As with other published pollinator models in ALMaSS (Duan et al. 2022; Ziółkowska et al. 2023), we use an agent-based modelling approach (Grimm and Railsback 2005). Each hoverfly is represented as a separate object of a particular state with given properties, such as age, location and pesticide loading.

Our model is spatially explicit; thus, the position of each individual hoverfly (object) within a landscape is simulated. The ALMaSS modelling environment provides a detailed spatio-temporal representation of the landscape from which individual hoverflies obtain the information necessary to simulate their behaviour. This representation describes spatial landscape heterogeneity through a detailed raster land-cover map with a spatial resolution of 1 m². Farmed areas are represented as accurate maps of fields grouped into farm units of different types (e.g. cattle or arable farms). The temporal component of agricultural landscape heterogeneity refers to year-round crop management, described through individually tailored management plans for each crop. The cropping system is understood as a pluri-annual crop rotation. Crop management plans consist of combinations of farm activities (including pesticide treatments), time windows and probabilities of carrying out these activities. The temporal component includes weather conditions and vegetation growth models for all modelled vegetation types and crops; this is all updated daily. Such an approach gives a highly realistic dynamic landscape simulation with vegetation growing in response to the weather and the pattern of farming activities related to each crop, farm and field (Topping et al. 2016). The resource-providing units are the vital elements of the landscape simulation for all pollinators simulated within ALMaSS. These define the quantity and quality of pollen and nectar for each habitat patch in the landscape.

The model will consider the behaviour of all hoverfly life stages daily. The model"s parameters are based on the field and laboratory data on E. tenax available in the literature. When the necessary data were not available for E. tenax, we estimated parameters based on published data from other hoverfly species with similar life histories, particularly other species from the Eristalis genus, such as Eristalis pertinax (Scopoli, 1763) (similar size species found in Europe and Asia) and Eristalis arbustorum (Linnaeus, 1758) (the European drone fly of smaller size, abundant in Northern Hemisphere). The internal development of E. tenax individuals and interactions with each other and the environment are covered in each section below.

This section mentions some of the key problems and assumptions associated with the design of this model, and its linkage within the social-ecological systems and simulations where it will be used. The model aims to describe population changes in time and space and does not include all possible known behaviours unless these significantly influence population changes, with the aim to balance false inclusions and exclusions (Topping et al. 2015). Key assumptions are noted here, but for details see the discussion or details of the individual processes below.

The E. tenax model is built on extensive literature, though many processes have uncertain parameters and mechanisms, primarily derived from laboratory studies and other hoverfly species. Key assumptions and simplifications include:

Other decisions are made primarily as a consequence of the available data in the supporting ALMaSS landscape framework (Topping and Duan 2024a). These include simplifications regarding the prediction of soil moisture and its calculation at the patch level and the fact that population genetics and behaviour related to genetic makeup are not considered, even though they may affect population processes in the long term. This is a consequence of an important simplification for efficiency: we assume that females can be mated and do not model males individually.

In this formal model, we have defined a simple toxicological model. This description is intended as a starting point only for future carefully developed toxicological models for regulatory use. Future versions may include expanding the current model to include sub-lethal effects and more detailed internal toxicology, such as implementing toxicokinetic/toxicodynamic models. The form that these models take will depend on the regulatory scenarios developed and, in particular, the design of exposure, which will be affected by the limitations in mapping larval habitats.

Framing the E. tenax model within the social-ecological systems it is a part of is largely a function of the ALMaSS landscape model and the scenarios designed for the model use. Key elements of this are related to the assumptions regarding mapping details, pollen and nectar modelling, and human management, particularly farming. A discussion of these is outside the scope of this formal model. The main exclusion of those drivers that might change the model within the scope is the potential for different genetic traits and evolution. This omission is particularly important if local adaptation can alter key mechanisms represented in the model. However, including genetics will always be problematic due to the potential for seasonal long-distance dispersal.

The common drone fly E. tenax is one of the most widespread syrphid species in the world. Due to its highly synanthropic nature, the species is found on all continents except Antarctica. Still, it is not common in extremely southern latitudes outside the arid areas of Europe, Asia, and Africa. It is especially abundant near human activity centres (Speight 2018). It is a large hoverfly with the mean mass of females around 140–150 mg, and males around 110–120 mg (Table 1). The colour patterns of this species are a clear example of Batesian mimicry with Apis mellifera. Eristalis spp. do not only mimic honeybees visually but also behaviourally during flight and feeding; they also produce a similar buzzing sound. E. tenax, or the drone fly, as its name suggests, is similar in appearance to male honey bees (drones) (Heal 1982; Bressin 1999; Golding and Edmunds 2000).

In northern and central Europe, E. tenax exhibits partial migratory behaviour, characterised by both seasonal migration and overwintering strategies. The expression of migratory activity is modulated by climatic variables, the spatial-temporal availability of resources and reproductive strategy imperatives (Tomlison and Menz 2015; Reynolds et al. 2024). The species is not found active in the field from around October/November to February/March, depending on the region (Kendall and Stradling 1972; Bressin 1999). As hibernation is mainly controlled by temperature and photoperiod, it starts earlier and ends later in colder regions further north compared to warmer regions in southern Europe. However, in Mediterranean climates, hoverflies can be found throughout the year (Pérez-Bañón et al. 2013), but the number of active hoverflies is much lower in winter.

The development of the dronefly can be categorised into four main life stages: egg, larva, pupa, and adult. In central and northern Europe, there are usually two generations, spring/summer and autumn. A spring/summer generation begins in early spring with the emergence of females that were inseminated in the autumn, but the eggs did not develop until spring (Gilbert 1986; Bressin 1999; Upchurch et al. 2023). When the eggs are ready to be laid, a female searches for a suitable habitat for the larvae: stagnant water of various types (Campan and Campan 1979). The larvae, a type called a “rat-tailed maggot”, are saprophagous and filter-feed on microbes associated with decomposing organic material, using a siphon (the “tail”) to breathe (Kamdem and Otomo 2023). For pupation, the larvae crawl to a drier area and form a capsule to protect them from the environment during metamorphosis. The adults of the spring/summer generation mate and produce the autumn generation, which is much more abundant. As winter approaches, they begin to seek out moist crevices in rocks or buildings to hibernate (Bressin 1999). More details about each developmental stage are given below.

E. tenax eggs weigh around 180–189 µg (Table 1) and are laid side by side, perpendicular to the ground, in clusters, near or on the surface of stagnant water such as reservoirs, liquid manure or sewage. The duration of the egg phase is affected by temperature and humidity (Bressin 1999), but it is usually very short (2–3 days; Hartley 1961; Heal 1979; Gladis 1994; Basley et al. 2018; Campoy et al. 2020), mostly synchronised within one cluster. Egg cluster viability is rather high, estimated to be 86–100% by Bressin (1999).

Data on temperature-dependent development time at the egg stage were derived from five publications (Table 2). In Upchurch et al. (2023), the test was carried out at six temperatures, and unpublished data from Kendall (1971) are available for five temperatures, while in the other three studies, only one temperature (“room temperature”) was used. The data collected follows a quadratic relationship (Equation 1), as shown in Fig. 1, but there was not enough data to estimate the thermal performance curve for egg development.

D_egg (T) = 0.025 × T² – 1.421 × T + 20.579 Equation 1

where D_egg (T) is the developmental time of the egg stage, and T is temperature.

The larva is saprophagous, called a rat-tailed maggot and is typical for syrphids, with a characteristic telescoping siphon that helps the larva breathe in its oxygen-deprived, organic-rich habitat. There are three subsequent larval stages in development, divided by moulting periods. Very high mortality at the beginning of the first instar larva stage is caused by difficulty in finding a suitable part of the habitat (Hartley 1961; Campoy et al. 2020). Another critical moment within a life cycle is the end of the third larval instar (Campoy et al. 2020). During this phase, the larva changes habitat to find a drier place, more suitable for pupation and metamorphosis. The larva crawls using the pseudopodia (Buckton 1895) even several metres to find an optimal place for pupation (Hurtado 2013), and this movement can be quite rapid (>1 m/min) (Wilson et al. 2009). Larval development in E. tenax is temperature-dependent (Ottenheim et al. 1996; Hurtado 2013). It can also be impacted by the larval density and habitat quality, i.e., under nutritional stress, the larval developmental time increases (Equation 2), and the survival decreases (Equation 3) (Hurtado 2013; Fig. 2). These relationships may be described by linear functions:

D_larva (L_d) = 0.158 × L_d + 15.673 Equation 2

S_larva (L_d) = –0.475 × L_d + 74.350 Equation 3

where D_larva is the developmental time of the larval stage, S_larva is the larval survival and L_d is density of larvae.

