Pursuit stabilising the direction of a trajectory. Pursuit behaviour results from the movement of an attractor. In this case, both attractor and trajectory progress in the same direction, which is therefore stabilised even before the system enters any asymptotic regime. Upper panels show (quasi-)potential surfaces, lower panels phase portraits as in Figure 3C. The progress of time is shown through increasingly dark shading, and by the arrow at the bottom of the figure. (A) Pursuit behaviour shows similarities to autonomous dynamics, since the location of the initial condition determines the attractor towards which the system will converge. In this example, parameter changes do not alter the position of the separatrix. (B–D) However, the approach of the system towards the attractor is very different than in the autonomous case, since both attractors move away from the origin as activation strength (represented by α
in equation 1) is increased over time. This leads to an enduring pursuit by the trajectory of its moving attractor target. See also Additional file 2, Supporting Movie S2.