Fig 1.
Bifurcations are predicted using data from the pre-bifurcation regime.
The method predicts the edges of the bi-stable region using data either from outside or inside of this zone. In this example, the bifurcation at A is predicted from data collected at A1 and A2 (or and
), while the bifurcation at B is predicted from data at points B1 and B2 (or
and
).
Fig 2.
Overview of the method to forecast bifurcations using recoveries from perturbations.
The bifurcation diagram with both the stable (solid line) and unstable (dotted line) branches of the system dynamics are shown. The methodology predicts the post-bifurcation regime (A) using transient recoveries from perturbations from two locations outside the bi-stable region (A1 and A2). It can also predict the bifurcation (B) using transient recoveries within the bi-stable region (B1 and B2). The recovery trajectories are shown in the four insets. The areas where slowing down is observed align with the value of the equilibrium state of the bifurcation point. The method takes advantage of the changes in the recovery rates, and of the CSD generated by the presence of ‘ghost’ equilibrium, to extract information on the location and kind of bifurcation.
Fig 3.
Description of the method to forecast the bifurcation point μc using recoveries at two bifurcation parameter values (μ1 and μ2).
Using Eq (4), η curves are generated for each recovery, and their corresponding peaks are used to predict the bifurcation point μc. Additional points along the η curves can also be used to predict the amplitude in the post bifurcation regime.
Fig 4.
The inertial manifold (i.e., the invariant set in which the dynamics is slowest in time) is low dimensional and varies slowly with the bifurcation parameter.
Table 1.
Parameter values of the vegetation grazing model used to produce simulated measurement data.
Table 2.
Parameter values of the model of the feedback between macrophytes and phytoplankton in a lake used to produce simulated measurement data.
Fig 5.
Analysis of recovery from perturbations to forecast a bifurcation.
(a) Noisy observations at three conditions (i.e., three bifurcation parameter values) for a vegetation grazing ecosystem recovering from perturbations. Labels A, B, and C indicate regions where the dynamics slows down. The plot for region A corresponds to the parameter value closest to the bifurcation, and the plot for region C corresponds to the parameter value farthest from the bifurcation. (b) A variable η is obtained from the transient response data near the slowing down region and plotted versus the vegetation biomass. Curves are fit for each condition. Note that labels A, B, and C correspond to the peaks of each curve. Additional points are indicated to the left (subscript L) and right (subscript R) of the peaks on each curve. (c) One η value obtained from each curve (A, B and C) is plotted versus the bifurcation parameter c. The location of the bifurcation is forecasted by approximating where the line connecting these points intersects the x-axis. The post-critical regime can be forecasted in a similar manner by approximating where lines defined by other sets of points such as AL, BL and CL, and AR, BR and CR cross the x-axis.
Fig 6.
Predicted bifurcation diagrams for the vegetation model using 100 recoveries at three pre-bifurcation conditions.
The exact bifurcation diagram with both the stable (solid line) and unstable (dashed line) branches are shown in each plot. Transient data were collected at three locations (indicated by labels X) in the pre-bifurcation regime. The predicted post-bifurcation regime is shown together with the standard deviation error bars for 100 separate noisy realizations for (a) 10% noise, (b) 20% noise, and (c) 50% noise.
Fig 7.
Predicted bifurcation diagrams for the feedback system between macrophytes and phytoplankton in a shallow lake using 100 recoveries at two pre-bifurcation conditions.
The exact bifurcation diagram with both the stable (solid line) and unstable (dashed line) branches are shown in each plot. Transient data were collected at two locations (indicated by labels X) in the pre-bifurcation regime. The predicted post-bifurcation regime is shown by the standard deviation error bars for 100 separate noisy realizations for macrophyte cover with (a) 10% noise, (b) 20% noise, and (c) 30% noise, and vertical light attenuation with (d) 10% noise, (e) 20% noise, (f) 30% noise.
Fig 8.
Predicted bifurcation diagrams for the vegetation model using a single recovery at three pre-bifurcation conditions.
The exact bifurcation diagram with both the stable (solid line) and unstable (dashed line) branches are shown in each plot. Transient data were collected at three locations (indicated by labels X) in the pre-bifurcation regime. The predicted post-bifurcation regime is shown together with the standard deviation error bars for 100 separate noisy realizations for (a) 2% noise, (b) 5% noise, and (c) 10% noise.