Density-dependent larval development in Eristalis species was earlier demonstrated by Kobayashi (1979), who investigated Eristalis cerealis (Fabricius, 1805) and found that, akin to E. tenax, larval development time in E. cerealis is prolonged under conditions of nutritional stress, accompanied by a concomitant decline in survival rates (Fig. 3). Furthermore, Kobayashi (1979) reported that nutritional limitations negatively affect larval biomass and body length in E. cerealis (Fig. 4).

Data on temperature-dependent larval development time (including stages 1–3) were derived from six publications (Table 2, Fig. 5). In two of them, larval development was tested in the normal thermal range, and additionally, it was tested either under different larval density conditions or food conditions. In three papers, only one temperature was used for the tests. Unpublished data from Kendall (1971) provide the only available information across a broader temperature range. The data collected follows a quadratic relationship (Equation 4), as shown in Fig. 5, but there was not enough data to estimate the thermal performance curve for larva development.

Data on temperature-dependent larval development time is also available for a species of similar biology, Eristales arbustorum (Linnaeus, 1758), a European drone fly (Fig. 6; Ottenheim et al. 1996). E. arbustorum is a smaller hoverfly than E. tenax (9–11 mm length compared to 12–16 mm length of E. tenax). It appears to exhibit a shorter developmental duration at its thermal optimum (14 days versus 18 days) and a higher thermal optimum for development relative to E. tenax (27.0 °C versus 23.6 °C; according to Equations 4 and 6).

We found no data on the developmental time for individual instars for E. tenax. Such data is available for Eristalinus arvorum (Fabricius, 1787), an Asian hoverfly with saprophagous larva. E. arvorum is smaller than E. tenax (8–10 mm adult length compared to 12–16 mm adult length of E. tenax) and has a slightly shorter larval developmental time (11 days compared to 14 days at 25 °C for E. tenax) (Cao et al. 2022). The total larval developmental time for this species at 25 °C was divided into three instars: 31% for the 1^st instar, 36% for the 2^nd instar and 38% for the 3^rd instar (Cao et al. 2022).

D_larva (T) = 0.272 × T² – 12.886 × T + 170.493 Equation 4

D_mlarva (T) = 0.043 × T² – 2.136 × T + 40.680 Equation 5

D_flarva (T) = 0.032 × T² – 1.728 × T + 37.361 Equation 6

where D_mlarva and D_flarva is the developmental time of the larva stage for males and females, respectively, and T is temperature.

The pupal weight is approximately 170–180 mg, with males being smaller than females (Table 1). It is an inactive stage, the duration of which depends on temperature. Some authors identify this stage as one of the most critical moments within the life cycle (Campoy et al. 2020), with mortality depending on temperature (Daňková et al. 2023). Two particularly vulnerable transitions within this stage include (1) the emergence of the pupal spiracles, a process that frequently fails and may compromise successful development, and (2) adult eclosion followed by wing expansion, which is also prone to failure (personal communication with F. Gilbert).

Temperature-dependent pupal development time data were derived from nine publications, four of which reported data from the wide thermal range (Table 2). Thermal performance curves for pupal development time were estimated using a set of models provided by the rTPC R package (Padfield et al. 2024) as described in Padfield et al. (2021). Such an estimation required the use of an inverse of development as the dependent variable. Table 3 shows the estimations of critical thermal minimum (CT_min), and critical thermal maximum (CT_max) across tested models. The graphical representation of the five best-performing models is shown in Fig. 7.

The model will consider E. tenax behaviour of all four life stages (i.e., egg, larva with three instars, pupa and adult) on a daily basis. We assume a ratio of 1:1 of females to males at the population level (following Ibrahim 1976, who investigated a large field-sample of saprophagous hoverfly pupae of Eristalinus taeniops (Wiedemann, 1818) and thus, males will not be included in the model. It is further assumed that all adult females are fertilized; therefore, mating behaviour is not explicitly modelled, although it is implicitly incorporated into the timing of the reproductive process.

The development will be modelled according to the life-stage state machine and transition path (Fig. 8). Following emergence from hibernation in spring or arrival via migration, adult fertilised females begin interacting with their environment. This interaction includes dispersing in search of food resources, foraging for pollen and nectar, and reproducing (see section "Adults" for more details). Eggs are laid in clutches near a suitable habitat for the larvae, and their only behaviour is temperature-dependent development. The eggs hatch into larvae. On hatching, we assume that 50% are male and these are then removed from the model. Larval development occurs in stagnant water and depends on temperature and larval density (which is derived from larval habitat quality). The third larval instar crawls out onto dry land. We assume that this movement is rapid and takes place within one day. The length of the movement will be drawn from a uniform distribution with a maximum distance of D_l meters, and a minimum distance of 1 meter from the edge of the water body. As the exact movement capabilities of E. tenax larvae are not known, we assume D_l to be 10 m, subject to adjustment during the model calibration phase. The larvae transition into immobile pupae, and their development is temperature-dependent. The pupae eventually develop into adult females, forming the first, spring/summer generation. These spring/summer-generation adult females mate and produce a more abundant autumn generation, which subsequently hibernates. At each developmental stage, there is a probability of mortality (see section “Mortality”).

In the model, the temperature-dependent development on the egg stage will be modelled using a quadratic function (Equation 1) fitted to the data for E. tenax (Fig. 1) with an optimum at T_opt = 26.9 °C, at which the maximal developmental speed of 1.5 days is reached. Unfortunately, there was not enough data available to calculate the critical minimum (CT_{egg_min}) and maximum (CT_{egg_max}) temperatures for egg development. Thus, these values will be calibrated.

We will combine the dependence of larval development time on both temperature and larval density. The temperature dependence will be modelled using a quadratic function (Equation 4) fitted to the data for E. tenax (Fig. 5) with an optimum at T_opt = 23.6 °C, at which the maximal developmental speed of 18.1 days is reached. Unfortunately, there was not enough data available to calculate the critical minimum (C_{mlarva_min}) and maximum (CT_{larva_max}) temperatures for larval development. Thus, these values will be calibrated.

We will assume that the dependence on larval density (Fig. 2), which shows that larvae require more time to develop at higher larval densities, is valid for a temperature of 20 °C (as this was the constant temperature in the laboratory during Hurtado"s (2013) experiment). We will use a linear equation fitted to the three data points from Hurtado (2013) (i.e., development times measured for densities of 20, 60 and 100 larvae in a 10.5 × 10.5 × 5 cm container; Fig. 2) and a data point derived from the study of Ottenheim et al. (1996) by calculating the developmental time at 20 °C from Equation 5. In the study by Ottenheim et al. (1996), a maximum of 20 individuals of E. arbustorum were reared in 18 × 13 × 4.5 cm containers. As E. tenax is 1.3 times larger than E. arbustorum (12–16 mm length compared to 9–11 mm length of E. arbustorum), it can be assumed that the maximum larval density of E. arbustorum tested by Ottenheim et al. (1996) corresponds to 15.4 larval densities of E. tenax. Furthermore, considering that the containers used for larval rearing in both studies by Hurtado (2013) and Ottenheim et al. (1996) were similar in terms of depth, and that Eristalis larvae stay close to the water surface to breathe, we can assume that the reported densities are related to the water surface area of the containers, i.e. 0.011 m² in the study by Hurtado (2013) and 0.0234 m² in the study by Ottenheim et al. (1996). We can then recalculate the densities reported in both studies to the number of larvae per m² of water surface and fit a linear equation relating larval density to development time at 20 °C (Equation 7).

D_larva (T = 20 °C, L_d) = 0.002 × L_d + 14.966 Equation 7

where D_larva is the developmental time of the larval stage, L_d is larva density and T is temperature.

By linking Equation 4 and Equation 7, we will then create a family of quadratic curves (Fig. 9), parameterised so that their values reached at the temperature 20 °C match the larval density dependence (Equation 8).

D_larva (T, L_d) = 0.272 × T² – 12.886 × T + 0.002 × L_d + 163.886 Equation 8

In addition, an individual variation will be included around a maximal developmental speed (+/- 10%). The total developmental time for larvae will be divided into three instars using the proportions reported by Cao et al. (2022) for E. arvorum.

