Anticipation induces polarized collective motion in attraction based models

In most models of collective motion in animal groups each individual updates its heading based on the current positions and headings of its neighbors. Several authors have investigated the effects of including anticipation into models of this type, and have found that anticipation inhibits polarized collective motion in alignment based models and promotes milling and swarming in the one attraction-repulsion model studied. However, it was recently reported that polarized collective motion does emerge in an alignment based asynchronous lattice model with mutual anticipation. To our knowledge this is the only reported case where polarized collective motion has been observed in a model with anticipation. Here we show that including anticipation induces polarized collective motion in a synchronous, off lattice, attraction based model. This establishes that neither asynchrony, mutual anticipation nor motion restricted to a lattice environment are strict requirements for anticipation to promote polarized collective motion. In addition, unlike alignment based models the attraction based model used here does not produce any type of polarized collective motion in the absence of anticipation. Here anticipation is a direct polarization inducing mechanism. We believe that utilizing anticipation instead of frequently used alternatives such as explicit alignment terms, asynchronous updates and asymmetric interactions to generate polarized collective motion may be advantageous in some cases. ∗Corresponding author: [email protected] Introduction Self-propelled particle (SPP) models have been used to model collective motion in a large variety of animal groups including fish schools, bird flocks, herds of sheep and human crowds [1]. In most SPP models each particle calculates its new heading and position based on the current positions and/or headings of its neighbors. Recently a number of studies have highlighted the potential importance of anticipation in models of this type [2–5]. In models including anticipation each particle uses the future anticipated positions and/or headings of its neighbors to calculate its new heading, rather than their current positions and headings. The effects of anticipation on group formation in alignment based Vicsek type models [6] has been described in [2, 4] and the effects on an attraction-repulsion model [7] has been described in [3]. Interestingly, in both types of models it was established that including anticipation inhibits polarized collective motion and promote, or stabilize, milling and swarming. This is particularly surprising in the case of alignment based models because the production of polarized collective motion is a key feature of these models, whereas production of mills and swarms is not. In contrast it was recently shown that mutual anticipation allowed for polarized collective motion to emerge in an alignment based asynchronous lattice model [5]. How general are these findings? In particular, does anticipation tend to promote swarming and milling in SPP models unless specific additional implementation choices are made? For example, is asynchrony, motion restricted to a lattice or mutual anticipation necessary requirements for polarized collective motion to emerge in models with anticipation? To ad-