An energetics-based honeybee nectar-foraging model used to assess the potential for landscape-level pesticide exposure dilution

Estimating the exposure of honeybees to pesticides on a landscape scale requires models of their spatial foraging behaviour. For this purpose, we developed a mechanistic, energetics-based model for a single day of nectar foraging in complex landscape mosaics. Net energetic efficiency determined resource patch choice. In one version of the model a single optimal patch was selected each hour. In another version, recruitment of foragers was simulated and several patches could be exploited simultaneously. Resource availability changed during the day due to depletion and/or intrinsic properties of the resource (anthesis). The model accounted for the impact of patch distance and size, resource depletion and replenishment, competition with other nectar foragers, and seasonal and diurnal patterns in availability of nectar-providing crops and wild flowers. From the model we derived simple rules for resource patch selection, e.g., for landscapes with mass-flowering crops only, net energetic efficiency would be proportional to the ratio of the energetic content of the nectar divided by distance to the hive. We also determined maximum distances at which resources like oilseed rape and clover were still energetically attractive. We used the model to assess the potential for pesticide exposure dilution in landscapes of different composition and complexity. Dilution means a lower concentration in nectar arriving at the hive compared to the concentration in nectar at a treated field and can result from foraging effort being diverted away from treated fields. Applying the model for all possible hive locations over a large area, distributions of dilution factors were obtained that were characterised by their 90-percentile value. For an area for which detailed spatial data on crops and off-field semi-natural habitats were available, we tested three landscape management scenarios that were expected to lead to exposure dilution: providing alternative resources than the target crop (oilseed rape) in the form of (i) other untreated crop fields, (ii) flower strips of different widths at field edges (off-crop in-field resources), and (iii) resources on off-field (semi-natural) habitats. For both model versions, significant dilution occurred only when alternative resource patches were equal or more attractive than oilseed rape, nearby and numerous and only in case of flower strips and off-field habitats. On an area-base, flower strips were more than one order of magnitude more effective than off-field habitats, the main reason being that flower strips had an optimal location. The two model versions differed in the predicted number of resource patches exploited over the day, but mainly in landscapes with numerous small resource patches. In landscapes consisting of few large resource patches (crop fields) both versions predicted the use of a small number of patches.

Parameter: definition value T 1 : time from start of unloading to start of following, dancing, or foraging, A foragers 1.0 T 2 : time from start of dancing to start of foraging, A foragers 1.5 T 3 : time from start of foraging to start of unloading, A foragers 2.5 T 4 : time from start of following dancers to start of foraging, A and B foragers 60 T 5 : time from start of unloading to start of following, dancing, or foraging, B foragers 3.0 T 6 : time from start of dancing to start of foraging, B foragers 2.0 T 7 : time from start of foraging to start of unloading, B foragers 3.5 f The rates p 1 to p 7 were obtained from 1/T 1 to 1/T 7 . The recruitment rate of followers becoming foragers for A or B, f l A and f l B were defined as the likelihood of encountering a dancing dancer with τ i representing the proportion of time at the dance floor spent in dancing. It was calculated as the product of the average number of dance circuits and the average circuit time, divided by the total time a bee is in compartment D i (thus T 2 or T 6 ).
Our implementation produced the same behaviour as depicted in Camazine & Sneyd 1991 figure 4, when after 4 hours the sugar solutions in the feeders are switched (Fig. R1). Figure R1. Number of bees foraging on each of the two feeders (dancers + unloaders + foragers at a source), and the number of dance followers.

