Optimal recruitment strategies for groups of interacting walkers with leaders

We introduce a model of interacting random walkers on a finite one-dimensional chain with absorbing boundaries or targets at the ends. Walkers are of two types: informed particles that move ballistically towards a given target and diffusing uninformed particles that are biased towards close informed individuals. This model mimics the dynamics of hierarchical groups of animals, where an informed individual tries to persuade and lead the movement of its conspecifics. We characterize the success of this persuasion by the first-passage probability of the uninformed particle to the target, and we interpret the speed of the informed particle as a strategic parameter that the particle can tune to maximize its success. We find that the success probability is nonmonotonic, reaching its maximum at an intermediate speed whose value increases with the diffusing rate of the uninformed particle. When two different groups of informed leaders traveling in opposite directions compete, usually the largest group is the most successful. However, the minority can reverse this situation and become the most probable winner by following two different strategies: increasing its attraction strength or adjusting its speed to an optimal value relative to the majority's speed.

Figure 1

Illustration of the model's dynamics. Informed particle $I$ jumps rightwards with rate $k_i$ until it hits the food target at site $N$. (a),(b) When the uninformed particle $U$ is inside $I$'s attraction range $R$ (red shaded region), it jumps with rate $k_u=1+k_0$ towards $I$ and with rate $k_u=1$ away from $I$. (c) When it is outside that range, $U$ jumps right and left with rate $k_u^{\pm }=1$. $U$ stops when it reaches either the absorbing site 0 or $N$.

Figure 2

Probability $F^{\mathrm{N}}$ that the uninformed particle $U$ reaches target $N$ vs the jumping rate $k_i$ of the informed particle, for internal strengths $k_0=0.2(•),k_0=0.5\left(⧫\right)$, and $k_0=1.0\left(\blacksquare \right)$. $F^{\mathrm{N}}$ is maximum at $k_i^{*}$. Simulations correspond to a chain of length $N=100$ and an attraction range of the informed particle $R=10$.

Figure 3

Individual realizations of the searching process representing typical trajectories of the particles in three different regimes of the model: (a) small $k_i$, (b) optimal search $k_i=k_i^{*}=0.06$, and (c) large $k_i$. Space and time are displayed on the $x$ and $y$ axes, respectively, and the target at site $N=100$ is denoted by a red ellipse. Parameters: $N=100,R=10$, and $k_0=0.2$. In all plots black dashed lines represent the interaction range, which is centered at the thick solid orange line representing the trajectory of the informed particle, whereas the remaining blue solid line shows the trajectory of the uninformed particle.

Figure 4

First-passage probability of the uninformed particle to target $N,F^{\mathrm{N}}$, vs the speed of the informed particle $k_i$ on a log-log plot, for chains of length $N=50(•),N=100\left(\blacksquare \right),N=200\left(⧫\right)$, and $N=300\left(▾\right)$, with $k_0=0.2$ and $R=10$. $F^{\mathrm{N}}$ is shifted by $1/2$ to show the asymptotic behaviors $F^{\mathrm{N}}-1/2\sim k_i$ and $F^{\mathrm{N}}-1/2\sim k_i^{-1}$ in the small and large $k_i$ limits, respectively. Solid lines represent the high speed approximation from Eq. (6), while the dashed line corresponds to the low speed approximation (17). Vertical dashed lines denote the positions of the maximum $k_i^{*}$ and of two arbitrary speeds $k_i^1$ and $k_i^2$ with the same probability $F_{1,2}^{\mathrm{N}}$.

Figure 5

Scaling of the FPP of the uninformed particle $F^{\mathrm{N}}$ in the limit of low speed $k_i$ of the informed particle for a chain of length $N=100$ and four values of the attraction range: $R=10\left(▾\right),R=14\left(⧫\right),R=18\left(\blacksquare \right)$, and $R=22(•)$. Each panel corresponds to a different $k_0$: (a) $k_0=0.2$, (b) $k_0=0.5$, and (c) $k_0=0.8$. The $y$ and $x$ axes were rescaled in order to compare simulation results with the analytic expression (17) (straight solid lines), which corresponds to the straight line $y=x$ in these rescaled axis. Agreement between simulation and theory improves as $k_0$ and $R$ increase.

Figure 6

(a) Speed of the informed particle $k_i^{*}$ that maximizes $F^{\mathrm{N}}$ vs system size $N$ on a double logarithmic scale, obtained from numerical simulations, with $R=10$ and $k_0=0.2$. The solid line corresponds to Eq. (18), while the dashed line shows the scaling $k_i^{*}\sim N^{-1/2}$. (b) $k_i^{*}$ as a function of the free jumping rate $k_u$ of the uninformed particle on a chain of length $N=100$. The dashed line is $k_i^{*}=k_u-2$. (Inset) In the low $k_u$ regime, data are well fitted by $k_i^{*}=0.062k_u$ (dashed line). $k_i^{*}$ is constant in the $1\le k_u\le 2.062$ interval.

Figure 7

Dynamics of two competing informed particles $I_r$ and $I_l$ traveling in opposite directions at speeds $k_r$ and $k_l$, respectively. The uninformed particle $U$ displays a bias $k_0$ towards $I_r$ and/or $I_l$ when it is inside their respective attraction ranges, depicted by the red and green regions, respectively. $U$ symmetrically diffuses with rates $k_u^{\pm }=1$ outside that ranges until it hits either right or left target.

Figure 8

(a) Color-scale plot showing the probability $F^{\mathrm{N}}$ that the uninformed particle arrives at the right target $N$ in the presence of right- and a left-moving particles, as a function of their speeds $k_r$ and $k_l$. (b) Phase diagram showing the regions in the $k_l-k_r$ space where the uninformed particle most likely reaches the right (white, $F^{\mathrm{N}}>1/2)$ or left (black, $F^{\mathrm{N}}<1/2)$ target.

Figure 9

(a) Probability $F^{\mathrm{N}}$ that the uninformed particle reaches the right target $N$ as a function of the ratio $k_r/k_l$ between the speeds of the right and left moving particles. Each curve corresponds to a fixed speed $k_l$ [$k_l=1.0(•),k_l={10}^{-1}\left(\blacksquare \right),k_l={10}^{-2}\left(⧫\right)$, and $k_l={10}^{-3}\left(▾\right)$]. Vertical dashed lines denote the crossing points where $F^{\mathrm{N}}=1/2$. (b) Optimal speed of the right-moving particle $k_r^{*}$ that maximizes $F^{\mathrm{N}}$ as a function of the speed $k_l$ of its competing partner. The horizontal dashed line indicates the asymptotic value $k_r^{*}=0.062$ in the $k_l\to \infty$ limit.

Figure 10

(a) Probability (color scale) that the uninformed particle arrives at the minority (right) target, under the influence of two informed particles moving to the left at speed $k_{\mathrm{M}}$ and one to the right at speed $k_m$. All three informed particles have the same internal strength $k_0=0.2$ and interaction range $R=10$. (b) Regions in the $k_{\mathrm{M}}-k_m$ speed space where the uninformed particle has the largest chances to reach the minority (right) or majority (left) target. The largest region corresponds to the majority target favored. The green dot indicates the optimal speed $k_{\mathrm{M}}=k_m=0.062$.

Figure 11

Same as in Fig. 10 but for the case in which the strength $2k_0=0.4$ of the right-moving particle is twice the strength $k_0=0.2$ of each of the two left-moving particles. The probability that the uninformed particle reaches the minority (right) target is largest for most combinations of speeds $k_{\mathrm{M}}$ and $k_m$ (white region).