The duration of the pupal stage (inversion of developmental rate) will be related to the temperature using a thermal performance curve model of the Lobry–Rosso–Flandrois (LRF; lrf_1991). This model named by Ratkowsky and Reddy (2017) was proposed to represent microbial growth by Lobry et al. (1991) and Rosso et al. (1993). The four parameters in the model represent three cardinal temperatures (the maximum temperature at which development can occur T_max, the minimum development temperature T_min, and the optimum development temperature T_opt) and the specific growth rate at the optimum (U_opt). U_opt is independent of the three cardinal temperatures, and T is the temperature (Equation 9).

r(T)=uopt(T-Tmax)(T-Tmin)2(Topt-Tmin)[(Topt-Tmin)(T-Topt)-(Topt-Tmax)(Topt+Tmin-2T)] Equation 9

Among similarly performing thermal performance curves we selected the LRF model as it was previously proposed to describe growth and temperature relationships in the agent-based model of another pollinator, the green-veined white butterfly Pieris napi (Topping and Duan 2025). After fitting the LRF model to our data, the parameters T_min, T_opt, T_max and U_opt were set to 2.16 °C, 26.00 °C, 31.59 °C and 0.13, respectively.

We incorporate weather effects on hoverfly activity through an hourly weather assessment. Based on Gilbert (1985a), we assume that 50% of the population becomes active when the hourly daytime temperature reaches 11 °C, and 100% of the population is active when the temperature reaches 15 °C (as per Kato 1943, after Bressin 1999). This enables the calculation of the fraction of the population that is active (F_activ) at specific hourly temperatures (T_h) during the day, assuming a linear relationship (Equation 10).

F_activ (T_h) = 12.5 × T_h – 87.5 Equation 10

This activity is further restricted to daylight hours only (Fig. 10), and to specific rainfall and wind conditions (maximum acceptable rainfall R_max and wind speed W_max which will be calibrated). Individuals that become active at a given temperature are selected randomly from the population, reflecting the influence of varying local microclimatic conditions. From Equation 10 we can calculate the minimum temperature for activity to be 7 °C, which is consistent with the range of temperatures provided in the literature. In addition, after Jarlan et al. (1997), we will assume that the maximum temperature for activity is 30 °C.

Knowing when a given hoverfly starts and ends to be active during the day (based on temperature, rainfall and wind profiles during the day), we can calculate the total number of activity hours per day (h_tot). The total activity hours per day will then be divided into time spent on feeding (h_f), resting (h_r) and flying (h_m) based on proportions provided by Gilbert (1985a). These proportions were derived from instantaneous behavioural observations recorded at the moment of first sighting, with the aggregated data interpreted as representative of the species" energy budget. For the purposes of this study, we assume that "feeding" time includes all local movements within the flower patch, whereas "flying" time covers movements between flower patches. This would imply that there is a local movement distance threshold d_l (to be calibrated) above which a movement bout is considered to be long enough to allow a hoverfly reaching flight speed to find a new flower patch (see "Dispersal" for details). We further assume, after Thyselius (2022), that the activity pattern of E. tenax does not change with the hoverfly"s age.

Gilbert (1985b) analysed dietary preferences (estimated feeding-time-budgets) of hoverflies based on field observations and categorised hoverflies according to the proportion of time spent feeding on nectar versus pollen. In this study, E. tenax was categorised as a mixed-feeder, feeding on both pollen and nectar. Hoverflies participate in pollination as a result of the pollen getting stuck to their hairs. When foraging, dipterans use visual cues for long distances and olfactory ones for short distances (Willmer 2011).

To our knowledge, there is only limited information on the energetics of E. tenax and other hoverflies. Existing studies focus on examining the diets of hoverflies by analysing dissection of their intestines or their excrements (which allows identification of the pollen types ingested; e.g. Halloway 1976; Irvin et al. 1999), pollen transport by hoverflies (Lucas et al. 2018), or impact of type of food consumed on hoverfly"s performance (i.e., longevity, activity or survival; i.e., Pinheiro et al. 2015; Thyselius and Nordström 2016). The only detailed analyses of energetics of the foraging behaviour of E. tenax was presented by Sotavalta and Laulajainen (1961) and Gilbert (1983). Gilbert (1983) found that the energy gain while foraging on Aster is low (0.01 W) and that when continuously foraging on Aster nectar, the hoverflies can fill their crop in about 75–220 min, assuming the average and maximal crop volumes of 5.8 and 17.2 µl respectively. Gilbert (1983) suggested that E. tenax can thus make a sizeable energetic profit, even at low nectar availabilities.

Both E. tenax males and females usually eat pollen at the beginning of their life because adults emerge with immature reproductive organs and gametes, and pollen proteins are essential for maturation (Gilbert 1986). Subsequently, females consume more pollen than nectar because it is needed for egg maturation (Gilbert 1985b). Males that hover have been found to have their abdomen partly filled with air, probably to reduce their mass and save energy, and tend to feed on nectar only (Gilbert 1986). These distinct food requirements between female and male hoverflies were confirmed by several studies on the gut content of hoverflies in which the amount of pollen was larger in the gut content of gravid females compared to non-gravid females and males (Haslett 1989; Irvin et al. 1999; Villa et al. 2021).

The detailed study of the gut contents in Rhingia campestris (Meigen, 1822) hoverfly (body length of 8–11 mm, wingspan of 12–18 mm) showed a relationship between diet and the degree of ovarian development in females. Females switch from nectar to pollen feeding during the maturation of ovaries and back to nectar feeding once eggs are laid (Haslett 1989; Fig. 11). For females, nectar-feeding is thus associated with mating and oviposition, activities that have substantial energy costs (Gilbert 1986). Based on personal observation, Haslett (1989) suggested that a similar pattern can be observed in other hoverfly species, including E. tenax.

E. tenax forage on a wide range of flowers. Flower choice is made according to the nutritional value of the pollen provided by the flower and according to the colour of the petals (Bressin 1999).

E. tenax exhibits distinct floral preferences influenced by the colour, size, and UV properties of flowers. Studies have shown that yellow is the most preferred colour for both landing and proboscis extension, with larger yellow flowers being particularly attractive (Matoušková et al. 2023). The preference for yellow is so strong that it triggers landing regardless of UV reflection, while proboscis extension is specifically triggered by yellow colours that absorb blue and UV light (Lunau and Wacht 1994; Lunau and Wacht 1997; An et al. 2018). Despite their ability to learn to land on other colours, E. tenax still shows a significant preference for yellow and UV-absorbing colours over UV-reflecting ones. Additionally, these flies tend to avoid dark colours, except for yellow, and prefer bright colours for landing (An et al. 2018). The spectral sensitivity of their proboscis reaction is finely tuned to a small range of wavelengths (520–600 nm), which aligns with the yellow colour of pollen and floral guides in natural flowers (Wacht et al. 1996). This innate preference for yellow and the limited ability to condition their proboscis reflex to other colours suggest that E. tenax has evolved specific perceptual dimensions for yellowness and blueness, optimizing their foraging efficiency on yellow flowers (An et al. 2018).

In the case of E. tenax, the plant-pollinator interactions were analysed using different methods: direct visual observations (Cowgill 1991; Hoffmann 2005; Klecka et al. 2018; Ouvrard et al. 2018; Plume et al. 2024), analysis of the gut content (Irvin et al. 1999), analysis of pollen carried by hoverflies on their body (e.g., Holloway 1976) or DNA metabarcoding (Lucas et al. 2018). Based on these publications, a summary of plants visited by E. tenax is presented in Table 5. The table does not distinguish between plants visited by female and male hoverflies, as according to Klecka et al. (2018) differences in flower preferences between sexes in E. tenax are negligible.

Since the foraging movements are intertwined with reproduction and dispersal, we consider foraging movement under the heading "Dispersal" below.

Following Haslett (1989), we assume that a female hoverfly switches the ratio of pollen to nectar in her daily diet depending on her behavioural state, and that she can forage on pollen and nectar at the same time (from the same habitat patch if both resources are available). Thus, depending on the behavioural state, the female will have a specific foraging goal expressed in terms of the ratio of pollen to nectar. During ovarian development and egg maturation, the female will feed mainly on pollen (foraging goal 90:10 pollen to nectar ratio), whilst during mating and oviposition, the female will feed mainly on nectar (foraging goal 20:80 pollen to nectar ratio).

We further assume (following Gilbert 1983) that E. tenax can make a sizeable energetic gain even at low nectar availability, thus replenishing the energy lost during flight very quickly. Therefore, we do not intend to model the energetic budget of the hoverfly (linking energy losses with flying and mating activities and energy gains with nectar foraging). Instead, the percentage of gut filling will be tracked in the model separately for pollen and nectar resources (according to the foraging goals). We assume that the gut volume (G_v) corresponds to maximum crop volumes measured by Gilbert (1983), i.e., vary between 5.8 and 17.2 µl, depending on the hoverfly"s mass (M). Here, a linear relationship between gut volume and mass will be assumed (Equation 11), considering the mass range of females from 109 to 187 mg (as measured for hoverflies from the field; Nicholas et al. 2018).

