Stable swarming using adaptive long-range interactions

Sensory mechanisms in biology, from cells to humans, have the property of adaptivity, whereby the response produced by the sensor is adapted to the overall amplitude of the signal, reducing the sensitivity in the presence of strong stimulus, while increasing it when it is weak. This property is inherently energy consuming and a manifestation of the nonequilibrium nature of living organisms. We explore here how adaptivity affects the effective forces that organisms feel due to others in the context of a uniform swarm, in both two and three dimensions. The interactions between the individuals are taken to be attractive and long-range and of power-law form. We find that the effects of adaptivity inside the swarm are dramatic, where the effective forces decrease (or remain constant) with increasing swarm density. Linear stability analysis demonstrates how this property prevents collapse (Jeans instability), when the forces are adaptive. Adaptivity therefore endows swarms with a natural mechanism for self-stabilization.

Figure 1

(a) The effective force (thick line) and the stimulus (dashed line) for $n=2$. Both are functions of $z_0$ and in units of $C_2/R_{\text{ad}}^2$. The red dot-dashed line indicates the radius of the swarm at $r=R_s$. The effective force is plotted in the “perfect adaptivity” regime, where $R_s\ll N^{\frac{1}{n}}{R}_{\text{ad}}$. The green line is the same effective force but with $R_{\text{ad}}\sim \sqrt{3}/\sqrt{4\pi \rho R_s}$. (b), (c) The stimulus for $n=3,4$, respectively. (d), (e) The effective force for $n=3,4$, respectively. The gap around $r=R_s$ is due to the cutoff that we use here $r_c=0.03R_s$. The insets show the same plots in log-log scale.

Figure 2

The cylindrical coordinates $(r,z)$ (and $\phi$) that we use for the calculation of the effective force at a point A in a uniform-density spherical swarm. The cutoff at the radius $r_c$ from A is indicated as well. The green lines separate the three regions of integration that we used.

Figure 3

(a) The Cartesian coordinates $(x,y)$ that we use for the calculation of the effective force at a point A in a uniform-density circular disk. (b) The new polar coordinates $(\tilde{r},\phi )$.

Figure 4

The normalization factor, $N_{\text{tot},\left(2\right)}$ in 2D using numerical integration for three different values of cutoffs $r_c/R_s=0.1,0.05,0.01$ (a)–(c). The red dot-dashed line indicates the radius of the swarm at $r=R_s$. The green points are calculated from the lattice model approximation for comparison.

Figure 5

The normalization factor, $N_{\text{tot},\left(n\right)}$ in 2D for three different values of $n$ with the same value of cutoff $r_c/R_s=0.03$: (a) $n=2$ in units of $C_2/R_{\text{ad}}^2$, (b) $n=3$ in units of $C_3/R_{\text{ad}}^3$, (c) $n=4$ in units of $C_4/R_{\text{ad}}^4$. The red dot-dashed line indicates the radius of the swarm at $r=R_s$.

Figure 6

The total stimulus in 2D as a function of the distance from the center of the disk $r$ (from the numerical integration): (a) $n=2$ in units of $C_2/R_{\text{ad}}^2$, (b) $n=3$ in units of $C_3/R_{\text{ad}}^3$, (c) $n=4$ in units of $C_4/R_{\text{ad}}^4$. The effective force in 2D as a function of $y_0$: (d) $n=2$ in units of $C_2/R_{\text{ad}}^2$, (e) $n=3$ in units of $C_3/R_{\text{ad}}^3$, (f) $n=4$ in units of $C_4/R_{\text{ad}}^4$. The red dot-dashed line indicates the radius of the swarm at $r=R_s$. (A cutoff value of $r_c/R_s=0.03$ is used). The insets show the same plots in log-log scale.