Moment approximation of individual-based models. Application to the study of the spatial dynamics of phytoplankton populations

Abstract

The objective of this paper is to analyze the efficiency of a spatial moments model (SMM) in approximating an individual-based model (IBM) dedicated to the study of the aggregation phenomenon in a phytoplankton population. The dynamic system of spatial moments consists of a system of integro-differential equations derived from a phytoplankton IBM. The later is built on the basis of stochastic processes describing the dynamics of phytoplankton cells and their interactions. These processes are: (1) the movement of cells, which takes into account the random dispersion of cells in water and the attraction between cells due to their chemosensory abilities, (2) the demographical process (cell division or cell death) in which, the effect of local competition for nutrient resources on the cell’s division process is taken into account through the use of density dependent division rate.We solve numerically the SMM and then simulate the two models (IBM and SMM), for a set of scenarios chosen to analyze the aggregation and the different interactions in the population. The numerical results have led to the conclusion that in contrast to the mean-field model (MFM), the method of spatial moments is efficient to capture the dynamics of the IBM. Further, the SMM easily permits the prediction of long term behavior of phytoplankton cells and their spatial structure, namely that the aggregation behavior of the cells is very much related to their cellular diffusion and the perceptual radius.Concerning competition on resources, this process has a short and medium-term effect on aggregation, since regardless of its intensity, in the long term the population tends to be stable with formation of aggregates, as a result of the equilibrium between different mechanisms, including reproduction, competition, diffusion and cells attractions due to chemosensory abilities.