Complex spatial group patterns result from different animal communication mechanisms

Abstract

We present previously undescribed spatial group patterns that emerge in a one-dimensional hyperbolic model for animal group formation and movement. The patterns result from the assumption that the interactions governing movement depend not only on distance between conspecifics, but also on how individuals receive information about their neighbors and the amount of information received. Some of these patterns are classical, such as stationary pulses, traveling waves, ripples, or traveling trains. However, most of the patterns have not been reported previously. We call these patterns zigzag pulses, semizigzag pulses, breathers, traveling breathers, and feathers.nonlocal hyperbolic system ͉ signal reception ͉ spatial pattern ͉ zigzag P attern formation is one of the most studied aspects of animal communities. Here we present 10 complex spatial patterns that emerge in a one-dimensional mathematical model used to describe the formation and movement of animal groups.Some of the most remarkable examples of patterns observed in animal groups are related to the behavior displayed by these groups (1). Stationary aggregations formed by resting animals, migrating herds of ungulates, zigzagging flocks of birds, and milling schools of fish are only a few of the patterns. To understand the underlying mechanisms, scientists use mathematical models to simulate these observed biological patterns. The most spectacular examples of group patterns shown by numerical simulations are obtained with individual-based models: swarms, tori, and polarized groups (2, 3). A second mathematical modeling approach is based on continuum models, which are usually described by partial differential equations. In many areas, the continuum models have been successful at deducing conditions that give rise to biological patterns [e.g., morphogenesis (4)], even in one spatial dimension (5). However, this has not been the case for animal grouping models. The one-dimensional continuum models that investigate animal aggregations fail to account for the multitude of complex patterns that one can observe in nature. Generally, the patterns exhibited by these models are simple: local parabolic models do not support traveling waves (6), and nonlocal parabolic models can give rise to stationary pulses (7) or to traveling waves, provided that diffusion is density-dependent (8). Hyperbolic models give rise to ripples (9) and aggregations (9, 10). Considering that one-dimensional models have not explained the complexity of the patterns observed in biological systems, scientists have directed their attention toward two-dimensional models. The results are more complex [e.g., ripples (11), stationary aggregations (7), vortex-like groups (12), patches of aligned individuals (13,14)], but they still cannot account for the multitude of observed patterns.One possible reason for this failure is that the assumptions considered by these models do not fully describe the social interactions between individuals governing group formation. More precisely, these models consider that...

Emergence of Lévy walks in systems of interacting individuals

“…The similar repulsion rate has been well used in the hyperbolic model [28] to obtain spatial patterns. In our non-Markovian model, the role of the collision and repulsion rate γ ± is drastically changed.…”

“…Note that the advantage of moving in a large group in the same direction is very similar to the "peloton" phenomenon in a road bicycle race. Similar nonlocal dependencies of the turning rate on the population density has been successfully used to describe the animal spatial group patterns and bacterial swarming in terms of the hyperbolic and kinetic models [25,[27][28][29][30]. The second term γ ± > 0 in the turning rate T ± , Eq.…”

“…There exist two main types of models used for a collective behavior: (1) Lagrangian models describing the movements of self-propelled particles in terms of nonlinear equations for the positions and velocities of all particles [20][21][22][23], and (2) kinetic models involving partial differential equations for the population densities [24][25][26][27][28][29][30]. In this Rapid Communication we are only concerned with the nonlinear kinetic and macroscopic equations.…”

“…However, it is still an open question how Lévy walks emerge in systems of interacting self-propelled particles. The current theory of Lévy walk [8] assumes noninteracting particles and power-law distribution of traveled distances from the inception.The collective behavior of large groups of interacting individuals such as bird flocks, fish schools, and the collective migration of cells or bacteria is another rapidly growing area of active matter research [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. There exist two main types of models used for a collective behavior: (1) Lagrangian models describing the movements of self-propelled particles in terms of nonlinear equations for the positions and velocities of all particles [20][21][22][23], and (2) kinetic models involving partial differential equations for the population densities [24][25][26][27][28][29][30].…”

Optimizing the Search for Resources by Sharing Information: Mongolian Gazelles as a Case Study

“…[7,8], and to pool information on resource locations when no single individual is sufficiently knowledgeable [9][10][11][12][13]. Communication among individuals frequently leads to group formation [14], which often has clear direct benefits such as reducing individual vulnerability to predators. Such strategies may, however, also have important incidental benefits.…”