A nonlocal model of phytoplankton aggregation

The mechanisms of grouping and the models revolving around these problems truly impassioned many mathematicians. Our main goal in this paper is the development and analysis of an aggregation model of phytoplankton. The model is the continuum limit of an interacting particle model describing a “long-ranged” aggregation mechanism among particles. It consists of an integro-differential advection-diffusion equation, with a convolution term responsible for the agreggation process. The nonlinearity in the equation is homogeneous of degree one, which introduces several complications. We prove that the Cauchy problem associated to this model is well posed, i.e., there exists a unique global positive solution and it satisfies the principle of conservation of mass. Further, we establish the existence of nonuniform stationary solutions using the topological degree theory, namely Leray-Schauder's fixed point theorem. This asymptotic result agrees with our beliefs that nonlinear interactions at small scales can produce some aggregating patterns at large scales.