Coarse-grained analysis of stochasticity-induced switching between collective motion states

Abstract

A single animal group can display different types of collective motion at different times. For a one-dimensional individual-based model of self-organizing group formation, we show that repeated switching between distinct ordered collective states can occur entirely because of stochastic effects. We introduce a framework for the coarse-grained, computer-assisted analysis of such stochasticity-induced switching in animal groups. This involves the characterization of the behavior of the system with a single dynamically meaningful "coarse observable" whose dynamics are described by an effective Fokker-Planck equation. A "lifting" procedure is presented, which enables efficient estimation of the necessary macroscopic quantities for this description through short bursts of appropriately initialized computations. This leads to the construction of an effective potential, which is used to locate metastable collective states, and their parametric dependence, as well as estimate mean switching times.

Fig. 1.

Nonoverlapping behavioral zones for the model: zone of repulsion Z_r, zone of orientation Z_o, and zone of attraction Z_a.

Fig. 2.

Positions of N = 100 agents for a 10⁴ step run, with parameters s = 0.75, τ = 0.1, r_r = 1, Δr_o = 0.6, Δr_a = 1, p = 0.001, blue (resp., red) indicates motion of an agent in the positive (resp., negative) direction. The black line shows the corresponding time series plot of the coarse observable A(t).

Fig. 3.

Probability distribution functions (a) and effective potentials (b) for 100 trials with 10⁴ steps per trial: N = 100, r_r = 1, Δr_a = 1, s = 0.75, τ = 0.1, p = 0.001, Δr_o = 0.6 (blue), Δr_o = 1 (red), and Δr_o = 1.1 (green).

Fig. 4.

Coarse bifurcation diagram showing the critical points of the effective potential as Δr_o is varied. The minima of the effective potential correspond to the stable branch (black) and the maxima correspond to the unstable branch (gray). Dots (resp., ×) show critical points of the long-time (resp., short-time) simulation estimate of the effective potential. In both cases, the unstable solutions were located by performing a quadratic fit of the data between the two wells and then locating the maxima of the fit.

Fig. 5.

Comparison of effective potentials for N = 100, r_r = 1, Δr_o = 0.6, Δr_a = 1, s = 0.75, τ = 0.1, and p = 0.001. Black line, Φ(A) = −log(P_s(A)) where P_s(A) was obtained from 100 trials with 10⁴ steps per trial. Red line, Eq. 10 with drift and diffusion terms estimated from the same database. Blue line, Eq. 10 with drift and diffusion terms estimated by using the lifting procedure to initialize ensembles of short trajectories with the same A₀.

Fig. 6.

Sample trajectories initialized using the lifting procedure approaching an apparent slow manifold parameterized by A. Trajectories (gray) evolve for 15 steps with arrows showing the direction of increasing time. Black dots represent a numerical approximation of the slow manifold in the (A, S) plane. Data were taken from an ensemble of five 10⁴-step simulations after steady state was reached (≈10³ steps) for Δ_ro= 0.6.