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.F ish, birds, and honey bees, as well as many other animal groups, display collective types of motion such as schooling, flocking, and swarming (1, 2). A single animal group can display different types of collective motion at different times, with Ͻ1 day of residence time in each state (3). Although such transitions could be due to changing behavioral rules or environmental factors, they also can occur entirely due to stochastic effects, as will be demonstrated for the model considered in this paper.One class of biologically motivated, individual-based models for group formation, frequently used for schooling fish, abstracts animal behavior by placing zones around individuals in which they respond to others through repulsion, alignment, and/or attraction (4-11). In the three-dimensional model of Couzin et al. (10), long-time steady-state computations revealed four different types of stable collective motion in different parameter regions: swarm, torus, dynamic parallel, and highly parallel. It was also shown that by changing the quantitative features of the behavioral rules (increasing or decreasing the radius of alignment), the collective state of the school could be changed.In ref. 10, stochasticity is incorporated by adding a small deviation to the heading of each individual obtained from the deterministic evolution algorithm. Our simulations show that if one instead considers relatively rare but substantial variations, namely that there is a small probability of each individual changing its direction substantially from that obtained from the deterministic algorithm, then for certain parameter regions multiple successive transitions between the torus and the dynamic parallel state can occur. See supporting information (SI) Fig. 7.In this paper, we study a one-dimensional individual-based model for group formation with stochasticity included along the lines of the variation described above. This system exhibits repeated stochasticity-induced switching between distinct ordered collective motion states. This switching appe...

For aquatic animals, turning maneuvers represent a locomotor activity that may not be confined to a single coordinate plane, making analysis difficult, particularly in the field. To measure turning performance in a three-dimensional space for the manta ray (), a large open-water swimmer, scaled stereo video recordings were collected. Movements of the cephalic lobes, eye and tail base were tracked to obtain three-dimensional coordinates. A mathematical analysis was performed on the coordinate data to calculate the turning rate and curvature (1/turning radius) as a function of time by numerically estimating the derivative of manta trajectories through three-dimensional space. Principal component analysis was used to project the three-dimensional trajectory onto the two-dimensional turn. Smoothing splines were applied to these turns. These are flexible models that minimize a cost function with a parameter controlling the balance between data fidelity and regularity of the derivative. Data for 30 sequences of rays performing slow, steady turns showed the highest 20% of values for the turning rate and smallest 20% of turn radii were 42.65±16.66 deg s and 2.05±1.26 m, respectively. Such turning maneuvers fall within the range of performance exhibited by swimmers with rigid bodies.

Schools of fish and flocks of birds are examples of self-organized animal groups that arise through social interactions among individuals. We numerically study two individual-based models, which recent empirical studies have suggested to explain self-organized group animal behavior: (i) a zone-based model where the group communication topology is determined by finite interacting zones of repulsion, attraction, and orientation among individuals; and (ii) a model where the communication topology is described by Delaunay triangulation, which is defined by each individual's Voronoi neighbors. The models include a tunable parameter that controls an individual's relative weighting of attraction and alignment. We perform computational experiments to investigate how effectively simulated groups transfer information in the form of velocity when an individual is perturbed. A cross-correlation function is used to measure the sensitivity of groups to sudden perturbations in the heading of individual members. The results show how relative weighting of attraction and alignment, location of the perturbed individual, population size, and the communication topology affect group structure and response to perturbation. We find that in the Delaunay-based model an individual who is perturbed is capable of triggering a cascade of responses, ultimately leading to the group changing direction. This phenomenon has been seen in self-organized animal groups in both experiments and nature.

We present a progression of three distinct three-zone, continuum models for swarm behavior based on social interactions with neighbors in order to explain simple coherent structures in popular biological models of aggregations. In continuum models, individuals are replaced with density and velocity functions. Individual behavior is modeled with convolutions acting within three interaction zones corresponding to repulsion, orientation, and attraction, respectively. We begin with a variable-speed first-order model in which the velocity depends directly on the interactions. Next, we present a variable-speed second-order model. Finally, we present a constant-speed second-order model that is coordinated with popular individual-based models. For all three models, linear stability analysis shows that the growth or decay of perturbations in an infinite, uniform swarm depends on the strength of attraction relative to repulsion and orientation. We verify that the continuum models predict the behavior of a swarm of individuals by comparing the linear stability results with an individual-based model that uses the same social interaction kernels. In some unstable regimes, we observe that the uniform state will evolve toward a radially symmetric attractor with a variable density. In other unstable regimes, we observe an incoherent swarming state.