Concentration Phenomena in a Nonlocal Quasi-linear Problem Modelling Phytoplankton I: Existence

We study the positive steady state of a quasi-linear reaction-diffusion system in one space dimension introduced by Klausmeier and Litchman for the modelling of the distributions of phytoplankton biomass and its nutrient. The system has nonlocal dependence on the biomass function, and it has a biomass-dependent drifting term describing the active movement of the biomass towards the location of the optimal growth condition. We obtain complete descriptions of the profile of the solutions when the coefficient of the drifting term is large, rigorously proving the numerically observed phenomenon of concentration of biomass for this model. Our theoretical results reveal four critical numbers for the model not observed before and offer several further insights into the problem being modelled. This is Part I of a two-part series, where we obtain nearly optimal existence and nonexistence results. The asymptotic profile of the solutions is studied in the separate Part II.