G_v (M) = 0.13 × M – 8.16 Equation 11

We further assume that the hourly rate of food (pollen or nectar) intake r_h in hoverflies is limited and depends on the gut volume (i.e., it can be represented as a fraction of the gut volume). Knowing the number of daily foraging hours, we can calculate the maximum daily food intake (Equation 12) and, hence, the maximum fraction of the gut which can be filled within a day. Thus, by changing r_h, we can change the minimum time it takes for a hoverfly to reach its reproduction goals (see "Reproduction" for details).

F_max = r_h × h_f × G_v (m) Equation 12

where F_max is the maximum daily food intake, h_f is the number of daily foraging hours, and r_h is the maximum hourly rate of food intake (0 < r_h < 1) to be calibrated.

The maximum daily food intake F_max will be a sum of pollen and nectar in a proportion in line with the current foraging goal. The amount of pollen and nectar needed to completely fill the gut, and thus the maximum daily food intake (F_max), will be calibrated to the amount of resources available in the environment, such that when resources are low, it is much more difficult to reach a full gut than when resources are high. This calibration will take into account the reproductive goals (see "Reproduction" below) as these are associated with certain levels of gut filling.

The amount of pollen and nectar available to E. tenax from each habitat patch in the landscape per day will be modelled in ALMaSS using the floral resource models. The floral resource models have been developed within the B-GOOD Horizon 2020 project (Ziółkowska et al., unpublished). The framework integrates data on the amount of pollen, nectar and sugars produced by a single flower/inflorescence of a given plant species with information on flower/inflorescence abundance (i.e. density of flowers/inflorescences per unit area), flowering phenology, plant composition of the habitat and habitat composition and configuration within the landscape. This approach makes it possible to predict the spatio-temporal availability of floral resources within different habitat types and across different landscapes.

The amount of pollen available to E. tenax in the landscape will be limited to those plants visited only by E. tenax. Such a list of plants will be available based on analysis of plant-pollinator interactions. Table 5 presents a preliminary list of plants visited by E. tenax, which could be further revised by additional literature searches and expert knowledge. In addition, the daily pollen attractiveness P_h of a given habitat patch h will be calculated by multiplying the amount of pollen resources P available to E. tenax from each plant p in the patch on day d by the plant"s pollen preference score p_s (Equation 13). The amount of pollen resources produced by plant p on day d in habitat patch h results from the daily pollen production of the plant, the proportion of the plant in the habitat patch and the area of the habitat patch. The pollen preference score p_s of a plant is a pollinator-specific measure on a semi-quantitative scale (from 0 to 3), calculated on the basis of plant-pollen interactions and the nutritional and chemical properties of a pollen (i.e. protein to lipid ratio and potassium to sodium ratio). The pollen preference score p_s is being developed in the Better-B Horizon Europe project and will be linked to the floral resource models in ALMaSS.

P_h = ∑_pP_p,d × p_s Equation 13

The daily pollen attractiveness P_h of a given habitat patch h in a landscape will be further scaled by the nectar attractiveness N_h modified by a scalar c (to be calibrated) to obtain the daily food attractiveness F_h of the patch (Equation 14). To our knowledge, detailed information on the nectar preferences of E. tenax, particularly regarding sugar concentration preferences, is lacking. Gilbert (1981) investigated the ingestion rates of sugar solutions ranging from 10% to 67% in various hoverfly species, including E. tenax. While complete data were available for Syrphus ribesii (Linnaeus, 1758), indicating an optimal energy intake rate at approximately 50% sugar concentration, the data for E. tenax were incomplete. However, the available results suggest that E. tenax exhibits lower ingestion rates compared to S. ribesii, ranging from 0.31–0.35 µl/s for 20–30% solutions and declining to 0.04 µl/s at 67% concentration. Given the limited empirical data, we assume that the daily nectar attractiveness of a given patch h is proportional to the total daily nectar production of all plant species within that habitat that are visited by E. tenax. This assumption can be changed later when more information on E. tenax nectar preferences is available.

F_h = P_h × N_h × c Equation 14

E. tenax exhibits bivoltinism, producing two generations annually (Gilbert 1986; Bressin 1999; Upchurch et al. 2023). Although details on number of mating events are lacking for E. tenax, insights can be drawn from related species. For instance, Gladis (1993, 1994b) investigated the reproductive behaviour of Helophilus pendulus (Linnaeus, 1758), another hoverfly species with saprophagous larvae. The study demonstrated that females of H. pendulus are capable of multiple mating events. Under controlled laboratory conditions, individual females underwent up to three copulation-oviposition cycles. Following the third cycle, reproductive activity declined. While occasional copulation was still observed, it was not accompanied by complete egg development, suggesting a reduction in reproductive potential after successive mating cycles.

Mating can last from seconds to hours. Males hover and wait or search for passing females with which to mate. Several authors have reported that male hovering activity is timed to coincide with female activity, as it is energetically demanding (Thyselius 2022). In Eristalis spp., the eggs are not immediately fertilized but are stored in the female spermathecae. The sperm are nourished by the secretions of the female"s special tube glands. These glands are particularly well developed in E. tenax, in which the sperm of the autumn (second) generation of females are nourished throughout the winter, and the eggs are not fertilized until the spring (Gilbert 1986). After hibernation, E. tenax females require a certain number of days for the development of their ovaries and maturation of eggs. This pre-oviposition period could vary considerably, from 10 to even 38 days, with a mean around 20 days (Table 9).

E. tenax lays eggs in clutches. The total reproductive potential (the total number of eggs) of a female, and the number and size of clutches differ considerably (Table 9). Honĕk (1993) found a linear relationship between fecundity and adult female dry weight, and between number of ovarioles and adult female dry weight for oviparous insects, including Diptera. Although E. tenax was not covered in this study, data were included for another hoverfly, Metasyrphus corollae (Fabricius, 1794) (Scott and Barlow 1984). Gladis (1993, 1994b) identified a consistent pattern in the reproductive behaviour of H. pendulus. Females typically deposited eggs in two to three (or more) adjacent clutches on the same day, with occasional additional egg deposition occurring the following day. The initial oviposition generally took place approximately one week after mating. Subsequent mating events occurred immediately following oviposition, with intervals between mating and the next oviposition ranging from 10 to 14 days. In contrast, E. tenax exhibits more variable inter-oviposition intervals, ranging from 2 to 12 days (see Table 9). As egg maturation within the ovaries is a gradual process, these intervals allow the female to accumulate sufficient nutrients and energy reserves. At the population level, in the lab conditions, Upchurch et al. (2023) found that the peak of egg production in E. tenax occurs around the 5^th (22% of total egg production) and 8^th (20%) weeks of the hoverfly"s life. The variability in E. tenax reproduction parameters depends on various factors, including energy reserves, nutrient availability and environmental factors (Gladis 1994).

During oviposition, E. tenax females are attracted by the odour of manure, stagnant water, decaying matter, etc. (Bressin 1999). On farms, females can lay eggs in cracks in the walls of septic tanks and even on the dry crust of slurry. Studies of larval densities in different types of media are scarce. Eristalis eggs are laid in clutches and hatch more or less synchronously within a clutch (Gladis 1994), resulting in a patchy appearance of larvae and making it difficult to obtain useful density measurements. In the study by Mahmoud (1997), mean monthly Eristalis larval densities varied from about 3 to about 60 (with a mean of about 24) larvae per 900 cm² surface area in a gravel-bed hydroponic system in Abu-Attwa (Egypt), where the water was 30 cm deep, (approximately 30 to 650, mean 275, larvae per m² of water surface). In a much more extensive study by Ibrahim (1976), Eristalis larval densities were investigated in eight different habitat types at 22 study sites throughout Egypt. The highest mean and maximum larval densities of approximately 300 and 940 larvae per m² of water surface, respectively, were found in drain water bodies (Fig. 12).

Study of Campan and Campan (1979) showed that the approach to the oviposition site in E. tenax involves a complex interplay of olfactory and visual cues, with different behaviours and reactivities at various stages of ovarian development. The hoverflies response to the smell of manure varied with the stage of oocyte development. Immature females were repelled by the smell, developing females were indifferent, and mature females were attracted to it. The field observations of Campan and Campan (1979) showed that egg-laying in E. tenax is not typically performed during a single flight. The study suggests that the approach to the oviposition site involves multiple visits over several days. During these visits, the flies perform various activities such as flying over the manure, exploring the site, and adopting preparatory postures before finally laying eggs. This sequence of behaviours reflects a gradual approach to the egg-laying site, rather than a single, direct flight. In addition, Campan and Campan (1979) observed that not all females that explored the oviposition site ended up laying eggs. Many females performed exploratory behaviors, but then flew away without laying eggs. In addition only a small percentage of marked females returned to the manure site, and those that did often performed the same type of visit multiple times before laying eggs.

