Front Instabilities Can Reverse Desertification

Abstract

Degradation processes in living systems often take place gradually by front propagation. An important context of such processes is loss of biological productivity in drylands or desertification. Using a drylandvegetation model, we analyze the stability and dynamics of desertification fronts, identify linear and nonlinear front instabilities, and highlight the significance of these instabilities in inducing self-recovery. The results are based on the derivation and analysis of a universal amplitude equation for pattern-forming living systems for which nonuniform instabilities cannot emerge from the nonviable (zero) state. The results may therefore be applicable to other contexts of animate matter where degradation processes occur by front propagation.

Continuum Modeling of Discrete Plant Communities: Why Does It Work and Why Is It Advantageous?

“…In that range we consider precipitation values below the Maxwell point, where desertification fronts exist (bare soil displacing uniform vegetation). In the vicinity of the bare-soil instability to uniform vegetation (P = P C in Figure 4) the two-variable model presented in Appendix B can be reduced to an amplitude equation for the uniform mode that begins to grow at this instability [85]. Analysis of this equation reveals a transverse front instability [86,87], whereby small bulges along the front line are first enhanced, then develop into growing fingers that avoid one another, and ultimately fill up the system domain with a stationary labyrinthine pattern.…”

“…According to the analysis of the amplitude equation, the instability occurs as the soil-water diffusion coefficient, D W , exceeds a threshold value, and that threshold is inversely related to the root-to-shoot ratio, E, which controls water uptake by plants' roots [85]. This result uncovers the mechanism of the instability-fast soil-water diffusion (relative to biomass expansion) towards incidental bulges along the front line that locally deplete the soil-water content.…”

The Lefever–Lejeune nonlinear lattice: Convergence dynamics and the structure of equilibrium states

“…Of exceptional interest may be the study of the higher-dimensional lattice, as exciting pattern formation dynamics may emerge due to the interplay of discreteness and higher-dimensionality, see [25,42]. Another interesting direction is leading to the study of relevant coupled lattice systems [24,25,[43][44][45]. Studies revolving around the above themes are in progress and will be reported in future works.…”

“…An important, yet incomplete issue, concerns a detailed bifurcation analysis for the emergence of the aforementioned spatial-inhomogeneous states and the potential detection of snaking effects [39][40][41]. Of exceptional interest may be the study of the higher-dimensional lattice, as exciting pattern formation dynamics may emerge due to the interplay of discreteness and higher-dimensionality, see [25,42]. Another interesting direction is leading to the study of relevant coupled lattice systems [24,25,[43][44][45].…”

“…In the first case, we identify several parametric regimes for the parameters α and β, for the global stability of the trivial steady-state. This case is of particular importance since it can be associated with parametric conditions for the spatial extinction of vegetation densities, and consequently, with the emergence of desertification [24,25]. The second one, which is the generic case of uniform in time bounds is associated with the existence of an attracting set, for all the cases of the boundary conditions considered, and the potential convergence to non-trivial equilibrium states.…”