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

Abstract

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.

Fig 1. Social interaction zones.

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

Fig 2. The social interaction kernels, K_j(s), for j = r, al, a.

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.

Fig 3. Cartoon depiction of the direction-dependent interaction rules in submodels M1 through M5.

A right-moving reference individual at x receives signals from distant individuals, to the right at x + s, and from the left at x − s. Arrows at x + s and x − s indicate whether the reference individual will receive stimuli from distant right-moving (right arrow) and distant left-moving (left arrow) neighbours. For example, the direction-dependent interaction rules in submodel M1 describe how a right-moving reference individual at x receives signals from distant individuals, to the right at x + s and from the left at x − s. For attraction and repulsion (a), this individual uses all information from neighbours regardless of their direction of travel. Arrows pointing left and right indicate this. For alignment (b), the reference individual only uses information from neighbours heading toward it, as indicated by the arrows heading toward the reference individual. The submodels illustrated here are the same as those considered in [–24].

Fig 4. Examples of patterns obtained by various direction-dependent interaction rules.

Parameters and rules (submodel) are described in Table 3, and boundary conditions are periodic.

Fig 5. Density plots of patterns obtained by various direction-dependent interaction rules.

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 Fig 4. Subfigures correspond to the patterns shown in Fig 4. Parameters and rules (submodel) are described in Table 3.

Fig 6. Density plot of a stationary pulse formed using the revised repulsion kernel, K˜r(s).

Bright colours indicate high numbers of individuals. The number density has been normalized to 1. Parameters are identical to those for the stationary pulse observed in Fig 5(a) (M1, λ₁ = 0.2, λ₂ = 0.9, q_r = 2.4, q_al = 0, q_a = 2). Note the loss of high-density subgroups within each stationary pulse. The revised repulsion kernel does not permit conspecifics to remain together as a small group without exerting a large repulsive force on each other.

Fig 7. Splitting and merging behavior in the IBM with density-dependent speed results with or without alignment.

Here, submodel M1 is used with N = 500, L = 10, Δt = 0.05 and the boundary conditions are periodic. Other parameters are given in Table 5. Corresponding density plots in are shown in Fig 8. In (a) and (b), stationary pulses form. In (c) and (d), splitting and merging behavior is observed with (c) or without alignment (d).

Fig 8. Density plots of patterns produced by the IBM with density-dependent speed.

Bright colours indicate high numbers of individuals. The number density has been normalized to 1. Corresponding trajectories are shown in Fig 7.