Turing vegetation patterns in a generalized hyperbolic Klausmeier model

Abstract

The formation of Turing vegetation patterns in flat arid environments is investigated in the framework of a generalized version of the hyperbolic Klausmeier model. The extensions here considered involve, on the one hand, the strength of the rate at which rainfall water enters into the soil and, on the other hand, the functional dependence of the inertial times on vegetation biomass and soil water. The study aims at elucidating how the inclusion of these generalized quantities affects the onset of bifurcation of supercritical Turing patterns as well as the transient dynamics observed from an uniformly vegetated state towards a patterned state. To achieve these goals, linear and multiple-scales weakly nonlinear stability analysis are addressed, this latter being inspected in both large and small spatial domains. Analytical results are then corroborated by numerical simulations, which also serve to describe more deeply the spatio-temporal evolution of the emerging patterns as well as to characterize the different timescales involved in vegetation dynamics.

Eckhaus instability of stationary patterns in hyperbolic reaction–diffusion models on large finite domains

“…However, none of those works has inspected the EI instability of stationary and quantized Turing patterns in the context of hyperbolic models. As widely reported in previous works [19][20][21][22][23][24][25][26][27], hyperbolic systems bring many advantages. Firstly, they take inertial effects explicitly into account and, thus, allow to overcome to paradox of propagation of disturbances at infinite speed, typical of parabolic models.…”

“…Since the eigenvalue λ (2) n is always negative, the stability of the associated mode depends on λ (1) n which, taking into account (20), can be expressed as:…”

“…In particular, the analysis will be focused on its modified version [29][30][31] that has been introduced in the literature to model pattern dynamics over flat terrains. Moreover, motivated by the observation of inertial effects [46][47][48][49][50] and long transient regimes [51][52][53] in vegetation patterned dynamics, an hyperbolic generalization was proposed in [19,20].…”

“…In particular, the present manuscript addresses the analysis of one-dimensional stationary Turing patterns emerging in a class of two-compartments hyperbolic reaction-diffusion models, where both species undergo self-diffusion, defined on a large finite domain. In some previous works, linear and weakly-nonlinear stability analyses on the homogeneous steady states were carried out over small (in the presence of cross-diffusion [25]) or extended domains (with constant [19] and non-constant [20] inertial times). Here, we aim at addressing the stability and quantization of spatially-periodic patterns, in both supercritical and subcritical regimes.…”

“…Theoretical predictions here developed are then validated through a comparison with numerical simulations. To this aim, an illustrative example in which EI takes place in the context of dry-land ecology is here considered: the hyperbolic generalization of the modified Klausmeier model [19,20,[28][29][30][31]. It represents one of the simplest two-component systems able to predict the formation of vegetation bands in flat arid environments as the result of competition between soil water and vegetation biomass.…”