A nonlocal model of phytoplankton aggregation

Abstract

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.

Mathematical models for cell migration: a non-local perspective

“…A detailed analysis of a nonlocal chemotaxis system was carried out in [56]. Several studies, in particular [1,25,34,35,39,104], address equations or systems featuring the adhesion operator A r or its extension to a possibly unbounded sensing region [57]. Some works exploit specific solutions which are particularly relevant for applications, including steady states and their stability, existence of travelling waves, etc., see [115] and [26,34,35,56,94] for models with∇ r and A r , respectively.…”

“…This corresponds to direct, long-range intraspecific interactions. Line 2 inTable 1refers e.g., to the case of individuals (cells, ants,...) moving in a collective way, thereby perceiving and correspondingly adapting to regions with large crowd density 1. Concerning the remaining lines ofTable 1, an operator M tA r ,∇ r , T r ∇, S r ∇u can be used to include a nonlocality of first order.…”

Numerical Simulations of a Nonlinear Stochastic Partial Differential Equation Modeling Phytoplankton Aggregation

“…The major difficulties come from the two nonlinear terms: the chemotaxis term with a convolution and the stochastic branching term with the nonlinear multiplicative noise. We point out that previous works ( [2], [13]) have concerned the study of the deterministic part of the SPDE (1) (i.e. equation (1) without the noise term) with the goal of analyzing the influence of the spatial interactions between phytoplankton cells on the aggregation process.…”

Boundary-value problem for density–velocity model of collective motion of organisms

“…The collective motion is a common phenomenon inherent to a variety of biological species at different spatial scales, from microscopic bacterial colonies [7,14], phytoplankton aggregations [2], insect swarms [6,15,17,23], to macroscopic fish schools [1,3,5,17,18,22,27], bird flocks [17], and others. The mutual separation distance and the alignment between the neighbors in a group of organisms are the main factors in their collective motion [16,18,22].…”

“…Individual-based models [5,11,16,26] provide a useful tool for studying relatively small groups, but become impractical when the number of individuals approaches the sizes of real biological groups. On the other hand, population-based models [1][2][3][4]6,[12][13][14][15][23][24][25] are particularly useful when the number of individuals is large. They can be regarded as the continuum limit of individual-based models.…”