A nonlocal model of phytoplankton aggregation. (English) Zbl 1077.35059
Summary: 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.
MSC:
| 35K57 | Reaction-diffusion equations |
| 92D40 | Ecology |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 47N60 | Applications of operator theory in chemistry and life sciences |
Keywords:
interacting particles; Lagrangian model; Eulerian model; McKean-Vlasov interaction; semigroups; Leray-Schauder’s fixed point theorem; integro-differential advection-diffusion equation; Cauchy problemReferences:
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