type of plant", described by flower density, amount of pollen per flower, and the proportion of pollen that is available per pollinator visit (pollen availability, limits instant flower depletion) with parameter values from published data. This generic habitat with flowers represents a natural situation for wild bees (patches with flowering crops, which are extremes in the sense of flower densities and flower sizes, are not further considered here). The vegetated habitat is further split up in edge (width set by landscape detail, hence 5 m wide strips) and interior to represent two types of nesting habitat (Fig 1d panel A). Edge and interior are both used for nesting (and foraging) by soil-nesting bees (nest at bare spots in the vegetation, Fig 1d panel C), while wood-nesting bees use field edges only (where in real landscapes shrubs and trees grow, Fig 1d panel B) to nest in. The landscape quality for bees (maximum potential number of brood cells per individual per bee, here with a value of 30, representing a clear excess of pollen in the landscape) is used to regulate initial bee densities in the landscape (see also section 2.2.3 and details in section 5 of ODD in S1 Appendix). We track the remaining pollen volume and the number of bee visits per grid cell during simulation. The time horizon (flytime) is 14,400 time steps, each representing one second (foraging period of four hours).

Bees are characterized by body length and nesting preference, i.e. nesting in dead wood and other cavities (hereafter ‘wood nesting’) or in the upper soil layer (hereafter ‘soil nesting’). Nesting preference determines the fixed nest location for each individual bee. Empirical allometric rules (S1 Appendix, Table C) are applied to link foraging traits (pollen per brood cell, general return distance, far return distance, velocity medium/low, velocity high, handling time per flower), and subsequently derived traits (pollen capacity per bee, perception distance, length of flight units) to body size. Other behavioural traits needed for foraging (flower memory, habitat cell memory, upper patch leaving threshold, lower patch leaving threshold, flight path tortuosity, ignorance, time at the nest) were also parameterized with literature data [37]. All traits of bees remain constant during simulation (calculated values given in Table 1). Specification of state variables of bees can be found in the supplementary material (S1 Appendix, ODD section 2).

The mean number of brood cells per bee (potential brood success based on collected pollen) represents a proxy for number of offspring and fitness. To calculate this number from collected pollen, we correct for body size (dividing pollen volume by pollen per brood cell, see values in Table 1). As proxies for pollination performance we track the number of flower visits and the mean distance from the nest at return. These three responses at the bee level are given as mean averaged over all bees in the community at the end of the simulation. At the landscape level, we calculate at the end of the simulation the percentage of foraging habitat that was visited by at least one bee (percentage of grid cells with one or more visits) which we also use as indicator for pollination performance. No further population level variables are considered.

2.2.3. Process overview and scheduling.

The initialization phase consists of generating a landscape with foraging habitat and defining a bee community (with individuals of one type) and setting individual nest locations (Fig 1b). To allow direct comparison of the performance of the six bee types, we assumed equal resource availability per individual, having for a certain landscape either many small or few large bees. To focus on landscape fragmentation and avoiding to model food loss, we decided to have foraging habitat availability also increase the number of individuals. This assumption of bee population being in balance with the available resources, allows to focus on spatial allocation differences (of pollen, and bee nests). The calculation of the number of bees within a landscape includes all these variables and uses pollen volume (mm³) as common unit. In short is calculated by dividing the pollen volume present in the landscape by the pollen volume that should be available per individual. The numerator is calculated by multiplying foraging habitat availability, landscape area (1 km²), flower density and pollen per flower (i.e. x flowers multiplied by pollen volume) and the denominator by multiplying landscape quality and pollen per brood cell (volume for x brood cells). The landscape quality defines the maximum potential number of brood cells per individual, as being equal for all bee types in all landscapes. The initially calculated number of individuals (per bee community of bees of one type) remains constant during simulation (one foraging day without population dynamics). After initialization, a foraging day (time horizon flytime) starts for each individual with five types of behaviour in a foraging cycle (modified after [47]). The behaviour of one individual is strictly sequential in time, and lasts at least for one second (discrete time steps, ‘time penalties’, Table 2). The five behaviours in the foraging cycle (Fig 1c, details in S1 Appendix) are: FORAGE FLOWERS (collect pollen on flowers); NEIGHBORING CELL (fly to a suitable neighbouring landscape grid cell); FLY AROUND (fly around and visually search for unknown foraging areas); FLY BACK (fly back to nest); NEST REACHED (deliver the pollen to the nest). When an individual performs a behaviour, foraging performance state variables are changed in the first second (and for simplification the bee is set to a "still performing" pause mode). Individuals are processed in random order each model second and start a new behavioural element when they are not in "still performing" mode. The sequential approach assumes that always a single individual reaches a flower as first being allowed to remove resources. The randomization ensures that 'being first' is a process of chance. This sequence is repeateduntil a certain foraging time (flytime) is completed after which all activities are stopped instantaneously and values are averaged per bee and written to an output file (Fig 1b).

