Investigation of pollen release by poricidal anthers using mathematical billiards

Buzz pollination is described using a mathematical model considering a billiard approach. Applications to a rough morphology of a typical poricidal anther of a tomato flower (Solanum lycopersicum) experiencing vibrations applied by a bumblebee (Bombus terrestris) are made. The anther is described by a rectangular billiard with a pore on its tip while the borders are perturbed by specific oscillations according to the vibrational properties of the bumblebee. Pollen grains are considered as noninteracting particles that can escape through the pore. Our results not only recover some observed data but also provide a possible answer to an open problem involving buzz pollination.

Figure 1

Sketch of (a) a box-shaped poricidal anther proposed by Buchmann and Hurley [8] to describe the buzz pollination process and (b) the rectangular billiard adopted as a model with a few collisions of a common point particle.

Figure 2

The reproduction via beat type vibrations of the (a) displacement, (b) velocity, and (c) acceleration in a poricidal anther in contact with a bumblebee. Such a reproduction has very good agreement with Vallejo-Marín's work [1].

Figure 3

The 3D parameter space for (a) peak velocity of the wall and (b) the fraction of pollen released, when several combinations of frequencies and displacements are taken into account. As one can see comparing both pictures, the regions where the fraction of pollen released is high are linked with regions where $V_{\mathrm{peak}}$ also assumes high values. These results also show that for small displacements and low frequencies, the fraction of pollen released increases very fast with the peak velocity. However, when the $V_{\mathrm{peak}}$ is too high, such an increase is slowed down, and the pollen released approaches to a constant profile.

Figure 4

The behavior of the pollen released when the peak velocity of the wall is kept constant while (a) frequency, (b) peak displacement, and (c) peak acceleration change. Here, the blue and cyan lines indicate the average behavior for the pollen released in our simulations, where it is possible to observe that on average the pollen released almost does not change when the peak velocity is constant. For the cases (d)–(f) we present the behavior of the pollen released when the $V_{\mathrm{peak}}$ is increased while the frequency, peak displacement, and peak acceleration are kept constant, respectively. As one can see, the pollen released follows the behavior of $V_{\mathrm{peak}}$, which might indicate that this observable is the best to describe the evolution of pollen release during buzz pollination.