Introduction

Flowering plants (angiosperms) comprise about 90% of all terrestrial plant species (Govaerts et al. 2021). Plants pollinated by both insects and wind are known as "ambophilous," those pollinated solely by wind are "anemophilous," and those pollinated exclusively by insects are "entomophilous." Most flower-visiting insects have diversified alongside angiosperms (Grimaldi 1999). Around 82% of angiosperms are pollinated by insects, 6% by vertebrates, and the remainder by wind (Ollerton et al. 2011). Approximately 33% of angiosperms do not produce seeds without pollinators, and for 50% of them, 80% of their seed production relies on pollinators (Rodger et al. 2021).

For social bees like honeybees (Apis mellifera L.), successful foraging and colony growth depend on their ability to explore the landscape for food sources, which are often patchily distributed (Hussein et al. 2014; Reynolds et al. 2009). Typically, honeybees receive information about good nectar and pollen sources through the waggle dance performed by nest mates who have scouted the landscape. However, when this information is unavailable, or inaccurate, or when a particular food source becomes depleted, bees employ alternative search strategies (Reynolds et al. 2007b). Advancements, such as harmonic radar technology, enabled researchers to track individual flying bees during foraging activities (Capaldi et al. 2000; Reynolds 2006; Riley et al. 2005). Field experiments using harmonic radar showed that after the removal of a feeder, honeybees initially fly to the vicinity of the removed feeder before engaging in extensive, looping flights indicative of searching (Reynolds et al. 2007b). These flight patterns exhibit a scale-free (Lévy-flight) characteristic, representing an optimal search strategy for relocating the original feeder location.

Lévy flights (LFs) are a stochastic process and do not necessitate intricate computations for execution (Reynolds and Rhodes 2009). Shlesinger and Klafter (1986) were the first to propose Lévy flights as the movement patterns of certain animals. Lévy flights involve sequences of independent steps with random orientations and lengths, where the lengths are drawn from a probability distribution function characterized by a power-law tail (p(l) ~ l^(-μ), where μ is typically between 1 and 3). These flights lack a characteristic scale due to the divergent variance of the probability distribution function and are thus described as "scale-free" (Reynolds et al. 2007a). Lévy fliers, however, lack prior knowledge regarding the probable locations of their search targets. Consequently, their search patterns involve unrestricted roaming, without looping back to the starting point after each straight-line movement (Reynolds and Rhodes 2009).

Honeybees employ Lévy flights not only when searching for feeders but also when attempting to locate their hive or when deprived of navigational cues (Reynolds 2018; Reynolds et al. 2007a). This strategy is especially advantageous when honeybees lack access to waggle dance information from nest mates or when their navigational mechanisms are impaired (Vallaeys et al. 2017). This flight pattern also results in significantly higher rates of pollen spread compared to those predicted by Brownian motion (Vallaeys et al. 2017). Previous studies have traditionally relied on the assumption that honeybee movement resembles Brownian motion when predicting outcrossing rates for bee-pollinated crops. In this framework, an individual's movement trajectory in space is depicted as a series of discrete, randomly oriented step lengths drawn from a Gaussian distribution (Reynolds and Rhodes 2009).

Expansions of Lévy flight theory also provide insights into the movement behaviors of central place foragers and those with preferred feeding grounds. Bees exhibit central place foraging behavior (Bell 1990), and research indicates their ability to distinguish between high-quality and low-quality resources, strategically locating their nesting sites accordingly (Collett et al. 2013; Olsson et al. 2015). This creates a trade-off between proximity and resource quality, where farms closer to nesting habitats may be visited more frequently than those farther away but with higher-quality resources (Fernandes et al. 2020; Rahimi et al. 2021c). Lévy flight patterns in foraging bumblebees are uncommon and limited to certain flower species. Furthermore, if such patterns do occur, they are not inherent in an evolved optimal searching strategy (Reynolds 2009).