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Table 2. Overview of time penalties (time budgets) during the foraging cycle.

https://doi.org/10.1371/journal.pone.0188269.t002

4.1.1. Fragmentation effects on fitness.

By design, all bees had the opportunity to build the same number of brood cells within a day assuming spatial allocation of nests and pollen producing flowers would not play a major role. This was indeed the case for soil-nesting bees of the same size, which did not respond to landscape fragmentation (Fig 2a) or to the ratio of nest to foraging habitat (Fig 3c). They had optimal access to foraging resources from their nest, because nests were distributed randomly over the foraging habitat (Fig 1c). Wood-nesting bees, in contrast, responded positively to landscape fragmentation. This supports the hypothesis that fragmented landscapes increase bees diversity by providing increased nest-site availability [26, 50], which applies to wood-nesting bees restricted to edge habitat. In our model, bees were indeed positively affected by nest-habitat availability (Fig 3a). Although the interaction effect between foraging habitat availability and landscape fragmentation was low, wood-nesting bees seem to respond stronger in landscapes with high foraging habitat availability (Fig 2a).

The apparent decrease in brood cell number with increasing foraging habitat availability (Fig 2a) is a side-effect of the model setup, highlighting the importance of time and time-budgets. The number of individuals per landscape was by definition linked to foraging habitat availability. The resulting local bee density (and hence competition around nest sites for resources) had a strong negative effect on brood cell number (Fig 3b). This supports the idea that time can be more limiting for wild bees than foraging resources [47]. Spatially induced time constraints deserve more attention in pollinator research. It can for example be argued that oligolectic and monolectic bees (highly specialized on certain plant species, e.g. Andrena hattorfiana [51]) face very low plant densities at the landscape scale and have to deal with strong time constraints, an additional competitive disadvantage compared to polylectic bees.

We showed that traits (nesting preference and body length) affect the response to fragmentation, explaining why some studies find no effects of fragmentation on bee performance [20, 26], while others did [19, 27]. Lack of response to fragmentation in field studies could relate to dominance of certain traits in the community (e.g. a large proportion of bumblebees). Nevertheless, we also recognize that the effect of landscape fragmentation can be low compared to other landscape-level parameters (compare Tables 3 and 4) and may remain undetectable in many field studies, suggesting a need for alternative measures. Finally, also an increase in foraging habitat availability above an intermediate amount reduced the amount of edge structures (merging of patches), which is independent of landscape fragmentation (e.g. nest habitat availability is frequently low beyond low landscape fragmentation, Fig 3a), again demonstrating that landscape fragmentation on its own is a poor measure.

4.1.2. Fragmentation effects on pollination.

The pollination measures (number of flower visits, percentage visited foraging habitat and mean foraging distance) largely followed the patterns for the number of brood cells. Pollination by soil-nesting bees remained largely unaffected by fragmentation. However, the pollination performance of wood-nesting bees was affected by fragmentation. Wood-nesting bees visited fewer flowers (Fig 2b, small and intermediate sized wood-nesting bees) and flew shorter distances (Fig 2e) with increasing landscape fragmentation. Under high local bee density (low landscape fragmentation), bees inevitably encountered empty flowers more frequently and were forced to visit and probe more flowers (Fig 3e) and fly longer distances (Fig 3k). This means that fragmentation can both reduce the frequency of pollen transfer between flowers and reduce the mean distance over which the pollen is transferred. Landscape fragmentation increased the landscape-level coverage with pollinators (percentage visited foraging habitat, Fig 2c) by a better distribution of nest habitat over the landscape.

