Markov chain model of phytoplankton dynamics

Phytoplankton, as the first level of food accessible to animals, is the main source of nutrient in the ocean. That is why the understanding of its behaviour becomes so important. Properties of phytoplankton have been widely investigated by researchers from various branches of science. Phytoplankton cells have the ability to form aggregates— groups of cells bonded together. Plankton cells in such an aggregate, are joined with a kind of organic glue, called TEP (Passow and Alldredge, 1995). The aggregates undergo diffusion, currents and turbulence, which lead to a dispersion and patchy distribution of phytoplankton in the water. Since numerical and mathematical modelling plays a crucial role in the understanding of plankton dynamics, there have recently appeared different approaches to the description of plankton with the use of various mathematical methods. One of the approaches uses advectiondiffusion-reaction equations, which describe the spatial densities of cells concentrations (Franks, 2002; Levin and Segel, 1976). In some models the process of the coagulation of plankton cells is included (Laurençot and Mischler, 2002), and these are of great interest for us. An extensive survey of mathematical models of coagulation is given by Aldous (1999). Another approach is based on individual behaviour of cells. It assumes that single cells undergo some random movement and they somehow interact with others. This may lead to the so-called superprocesses (Adler, 1997; Young et al., 2001)), but in our case such a model is related to a description by means of fragmentation-coagulation equations. We consider here an individual-based model, where a plankton aggregate plays the role of an individual unit. It is a discrete-time, simulation-oriented version of a model presented by Rudnicki and Wieczorek (2006b). It describes a population of aggregates of plankton cells structured by mass and location in the water. The movement of the aggregates is described by a random walk. The proliferation of cells in the aggregates results in the growth of the latter. The fragmentation of the aggregates, as well as their coagulation (i.e., joining together), is included. The model is introduced in Section 2 (and Appendix A). The idea of structuring the plankton population according to the size of aggregates comes here from Arino and Rudnicki (2004), but it was used before, e.g., by Jackson (1990). We present also a macroscopic model in which the mass-spatial distribution of plankton aggregates is described by an evolution equation of the fragmentationcoagulation type. The connection between the microand macroscopic models is stressed, namely, we show (in Section 3 with Appendices B and C) the limit passage from the first model to the second one as the number of aggregates tends to infinity. An extensive survey of phytoplankton models may be found in the work of Rudnicki and Wieczorek (2008).