Applied and Computational Mathematics
Volume 8, Issue 1, February 2019, Pages: 3-8
Received: Dec. 14, 2018;
Accepted: Jan. 11, 2019;
Published: Feb. 21, 2019
Views 771 Downloads 153
Antoanela Terzieva, Department of Probability, Operations Research and Statistics, Sofia University “St. Kliment Ohridski”, Sofia, Bulgaria
Georgi Terziev, Department of Probability, Operations Research and Statistics, Sofia University “St. Kliment Ohridski”, Sofia, Bulgaria
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.
Model Diatom Population by Branching Stochastic Processes, Applied and Computational Mathematics.
Vol. 8, No. 1,
2019, pp. 3-8.
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Bowler Ch., De Martino A., Falciatore A. (2010) Diatom cell division in an environmental context. Elsevier/Plant Biology. Vol. 13, pp. 623-630.
Harris T. E. (2002) The Theory of Branching Processes. Dover Publications.
Denaro G., Valenti D., La Cognata A., Spagnolo B., Bonanno A., Basilone G., Mazzola S., Zgozi S. W., Aronica S., Brunet C. (2013) Spatio-temporal behavior of the deep chlorophyll maximum in Mediterranean Sea: Development of a stochastic model for phytoplankton dynamics. Ecological Complexity, 13 21-34.
Silvestrov S., Malyarenko A., Rancic M. (2018) Stochastic Processes and Applications. Springer Proceedings in Mathematics & Statistics.
Kasprzak P., Padisak J., Koschel R., Krienitz L., Gervais F. (2008) Chlorophyll a concentration across a trophic gradient of lakes: An estimator of phytoplankton biomass? Elsevier/Management of Inland Waters. Vol. 38, pp. 327-338.
Terzieva A. June (2016) Model phytoplankton population by branching processes, In: Biomath Communications, Vol. 3, Issue 2, ISSN 2367-5233 (Print) 2367-5241 (Online).
Haccou P., Jagers P., and Vatutin V. A. (2005) Branching Processes: Variation, Growth, and Extinction of Populations. Cambridge University Press.
Terzieva A. (2013) Modeling of the amount of chlorophyll-a, contained in the phytoplankton population by branching process, MIE 2013: Doctoral Conference in Mathematics, Informatics and Education, (2013) Sofia, Bulgaria.
B. A. Sevastyanov (2016) A class of subcritical branching processes with immigration and infinite number of types of particles. In: Discrete mathematic and applications. Vol. 17. Issue 1. Schmidt Periodicals GmbH, Bad Feilnbach https://doi.org/10.1515/DMA.2007.001.
A. Terzieva (2015) Model population dynamics by branching stochastic processes with three types of particles, International conference: Mathematics as a fundamental and applied science, Varna, Bulgaria.
Athreya K. B., Ney P. E. (2004) Branching processes. Dover Publications Inc., ISBN 10: 0486434745 - ISBN 13: 9780486434742.
Raven P., Evert R., Eichhorn S. (2006) Biologie der Pflancen/15.6.2 Diatomeen (Kieselalgen). Walter de Gruyter, Berlin. New York. (In German).
Terzieva A. (2015) Model of phytoplankton by branching processes, 4-th Advanced Research in Scientific Areas ARSA, Zilina, Slovakia.
Terzieva A., Terziev G. (2015) Multitype branching process as model for phytoplankton population and chlorophyll-a contained therein, In: American Review of Mathematics and Statistics, Vol. 3, Issue 1, ISSN 2374-2348 (Print) 2374-2356 (Online).
Miguel González Velasco, Ines Maria Garcia del Puerto, George Yanev Petrov (2017) Controlled branching processes. Elsevier. Vol. 2.