4.1.3. Linking fragmentation, nest habitat, bee density and the ratio of nest to foraging habitat.

Increased landscape fragmentation resulted in higher nest availability, a lower local bee density and a higher ratio of nest to foraging habitat. Local bee density and the ratio of nest to foraging habitat were both better predictors for brood cells, visited foraging habitat and foraging distance than landscape fragmentation itself. The nest to foraging habitat seems a suitable landscape-level parameter to replace landscape fragmentation, foraging habitat availability and nesting preference by a single measure, giving an informative gradient of habitat complementarity for wild bees [11], capturing local landscape quality well (Fig 3c, 3f, 3i and 3l). Although local bee density was an equally good predictor—and can in principle be measured in the field—it is less practicable because it requires high sampling effort. The ratio is probably easier to approximate with local information about potential bee habitats.

The result that bees need an appropriate ratio of nest habitat relative to foraging habitat can be considered a novel, but also logical, insight gained by modelling practice. We believe that the general observation that wild bee communities are mainly affected by foraging habitat availability [18–20] is therefore incomplete. The hypothesis that solitary bees are most limited by nest sites [20–22] applies therefore, according to our results, only when foraging resources are constant. The here proposed ratio of nest to foraging habitat availability could satisfy the need for a landscape measure that is more suitable in describing the resource needs of solitary bees [36].

4.2.1. Body size.

One would expect a large bee to visit more flowers (required for a full load) and fly longer distances than a small bee. This was in general true (Fig 2b and 2d). As a result, large bees built fewer brood cells than small bees, which is also supported by field observations (reviewed in [37]). The expectation that large bees perform better than small bees in fragmented landscapes because of their faster movement [50] and shorter flower handling times, turns out to be wrong, at least under the assumptions made.

The number of visited flowers was the response most affected by body length. Obviously, large bees need to visit more flowers for their pollen requirements than small bees (Fig 2b), in accordance with field studies [52, 53]. The difference in flower requirements per brood cell between large and small bees is even higher than the difference in daily requirements, given that large bees also build fewer brood cells. Time allocation diagrams for the different bees (S2 Appendix Fig B) make clear that model bees spent far more time visiting flowers than for flying between them or back to the nest. Therefore, flower-handling time is the most important trait for compensating high pollen requirements. This trait was in the sensitivity analysis indeed the most influential one among other size-related traits [37].

When flowers are larger, the explained variance by body length for brood cells drops more than half, the most prominent difference for altered vegetation parameters (S3 Appendix Table A). Larger flowers bring an advantage for large bees when they do not have to visit so much more flowers than smaller bees (indeed reduced explained variance by body length for flower visits, S3 Appendix Table A).

Small bees have an advantage compared to large bees, but they are also more sensitive to changes in the landscape. Small and intermediate-sized bees (wood nesting) visited fewer flowers with increasing landscape fragmentation, while large bees (wood nesting) visited more flowers (Fig 2b). In addition, the crossing lines in Fig 3c show that there are also landscapes in which small bees perform worse than larger bees. In landscapes with much foraging habitat and little nest habitat (low nest to foraging habitat ratio, Fig 3c) local nest density of bees is very high with small bees having the highest nest densities (Fig 3b). This nest density problem is most problematic for small bees due to the mass proportional relation between body size and bee numbers in the landscape (S1 Appendix Table D). At some point the density of small bees becomes so large that flying farther from the nest for foraging resources (Fig 3k) cannot compensate this, leading to a very steep decrease of brood cell number which is absent for large bees (Fig 3b). Also does an increase in local bee density imply more visits of empty flowers, causing an increase of total flower visits for small bees, while large bees show a slight decrease (Fig 2e), explaining the different responses to landscape fragmentation (which reduces local bee density). Large bees seem to escape local overpopulation more easily (steeper increase in mean foraging distance with local bee density, Fig 3k). Consequently, especially small bees could be driven to evolve mechanisms for optimizing their foraging behaviour and to quickly sense high local bee densities and empty flowers (e.g. by smart flower probing rules), to justify the investment in farther flights.

Larger bees seem in general to be better pollinators: they visit more flowers, cover more foraging habitat and fly longer distances. For visited foraging habitat is the effect of body size lowest, and small bees cover the foraging habitat almost as much as large bees (Fig 3g–3i) although they fly on average less far from the nest (Fig 3j–3l). The pollination potential of small insects is often underestimated and they transfer enough pollen for sufficient seed set [54].

