Figure 4

Transition. A transition indicates the switch of a non-autonomous system from one attractor state to another. 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. The trajectory’s initial conditions coincide with the attractor at high x, low y (dark blue). The trajectory is therefore at steady state at the outset. (B) Changes in auto-activation thresholds a and c (equation 1) over time cause the system to undergo a subcritical pitchfork bifurcation and enter the tristable regime (see also Figure 2). (C, D) The trajectory does not switch attractors immediately after the bifurcation occurs. However, it does not remain anchored to its current attractor either. Instead, it is left behind by the moving attractor, which it starts to pursue (see also Figures 5 and 6). (E) The system enters the monostable regime as the two bistable attractors disappear via two simultaneous saddle-node (or fold) bifurcations. The trajectory suddenly finds itself in an alternative basin of attraction, and eventually converges to the new, monostable attractor with low x and y. This change in basins of attraction is represented by a change in colour of the trajectory shown in (E). See also Additional file 1, Supporting Movie S1.