Following Honĕk (1993), we assume that the E. tenax potential fecundity (potential total number of eggs to be laid) depends linearly on the female"s mass and thus is set at the female"s emergence. As data on individual mass and fecundity were not available for E. tenax, we based the linear relationship on the minimum and maximum values reported in the literature (Tables 1, 9), assuming that the minimum reported mass corresponds to the minimum reported fecundity, and the maximum reported mass corresponds to the maximum reported fecundity (Equation 15). Female mass data were taken from the field measurements of Nicholas et al. (2018) (Table 1). As field data on fecundity was not available for E. tenax, we decided to use fecundity data measured in the laboratory from Hurtado (2013), as the percentage of ovipositing females in this study was much higher than in Campoy et al. (2020) (Table 9).

N_Tegg = 8.782 * M_f – 867.244 Equation 15

where N_Tegg is the potential fecundity of a female, and M_f is the mass of a female.

After hibernation (autumn generation) or emergence (spring/summer generation), a female enters a pre-oviposition period. We will divide the pre-oviposition period into two distinct phases: (1) the time required for ovary development and (2) the time required for egg maturation (at the beginning of which mating occurs). The duration of these phases will be tracked separately. This tracking can be achieved by monitoring when a female hoverfly accumulates a specific amount of nutrients (i.e., floral resources) in its gut (i.e., achieves the “reproduction goals”). This artificial construct ensures a consistent tracking mechanism for the gut content associated with each clutch produced by a female.

We assume that the “egg maturation” phase must be repeated for each clutch, while ovary development occurs only once during a female"s lifetime. Following the findings of Gladis (1993, 1994b), we hypothesise that, as in H. pendulus, mating in E. tenax may occur multiple times, on the day following previous oviposition event. Consequently, mating events are considered an integral part of the “egg maturation” phase. Thus “egg maturation” phase will start with the mating event and end with the oviposition event. Both processes depend on nutrient accumulation, with ovary development requiring a one-time nutrient threshold and egg maturation requiring a nutrient threshold for each clutch. We assume that the same foraging-based resource counter (linked to the rate of gut filling) can be applied to each clutch implying that the target clutch size remains unchanged over time and does not vary with the age of the female. This is a simplifying assumption, as empirical evidence from Campoy et al. (2019) indicates that fecundity in E. tenax may vary with age. For the purposes of our model, we further assume that each oviposition event results in the deposition of a single clutch of eggs. According to Gladis (1993, 1994b), oviposition in H. pendulus typically concludes with the laying of two to three (or more) clutches of eggs, all deposited on the same day and in close proximity. To simplify our modeling framework, these multiple clutches are treated as a single oviposition unit, and thus considered as one clutch.

Following Honĕk (1993), we assume that the nutrient threshold P_odev for the “ovary development” phase depends linearly on the female"s mass, as the bigger the female, the more ovaries it has, and the more nutrients she needs to accumulate for the ovary development. We further assume that each female, depending on her mass, has a certain minimum and maximum target number of eggs per clutch to oviposit, which correspond with the minimum and maximum nutrient thresholds, P_eggmin and P_eggmax, needed for the maturation of eggs. Nutrient thresholds P_odev, P_eggmin and P_eggmax will be expressed as percentages of female"s gut filled with pollen, and set during the calibration process.

The actual clutch size will vary from the targeted maximum due to environmental factors such as spatial and temporal distribution of floral resources and weather conditions that influence daily foraging activities, and thus the daily intake of nutrients needed for the maturation of eggs (see "Foraging" section for details). These factors, along with the potential fecundity of the female, will determine the number of clutches produced. Under permitting weather conditions, the hoverfly female will gather a certain amount of pollen per day (from various resource patches reached in multiple movement bouts) depending on the availability of food resources around its location. She will continue feeding on pollen until the minimum nutrient threshold P_eggmin is achieved and the probability of continuing foraging p_f is above zero or until the maximum nutrient threshold P_eggmax is achieved. The probability of continuing foraging p_f or, in other words, the willingness to continue foraging will be set to 1 until the optimal duration of maturation of eggs period (D_Mopt) is reached and will drop afterwards to reach zero when the maximum duration of maturation of eggs period (D_Mmax) is reached (Fig. 13). The optimal and maximum duration of maturation of eggs period, D_Mopt and D_Mmax, will be initially set based of reproductive parameters of E. tenax measured in the laboratory conditions (Table 9). The shape of the probability curve will be calibrated to derive distributions of egg-laying patterns. The calibration of nutrient thresholds and probability p_f distribution will need to consider the duration of the pre-oviposition period, intervals between clutches, and egg clutch size, representing the field/laboratory situation (Table 9). Here, we assume that the period needed to develop the ovaries can be calculated by identifying the patterns of the duration of pre-oviposition periods and the interval between clutches.

Once the maturation of eggs is finished (according to the rules described above), the female will move to the egg-laying habitat, and the oviposition will be triggered. As this is an energetically demanding activity, it is only triggered if enough nectar is collected in the gut (nutrient threshold N_ovi is reached; to be calibrated). If not, the female will continue feeding on nectar till the required amount is reached. The size of a clutch will depend on the percentage of the gut filled with the pollen, assuming the linear relationship. We assume that if the size of a clutch is smaller than the maximum clutch size, the remaining egg load is “lost” (i.e. we assume the phenomenon of egg resorption in E. tenax). This means that the total number of eggs laid by a female in all clutches may differ from her potential fecundity. Once the clutch has been laid, foraging activity is resumed and the reproductive behaviour is repeated until no eggs remain to be laid.

For oviposition, a map of larval habitats is needed, as eggs are laid close to these habitats. We will assume that each patch in a landscape is characterized by a maximum density of E. tenax larvae L_dmax, which corresponds to a maximum egg density E_dmax, above which a clutch oviposition is no longer possible. Unfortunately, studies on the effect of habitat type and quality on Eristalis larval densities are very scarce and practically limited to laboratory studies analyzing the survival and development time of larvae cultured in different growing media. In addition, it is very difficult to properly map possible larval habitats in a landscape because Eristalis larvae are quite versatile and can adapt to different environments as long as there is sufficient organic matter for them to feed on. They can be found in drainage ditches, lagoons and manure pits on farms, stagnant water of various origins, compost heaps or rooting vegetation. Therefore, instead of directly mapping Eristalis larval habitats, we propose to characterize each landscape element by a larval habitat quality index L_q, based on a general classification of landscape elements according to their suitability as habitats for Eristalis larvae (Table 6), but also taking into account the moisture conditions resulting from the properties of the soils and the local topography, as well as the current meteorological conditions (i.e. the deviation of the precipitation regime from normal conditions). Thus, the larval habitat quality index L_q of a habitat patch p will be calculated as a combination of larval habitat score h_s defined based on habitat classification in terms of suitability as a larval habitat, and soil moisture score m_s (averaged across the patch area), modified by the precipitation regime score p_s at day d (Equation 16).

Lq,p=hs×∑i=1nmsn×ps,d Equation 16

The soil moisture score m_s (initially ranging from 1 to 5, with 1 representing very low soil moisture and 5 representing very high soil moisture) will be calculated for each pixel of a raster map of the landscape, and then averaged over all n pixels of a given habitat patch to represent the patch-level value. The score will be derived from the soil moisture proxy model by integrating three components, initially with equal weights: slope, total available water of a soil, and solar radiation (as proposed in Brown 2016). Slope is inversely related to soil moisture in the model, as the steeper the slope, the more runoff and the less water infiltration. Solar radiation is an alternative measure of aspect and may serve as a proxy for varying levels of potential evapotranspiration during a growing season (Jensen and Haise 1963; Nikolaou et al. 2023). Solar radiation is also inversely related to soil moisture in the model. Total available water (TAW), or plant-available water, quantifies the amount of water available to plants that the soil can hold (de Melo et al. 2023). The bigger the TAW, the wetter the soil and better plant growth conditions. TAW values for different soil particle-size classes (according to FAO 2006) used in ALMaSS will be defined based on the soil textural triangle of TAW predicted by the FAO method by de Melo et al. (2023) (Table 7).

