Fig 1.
Illustrative examples of agroforestry practices commonly used in sub-Saharan Africa.
From [9].
Fig 2.
A schematic representation of the model.
Along with the four dynamics variables B1,B2,s1, and s2, the model takes into account the mean annual precipitation P, the evaporation from the upper soil later Ev, infiltration from upper soil layer to lower one L1, and the infiltration from the lower soil layer to deeper layers L2. The transpiration terms T1 and T2, of the herbaceous and woody species respectively, are dependent on the aerial biomass density of each species, together with the relative soil moisture in each soil layer.
Table 1.
Model’s parameters and their values.
Fig 3.
Bifurcation diagram for the model.
Partial bifurcation diagram over the 2D domain Ω = (3.00 × 1.75)m. Stable (unstable) solutions are denoted by solid (dashed) curves. The bifurcation point BP1 marks the appearance of a solution branch describing homogeneous herbaceous vegetation(red curve) from the bare-soil solution (black curve). This herbaceous-only branch loses its stability in a subcritical bifurcation to a homogeneous mixed woody–herbaceous branch (magenta curve) at point BP2. At BP3 three patterned mixed woody-herbaceous solution branches bifurcate from the homogeneous mixed woody–herbaceous branch: gap patterns (cyan curve), stripes (green curve), and spots (turquoise curve). At BP4 the mixed-spot solution branch bifurcates to a woody-only spot branch (orange curve). An additional mixed-spot solution branch of longer wavelength (olive curve) bifurcates at PB5 to a stable woody-only long-wavelength spot branch (brown curve). A uniform woody-vegetation bifurcates from bare soil at a relatively high precipitation value (blue curve) and remains unstable throughout the range we explored.
Fig 4.
The three alternative patterns at tristability range.
Three alternative states for P = 600mm/y over a domain of Ω = (30.0 × 30.0)m. The upper row depicts the herbaceous species, and the lower row the woody species.
Fig 5.
Results of direct numerical simulation at P = 170mm/y, starting from an initial conditions near BP4.
Fig 6.
Performance metrics for the model.
Performance metrics for the three alternative states, gaps, (light blue curve), stripes (light green curve), and spots (turquoise curve), along with the precipitation range, where only the stable parts of the corresponding solution branches are plotted.
Fig 7.
Continuation over the niche-separation parameter.
Continuation over the parameter θ of the three patterned states, gaps(light blue curve), stripes (light green curve), and spots (turquoise curve), for P = 600 mm/y. Solid (dashed) lines denote stable (unstable) solutions.
Fig 8.
Precipitation downshift for the stripes patterns.
Time integration of the stripes patterns for a downshift of the mean annual precipitation from P = 600mm/y (in (A)), to P = 400mm/y (in (B)), and to P = 200mm/y (in (C)), over a period of T = 100y.
Fig 9.
Precipitation downshift for the spots patterns.
Time integration of the spots patterns for a downshift of the mean annual precipitation from P = 600mm/y (in (A)), to P = 400mm/y (in (B)), and to P = 200mm/y (in (C)), over a period of T = 100y.