2.2. Flight dynamics

Flight dynamics of all foragers were modelled as a random walk with drift [55,56]. At the beginning of each simulation (t = 0), the position of each bee i was x_i(0) = (0,0), i.e. all bees were at the hive. Each bee was assigned a different drifting vector v_i, which determined its flight direction when leaving the hive, and its flight pattern is described as

where σ_i dW_t is the random contribution to the distance and angle a bee moved. This term has a normal distribution with a mean of zero and variance of σ_i, thus closely resembling a diffusion process [57]. Specifically, 1/σ_i measures the precision of the flight. Because E[dW_t] = 0, the average velocity of the ith bee was v_i, and its magnitude v_i = |v_i| defined the average flight velocity. The stochastic dynamics in equation (2.1) produce slight variation among bees in their flight patterns to avoid an unrealistic scenario in which bees take a straight line between two points. Using the Euler–Maruyama method [58], equation (2.1) can be solved numerically using

with Δt being a fixed time step, and σ~i=Δtσi. At the beginning of each simulation (t = 0), all scouts left the hive, with drifting vectors v_i assigned from a uniform distribution, and continued flying until they found a resource. Once a scout detected a resource, it returned to the hive to recruit other foragers, referred to as ‘recruits’. Scouts and recruits differed in the precision of their flight: σ~i of scouts was larger (σ~i=5) than that of recruits, resulting in flight paths that covered a larger area than recruits (figure 1). The dispersion of recruited bees (σ~i=2) was fitted using data from experiments with feeders positioned at distances varying from metres to kilometres [33]. To differentiate between the flight patterns of bees that are exploring the environment and those that are exploiting a resource patch, are familiar with their location, and are therefore faster and more precise, we assigned v_i = 1 to scouts and v_i = 1.5 to recruits, following [59]. Foragers that reached the limit of the simulated area were set back to the hive instantly to start foraging again.

Flight dynamics of scouts and recruits. (a) Scouts left the hive at the beginning of the simulation and once they found a resource, they recruited other foragers, referred to as ‘recruits’. (b) Variance of the scouts' deviations from a straight path on outgoing trips (σ~=5, red) was larger than that of the recruits and persistent scouts (σ~=2, blue), resulting in greater spatial dispersion. (c) System dynamics approach based on a compartmental model, with square boxes representing the states of foragers and the green circle representing the amount of food retrieved by all foragers. Black arrows are state-transition rates (see equations (2.6) and (2.7)); the blue dashed arrow represents the recruitment of foragers by scouts; the green double arrows represent foragers delivering food to the hive.

During recruitment, scouts communicated the location and distance of the newly found resource. The recruiting scout remained at the hive for 1 min (approx. 50 time steps in the numeric simulations) to simulate the time it would take to recruit foragers using the waggle dance [33]. During this period, an average of five randomly selected recruits left the hive in the direction of the resource. Recruiting on average 1, 5 or 10 foragers by each scout did not qualitatively change the results of our simulations. For simplicity, only the recruitment by scouts is considered here, and we examine the effect of adding recruitment by recruits in the electronic supplementary material, figure S1. Distance and quality of food patches are also communicated in the waggle dance [33,60], and variation in distance and quality could be easily incorporated in further investigations of our model by varying the number of recruits that respond to each recruiting forager and the time that each scout spends recruiting.

Each of the newly recruited bees left the hive with their drifting vectors pointing exactly towards the location reported by the recruiting scouts, analogous to previous experiments [61]. The direction of this drifting vector is the deterministic portion of the flight dynamics (see v_idt in equation (2.1)), which is accompanied by a stochastic contribution from σ_i dW_t. Recruited bees exploited the first resource they found during their trips. The dispersion of recruited bees (σ = 2) was fitted using data from experiments with feeders [32]. Because the stochastic element of the flight of a recruited bee is very small compared with the size of the resource patches in our simulations, bees always exploited the same resource patch that was reported to them. The effect of communicating the distance to the source was modelled by slightly changing the dynamics in equation (2.1) to

where α(x) is any function that goes to zero when x → 0 and x_r is the location of the resource reported. This turned the flight dynamics into a purely random walk (i.e. without bias) near the location of the reported resource. For simplicity, we used a Heaviside function that removed all bias in the flight dynamics when the forager was less than 2 m from the resource:

During our simulations, scouts and recruits obtained resources for the colony. Upon obtaining a resource, foragers (both scouts and recruits) returned to the hive in a straight line, with constant velocity v_i, carrying one resource unit, equivalent to 1.0 ± 0.3 µl [33]. If a forager reached the boundaries of the area considered in the simulation, it was reassigned to the hive, without bringing food, to begin foraging again. For simplicity, this reassignment was instantaneous, but adding a return trip or changing the distance explored by these foragers before they return to the hive did not change our findings (electronic supplementary material, figure S2).

Each forager, scout or recruit, was assigned a persistence value π_i, defined as the number of consecutive trips it performed to each resource location. If the persistence of a scout was greater than 1, its v_i and σ_i after the first trip were set to those of recruits and its flight dynamics was adjusted to follow equation (2.2). Scouts that completed π_i trips to the same location randomly changed their drifting vector and began scouting again. Recruits that completed π_i trips remained at the hive until they were recruited again.