Model Diatom Population by Branching Stochastic Processes

Abstract

The phytoplankton is one of the most ancient inhabitants of our planet. It consists of mostly unicellular aquatic species, both fresh water and marine. The purpose of this work is to model the dynamics of a diatoms population because it is a predominant phytoplankton kind and plays a key role at the base of the food chains, climate regulation and ecology. The formulated mathematical model would give a better idea about the expected population size in the near and further future. As a modelling tool we propose the branching stochastic process of Bellman-Harris (BPBH) Z (t). In general, the generating function (g.f.) F (t) for non Markov multidimensional BPBH is difficult for explicit expression. Impossibility for simultaneous birth and death of the BPBH-particle together with producing offspring would correspond to the biological side. Only after completion of the whole cycle the cell is capable of dividing and every particle is of zero age at birth, which corresponds to the condition of right continuity at the zero point of the distribution function (d.f.) G (t). It makes the multidimensional g.f. F (t) more suitable for research and analytical expression, allowing the use of basic theorems. The matrix U (t) of means meets the requirements and satisfies the basic matrix equation for a multidimensional non Markov branching processes. The matrix equation, corresponding to the system of sixteen integral equations is determined. The moments of Z (t) are expressed. The most characteristic feature of the diatoms is their cell wall - the cause of mitosis to result in one of the two daughters decreasing in size. This again directs the authors to determine the particle's type by its initial size and model by suggesting a decrease in the offspring size. The diatom's cell stops dividing when their size drops below the minimum. Accumulating sufficient critical mass, cells that have ceased to divide begin to merge with each other, generating a new cell. In contrast to the determined models the stochastic processes assess the probable future development. A certain fact is that the diatoms number is influenced by many factors of random nature in the environment.