Table 1.
A short glossary, defining the central terms used in this manuscript.
See S1 Appendix for more detailed glossary, with more detailed definitions and a more comprehensive list of terms.
Fig 1.
Managing grazing in grasslands.
(a) A partial bifurcation diagram obtained from Eq (1), where solid (dashed) lines represent stable (unstable) states. The diagram shows three states: bare soil (black line, BS), uniform vegetation (dark green line, UV), and periodic vegetation pattern (light green line, PP). The insets show spatial biomass distributions of these three states. The uniform vegetation state disappears in a saddle-node bifurcation (fold bifurcation) at precipitation Pc. (b,c) Phase space spanned by the self-organized (SO) variables A0 and Ak at precipitation P2 > Pc and at 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 invariant manifolds (stable or unstable) associated with two saddle points. At P = P2, where unstable uniform vegetation states still exist (two open circles on the horizontal axis in panel b), phase trajectories that emanate from a nearly uniform-vegetation state converge to a periodic pattern (green line in panel b). By contrast, at P = P1, where uniform vegetation states no longer exist, phase trajectories collapse to bare soil (red line in panel c). However, introducing a small component of the periodic mode Ak to the initial uniform state, which may represent non-uniform grazing, results in convergence to periodic pattern (green line in panel c). Units: precipitation [mm/yr], biomass and SO variables A0 and Ak [kg/m2].
Fig 2.
Response of the ecosystem to a drought under different grazing practices.
Vegetation undergoes grazing in either uniform (a) or non-uniform (b) ways, modeled by modifying the biomass-decay parameter M in Eq (1). Snapshots taken over time (left to right) show the vegetation profiles, for a system undergoing a strong drought at t = 0, from precipitation P = 115 to P = 90[mm/yr] (similarly to red and green lines in Fig 1c). Uniform grazing is enacted by keeping a constant M = 2, while non-uniform grazing is modeled by having the system’s domain split into 16 (8) segments in the top (bottom) rows of panel b, where only odd segments undergo grazing for the first five years of a drought (marked by gray shading in the panels). During non-uniform grazing, values of M for grazed segments and non-grazed segments are M = 2.2 and M = 1.8, respectively.
Table 2.
Model parameters.
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.
Fig 4.
Self-organized (SO) modes associated with rhombic pattern.
A nearly asymptotic rhombic pattern in the (x, y) plane (left), resulting from an instability of a stripe pattern, and the corresponding Fourier plane (kx, ky) showing the three SO modes that comprise it (right). See Fig A in S1 Appendix for earlier snapshots. These modes include the original stripe mode with wave-vectors ±(kf, 0) (red squares in right panel), representing periodicity in the x direction as the inset with vertical red stripes in the right panel show. The signature of this mode in the rhombic pattern (left panel) is illustrated by the red lines. The two additional SO modes are a pair of oblique modes with wave-vectors ±(kf/2, ky) and ±(kf/2, −ky) (blue and green squares, respectively, in right panel), which represent slanted stripe patterns as the insets in blue and green in the right panel show. The wave-vector component ky is determined such that the corresponding wave-vector sits on a circle of radius k0—the wavenumber (periodicity) that the natural pattern (in the absence of ground modulations) tends to form. The signatures of the oblique modes in the rhombic pattern (left panel) are illustrated by the slanted blue and green lines.
Fig 5.
Responses to precipitation downshifts under stochastic precipitation and different initial conditions of mixed vegetation states.
Left, middle and right columns correspond to negligible, weak and strong precipitation fluctuations, respectively. (a) Demonstration of noise level. (b) Asymptotic states (see color legend) for the grazing management system, where initial conditions consist of increasing portions of periodic pattern in uniform vegetation (pattern share). (c) Asymptotic states for the vegetation rehabilitation system, where initial conditions consist of increasing portions of rhombic pattern in stripe pattern (rhombic share). Each pixel in the parameter plane (mean precipitation—share) shows the asymptotic state obtained from averaging over 20 simulations with a unique randomization of temporal noise from a Gamma distribution per simulation, where the initial conditions correspond to mixtures of states calculated at P = 115[mm/yr] (P = 260[mm/yr]) for middle (bottom) row. Note that this vertical axis is logarithmic.
Fig 6.
Emergence of oblique self-organized (SO) modes in afforestation projects.
(a) A region of size 30x30 [m] containing four stripes of planted trees along bunds, taken from aerial images of the northern Negev region (Coordinates: 31.295N, 34.815E) in 2010 and 2019. (b) Model results of a comparable system, consisting of four initial vegetation stripes that has been subjected to a precipitation downshift from P = 205 to P = 180 [mm/yr] at t = 0, and simulated to t = 200[yr] (see full details in Materials and methods section and S1 Appendix). Note that similarly to Fig 3, but with different values of P, at high precipitation (P = 205) both stripe and rhombic patterns are stable, but at low precipitation (P = 180) stripe patterns are no longer stable. Top row shows spatial images, while bottom row shows spectral densities obtained from spectral (FFT) analysis, which demonstrates the periodicity of vegetation along different directions. The empirical spectral density in 2010 (a) shows the dominance of a stripe SO mode (yellow dots on x axis), representing the original planted pattern, while that in 2019 shows, in addition, the development of oblique modes (light-blue dots off the x axis), which represent vegetation mortality to form a spot-like pattern. The model simulations in (b) show a similar trend. During the transient dynamics towards a rhombic pattern the emerging pair of oblique modes are not symmetric (compare with Fig 4) both in the empirical data and the model simulations. However, simulations to longer times (t = 200[yr]) indicate the eventual emergence of a symmetric pair of oblique modes.