Parameter values produce spatial patterns.

<p>Parameter values produce spatial patterns.</p

Patterns produced by the IBM.

<p>Patterns produced by the IBM.</p

Examples of patterns obtained by various direction-dependent interaction rules.

<p>Parameters and rules (submodel) are described in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0198550#pone.0198550.t003" target="_blank">Table 3</a>, and boundary conditions are periodic.</p

Social interaction kernel parameter values.

<p>Social interaction kernel parameter values.</p

Splitting and merging behavior in the IBM with density-dependent speed results with or without alignment.

<p>Here, submodel M1 is used with <i>N</i> = 500, <i>L</i> = 10, Δ<i>t</i> = 0.05 and the boundary conditions are periodic. Other parameters are given in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0198550#pone.0198550.t005" target="_blank">Table 5</a>. Corresponding density plots in are shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0198550#pone.0198550.g008" target="_blank">Fig 8</a>. In (a) and (b), stationary pulses form. In (c) and (d), splitting and merging behavior is observed with (c) or without alignment (d).</p

Density plots of patterns produced by the IBM with density-dependent speed.

<p>Bright colours indicate high numbers of individuals. The number density has been normalized to 1. Corresponding trajectories are shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0198550#pone.0198550.g007" target="_blank">Fig 7</a>.</p

The social interaction kernels, <i>K</i>_<i>j</i>(<i>s</i>), for <i>j</i> = <i>r</i>, <i>al</i>, <i>a</i>.

<p>In (a), the repulsion kernel (red, solid) weights conspecifics close to the target individual strongly, the alignment kernel acts for intermediate distances (blue, dashed), and the attraction kernel acts on large distances (green, dotted). In (b), the repulsion kernel (red, solid) is centered over the target individual, adding biological realism as the conspecifics very close to the target individual are weighted most heavily for the repulsion social interaction force. The alignment (blue, dashed) and attraction (green, dotted) kernels remain unchanged.</p

Social interaction zones.

<p>Cartoon depiction of the three social interaction zones surrounding an individual at location <i>x</i>. Repulsion (<i>r</i>) acts over short distances from the reference individual at <i>x</i>, alignment (<i>al</i>) over intermediate distances, and attraction (<i>a</i>) over longer distances. These zones may be disjoint, as illustrated, or may overlap.</p

Density plots of patterns obtained by various direction-dependent interaction rules.

<p>Bright colours indicate high numbers of individuals, where the number density of individual is normalized by the total number of individuals. Density estimates are obtained via a kernel smoothing estimate from the corresponding trajectories in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0198550#pone.0198550.g004" target="_blank">Fig 4</a>. Subfigures correspond to the patterns shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0198550#pone.0198550.g004" target="_blank">Fig 4</a>. Parameters and rules (submodel) are described in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0198550#pone.0198550.t003" target="_blank">Table 3</a>.</p

Direction-dependent interaction rules enrich pattern formation in an individual-based model of collective behavior

<div><p>Direction-dependent interaction rules are incorporated into a one-dimensional discrete-time stochastic individual-based model (IBM) of collective behavior to compare pattern formation with an existing partial differential equation (PDE) model. The IBM is formulated in terms of three social interaction forces: repulsion, alignment, and attraction, and includes information regarding conspecifics’ direction of travel. The IBM produces a variety of spatial patterns which qualitatively match patterns observed in a PDE model. The addition of direction-dependent interaction rules exemplifies how directional information transfer within a group of individuals can result in enriched pattern formation. Our individual-based modelling framework reveals the influence that direction-dependent interaction rules such as biological communication can have upon individual movement trajectories and how these trajectories combine to form group patterns.</p></div