High-integrity human intervention in ecosystems: Tracking self-organization modes
Fig 3
Vegetation rehabilitation using periodic ground modulations.
(a) A partial bifurcation diagram obtained from Eq (3), showing bare soil state (black line, BS), a stripe pattern (dark green line, SP) and a rhombic pattern (light green line, RP). Solid (dashed) lines represent stable (unstable) states. The insets show examples of two-dimensional spatial biomass distributions of the two patterned states. The stripe pattern disappears in a saddle-node bifurcation at precipitation Pc. (b,c) Phase space spanned by the self-organized (SO) variables A and a above the saddle-node bifurcation (P = P2 > Pc) and below it (P = P1 < Pc), respectively. Solid (open) circles represent stable (unstable) states. They correspond to the intersection points of the black vertical dotted lines in panel (a) with the various solution branches. The lines in blue represent stable and unstable manifolds. At P = P2, where the unstable stripe pattern still exist, phase trajectories that emanate from a nearly stripe-pattern state converge to the rhombic pattern (green line in panel b). In contrast, at P = P1, where the stripe-pattern state no longer exists, phase trajectories collapse to bare soil (red line in panel c). However adding small components of the oblique modes to an initial stripe pattern places the system above the stable manifold of the unstable rhombic pattern and results in convergence to the rhombic pattern (green line in panel c).(d-f) Snapshots of a 13 × 13 [m] domain, taken over time (left to right) showing response of ecosystem to a drought, corresponding to green line in panel b, and red and green lines in panel c, respectively.