Figure 8

Capture due to a change in the geometry of the phase portrait. A capture results from a trajectory being recruited into a new basin of attraction due to the movement of a separatrix. In this example, the relevant separatrix is caused to move by a break in the symmetry of the repressive interactions between X and Y. No new attractor states or separatrices are created. Therefore, the change in landscape topography is purely geometrical and does not affect the topology of phase space. 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) The system starts off in the bistable regime and the initial conditions place the trajectory in the basin of the low x, high y attractor (light blue). (B) Changing the threshold of one of the repressive interactions only (b in equation 1) shrinks the basin of attraction of the low x, high y attractor (light blue) causing the separatrix to move towards the upper left corner of the phase portrait. (C) The shifting separatrix catches up with the converging trajectory, recruiting it into the basin of the high x, low y attractor (dark blue). A capture event has taken place without any preceding bifurcation. This change in basins of attraction is represented by the colour coding of the trajectory on the phase portrait. (D) After the capture, the system converges to its new attractor (dark blue). Interestingly, the geometry of this capture has made the trajectory loop over itself. Such self-crossing trajectories are never observed in autonomous dynamical systems. The subsequent saddle-node bifurcation by which the saddle (red) and the light blue attractor annihilate each other does not affect the trajectory any further. See also Additional file 5, Supporting Movie S5.