Model of the best-of-$N$ nest-site selection process in honeybees

The ability of a honeybee swarm to select the best nest site plays a fundamental role in determining the future colony's fitness. To date, the nest-site selection process has mostly been modeled and theoretically analyzed for the case of binary decisions. However, when the number of alternative nests is larger than two, the decision-process dynamics qualitatively change. In this work, we extend previous analyses of a value-sensitive decision-making mechanism to a decision process among $N$ nests. First, we present the decision-making dynamics in the symmetric case of $N$ equal-quality nests. Then, we generalize our findings to a best-of-$N$ decision scenario with one superior nest and $N-1$ inferior nests, previously studied empirically in bees and ants. Whereas previous binary models highlighted the crucial role of inhibitory stop-signaling, the key parameter in our new analysis is the relative time invested by swarm members in individual discovery and in signaling behaviors. Our new analysis reveals conflicting pressures on this ratio in symmetric and best-of-$N$ decisions, which could be solved through a time-dependent signaling strategy. Additionally, our analysis suggests how ecological factors determining the density of suitable nest sites may have led to selective pressures for an optimal stable signaling ratio.

Figure 1

Dynamics of the complete decision system of Eq. (3) for the symmetric case $\kappa =1$ (i.e., $v_1=v_2=v_3=v$). Panel (a) shows the stability diagram as a function of the parameter $r$ and the quality $v$ for $N=3$ options. The two curves represents the two bifurcations $r_1$ (blue solid) and $r_2$ (red dashed) of Equations (4). There are three possible system phases: (I) decision-deadlock, (II) coexistence of decision deadlock and stable solutions for any option, and (III) decision for any option. Panel (b) shows the bifurcation diagram for $N=3$ and $v=5$ as a function of the parameter $r$. This illustrates the three system phases when varying the control parameter $r$. Note that, due to the 2D visualization, some equilibria overlap and thus the bottom branches in panel (b) correspond to the two overlapping equilibria for the options $x_2$ and $x_3$. Panel (c) shows a stability diagram that visualizes the dependence of the bifurcation points $r_1$ (solid lines) and $r_2$ (dashed lines) as a function of $N$ for varying $v\in \{1,2,3,5,10\}$, and reports the same three system phases.

Figure 2

Phase portraits of the complete system (3) for $N=3$ options in the symmetric case $\kappa =1$ (i.e., $v_1=v_2=v_3=v=5$). Blue dots represent stable equilibria, and green dots represent unstable saddle points. Saddle manifolds are shown as red (repulsive) and blue (attracting) lines. Panel (a) shows the system in a decision deadlock phase (i.e., phase I of panel (b), $r=1$); in fact, there is only one stable solution with all the three committed population with equal size. Panel (b) shows the coexistence of the decision deadlock and the decision for any option (phase II, $r=3$). Panel (c) shows the system for high values of $r$, in which the decision deadlock solution is an unstable saddle point, and therefore the only stable solutions are the decision for any option (phase III, $r=10$). The same phase portrait from another perspective is shown in panel (d), where a set of trajectories (red lines) are shown. Looking at panel (d), the central unstable saddle node is unstable on the displayed plane while is stable (i.e., attracting) on the direction orthogonal to the field of view of the plot (d) (i.e., the attraction manifold is the line $x_1=x_2=x_3$). The system does not possess any periodic attractors.

Figure 3

Dynamics of the complete decision system of Eqs. (3) for $N=3$ options for the asymmetric case ($\kappa <1$) and superior option's quality $v=5$. The left panel shows the stability diagram as a function of the parameter $r$ and the ratio between qualities $\kappa$. The parameter space is divided in five different areas (see Fig. 8 to see a representative 3D phase portrait for each area). In area A, the system has a unique solution corresponding to selection of the best option; in areas B and C, the system may select any of the possible options; in areas D and E the system may end in a decision deadlock. The underlying density map show the population size of the stable solution for the best option. For low values of $r$ and similar options (top-left corner), this population is relatively small and may be not enough to reach a quorum threshold. The right panels show three bifurcation diagrams as a function of the parameter $r$ for $\kappa \in \{0.5,0.9,0.97\}$. Note that, due to the 2D visualization, some equilibria overlap and thus the bottom branches of the bifurcation diagrams correspond to two overlapping equilibria for selection of options $x_2$ and $x_3$.

Figure 4

(a) Stability diagram for best option quality $v=5$ in the parameter space $r,\kappa$ for varying number of options $N\in \{2,\cdots ,7\}$. For each option, the system has five possible phases that are consistent with the phases described in caption the of Fig. 3. Here we label only areas A (monostability) and B (multistability) to facilitate readability. (b) Maximum value of $\kappa$ as a function of $N\in \{2,\cdots ,7\}$ and $r\in (0,20]$ for which the system has a unique attractor for the selection of the best-quality option, defined as the best option attracting commitment from at least 75% of the total decision-making population.