We assume that the habitat patches that are potentially suitable for E. tenax larval development (Table 6) may differ in terms of topography and soil properties, which influence the occurrence of microhabitats suitable for larval development within the patches. This can be further modified by the current meteorological conditions, i.e., if the precipitation regime is wetter or drier than normal, expressed by the precipitation regime score p_s in Equation 15. The precipitation regime score p_s will be derived from the Standardized Precipitation Index (SPI) (McKee et al. 1993), which measures precipitation anomalies (deviations from the mean) at a given location, based on a comparison of observed total precipitation amounts for an accumulation period of interest (e.g. 1, 3, 12, 48 months), with the long-term historical precipitation record for that period. It is a commonly used indicator for the detection and characterization of meteorological droughts. Here, we will use the short-term SPI (i.e., calculated for shorter accumulation periods) because it can be used as an indicator of immediate effects such as reduced soil moisture, and flow in smaller streams and ponds. Relative scores of SPI values will be used to modify the larval habitat quality, taking into account drier or wetter hydrological conditions as normal (Table 8). Raster maps of SPI (spatial resolution of 0.25°, temporal resolution of 10 days) can be obtained from the European Drought Observatory of the Copernicus programme (https://drought.emergency.copernicus.eu/).

The larval habitat quality index L_q (initially from 0 to 30) will be linked to maximum larval density L_dmax via a linear relationship, i.e., assuming a linear relationship between minimum and maximum larval densities observed in the field and (see Table 4) and larval habitat quality (Equation 17).

Ldmax,p(Lq,p)=[Ap×Ld(fieldmax)-Ld(fieldmin)Lqmax-Lqmin×(Lq,p-Lqmin)+Ld(fieldmin)]×r Equation 17

where A_p is area of habitat patch p, L_d(fieldmin) and L_d(fieldmax) are minimum and maximum larval densities per m² possible to be reached in larval habitats of different qualities (e.g., with different amount of organic matter), and L_qmin and L_qmax are minimum and maximum values of larval habitat quality. L_d(fieldmin) and L_d(fieldmax) will be initially set at 30 and 940 larvae per m² (according to studies by Ibrahim 1976 and Mahmoud 1997, which examined the densities of Eristalis species larvae in different microhabitats across Egypt; see Table 4), but may be changed in the future calibration process or if new field data become available. A scalar r will be used (0 < r < 1) to scale down the larval density values, as the densities measured in the field were not obtained using a random sampling method (i.e. sampling was targeted to those microhabitats or parts thereof where larvae were observed), and thus their direct conversion to densities per unit area of an entire habitat patch would result in a (rather significant) overestimation. Scalar r will be set during the calibration process.

Drone flies begin their adult stage with obligatory dispersal, during which they leave their place of emergence and virtually never return (Wellington and Fitzpatrick 1981). Males are territorial, but only in the summer generation and not in the autumn generation. This territoriality means “an increasing attachment to one particular site, e.g. a few leaves and flowers of one plant”. Territorial area is from one site of 0.36 m² to several sites scattered over 500 m² (Wellington and Fitzpatrick 1981).

Only limited information on distances travelled daily by hoverflies is available. In the capture-recapture studies, non-migrant syrphids travelled distances of up to 400 m (most flies dispersed up to 100 m after release) (Conn 1976; Wratten et al. 2003) with a speed of 12–18 km/h (Aubert et al. 1969). In a field study of Wratten et al. (2003), distances travelled by Melanostoma fasciatum (Macquart, 1850) (New Zealand hoverfly species of 6–7 mm length), Episyrphus balteatus (De Geer, 1776), and M. corollae (United Kingdom hoverfly species of 6–10.25 mm and 7–10.5 mm length, respectively) from a Phacelia strip were investigated (Fig. 14). This study was based on analysing the gut contents of hoverflies caught in traps placed at different distances from the food source. Thus, what was actually measured was the displacement from the food source, from which it is difficult to infer the actual distance travelled by the hoverflies per day, especially as the traps were only emptied every seven days.

Collett and Land (1978) measured the E. tenax fight behaviour and velocities in the field as being at least 10 m/s but for short periods of time only. However, only flies that are already airborne can reach these speeds; they are likely to fly much slower between flowers during foraging. Golding and Edmunds (2000) measured the time droneflies (E. tenax, E. pertinax, E. arbustoeum and E. nemorum) spend on individual flowers and the time spent on flying between flowers, finding that it was more similar to that of honeybees than to the times of other hymenopterans and dipterans. The time spent on individual flowers depended on the flower type and varied between 5 to 16 seconds. The time spent flying between flowers also depended on flower type and varied between 1.3 to 3.7 seconds (but mostly around 1.5 seconds).

Francuski et al. (2014) studied the landscape genetics of E. tenax in a spatially heterogeneous landscape of Durmitor Mountain (Montenegro). Their sampling points covered a broad range of altitudes and vegetation structures, from deciduous and coniferous forests to alpine meadows). They found a lack of population structure in the studied area, suggesting that E. tenax dispersal ability is sufficient to establish gene flow over a network of habitats involving the whole study area. The observed lack of isolation by geographical distance suggests that dispersal ability is not the key factor determining distributions of E. tenax at the spatial scale studied (15 × 15 km). Similar studies on E. tenax populations conducted on even larger spatial scales, encompassing populations from the Central-Eastern Mediterranean (Francuski et al. 2013a) and along a north-south European gradient from Finland through Switzerland to Serbia (Francuski et al. 2013b), also demonstrated the absence of large-scale geographic structuring. Therefore, it seems that the species is restricted in its distribution by the availability of suitable habitat rather than the ability of a species to disperse between available habitat patches.

Adults are capable of extensive, long-distance migration (Bailes et al. 2018) and can fly continuously over distances for at least 75 km across the open ocean (Krčmar et al. 2010). In Europe, they can migrate north to avoid hot summers before returning south in autumn (Francuski et al. 2014; Francuski and Milankov 2015). A comprehensive review by Reynolds et al. (2024) consolidates robust evidence for long-distance migration of E. tenax in Europe, North America, and Oceania. Despite the classification of E. tenax as a weakly migratory species by Hlaváček et al. (2022), long-term observational studies, such as the 12-year investigation by Aubert et al. (1976) and the extensive 50-year monitoring programme in Germany led by Gatter (Gatter 1976; Gatter et al. 1990, 2020), underscore the ecological relevance of migratory behaviour in this species. Further supporting this migratory capacity, Hawkes et al. (2022) demonstrated the potential for long-distance pollen transport by E. tenax, indicating functional connectivity between distant habitats.

Tomlison and Menz (2015) explored partial migration in hoverflies, linking migratory propensity to physiological traits such as metabolic rate, thermal tolerance, and rates of water loss. Their findings suggest that most individuals possess the physiological capacity for migration, with inter-individual variation, particularly among females, potentially reflecting differential adaptation to thermal environments and reproductive strategies. However, the relative proportion of migratory versus resident individuals within populations remains unknown. Reynolds et al. (2024) propose that this ratio is likely modulated by complex interactions between seasonal temperature regimes and photoperiodic cues.

We will treat E. tenax as a partially migratory species. Specifically, we assume that the proportion of females from the autumn generation will exhibit migratory behaviour, departing from the local landscape after mating. A given female from the autumn generation will initiate migration movement with a defined probability p_mig.

We will distinguish between local movements within the flower patch (considered under "feeding time") and movements between floral patches (considered under "flight time"; see "Diurnal activity" for more details). The former type of movement will not be tracked in the model. The maximum daily dispersal ability D_tmax (for movements between floral patches) will be calculated from the available daily flight time (h_m) and the flight speed v achieved by E. tenax hoverflies already in the air (Equation 18). Dispersal activity will be tracked individually for each hoverfly, both in terms of time spent and distance travelled, and once daily limits are reached a hoverfly will suspend its activity for that day.

D_tmax = h_m × v Equation 18

We assume mean v being set to 10 m/s, as measured by Collett and Land (1978). In addition, an individual variation will be included around this mean speed (+/- 10%).

Four types of dispersal actions will be considered:

We plan to represent the movement using the following rules:

The flow diagram for this behaviour is described in Fig. 15. This behaviour requires the specification of four functions: one for setting the forage goal, the second for determining the forage status (in relation to reproduction goals), the third for accepting a habitat patch for foraging, and the fourth for larval habitat acceptance. The function for setting the forage goal is expressed in terms of the ratio of pollen to nectar. Two foraging goals are possible, either 90:10 or 20:80 pollen to nectar ratio, depending on the behavioural state of the female (see "Foraging" for details). The function for determining the forage status tracks the percentage of gut filling in terms of pollen and nectar (see "Foraging" for details).

