The impact of different degrees of leadership on collective navigation in follower-leader systems

Abstract

In both animal and cell populations, the presence of leaders often underlies the success of collective migration processes, which we characterise by a group maintaining a cohesive configuration that consistently moves toward a target. We extend a recent non-local hyperbolic model for follower-leader systems to investigate different degrees of leadership. Specifically, we consider three levels of leadership: indifferent leaders, who do not alter their movement according to followers; observant leaders, who attempt to remain connected with the followers, but do not allow followers to affect their desired alignment; and persuadable leaders, who integrate their attempt to reach some target with the alignment of all neighbours, both followers and leaders. A combination of analysis and numerical simulations is used to investigate under which conditions each degree of leadership allows successful collective movement to a destination. We find that the indifferent leaders' strategy can result in a cohesive and target-directed migration only for short times. Observant and persuadable leaders instead provide robust guidance, showing that the optimal leader behavior depends on the connection between the migrating individuals: if alignment is low, greater follower influence on leaders is beneficial for successful guidance; otherwise, it can be detrimental and may generate various unsuccessful swarming dynamics.

Figures (8)

• Figure 1: Modeling different degrees of leadership, ranging from highly certain leaders to uncertain leaders. Indifferent leaders have no movement response to followers; observant leaders respond to follower position through attraction; persuadable leaders respond to both follower position and alignment.
• Figure 2: Dynamics of M1. (A) Second decile, median and eighth decile of the follower distribution for different parameter regimes: P1 weak attraction and strong alignment ($q_a=0.5, q_l=2$); P2 strong attraction and strong alignment ($q_a=2, q_l=2$); P3 strong attraction and weak alignment ($q_a = 2, q_l=0.1$); P4 weak attraction and weak alignment ($q_a = 0.5, q_l=0.1$). Results are shown for variations of key parameters describing speed differential between leaders and followers (i.e. $\beta > \gamma$) and leader influence on followers (i.e. $\alpha$). Specifically, we focus on: (i) $\beta=\gamma=0.1$, $\alpha=1$; (ii) $\beta=0.5$, $\gamma=0.1$, $\alpha=1$; (iii) $\gamma=\beta=0.1$, $\alpha=5$. (B-D) Space-time evolution of densities for $\beta=\gamma$, $\alpha=1$, under (B) P1, (C) P2, (D) P3, P4. Other parameter values are set as $\lambda_1=0.2$, $\lambda_2=0.9$, $M_u=M_v=12.61$, and $x_0=6.5$.
• Figure 3: M2 and M3, Proportion of right-moving populations at steady state(s). (A), (B), (C) Effect of $q_l$ on position and number of equilibrium points, for $\eta=0$, $\eta=0.2$, $\eta=0.4$. (D) Effect of information level of leaders $\eta$ on position and number of equilibrium points, for $q_l=0.8$. (E), (F) Effect of the influence of right-moving (left-moving) leaders on followers $\alpha^+$ ($\alpha^-$) on position and number of equilibrium points, for $\alpha^-$ = 1 ($\alpha^+$ = 1) and $q_l=0.5$. Other parameter values fixed at $A_u = A_v = 1$, $\lambda_1 = 0.8$, and $\lambda_2 = 3.6$.
• Figure 4: M2 and M3, Proportion of right-moving populations at steady state(s). (A-B) Effect of $\eta>0$ on position and number of equilibrium points: (A) $\frac{\alpha^+}{\alpha^-} = \frac{1}{1.5}$, (B) $\frac{\alpha^+}{\alpha^-} = 0.5$, with $q_l=0.8$. (C-D) Effect of $\frac{\alpha^+}{\alpha^-} > 0$ on position and number of equilibrium points: (C) for $\eta=-1$, (D) $\eta=-5$, with $q_l=0.5$. Other parameter values fixed at $A_u = A_v = 1$, $\lambda_1 = 0.8$, and $\lambda_2 = 3.6$. Note that the yellow square with a red star indicates the position where the steady state is symmetric with respect to the proportion of rightward v leftward oriented individuals.
• Figure 5: M2 and M3, Effect of Obias, Sbias and Cbias leader strategies on the swarm dynamics as $q_l$ increases under low attraction regimes and clustered initial configuration. (A-C): swarm dynamics is evaluated in terms of the speed (in blue) and the cohesion index (in orange) of the follower population. We highlight in yellow the $q_l$ values for which optimal swarming is obtained for M3, in green those for which optimal swarming is obtained for M2. Red regions denote unsuccessful swarming for both M2 and M3. Numerical simulations are obtained for (A) $\eta=10$, $\alpha^\pm=1$, $\beta_\pm=0.1$, (B) $\beta_+/\beta_-=0.5/0.1$, $\eta=1$, $\alpha^\pm=1$, (C) $\alpha^+/\alpha^-=5/1$, $\beta_\pm=0.1$, $\eta=1$. (D-F): examples of successful swarming patterns displayed by M2 and M3 (see correspondence with panels in the first row). (G-J): examples of unsuccessful swarming patterns displayed by M2 and M3 (see correspondence with panels in the first row). Other parameter values are set as $q_a=0.5$, $\lambda_1=0.2$, $\lambda_2=0.9$, $M_u=12.61$, $M_v=12.61$, and $x_0=5$.