Tamtam Proceedings - lamsin

SPDE for phytoplankton aggregatios 529its length ([3],[10],[11]).Taking into account these biological considerations, we assume:1) there are interactions of each particle with other particles in the system whichbelong to a suitable neighborhood.2) particules tend to aggregate within a range (range of sensitivity for aggregation).So, we define pair interaction forces as follows:let r 0 and r max in R ∗ +. The interaction between two particles at positions x and y dependson the distance between the two particles and is determined by:where F a is an attractive force defined by:K(x, y) = 1 N F a(x − y)F a (z) ={− |z| 2 + (r 0 + r max ) |z| − r 0 r max if r 0 < |z| < r max0 otherwise2.4. BranchingNow, we add a complication: the particles branch.In phytoplankton, the most common mean of reproduction is asexual cell division (mitosis).This process splits the organism into two identical copies. Therefore, we describethe dynamics of the system as follows:a particle performs a motion in some region in R according equation (1). In a time interval[t, t + h], the particle has a probability µh + o(h) of branching: it either splits into 2identical particles or it dies out with probability 1 2 each.If it splits, the 2 daughters begin their life at the branching point.They continue theirmotion following equation (1) until the time themselves branch and so on .We suppose that the initial configuration of the system is described by a measureν ∈ M F (R) and that the initial number of particles, N, is large but finite.3. Spatial and temporal distribution of a finite system ofphytoplankton particlesThe spatial distribution of particles at time t is described by the empirical randommeasure process {η t } (the interacting branching-diffusion process) on R:η t =N(t)∑i=1δ Xi(t),TAMTAM –Tunis– 2005