A stochastic modelling of phytoplankton aggregation

Abstract

International audience The aim of this work is to provide a stochastic mathematical model of aggregation in phytoplankton, from the point of view of modelling a system of a large but finite number of phytoplankton cells that are subject to random dispersal, mutual interactions allowing the cell motions some dependence and branching (cell division or death). We present the passage from the ''microscopic'' description to the ''macroscopic'' one, when the initial number of cells tends to infinity (large phytoplankton populations). The limit of the system is an extension of the Dawson-Watanabe superprocess: it is a superprocess with spatial interactions which can be described by a nonlinear stochastic partial differential equation. L'objectif de ce travail est de fournir un modèle mathématique stochastique qui décrit l'aggrégation du hytoplancton,à partir de la modélisation d'un système de grande taille, mais finie, de cellules de phytoplancton sujettes à une dispersion aléatoire, des interactions spatiales qui donnent aux mouvements des cellules une certaine dépendance et un branchement (division cellulaire ou mort). Nous présentons le passage de la description microscopique à une description macroscopique, lorsque le nombre de cellules devient très grand (grandes populations de phytoplancton). La limite du système est une extension du superprocessus de Dawson-Watanabe: c'est un superprocessus avec interactions qui peut être décrit par une équation aux dérivées partielles stochastique non linéaire.

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

“…(1) in [0, T ] × Ω, where Ω = ]0, L[ is a bounded domain with boundary ∂Ω in R, x is a one dimensional coordinate, t is time. The motivation in studying this SPDE is that equation (1) arises as a model of phytoplankton aggregation ( [9], [10], [11], [12]). In fact, the authors in ( [9], [10], [11], [12]) have investigated an individual-based model (IBM) of a population of phytoplankton that takes into account small scales biological mechanisms for phytoplankton cells.…”

“…The motivation in studying this SPDE is that equation (1) arises as a model of phytoplankton aggregation ( [9], [10], [11], [12]). In fact, the authors in ( [9], [10], [11], [12]) have investigated an individual-based model (IBM) of a population of phytoplankton that takes into account small scales biological mechanisms for phytoplankton cells. These processes are: random dispersal of phytoplankton cells due to turbulence, spatial interactions between phytoplankton cells caused by chemical signals and random division and death of phytoplankton cells.…”

“…The aim of such modelling was to capture the main features of the individual dynamics at small scales that are responsible at a larger scale for the formation of aggregating patterns. An Eulerian version of the IBM has been derived in ( [9], [11]) and the SPDE (1) is obtained as a continuum limit of this IBM. u(t, x) represents the spatio-temporal distribution density of phytoplankton on the vertical water column.…”

A spatially explicit Markovian individual-based model for terrestrial plant dynamics

“…Within this framework, various kinds of dynamics have been considered: terrestrial plants [15], Darwinian evolution [5,4,6], phytoplankton aggregation [12], age-structured population [23].…”