Pattern formation in hyperbolic reaction-transport systems and applications to dryland ecology

Pattern formation and modulation is an active branch of mathematics, not only from the perspective of fundamental theory but also for its huge applications in many fields of physics, ecology, chemistry, biology, and other sciences. In this thesis, the occurrence of Turing and wave instabilities, giving rise to stationary and oscillatory patterns, respectively, is theoretically investigated by means of two-compartment reaction-transport hyperbolic systems. The goal is to elucidate the role of inertial times, which are introduced in hyperbolic models to account for the finite-time propagation of disturbances, in stationary and transient dynamics, in supercritical and subcritical regimes. In particular, starting from a quite general framework of reaction-transport model, three particular cases are derived. In detail, in the first case, the occurrence of stationary patterns is investigated in one-dimensional domains by looking for the inertial dependence of the main features that characterize the formation and stability process of the emerging patterns. In particular, the phenomenon of Eckhaus instability, in both supercritical and subcritical regimes, is studied by adopting linear and multiple-scale weakly-nonlinear analysis and the role played by inertia during the transient regime, where an unstable patterned state evolves towards a more favorable stable configuration through sequences of phase-slips, is elucidated. Then, in the second topic, the focus is moved to oscillatory periodic patterns generated by wave (or oscillatory Turing) instability. This phenomenon is studied by considering 1D two-compartment hyperbolic reaction-transport systems where different transport mechanisms of the species here involved are taken into account. In these cases, by using linear and weakly nonlinear stability analysis techniques, the dependence of the non-stationary patterns on hyperbolicity is underlined at and close to the criticality. In particular, it is proven that inertial effects play a role, not only during transient regimes from the spatially-homogeneous steady state toward the patterned state but also in altering the amplitude, the wavelength, the migration speed, and even the stability of the travelling waves. Finally, in the last case, the formation and stability of stationary patterns are investigated in bi-dimensional domains. To this aim, a general class of two-species hyperbolic reaction-transport systems is deduced following the guidelines of Extended Thermodynamics theory. To characterize the emerging Turing patterns, linear and weakly nonlinear stability analysis on the uniform steady states are addressed for rhombic and hexagonal planform solutions. In order to gain some insight into the above-mentioned dynamics, the previous theoretical predictions are corroborated by numerical simulations carried out in the context of dryland ecology. In this context, patterns become a relevant tool to identify early warning signals toward desertification and to provide a measure of resilience of ecosystems under climate change. Such ecological implications are discussed in the context of the Klausmeier model, one of the easiest two-compartment (vegetation biomass and water) models able to describe the formation of patterns in semi-arid environments. Therefore, it will be also here discussed how the experimentally-observed inertia of vegetation affects the formation and stability of stationary and oscillatory periodic vegetation patterns.