Social organization is one of the fundamental aspects of animal behavior, and has received attention both from experimental and theoretical perspectives. Examples of social groups appear at every size scale from the microscopic aggregates of mammalian cells (such as fibroblasts) to macroscopic herds of wildbeast, flocks of birds, and fish schools. There are two general frameworks when modeling such problems: the Lagrangian viewpoint and the Eulerian one. In this paper, we use both the approaches in the study of fish alignment. An individual-based model (IBM) (Lagrangian) provides a virtual world where fish forming a fish school try to adopt a common angular position. Fish are assumed to lie in horizontal planes, an individual angular position is the angle made by the oriented axis associated with the individual (tail to head) with a fixed direction. Two main forces are acting, a force of alignment, whose strength is assumed to be fixed in a given experiment but may be modified, and a force of dispersion, accounting for all disturbances. A transition from dispersion-dominant to alignment-dominant can be observed in the IBM experiments. A related PDE model (Eulerian) is used to determine the transition with sufficient accuracy.

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.

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