Enhancing our comprehension of honey bee foraging behavior holds the potential for sustaining crop pollination services (Baensch et al. 2020). Apis mellifera L., commonly known as the honey bee, plays a vital role as a pollinator in crop production worldwide (Badenes-Pérez 2022; Hung et al. 2018). However, the spatial aspects of honeybee foraging concerning movements in agricultural landscapes remain insufficiently understood. For example, Olsson et al. (2015) compared the Lonsdorf model (Lonsdorf et al. 2009), and a CPF model to assess bee visitation rates in heterogeneous landscapes. They found that the Lonsdorf model anticipates that any enhancement in habitat will improve crop pollination. Conversely, according to the CPF model, the hedgerow needs to offer favorable nesting sites, not solely foraging opportunities, to positively impact crop pollination.

Other studies have also examined the impact of landscape structure on bees and pollination in simulated landscapes. For instance, Rahimi et al. (2021b), using the Lonsdorf model, demonstrated that the effects of forest fragmentation on pollination varied significantly between landscape and farm levels. While fragmentation harmed pollination at the landscape level, farms within landscapes with high fragmentation exhibited maximum pollination rates. In another study by Rahimi et al. (2021a), based on the Mitchell model (Mitchell et al. 2015), it was found that two key factors contributed to increased pollination: the amount of habitat and the ability of small patches to contribute to pollination. When the capacity of small patches to support pollination was low, fragmented forest patch patterns reduced pollination. However, as the capacity increased, landscapes with high levels of forest fragmentation showed the highest levels of pollination.

Therefore, given the current understanding of Lévy movements in honeybees, there is a need for new assessments of flower visitation rates by honeybees in agricultural landscapes comprising farms and forests. If honeybees follow a Lévy flight pattern to locate resources, (1) where would honeybees exhibit the highest visitation rates of flowers? under fragmented or aggregated forest patterns? Addressing this question helps us determine the optimal locations for installing beehives in agricultural landscapes. This facilitates honeybees in visiting a greater number of flowers for nectar or pollen collection.

Methods

Generating simulated landscapes

Figure 1 visually represents some of the simulated landscapes created for our study, which primarily delves into landscapes hosting multiple forest patches amid a matrix adorned with flowers. In crafting these landscapes, we used the NLMR package in Rv4.3 software. We created 50*50 cell landscapes where forest patches coexist with farms or pastures featuring diverse flowering plants. Notably, we manipulated forest coverage and fragmentation levels across various scenarios. Forest proportion varied from 10 to 50%, yielding five sets of simulated landscapes representing distinct forest proportions (10, 20, 30, 40, and 50 of the entire landscape). Simultaneously, the degree of fragmentation per se, controlled by the parameter 'p' in the NLMR package, spanned from the highest (0.01) to the lowest (0.55). A fragmentation degree of 0.01 denotes landscapes with maximal fragmentation per se, resulting in highly fragmented patterns of forest patches.

Fig. 1
figure 1

Simulated landscapes in different forest proportions (black patches) and degree of fragmentation

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Assigning bees and flowers to simulated landscapes

In this study, we examined a fundamental premise concerning the interactions between honeybees and flowers, where we assumed both honeybees and flowers to be randomly dispersed throughout the landscape and flowers can only be found in farmland. We acknowledge the complexity of floral resource distribution across different landscapes, including both farmland and forests (Dixon et al. 2021; Gray and Ewers 2021). However, due to modeling limitations and the need to maintain consistency within our study's scenario-based approach, we focused solely on floral resources within farmland. Introducing forest cells as potential flower locations would alter the forest proportion, thereby affecting the study's scenario.

Under this assumption, we hypothesized that honeybees can visit flowers within a radius of up to 5 cells from their location. Each cell was considered as 400 × 400 m land, meaning that the average foraging range of Apis mellifera honeybee was set as a 2 km radius (Couvillon et al., 2015). To simulate this scenario, we assigned a numerical value to each cell within a farm, representing a distinct flower species. Each simulated landscape in our study consisted of 2,500 cells, with a specific portion designated as forest patches. Consequently, the number of flowers was influenced by four scenarios corresponding to the extent of farmland within each landscape. For example, if a landscape contained 250 forest cells, with 10% of them covered by forest, we considered 225 unique flower species for 2250 cells. Hence, the outcomes of each scenario should be interpreted independently of the others, as we have assigned a different number of flowers to each scenario. Consequently, their results are not directly comparable. Table 1 illustrates the distribution of flowers across different proportions of forest cover.