4.2.2. Nesting preference.

Nesting preference was the most important predictor for foraging distance and for visited foraging habitat and second important for the number of brood cells. Soil-nesting bees were evenly distributed over the foraging habitat and found foraging resources near the nest, leading to short foraging distances (Fig 2d) and optimal coverage of the foraging habitat (Fig 2c). The maximum number of brood cells within a day (Fig 2a) was hardly affected by the gradient of landscape fragmentation. Wood-nesting bees, in contrast, responded strongly to different landscapes. They had a lower brood cell number in landscapes with a low degree of landscape fragmentation (Fig 2a) where they had longer foraging distances (Fig 2d). This makes sense in relation to lower nest habitat availability (Fig 3a), higher local bee density (Fig 3b) and low ratio of nest to foraging habitat availability (Fig 3c). Wood-nesting bees also covered less foraging habitat when there was more foraging habitat to cover in the landscape (down as low as 25% for high foraging habitat availability, Fig 2c). Bees nesting at field edges did not reach the interior of the fields in those cases. Nesting preference hardly affected the number of flower visits (Fig 2b), which seems rather being affected by body-size related traits alone.

Soil and wood nesting is often considered as a factorial contrast between bees, but the model shows otherwise. There seems to be a gradient of habitat use, where both nesting preferences are part of the same relationship for a bee of a certain size (Fig 3, all panels). Most models explained the variance sufficiently without nesting preference (Table 4). Soil-nesting bees in natural bee communities often nest in very high densities [55] and occur in much higher numbers on fallow land than wood-nesting bees [19], suggesting that soil-nesting should be modelled more restricted in nest habitat availability. This would likely complete the visible gap (especially in the ratio of nest to foraging habitat, Fig 3c, 3f, 3i and 3l).

4.3.1. Trade-offs between pollination measures.

We found a trade-off between mean foraging distance and coverage of the habitat with pollinators at the landscape level (negative relationship, Fig 4c and 4d). This means that either pollen is transported over larger distance or that pollen is transported at all places in the foraging habitat, not both. In the best landscapes for bees (e.g. high ratio of nest to foraging habitat), bees were spatially optimally distributed leading to a high percentage of visited foraging habitat (Fig 3i), also leading low mean foraging distances (Fig 3l). Oppositely in bad landscapes, bees are forced to fly farther distances, but do not cover all foraging habitat remaining certain areas unpollinated. In agricultural landscapes it is common that bees are most abundant in field edges [56, 57]. Field interiors often show low abundance of solitary bees, and bees foraging there may be soil-nesting bees provided locally with nests [58].

4.3.2. Up to 20% suitable as nesting habitat.

The simulation results suggest that visited foraging habitat (coverage with bees) increases steeply up to a nest to foraging habitat ratio of 0.2 (i.e. 20% nest habitat compared to foraging habitat) and beyond this levels off to optimal coverage (close to 100%, Fig 4a). The value seems robust for vegetation type (Fig 4a and 4b) and bee type. It may suggest a general pattern applying to bees as pollinators in general, but needs deeper investigation and field validation for broader application. This value reminds of the recommendation that about 25% of the landscape should remain refugee area for wild bees to maintain sustainability [59]. A recent review [60] estimates that 2 to 44% of the landscape with high quality flower strips is required for the sustainability of pollinator communities. However, those numbers focus on a different aspect of the bees' ecology (sustainable populations) and do not consider the required amount of nest habitat (and ratio of bee habitats) to pollinate a certain area. Hence, data-based estimations of the optimal ratio remain a future concern.

A simple calculation illustrates that most agriculturally dominated landscapes are really poor and a hostile environment for solitary bees, despite mass flowering crops. For example, providing a one ha crop field with a five-meter wide natural strip yields a desired ratio of 0.19. The same five-meter strip at a 49 ha field yields a ratio of 0.03. Practically considered, to improve the local ratio, the nest habitat for bees should be sufficient (increase of unmanaged field strips) of good quality (reduced pesticide application) and spatially well distributed within an area (fragmented spatial distribution). We recognize that 20% field strips is not economically feasible in modern agriculture, but we agree with others that management encouraging a mosaic of smaller fields and increase of the edge:area ratio would benefit pollinators [61]. For example, up to 8% of the field edge can be converted to bee-friendly habitat without losing yields [62], because the positive effect on pollinators (increased pollination) compensates for the smaller crop area.