The function for determining accepting a given patch for foraging at a given food attractiveness is given by a probability function between two thresholds such that a hoverfly will not accept a given patch for foraging below F_l, a minimum food attractiveness threshold (e.g. 0), and the probability of accepting a patch for foraging then increases linearly to an upper threshold F_h. The values for the probabilities at F_l and F_h would be given as F_lp and F_hp. The food attractiveness value to drive this function will come from Equation 15. Given the frequent movement of hoverflies between patches, we assume that the probability of leaving a given habitat patch (p_move) increases over time with the base probability inversely related to the food attractiveness of the patch (Equation 19). Consequently, patches with higher food attractiveness will have longer occupancy times compared to those with lower food attractiveness, compelling hoverflies to move more frequently when patches are less resource-rich.

p_move (t, F_h) = p_o + k × t × F_h^-1 Equation 19

where p₀ is the base probability of leaving the habitat, k is a constant that determines how the probability increases with htime, t is the time spent in the habitat patch, and F_h is the food attractiveness of the patch.

The function for determining clutch laying at a given habitat patch is given by a probability function between two thresholds that clutch is not laid below L_l, a minimum larval habitat attractiveness L_a (e.g. 0), and the probability of laying a clutch then increases linearly to an upper threshold L_h. The values for the probabilities at L_l and L_h would be given as L_lp and L_hp and will be calibrated. The larval habitat attractiveness value L_a to drive this function will come from larval habitat quality (Equation 16) modified by factor c (0 < c < 1) indicating the presence of other clutches in the same habitat patch p (Equation 20).

L_a,p = L_q,p × c_p Equation 20

Factor c allows to account for density-dependent oviposition, as clutch-size optimality models (e.g., Iwasa 1984; Wilson and Lessells 1994) suggest that insects may avoid laying clutches in areas with high egg densities to reduce competition for food and space for their offspring and to reduce the risk of predation. We are unaware of any studies that confirm that such strategy exists in saprophagous flies, so this factor will be set to 1 in the initial version of the model. Other settings may be used to test the effect of density-dependent oviposition on the results of the model, or if new evidence becomes available for saprophagous hoverflies.

Fig. 13 does not include hibernation and migration behaviours. These behaviours are triggered by the assessment of the reproduction state. After mating, if this is the autumn generation, the female initiates hibernation or migration movement. In case of hibernation, the behaviour follows the same procedure as Fig. 15 but replacing "larval habitat" with "hibernation habitat" and "lay egg clutch" with "enter the hibernation state". Once in hibernation, the female switches the state to the development of ovaries only after the emergence from hibernation. If it is the spring/summer generation, the female goes directly from mating to the development of ovaries. In the case of migration, a female is assumed to leave the landscape and is consequently removed from the modelled population. There is a certain probability p_mig of a certain female migrating instead of overwintering.

Part of the autumn (second) generation of E. tenax not migrating, overwinters. These individuals are predominantly adult females that have been inseminated and subsequently enter a facultative reproductive diapause. This diapause phase is characterized by a cessation of ovarian development and a marked hypertrophy of the fat body (Hondelmann and Poehling 2007; Tomlinson and Menz 2015; Campoy et al. 2022). Overwintering males are rare, as most die in the autumn (Kendall and Stradling 1972). Hibernation lasts from late autumn to early spring. In nature, fertilised females hibernate in groups (up to 20 individuals) in dark and humid places such as caves or crevices in rocks and buildings. These provide shelter from adverse climatic conditions in winter, i.e. stable humidity and temperature conditions that are less affected by the wide variations in air temperature outside. Laboratory experiments by Bressin (1999) showed that overwintering E. tenax hoverflies are water-stressed, so selecting overwintering sites with high humidity allows them to limit water losses, as does overwintering in clusters. Thus, certain high levels of humidity (Kendall and Stradling 1972) and darkness are required for a site to be suitable for overwintering. The relative humidity of hibernacula observed by Bressin (1999) was over 90%.

It appears that both the onset and termination of overwintering are primarily controlled by temperature, although day length may also play a role, with decreasing photoperiod inducing overwintering and increasing photoperiod signalling the time to leave the site (Siuda 1963, after Bressin 1999; Moog and Ernst 1978; Davenport 1992). However, daylight is not itself the trigger; it acts as a secondary stimulus to prevent flies from leaving hibernacula too early if the temperature is high enough (Bressin 1999).

In Scotland, Bressin (1999) observed that the onset and end of hibernation were not short periods but extended over several weeks (Figs 16, 17). While some individuals began hibernation in mid-September, others were still active, feeding and foraging until mid-November. The increase in the number of hoverflies settling in the overwintering site corresponded to a decrease in temperature. Under laboratory conditions, hoverflies enter hibernation at 8–10 °C (Bressin 1999). In Scotland, the number of overwintering E. tenax began to decline sharply from mid-February, corresponding to a significant rise in temperature. Bressin (1999) suggested that a sustained high temperature (i.e. for at least a few days) is required. This suggestion is supported by the slow response of crevice temperatures to outside temperatures. Different "trigger" ambient temperatures for the end of hibernation have been reported in the literature. Bressin (1999) reported 13 °C in Scotland, while Siuda (1963) reported a much lower temperature of 5 °C in Poland.

The landscape features that hoverflies are looking for in the search for hibernation places are types of landscape elements connected with rocks and caves, forest edges (tree bark and crevices), hedgerows and shrubs with dense vegetation providing good hiding spots, and single-housing settlement areas with gardens, particularly those with dense vegetation.

A certain number of days (H_days) with minimum night temperatures below a threshold (Th_start) will be required to initiate hibernation of E. tenax. The distribution of the number of individuals entering the hibernation state will be implemented according to field data from Bressin (1999) (Fig. 16). In addition, to account for the effect of photoperiod, a threshold date H_date after which the hibernation is mandatory will be applied. Based on field observations of Bressin (1999), H_days, Th_start will be initially set to 5 days and 10 °C, respectively, and H_date will be set to 1^st of November.

Emergence from hibernation will also be related to temperature. A certain number of days (E_days) with maximum day temperatures above a threshold (Th_end) will be required; however, emergence will also be restricted by date, not being earlier than a given day (E_date). The distribution of the number of individuals leaving the hibernation state will be implemented according to field data from Bressin (1999) (Fig. 17). Based on field observations of Bressin (1999), E_days and Th_end will be initially set to 5 days and 10 °C, respectively, and E_date will be set to 10^th of February.

It has been shown that developmental environments with low nutrient quality or quantity (i.e. nutritional stress) cause increased duration of preimaginal stages, decreased survival, and reduced adult size and fecundity in many dipteran species (Stearns 1992; Gleiser et al. 2000). For example, nutritional stress in Drosophila melanogaster caused a 20% reduction in viability, a 70% increase in total egg-to-adult development time, and a 50% reduction in adult weight (Kolss et al. 2009). Hurtado (2013) demonstrated that both larval developmental medium (with constant larval density) and larval density can impact the duration and survival of larval stage, as well as mass of adults in E. tenax, confirming that nutritional stress can occur in this species and impact larva"s survival and mass of adults. When tested with pig slurry as a rearing medium, mass of female E. tenax adults decreased significantly with increasing larval density (Fig. 18; Equation 21).

M_f (L_d) = -0.175 × L_d + 29.833 Equation 21

where M_f is mass of a female and L_d is larval density.

From Ludoski et al. (2023), we know that the pupa mass is linearly related to the adult female mass. Thus, if we know that the nutritional stress impacts larval and pupal mass, then we can assume that it directly affects adults mass without calculating the intermediate masses. This is possible since pupal mass is not used directly in the model.

In the model, mass variability will be introduced at the larval stage in the sense that different larval habitats will be assigned different qualities (see section "Reproduction and oviposition"). The quality of larval habitats will be expressed as possible maximum larval densities, and if those are reached, further eggs will not be laid in these habitats. The impact of larval density L_d on the mass of adult female M_f will be determined according to Hurtado (2013) (Fig. 18; Equation 21). This relationship will be scaled to the maximum larval density associated with the habitat (Equation 22):

Mf(Ld)=(a+b×Ld)×Ldmax,h×MfmaxLdmaxtotal Equation 22

where a, and b are from a linear regression fitted to Fig. 18 (Equation 21), L_dmax,h is the maximum larval density per habitat h, L_dmaxtotal is the maximum larval density for all habitats, and M_fmax is the maximum adult female mass.

Information on egg survival for E. tenax is scarce, but the egg cluster viability is rather high, estimated to be about 86–100% by Bressin (1999). There is also limited information on larval survival, but it seems that this is impacted mostly by the temperature (Heal 1989; Ottenheim et al. 1996), and to a lesser degree by larval density and food/environment type (Hurtado 2013). The data summarising the dependence of larval survival on the temperature are presented in Table 10 and Fig. 19.