Table 1 The number of flowers per unit cell of agricultural landscape relative to forest proportion scenario
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Modeling bee visitation rates

To model bee visitation rates, three distinct approaches involving random movement were employed: (1) moving window, (2) random walk, and (3) Lévy flight.

Moving window

In this method, a 5 × 5 window (representing the maximum flight range) is applied to each of the 25 randomly selected cells representing a bee within the landscapes. Within this window, all unique flower cells are counted. By excluding the central cell (which represents the bee itself), a total of 24 flowers can be anticipated around an isolated bee cell. To eliminate duplicated flower values, repetitions were removed, and the remaining distinct flower count was calculated. Ultimately, for every bee cell, the average count of unique flower values was computed across different forest proportion scenarios (Fig. 2).

Fig. 2
figure 2

Examples of simulated landscapes after assigning bees and flowers to cells

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Random walk

In a random walk (Chawla and Duhan 2018), the movement of the organism is entirely random and follows a simple probabilistic process. At each step, the organism moves with a fixed step size in a randomly chosen direction, such as forward, backward, left, or right. The step lengths and directions are uncorrelated and independent at each time step, resulting in a movement that appears like a sequence of random steps. To implement the random walk movement, a specific condition was established: the starting point must correspond to a cell inhabited by a honeybee. Within each landscape, 20 bee cells were randomly selected as initial points. The model was then directed to execute 24 random steps (as maximum flight range) for each chosen starting point. During these steps, the number of flowers visited by each simulated bee was counted. Ultimately, an average value was calculated based on the visits made by all the bees within a single landscape (Fig. 3).

Fig. 3
figure 3

Examples of random walks after selecting some forest cells (green cells) as starting points and applying the random walk algorithm (black dots)

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The Lévy flights

Lévy flights (LFs) are a type of non-Gaussian random process characterized by stationary increments distributed according to a Lévy stable distribution (Yang et al. 2013). The Lévy flight approach involves simulating bee movements inspired by the Lévy flight pattern, where occasional long steps are interspersed with shorter steps. This stochastic approach captures the intermittent long-distance flights often observed in animal foraging. In this approach, 15 honeybee cells were chosen randomly to serve as starting points for the Lévy flight. A maximum of 50 steps was set as the limit for each bee's movement. Subsequently, we counted unique flowers visited by each bee within the landscape. This process was then averaged across all the bees, yielding an average count of distinct flowers visited (Fig. 4).

Fig. 4
figure 4

Examples of the Lévy flight trajectories after one forest cell (blue cells) as starting points and applying the Lévy flight algorithm (black lines)

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Results

Honeybee visitation rates

Figure 5 illustrates the outcomes of the moving window approach across various forest proportion scenarios. In this approach, a 5 × 5 window was employed for each bee cell, and the count of unique flowers was recorded. The graph demonstrates that the highest average number of flowers per cell is expected when the landscape contains 10% forest coverage. This peak is associated with the highest level of fragmentation. Figure 6 displays the outcomes derived from the random walk approach across diverse forest proportion scenarios. This illustration indicates that the landscapes with lower forest cover and the highest fragmentation levels exhibit the highest average number of flowers per cell. Figure 7 shows the average number of visited flowers around chosen cells utilizing the Lévy flight approach. Similar to the preceding graphs, this representation demonstrates that landscapes featuring lower forest cover and the highest fragmentation levels tend to yield higher average numbers of visited flowers for randomly selected starting points.