In situations where the ratio remains far below 0.2 (e.g. large crop fields), a solution may be the provision of artificial nest sites. There are many examples where solitary cavity-nesting bees are employed as crop pollinators, by offering artificial nests [63, 64]. Also, at least one soil-nesting species (Nomia melanderi) is managed as crop pollinator by offering nesting beds in the soil [65, 66].

4.4.1. Foraging distances.

The mean foraging distance from the nest is a measure for how far pollen is transported and it is focused on the motivation of the bee. Assuming that bees do not fly farther than necessary can result in very short foraging distances for soil-nesting bees (< 50 m, Fig 2d) in accordance with natural ranges from field studies [37]. Their nests were evenly distributed over the foraging habitat and they found enough pollen within a short range from the nest in all landscapes. Mean foraging distances are higher when conditions are unfavourable, e.g. when bees face a high local nest and bee density in field edges (wood-nesting bees) and untouched foraging resources only farther away. Large bees flew farther (50–200 m) than small bees (30–100 m). These still relatively low distances are in agreement with the finding that that both large and small bees often forage below 200 m [67]. Some studies may overestimate foraging distances, especially when solitary and eusocial bees are not separated or landscape types are not considered [16]. Honeybees and bumblebees fly farther, as well as solitary bees without any foraging recourse near the nest. The mean foraging distance as measure for pollination does not provide information on extreme events (beyond the mean) that may lead to pollination as well. Nevertheless, it is clear that bees mainly exchange pollen between plants over short distances and that most activity is near the nest.

4.4.2. Matrix crossing.

Short foraging distances imply that bees did not often fly to more distance patches and did not often cross the matrix without foraging resources, despite a parameter that induces such behaviour. Matrix crossing behaviour was modulated by the model parameter ignorance (Table 1), which was earlier shown not to affect the model simulations much [37]. Low matrix crossing behaviour is in accordance with the idea that wild bees live on islands of foraging habitat [9] and is reported for various pollinators [68, 69]. Solitary bees have a high site fidelity with very conservative movement patterns [70], which does not seem to be improved by grassy field strip corridors [57]. The idea that solitary bees, as flying insects, often display matrix crossing behaviour and fly long distances needs to be further refined by future field and modelling studies. In ecosystem-service research, such concepts are trivial for estimating the spatial availability of pollination services.

4.4.3. Size and evolution.

The result that large solitary bees are on average worse performers than small ones may imply evolutionary consequences. The lower efficiency of large bees may be a driver to develop a social structure with higher efficiency and explain why large bees are more often social than small bees. In central Europe there are e.g. more species of eusocial bumblebees than large bees from the genus Xylocopa or Anthophora, in contrast to very small bees which are often solitary (e.g. Andrena) or only primitively eusocial (e.g. Lasioglossum). The performance constraint may thus be an additional driver for sociality in combination with other evolutionary drivers such as climate change [71] and time (lineage age [72]). At the same time it may explain why in regions where bees have been under pressure for decades, body size decreased over time at the species level [73].

4.4.4. Ratio of habitats as simplified measure.

Foraging habitat visitation increased with the ratio of nest to foraging habitat with a remarkably low effect of body length. This ratio seems a robust proxy descriptor for landscape structure, adapted to the perspective of bees and suitable for estimating pollination services. We expect that the ratio of nest to foraging habitat, after further study, can become an easy to calculate landscape simplification in field studies and a valuable addition to other indexes such as the LLI [74, 75]. When more field surveys include the identification of nest habitat in addition to foraging habitat, we expect that the positive effects of the ratio can be soon be confirmed.

