Random search with resetting as a strategy for optimal pollination

The problem of pollination is unique among a wide scope of search problems, since it requires optimization of benefits for both the searcher (pollinator) and its targets (plants). To address this challenge, we propose a pollination model which is based on a framework of first passage under stochastic restart. We derive equations for the search time and number of visited plants as functions of the distribution of nectar in the plant population and of the probability that a pollinator will leave the plant after examining a flower, thus effectively restarting the search. We demonstrate that nectar variation in plants serves as a driving force for pollination and establish conditions required for optimal pollination, which provides an efficient pollinator search strategy and the maximum number of plants visited by the pollinator.

Figure 1

A schematic description of the nectar search, which includes three stages: foraging, rummaging, and successful end of the search after finding nectar.

Figure 2

Mean turnover time, $\langle T_{\mathrm{turn}}\rangle$, as a function of $p_{\mathrm{off}}$ calculated for the bimodal nectar PDFs in Eq. (4). All calculations were performed for $A=0.5$. Other parameters: $x_1=0.05$, $x_2=0.225$ (orange solid line), $x_1=0.15$, $x_2=0.15$ (red dashed line), and $x_1=0.1$, $x_2=0.7$ (blue dot-dashed curve). Stars indicate the minima of $\langle T_{\mathrm{turn}}\rangle$ corresponding to the optimal values $p_{\mathrm{off}}=p_{\mathrm{off}}^{*}.$ Other parameter values: $\langle T_{\mathrm{on}}\rangle =1\mathrm{s}$ and $\langle T_e\rangle =2\mathrm{s}$.

Figure 3

Variation of $p_{\mathrm{off}}^{*}$, $\langle T_{\mathrm{turn}}\rangle$, $\langle n_V\rangle$, and $B\left(\langle T_{\mathrm{turn}}\rangle ,p_{\mathrm{off}}^{*}\right)$ in the $\left\{x_1,x_2\right\}$ space calculated using the bimodal nectar PDF given by Eq. (4) with $A=0.5$. Other parameters: $\langle T_{\mathrm{on}}\rangle =1\mathrm{s}$ and $\langle T_E\rangle =2\text{s}$. (a) Optimal escape probability, $p_{\mathrm{off}}^{*}$, as a function of $x_1$ and $x_2$. Dashed and dotted white lines are the contour lines of $p_{\mathrm{off}}=0$ and $p_{\mathrm{off}}=1$, respectively. These lines are shown also in panels (b) and (c). (b) and (c) Contour plots of the mean turnover time, $\langle T_{\mathrm{turn}}\rangle$, and the mean number of visited plants, $\langle n_V\rangle$, respectively, which were calculated for $p_{\mathrm{off}}=p_{\mathrm{off}}^{*}$. The values of $\langle T_{\mathrm{turn}}\rangle$ and $\langle n_V\rangle$ at the contour lines are displayed. The faded area of parameters [the bottom left corner in panel (c)] is prohibited under the constraint $\langle T_{\mathrm{turn}}\rangle \le T_{\lim }=10\mathrm{s}$. The red segments in panel (c) define the range of parameters where $\langle n_V\rangle$ achieves maximum under the constraints $\langle T_{\mathrm{turn}}\rangle \le T_{\lim }$. (d) Contour map of the normalized benefit function, $B/\max \left(B\right)$, defined by Eqs. (12, 13, 14). The calculations performed for $T_{\mathrm{turn}}^{\lim }=15\mathrm{s}$ and $p_{\mathrm{off}}^{\lim }=0.3$. The dotted and dashed black lines represent contours of $p_{\mathrm{off}}^{*}=p_{\mathrm{off}}^{†}$ and $\langle T_{\mathrm{turn}}\rangle =T_{\mathrm{turn}}^{†}$, respectively, where $p_{\mathrm{off}}^{†}$ and $T_{\mathrm{turn}}^{†}$ are the parameters for which $B\left(\langle T_{\mathrm{turn}}\rangle ,p_{\mathrm{off}}^{*}\right)$ achieves the maximum. The white stars, showing the intersections of these lines, correspond to the maximum of the benefit function, which defines the conditions for optimal pollination. The inset shows the same benefit function in coordinates of $\langle T_{\mathrm{turn}}\rangle$ and $p_{\mathrm{off}}^{*}$, which are functions of $x_1$, $x_2$, and $A$. The scale bar presents the magnitude of the normalized benefit function.

Figure 4

Mean turnover time, $\langle T_{\mathrm{turn}}\rangle$, as a function of $p_{\mathrm{off}}$ calculated for the bimodal nectar PDFs in Eq. (I13) (solid line). Other parameter values: $\langle T_{\mathrm{on}}\rangle =1\mathrm{s}$ and $\langle T_e\rangle =2\mathrm{s}$. The dotted horizontal line corresponds to $\langle T_{\mathrm{turn}}\rangle =T_{\mathrm{turn}}^{†}=18.30$ s and the dashed vertical line represents $p_{\mathrm{off}}=p_{\mathrm{off}}^{†}=0.267$.