Fig. 5
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Average visited flower numbers around selected cells in the moving window approach

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Fig. 6
figure 6

Average visited flower numbers around selected cells in random walk approach

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Fig. 7
figure 7

Average visited flower numbers around selected cells in the Lévy flight approach

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Discussion

As varying numbers of flowers were allocated to each scenario based on the amount of forest cover in this study, comparing the effects of different levels of forest cover on the visitation rate is not feasible. As a result, landscapes with fewer assigned flowers exhibit lower visitation rates compared to those with richer floral resources. Hence, irrespective of the specific allocation of flowers to each scenario, the presence of forest will occupy a larger portion of the available floral space, consequently reducing the area accessible for bee visitation. Nonetheless, this study aimed to investigate how fragmented or aggregated forest patterns influence honeybee visitation rates using three distinct approaches: moving window, random walk, and Lévy's flight. We found that the honeybee visitation rate is influenced by forest fragmentation degree in each scenario. In all visitation scenarios, the highest average number of visited flowers per cell was observed in landscapes with maximum fragmentation per se. These findings collectively suggest that honeybee visitation rates are impacted by fragmentation, emphasizing the significance of landscape structure in influencing pollinator behavior and flower visitation.

We employed three random movement approaches for honeybees. Consequently, under these conditions, certain patterns emerged within the fragmented landscapes with varying forest cover. Specifically: (1) low forest cover and high fragmentation: In landscapes with lower forest cover and higher fragmentation, Honeybees are more likely to visit a greater number of flowers. This tendency arises due to the increased probability of bees traversing across the landscape, encountering more flower cells, and visiting them without restrictions. (2) high forest cover and higher fragmentation: In landscapes with higher forest cover, when bees initiate random movement, they are more likely to traverse forest cells rather than visit flowers (Rahimi and Jung 2023). Consequently, these landscapes exhibit a reduced likelihood of bees visiting flowers, as they are more likely to pass through forested areas (Maurer et al. 2020). Additionally, in landscapes where cells are isolated due to high fragmentation, these isolated cells, which are surrounded by flowers, offer suitable options for nesting due to their relative isolation and floral availability (Lonsdorf et al. 2009; Mitchell et al. 2015; Rahimi et al. 2021a).

Upon initial examination, our findings may appear divergent from previous assertions, as fragmentation is commonly linked to adverse impacts on biodiversity. Hence, it is crucial to emphasize the distinction between fragmentation accompanied by habitat loss and fragmentation per se occurring independently of habitat loss. Fragmentation per se implies that the overall amount of habitat in a landscape remains constant, with only changes in the configuration of patches. Therefore, managing the provision of ecosystem services can be achieved by controlling the arrangement of natural patches (Fahrig et al. 2011). The impacts of fragmentation per se on biodiversity are generally less pronounced than those of habitat loss, and they often have beneficial outcomes. For instance, Fahrig (2017) reviewed 381 studies on the effects of fragmentation per se on biodiversity and found that 290 (76%) of them reported positive effects. Maurer et al. (2020) also highlighted the importance of investigating the effects of fragmentation per se pollinators to inform the spatial planning of landscapes aimed at enhancing pollination.

Therefore, the findings of this study imply that to enhance honeybee visitation rates, it is advisable to create landscapes with limited forest amounts and a high degree of fragmentation or consider siting beehives in fragmented landscapes with minimal forest coverage. However, this recommendation holds only if in forest patches, floral resources are scarce, and flowers are distributed across landscapes. This underscores the acknowledged significance of scattered trees as pivotal structural elements within ecosystems (Manning et al. 2006). Yet, it's worth noting that investigations have also explored the role of small patches in supplying pollination services by wild bees. For instance, empirical analyses have revealed that larger patches tend to yield more substantial services than their smaller counterparts (Tscharntke and Brandl 2004). Nevertheless, even diminutive forest patches spanning one hectare have demonstrated their capacity to accommodate pollinators and augment local pollination rates (Huais et al. 2020). In agricultural landscapes, compact forest patches have showcased their ability to harbor diverse pollinator communities, thereby exerting a localized impact on pollination dynamics (Proesmans et al. 2019). This phenomenon was confirmed by Proesmans et al. (2019), who identified that small forest patches had a positive influence on pollination within a 100-m radius surrounding the patches. Consequently, it becomes evident that even modest, scattered patches are competent contributors to bolstering pollination at a local scale.

