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Abstract Reaction-diffusion models describing interactions between vegetation and water reveal the emergence of several types of patterns and travelling wave solutions corresponding to structures observed in real-life. Increasing their accuracy by also considering the ecological factor known asautotoxicityhas lead to more involved models supporting the existence of complex dynamic patterns. In this work, we include an additional carrying capacity for the biomass in a Klausmeier-type vegetation-water-autotoxicity model, which induces the presence of two asymptotically small parameters:ɛ, representing the usual scale separation in vegetation-water models, andδ, directly linked to autotoxicity. We construct three separate types of homoclinic travelling pulse solutions based on two different scaling regimes involvingɛandδ, with and without a so-calledsuperslow plateau. The relative ordering of the small parameters significantly influences the phase space geometry underlying the construction of the pulse solutions. We complement the analysis by numerical continuation of the constructed pulse solutions, and demonstrate their existence (and stability) by direct numerical simulation of the full partial differential equation model. 

Abstract We construct far-from-onset radially symmetric spot and gap solutions in a two-component dryland ecosystem model of vegetation pattern formation on flat terrain, using spatial dynamics and geometric singular perturbation theory. We draw connections between the geometry of the spot and gap solutions with that of traveling and stationary front solutions in the same model. In particular, we demonstrate the instability of spots of large radius by deriving an asymptotic relationship between a critical eigenvalue associated with the spot and a coefficient which encodes the sideband instability of a nearby stationary front. Furthermore, we demonstrate that spots are unstable to a range of perturbations of intermediate wavelength in the angular direction, provided the spot radius is not too small. Our results are accompanied by numerical simulations and spectral computations.