Stationary and Oscillatory patterned solutions in three-compartment reaction–diffusion systems: Theory and application to dryland ecology

This work aims at elucidating the conditions under which stationary and oscillatory periodic patterns may emerge in a class of one-dimensional three-compartments reaction–diffusion models where one interacting species does not undergo any spatial dispersal. To this purpose, linear stability analysis is firstly employed to deduce the conditions under which the system undergoes a Turing or wave instability as well as to extract information on the main features that characterize the corresponding patterned solutions at onset. Then, a multiple-scale weakly nonlinear analysis is carried out to describe the time evolution of the pattern amplitude close to the bifurcation thresholds of the above-mentioned instabilities. Finally, to provide a quantitative estimation of the most relevant pattern features, an illustrative example in the context of dryland ecology is addressed. It deals with a generalization of the Klausmeier vegetation model for flat arid environments which describes the interaction among vegetation biomass, soil water and toxic compounds. Numerical simulations are also used to corroborate the theoretical findings as well as to gain some useful insights into the ecological response of ecosystems to variable environmental conditions.