Translating model coefficients
The coefficients listed above can be linked to the coefficients in the foraging model in the following way. The model of  can also be simplified without modifying its behaviour by taking the compartments A and H a (and B and H b , etc) together and summing the times T 1 and T 3 to get the new rates.
Thus, T 1 + T 3 = t trip + t UD -t D In our foraging model we assume t UD is constant. Simplifying the model of  we also assume that t D is constant, and thus, implicitly, that t U is constant. In reality t U may decrease with increasing sugar content of the resource, while t D may increase with resource quality. Because we are dealing mostly with attractive resources (high NEE) we set T 2 = t D to 2 minutes, the maximum value for the most attractive resource in . With a t UD of 3 minutes, this implies that time unloading t U is assumed to be 1 minute (the minimum value for the most attractive resource in . In our model NEE is assumed to define the attractiveness of a resource. Therefore we need to define f x , the probability of abandoning the resource, and f d , the probability of dancing for the resource, as functions of NEE: f x (NEE) and f d (NEE).
To obtain values for f x and f d we need an estimate of NEE for the feeders in the experiment. Concentrations were 0.75 mol/L and 2.5 mol/L. With molecular weight of succrose 342 g, these are equivalent to 257 g/L and 856 g/L, and approximately 0.20 and 0.46 g sugar per g nectar.
For the feeders at 400 m distance energy expenditure EE, ignoring costs at the feeder, is thus We further have to define functions for the dependency of f x and f d on NEE, fitting to these two data points. For the probability to dance for a resource we assume a Hill function: with exponent p=5 and h (the value for which f d =0.5) set to 13 (see Fig.   R2). For the probability to abandon a resource we assume an exponential function: ( ) = − • with a=0.325 (see Fig. R3). Clearly, with just two data points available, a linear relationship could be used as well, in particular for dancing probability (with truncation at 1). For abandoning probability we would in such case miss the likely steep increase with very low NEE.  Figure R3. The exponential function used for the relationship between abandoning probability and NEE. Camazine and Sneyd (1991) define τ i as the proportion of time a bee in compartment D i spends in dancing. This τ i is the product of (average nr of circuits) and (average circuit time) divided by the time in D i . The average nr of circuits N circ depends on resource quality. In (Becher et al. 2014) is assumed, referring to (Seeley 1994) that N circ (NEE) = 1.16 * NEE with a maximum value of 117 circuits (referring to (Seeley & Towne 1992)).
When average circuit time and time in D i are assumed to be constant, τ i scales linearly with the number of circuits danced, and it is simpler to formulate it as: With the relationships and coefficients as described above (unchanged T 1 , T 3 and T 2 =T 6 =2.0) the implementation based on NEE produces again (as should be expected) in the same dynamics (Fig. R4).

Dynamics over multiple hours
We applied the model over multiple hours. Each hour was a separate run, with the initial number of followers set to 100 (this was varied in a simple sensitivity analysis), and with the initial number of dancers (D) and foragers (F here representing foraging & unloading bees) copied from the final state of the previous run (if the resource was also present in the previous run). The relative values of bees exploiting resource i given by r i = D i +F i / Σ(D j +F j ) at the end of each run were used in the foraging model as the number of active foragers exploiting each resource (so r i multiplied by the assumed number of active foragers in this same hour). Figures R5 and R6 show how this could work out for r i in an actual simulation of the foraging model.
The constant number of followers at the start of each hour defines a kind of turn-over rate for foragers, with larger values speeding up the dynamics and thereby increasing the differences between resources faster. Non-constant numbers could be used, when data are available on the actual fluctuating numbers in a real hive. Comparison of the results for all scenarios with the number of followers set to 50 and to 200 instead of 100 showed that the impact of this parameter was small (Fig. S12), with slightly less patches exploited over the day when the parameter was large and the process of focussing on the best patches faster (Fig. S13)   figure R5) with the dots (only displayed for resource 2) representing the values that are actually used in the foraging simulation. With this approach the same number of active foragers each hour is used as in the "single-optimal" version of the model.

Futher assumptions
The minimum NEE for resources to be considered was set to 20, implying that energetic net profit (gain -cost) had to be 20 times the cost. For the "Alternative Fields" scenarios lower values worked well. However, for the other two scenarios, where numerous high quality field margins or off-field habitats could be present, this led to too large and unrealistic numbers of resources being considered in the recruitment model. Therefore a minimum NEE was set to 20 (as is also an option in the BEEHAVE model (Becher et al. 2014)).