Local versus nonlocal barycentric interactions in 1D agent dynamics

Abstract

The mean-field dynamics of a collection of stochastic agents evolving under local and nonlocal interactions in one dimension is studied via analytically solvable models. The nonlocal interactions between agents result from (a) a finite extension of the agents interaction range and (b) a barycentric modulation of the interaction strength. Our modeling framework is based on a discrete two-velocity Boltzmann dynamics which can be analytically discussed. Depending on the span and the modulation of the interaction range, we analytically observe a transition from a purely diffusive regime without definite pattern to a flocking evolution represented by a solitary wave traveling with constant velocity.

Imitation, Proximity, and Growth — A Collective Swarm Dynamics Approach

“…E.1 and E.2, the mean-¯eld population dynamics given by Eqs. (16) and (20) is already observed for the limited population of agents N ¼ 30; 100 and 1000. Table E.1 provides a characterization of the theoretical and simulated distributions obtained for the stable growing productivity regime displayed in Fig.…”

“…As shown by Eq. (20), and contrary to the reaction-di®usion evolutions such as the equation derived in [42], the collective log-productivity growth rate does not depend on the amplitude of the di®usion , which a®ects only the shape of the traveling wave m . It is interesting to observe the fundamentally di®erent dynamic behaviors emanating in the two regimes (A) and (B) exposed above, the solutions of which are m Due to the so-called Rankine-Hugoniot relation, it is known that for scalar hyperbolic conservation laws, to which the Burgers Eq.…”

“…When the imitation range U lies in-between the two limiting regimes solved in Eqs. (16) and (20), the propagation speed of the agent population density ðx; tÞ is observed in Fig. E.4 to monotonously increase with U. Augmenting the interaction range U enhances the average traveling velocity of the whole population.…”

“…Only a few rounds of observation and imitation processes between the agents are necessary to reach the stationary regime predicted by Eq. (20).…”

“…Like what is happening here, a reduction in the agents' observation range decreases the importance of the mutual interactions, which ultimately precludes the possibility to generate stable collective scenarios (i.e., for birds, the tendency to°ock). This illustration con¯rms that below a critical threshold, mutual interactions are too weak to sustain a°ocked evolution (corresponding here to a mirror image of a balanced growth path), [10,20]. (b) The role of the random environment.…”