Figure 5

Bifurcation diagram in 3D of the system (A3) with $N=3$ equal-quality options (i.e., $v_1=v_2=v_3=v$) as a function of $r=h/k\in (0,10]$ and $\beta \in (0,10]$. The vertical axis shows $x\in [0,1]$, which represents the proportion of bees committed to one of the three identical options. Blue surfaces represent stable equilibria, while green surfaces are unstable equilibria. We can see that for $r=1$, the decision deadlock is stable for any tested values of $\beta$. See Section pp2 for a formal proof of the decision deadlock for $r=1$ and $N=3$.

Figure 6

(a) Comparison of the stability diagrams in the binary and symmetric case (i.e., $N=2$ and $v_1=v_2=v$) of the newly proposed parametrization [Eq. (3)] and the previous work [6]. The bifurcation line that determines the two system phases is qualitatively similar, but the bifurcation parameter is different: In the previous work it is the cross-inhibition signal $\beta$, here it is the interaction ratio $r$. (b) Stability diagram of the system Eq. (3) as a function of the average quality $\overline{v}=(v_1+v_2)/2$ and the quality difference $\mathrm{\Delta }v=|v_1-v_2|$ for varying $r\in \{0.6,1,1.4,1.8\}$, in the binary decision case. The lines show the relationship between the minimum quality difference to have the system with an unique attractor for the best option and the quality mean. This relationship resembles the Weber's law observed in psychological studies, with $r$ determining the coefficient. The results are similar to the ones obtained in Ref. [6], but using a different coefficient (in the previous work the coefficient was the cross-inhibition, $\beta$).

Figure 7

Time-dependent solutions of the system of Eqs. (1) and (2) for $N=3$ options, spontaneous transitions strength $k=0.1$, interaction transitions strength $h=0.3$, best option quality $v=10$, and varying inferior alternatives' quality as $\kappa \in \{0.25,0.5,0.75\}$. The main plot displays the dynamics of the population committed to the best-quality option $x_1$; the inset shows the dynamics of all populations for $\kappa =0.5$; note that the populations committed for the inferior alternatives, $x_2$ and $x_3$, have overlaying trajectories. The horizontal dashed line shows an example quorum threshold [30].

Figure 8

Dynamics of the system Eq. (3) in the case of $N=3$ options. In the top-left panel, we report the stability diagram in the parameter space $r, \kappa$. The plot shows that there are five possible system phases, labeled with letters from A to E. The other panels show a representative 3D phase portrait for each phase. The letter in the bottom-right of each phase portrait indicates which phase they represent.

Figure 9

Stability diagrams for $v=5$ and $N\in \{4,5,6,7\}$, in panels (a)–(d), respectively. The area A indicates the systems phase with a single attractor in favor of the best option. Having an unique solution, in this area the system never converges for the selection of inferior options. The underlying density map shows the population size of the stable solution for the best option. In the dark area the population for the best option is not sufficient to reach a quorum to take a decision. For an increasing number of options, the dark area increases and low values of $r$ are not sufficient anymore to allow the swarm to take a decision for similar options (high $\kappa$). However, for sufficiently large values of $r$, the area A shifts toward higher values of $\kappa$. This effect is also shown in Fig. 4 of the main text.

Figure 10

Bifurcation diagrams of the complete system [Eq. (3)] in the symmetric case ($v=5$) for number of options $N=4$ in panel (a), $N=5$ in panel (b), $N=6$ in panel (c), and $N=7$ in panel (d). Blue (dark gray) curves represent stable equilibria and green (light gray) lines unstable saddle points. The vertical dashed lines are the bifurcation point predicted by the reduced system [Eq. (D4)]. These points always precisely match with the bifurcation point of the complete system.

Figure 11

Bifurcation diagrams of the complete system [Eq. (3)] in the asymmetric case for number of options $N=4$ in panel (a), $N=5$ in panel (b), $N=6$ in panel (c), and $N=7$ in panel (d). In all plots, the superior option's quality is $v_1=8$ while the inferior options' quality is $v_i=7.2,i\in [2,N]$, that is, $\kappa =v_i/v=0.9$. Blue curves represent stable equilibria and green lines unstable saddle points. Notice the increase of the range of values of $r$ in which the undecided state persists. Note also that the stable state at decision for the superior option appears earlier than the ones for the inferior alternatives. This supports a strategy to deal with the uncertainty in the decision-making scenario based on the gradual increase of $r$, which would initially bring the system into an indecision state and subsequently jump to the selection of the highest-quality option.