Dryland vegetation pattern dynamics driven by inertial effects and secondary seed dispersal

Vegetation Patterns in the Hyperbolic Klausmeier Model with Secondary Seed Dispersal

“…Indeed, the region of uphill migration shrinks from panel (a) to (c), whereas it is enlarged from (d) to (f). All of these results are in line with previous theoretical findings [51].…”

“…Moreover, the results of previous investigations [13,20,27,28,31,45,51] suggested that, at the onset, the sign of the imaginary part of the most unstable mode can be either positive or negative, denoting the (hypothetical) possibility of patterns migrating either downhill or uphill. Since the occurrence of downhill motion has been questioned in the literature, it is interesting to determine the boundary between these opposite regimes, namely, the scenario in which patterns become stationary.…”

“…Recently, motivated by the experimental evidence regarding the inertia of vegetation in the context of dryland ecology [46][47][48][49][50], a further generalization of the previous model was introduced in [51]. In particular, by following the Extended Thermodynamics theory [64,65], a hyperbolic version of the Klausmeier model was built.…”

“…In the following analyses, according to the values reported in the literature [6,8,15,66], the rainfall and plant loss are taken in the ranges B ∈ (0, 2) and A ∈ (0, 3), respectively, whereas the water advection speed depends on the gradient slope and is generally taken as ν 200. Moreover, since only a small percentage of seeds that falls onto the land are transported downhill by water flow, it is realistic to assume ψ ν [13,51]. In particular, to get more insights into the mechanisms giving rise to migrating or stationary patterns, let us now address linear stability analyses with the goal of determining the thresholds for wave and Turing instabilities, respectively.…”

“…In particular, in the model proposed by Consolo and Valenti [13], the addition of an advection term for mimicking anisotropic seed transport due to overland flow [20,27,28,31,45] was able to provide the occurrence of stationary periodic patterns over hill slopes. Recently, in order to account for the inertial effects that were experimentally observed in vegetation response [46][47][48][49][50], this model was further generalized through a hyperbolic framework [51]. The hyperbolicity introduced there was also able to overcome the unphysical paradox of the infinite propagation speed of disturbances, which is typical of parabolic models, and generally offered a better description of transient and wave propagation phenomena [52][53][54][55][56][57][58][59][60][61][62][63].…”