Limitations of this study

In this study, we created binary landscapes with two classes: farm and forest. We assumed that flowers are only found on farms, while forest patches serve as nesting habitats for honeybees, which start their foraging from it. However, we did not account for the fact that both canopy and understory vegetation in forest ecosystems provide flowers for pollinators like honeybees. For instance, canopy trees offer substantial floral resources, but their flower production varies significantly from year to year, affecting pollinator population dynamics (Inari et al. 2012). The biomass of entomophilous trees visited by bumblebees for example was estimated to be about 20% of the total tree biomass in a deciduous forest in northern Japan (Higashi et al. 1988). Moreover, the total plant biomass of canopy trees was approximately 1000 times greater than that of understory vegetation in a deciduous forest (Hiura 2001, 2005). Thus, canopy trees provide significantly more floral resources to bumblebees than understory vegetation (Inari et al. 2012).

In tropical forests, most angiosperms are allogamous, meaning they require the fertilization of an ovum from one individual by the spermatozoa of another, and they depend heavily on animals for pollination, with insects playing a crucial role in pollen transfer for cross-fertilization (Bawa et al. 1985; Fragoso and Varanda 2011). In addition, 54% of the 188 species of edible fruit plants in tropical forests rely on bee pollination (Paz et al. 2021). Forest edges are also ideal for pollinators such as honeybees, solitary bees, bumblebees, and butterflies (Kells and Goulson 2003; Rahimi et al. 2021b; Svensson et al. 2000; Zulian et al. 2013) as they provide suitable nesting sites and floral resources for pollinators (Ricketts et al. 2008; Svensson et al. 2000). Therefore, it is important to conduct similar simulations based on landscapes that consider flowers in both farms and forests to provide more realistic visitation rates of honeybees.

On the other hand, intensive agricultural practices, including the widespread use of herbicides, can have detrimental effects on the availability of floral resources for bees on farms. Herbicides are chemicals designed to kill unwanted plants, including weeds, but they can also impact non-target plants, including those that provide essential food sources for bees (Kovács‐Hostyánszki et al. 2017; Langlois et al. 2020; Ward et al. 2022). While in this study, we considered diverse farmlands with flowers for honeybees, the reality of intensive farming practices needs to be acknowledged. Incorporating the negative impacts of herbicide use on floral availability into future studies would provide a more accurate picture of the challenges faced by pollinators in agricultural landscapes.

Conclusion

We investigated the intricate relationship between the fragmentation of forest patches and their influence on honeybee visitation rates. We focused on how fragmented or aggregated forest patterns influence honeybee visitation rates using distinct approaches. We found that the degree of forest fragmentation influences honeybee visitation rates in each scenario, with landscapes exhibiting maximum fragmentation per se resulting in the highest average number of visited flowers per cell. Our study sheds light on the fascinating phenomenon of the Lévy flight pattern and its influence on honeybee visitation rates in fragmented landscapes. The Lévy flight pattern, characterized by sporadic long-distance movements interspersed with shorter movements, allows honeybees to efficiently explore their environment in search of floral resources.

Despite the theoretical expectation that fragmented landscapes may offer increased opportunities for honeybees to visit more flowers due to their dispersed nature, the application of the Lévy flight pattern in diverse landscapes with varying forest cover and fragmentation levels has not been extensively explored until now. This novel insight underscores the importance of considering honeybee movement patterns, such as Lévy flight, when evaluating the impact of landscape fragmentation on pollinator behavior. By adopting a Lévy flight strategy, honeybees can effectively navigate fragmented landscapes and capitalize on the dispersed floral resources scattered across the environment. This suggests that fragmented landscapes, often viewed as less conducive to pollinator activity, may offer valuable foraging opportunities for honeybees employing the Lévy flight pattern.