Rhombic and hexagonal pattern formation in 2D hyperbolic reaction–transport systems in the context of dryland ecology

In this paper the formation of rhombic and hexagonal Turing patterns in bidimensional hyperbolic reaction–transport systems is addressed in the context of dryland ecology. To this aim, exploiting the guidelines of the Extended Thermodynamics Theory, a hyperbolic extension of the classical modified Klausmeier vegetation model for flat arid environments is here proposed. This framework accounts explicitly for the inertial effects associated with the involved species and allows a more suitable description of transient dynamics between a spatially-homogeneous steady-state towards a patterned configuration. To characterize the emerging Turing patterns, linear stability analysis on the uniform steady-states is firstly addressed. Then, to determine the Stuart–Landau equations ruling the time evolution of the pattern amplitudes close to onset, multiple-scale weakly nonlinear analysis is performed for the two abovementioned geometries. Finally, to validate the analytical predictions as well as to provide more insights into the spatio-temporal evolution of the resulting pattern in both transient and stationary regimes, numerical simulations have been also performed.