4.5.1. Notes on biological assumptions in the model.

The combined fact that we tested all model parameters [37], that the model's output values come close to values from real systems (Fig 3), and that we ensured that our results hold for altered vegetation parameters, confirms that we have successfully covered a part of a complex multifactorial system and understand this part a little better. As with most models, some biological assumptions lie in the intrinsic structure of the model itself and are not critically tested, while at the same time simplifications are inevitable to speed up calculation. Some of our decisions and considerations should be mentioned for future model implementations. One example is the unrealistically high number of brood cells, which can easily be justified but also criticized. We assumed a short time at the nest after each trip and we neglected time needed for egg laying, cell closure and other activities (due to data deficiency and unnecessary complexity, see also [37] for a discussion). We further assumed that all pollen collected at the flowers reach the nest, while in reality pollen gets lost on the way [76] and solitary bees of the same species return to the nest with a high variability of pollen loads [77]. Also, in favourable landscapes bees may be egg-limited rather than pollen limited [38]. Some assumptions may have excluded important mechanisms from the model, such as the assumption that bees also fuel themselves with nectar on the same flowers when they forage for pollen. The time budget for nectar collection is relatively small compared to that for pollen collection (see [37]) and may be negligible, but we do not know the effect of reducing a decision sequence depending on two resource levels to a single one. Finally, considering the many different parameters that were parameterized with heterogeneous sources, the model may benefit systematic studies for each parameter to improve accuracy. To identify priorities, a future study could address how changes in the allometric rules affect time budgets (Table 2) and identify the rule requiring the highest quality data. Also the effect of potential correlations, such as a positive relation between body size and flower size preference (which probably would minimize the disadvantage for large bees that we found) or a positive relation between body size and time at the nest, could be of future interest, even when studies failed to prove a systematic difference in flower preference [78, 79] or time at the nest (reviewed in [37]) so far. In the end, the priory of considering such issues depends on the questions for which the model is used.

4.5.2. Towards simulation of real communities.

We chose to focus on fragmentation, kept other parameters constant and compared six bee types in a scenario-like way to reduce complexity and understand model processes. As may be desired by field ecologist working with pollinators, simulation of realistic plant and bee communities requires a data-driven approach, in which values are parameterized and combined in sets. For a simulation with multiple bee species one needs community data on species composition, local bee densities and nesting locations. It is already challenging to get such sufficiently detailed data in the field, but the model also requires definition of flower traits making up the vegetation, including pollen provision and flower density. As a minimum, the distribution of these traits in a community is needed to simulate realistic vegetation patches. At the same time, additional model rules may be required when bee species face different types of flowers or when the presence of other species affects foraging behaviour [80] or brood cell number [81]. A further advantage of such an attempt with real bee densities for each species separately is that it overrules the assumption that pollinator density scales negatively with body size and positively with foraging resources, which had a prominent effect on our results.

4.5.3. Applications with crop fields and semi-natural edge habitats.

The model system may roughly represent crop systems: fields with foraging resources and edge habitat with nesting resources for wood-nesting bees (and in some cases suitable soil within the crop field). However each crop has different properties that are not necessarily covered by our current simulations. Flower size (pollen production per flower) and flower density can be much more extreme, such as a clover field with very small flowers in high densities or a sunflower field with very large flowers in low densities. We did not model this explicitly for several reasons. We focussed on a more general system to understand wild bees as pollinators and general patterns, but specific crops systems could be focus of future studies. The current results help to think in terms of flower sizes and densities in real systems and to interpret other studies in this light. For example, in a recent study wild bees were hardly found in sunflower field interiors [82], which may be a result of enough pollen being offered near the nest, explaining that the quality of the edge habitat did not have an effect on the pollinator community in the interior of the field [82]. Our model suggests that an increase in fragmentation and the ratio of nest to foraging habitat at a very local scale should increase pollinator coverage largely independent of flower size.

A different challenge for the future is to see and treat an agricultural field as potential nest habitat for wild bees. When soil-nesting bees are limited to field edges they must fly farther distances from the nest and face more local competition with bees around the nest. In general, soil disturbance in crop fields is assumed to be too high for soil-nesting bees to survive [83–85], but recent studies suggest that it is possible [58, 86] for species that nest a meter below the surface. Agricultural practices that reduce the depth of mechanical 'action' and pesticide application will benefit the survival of soil-nesting bees within fields. This would give an optimal nest to foraging habitat ratio and lead to better pollination.

4.5.4. Combining strengths of pollinator models.

Several models with foraging bees in response to vegetation seem to have being developed in parallel [37, 87–90]. Pollination ecology and agricultural ecology could benefit from a kind of "master-model" in which strengths are combined. This requires important decisions on how to deal with different scales (requiring different model elements) and reduce complexity in favour of simulation time. We think that some elements in our model that are unpractical in application can be simplified. For example, our decision to use mass-scaled resource availability for each bee (resulting in many small bees being compared to a few large bees), is not very practical to combine with an IBM approach, since a high variance in individual numbers also results in a high variation in calculation times. Also, some of our parameters are elaborate to measure and may benefit clever proxies. Presently, each model has its own advantages and level of detail, applicable to selected research questions.