There was sufficient data to estimate the thermal performance curve for pupal survival, based on three publications: Heal (1989), Hurtado (2013) and Daňková et al. (2023) (Table 10). Thermal performance curves for pupal survival were estimated using 24 models with rTPC R package (Padfield et al. 2024) as described in Padfield et al. (2021). Table 11 shows the estimation for those models for which it was possible. The graphical representation of four best-performing models is shown in Fig. 20.

Background mortality will be defined separately for the egg-to-pupa stages. This mortality is a daily probability applied at the start of each daily time step. For the egg stage, due to limited data inputs, the daily background mortality will be drawn from a uniform distribution from a range between 0 and p_megg, where p_megg (based on Bressin 1999) is calculated as (Equation 23):

(p_megg)^d = 0.14 Equation 23

where d in the developmental time of the egg stage (in days).

For the larva stage, the daily background mortality p_mlarva will depend on both the temperature and the larval density (Equation 24).

p_mlarva = p_mlarvat × p_mlarvaLd Equation 24

The temperature-dependent mortality p_mlarvat will be calculated based on the linear relationship from Fig. 19, taking into account data from Ottenheim et al. (1996), Hurtado (2013), Basley et al. (2018), and Kandem et al. (2024) (Equation 25).

S_larva (T) = 8.16 × T – 85.50 Equation 25

where S_larva is larva survival, and T is temperature.

The larval density-dependent mortality p_mlarvaLd will be calculated based on the linear relationship presented in Equation 3.

The daily background mortality (inverse of survival) in the pupa stage will be related to the temperature using a thermal performance curve model of the Lobry–Rosso–Flandrois (LRF; lrf_1991) (same as the pupal growth). The four parameters in the model represent three cardinal temperatures (the maximum and minimum temperatures T_Smax and T_Smin at which pupa can survive, and the optimum temperature T_Sopt) and the specific rate at the optimum (U_Sopt). U_Sopt is independent of the three cardinal temperatures, and T is the temperature (Equation 26).

Spupa (T)=uSopt T-TSmax T-TSmin 2Tsopt -Tsmin Tsopt -TSmin TS-TSopt -TSopt -TSmax TSopt +TSmin -2T Equation 26

After fitting the LRF model to our data, the parameters T_Smin, T_Sopt, T_Smax and U_Sopt were set to 5.01 °C, 19.08 °C, 33.14 °C and 89.80, respectively.

Research on the effects of pesticides in Eristalis hoverflies remains limited, with existing studies primarily focusing on neonicotinoids due to their widespread agricultural use and well-documented risks to pollinators. Despite regulatory bans or restrictions on several neonicotinoids within the European Union, these compounds continue to be a central focus of ecotoxicological research, owing to their environmental persistence and ongoing use in certain regions.

In E. tenax, imidacloprid has been shown to exert stage- and route-specific toxic effects. Larvae are particularly sensitive to environmentally relevant concentrations, exhibiting dose- and time-dependent mortality, along with biochemical indicators of neurotoxicity and oxidative stress (Kamdem et al. 2024). In contrast, Basley et al. (2018), who investigated the effects of another neonicotinoid, thiamethoxam, reported that chronic exposure reduced larval survival only at high concentrations, with no significant behavioural effects observed in adults. Adult E. tenax appear to exhibit relatively high oral tolerance to imidacloprid compared to bees, with sublethal effects on feeding and survival emerging only at elevated doses (Nagloo et al. 2023). These effective doses exceed typical environmental concentrations, suggesting that adverse effects may only arise under extreme or cumulative exposure scenarios. However, when imidacloprid is administered directly to the brain, even sublethal concentrations can impair neural processing in motion-sensitive visual neurons, disrupting direction selectivity and contrast sensitivity, functions essential for flight and navigation (Rigosi & O"Carroll, 2021). Although such exposure routes are not environmentally realistic, these findings underscore the vulnerability of pollinator visual systems to cholinergic pesticides and highlight the need for further investigation into sublethal neurophysiological effects in non-bee pollinators. More recently, Martins et al. (2024) extended this line of research to another syrphid species, Eristalinus aeneus (Scopoli, 1763), demonstrating that imidacloprid exposure can also impair survival and behaviour in this ecologically important pollinator.

To date, the effects of insecticides beyond neonicotinoids on E. tenax remain largely unstudied, representing a critical gap in our understanding of the species" ecotoxicological sensitivity. This is particularly concerning given the wide array of insecticidal compounds currently used in agricultural landscapes. Findings from studies on neonicotinoid exposure suggest that E. tenax may exhibit toxicological responses that differ from those observed in bees, underscoring the importance of including hoverflies in environmental risk assessments.

To simulate pesticide responses in E. tenax, we adopt a modelling framework analogous to that used in the development of the ApisRAM honey bee colony model (Duan et al. 2022). Pesticide exposure can occur through the consumption of contaminated nectar, contact, and overspray. By considering these three exposure pathways, we provide an accurate representation of how model hoverflies may encounter pesticides in contaminated environments. P_B (t) is used to represent the accumulated pesticide burden for the individual until day t. P_B (t) is updated by all three exposure paths and is used to calculate the stress caused by pesticide exposure. The new pesticide exposure on day t is represented by P_N (t) which is calculated by Equation 27.

P_N (t) = P_I (t) + P_C (t) + P_O (t) Equation 27

where P_I(t) is the new contamination from intake, P_C (t) is from the contact exposure and P_O (t) is from the overspray exposure.

There are a number of other mortalities considered in the model, the important one being the daily background mortality of adult hoverflies. Little is known about the background mortality of adult hoverflies under natural conditions. Parasites on adults appear to be rare, but fungi of the genera Entomophthora and Enipusa are responsible for some mortality (Gilbert 1986). E. tenax can be also parasitised by wasps, Aspilota ruficornis, Rhembobius bifrons, and Rhembobius quadrispinus (Reemer et al. 2009), but little is known about the mortality rates. Hoverflies are also preyed upon by many animals such as birds, toads, wasps, etc., and by carnivorous plants (e.g. the sundew, Drosera rotundifolia) (Gilbert 1986). To some extent, adult hoverflies may be protected from predators by their resemblance to wasps or bees, but attacks by wasps are frequently observed (Bressin 1999). Nagloo et al. (2023), who investigated the acute and chronic toxicity of imidacloprid on adult E. tenax reared in laboratory conditions, reported that mortality in the control group increased from 0 to 20% with the duration of the experiment. This range of values will be used to adjust the background adult mortality in the model.

With regard to pre-imaginal stages, staphylinid beetles are recognised as natural enemies of syrphid flies and may prey on their egg clutches (Gladis 1994a). In mass-rearing experiments of E. tenax conducted by Gladis (1994a), several small staphylinid species, including Aleochara brevipennis, Aleochara lanuginosa, and Philonthus politus, were documented feeding on E. tenax larvae. These mortalities are, however, included in the general mortality rates of the egg-to-pupa stages.

The overwintering mortality test is carried out only once, at the end of the overwintering period (before emergence). Overwintering mortality may be related to severe winter conditions or predation. Siuda (1963) reported that the spider Tegenaria domestica preys on overwintering females of E. tenax. Certainly, dead hoverflies are often found entangled in spider webs in hibernacula (Bressin 1999). As little is known about the exact mortality rates during hibernation, this mortality would be an calibrated parameter in the model.

Mortalities will also be associated with management events. We hypothesise that if E. tenax eggs or larvae are present in the field, particularly within slurry deposits, at the time of soil cultivation, their probability of survival is minimal. Consequently, we assume 100% mortality for these stages during cultivation events. This assumption is based on the mechanical disruption of the soil structure by most cultivation methods, including light harrowing, which typically penetrates to depths of 2–5 cm, and the disturbance of technological lines by heavy machinery, where slurry tends to accumulate and larval development may occur. We do not expect E. tenax eggs or larvae to be present in the field during harvest, as these stages are associated with liquid environments. Therefore, mortality due to harvest operations is considered negligible. For pupae which use dry land, mortality values for soil cultivation can be parameterised using data for the ground beetle Bembidion lampros (Thorbek and Bilde 2004), and incorporated into the model for sensitivity testing. We do not expect E. tenax pupae to be present in the field during harvest. Other managements could be considered on a case-by-case basis as needed when applying the model. The potential impacts of various agricultural practices on E. tenax population dynamics can be evaluated using the model, and sensitivity analyses can be conducted to assess the robustness of these assumptions.

Finally, we assume that the mortality of non-diapause stages will occur with the onset of winter (here, we assume December 1^st) or with negative temperatures lower than a minimum temperature threshold (MinT), initially assumed to be -10 °C.