Model Diatom Population by Branching Stochastic Processes

Antoanela Terzieva; Georgi Terziev

doi:doi:10.11648/j.acm.20190801.12

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Model Diatom Population by Branching Stochastic Processes

Antoanela Terzieva, Georgi Terziev

Published in Applied and Computational Mathematics (Volume 8, Issue 1)

Received: 14 December 2018 Accepted: 11 January 2019 Published: 21 February 2019

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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.

Published in	Applied and Computational Mathematics (Volume 8, Issue 1)
DOI	10.11648/j.acm.20190801.12
Page(s)	3-8
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Cell Division, Diatoms, Phytoplankton, Bellman-Harris, Chlorophyll, Non Markov Multidimensional Stochastic Process

References

[1]	Bowler Ch., De Martino A., Falciatore A. (2010) Diatom cell division in an environmental context. Elsevier/Plant Biology. Vol. 13, pp. 623-630.
[2]	Harris T. E. (2002) The Theory of Branching Processes. Dover Publications.
[3]	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.
[4]	Silvestrov S., Malyarenko A., Rancic M. (2018) Stochastic Processes and Applications. Springer Proceedings in Mathematics & Statistics.
[5]	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.
[6]	Terzieva A. June (2016) Model phytoplankton population by branching processes, In: Biomath Communications, Vol. 3, Issue 2, ISSN 2367-5233 (Print) 2367-5241 (Online).
[7]	Haccou P., Jagers P., and Vatutin V. A. (2005) Branching Processes: Variation, Growth, and Extinction of Populations. Cambridge University Press.
[8]	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.
[9]	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.
[10]	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.
[11]	Athreya K. B., Ney P. E. (2004) Branching processes. Dover Publications Inc., ISBN 10: 0486434745 - ISBN 13: 9780486434742.
[12]	Raven P., Evert R., Eichhorn S. (2006) Biologie der Pflancen/15.6.2 Diatomeen (Kieselalgen). Walter de Gruyter, Berlin. New York. (In German).
[13]	Terzieva A. (2015) Model of phytoplankton by branching processes, 4-th Advanced Research in Scientific Areas ARSA, Zilina, Slovakia.
[14]	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).
[15]	Miguel González Velasco, Ines Maria Garcia del Puerto, George Yanev Petrov (2017) Controlled branching processes. Elsevier. Vol. 2.

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Antoanela Terzieva, Georgi Terziev. (2019). Model Diatom Population by Branching Stochastic Processes. Applied and Computational Mathematics, 8(1), 3-8. https://doi.org/10.11648/j.acm.20190801.12

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Antoanela Terzieva; Georgi Terziev. Model Diatom Population by Branching Stochastic Processes. Appl. Comput. Math. 2019, 8(1), 3-8. doi: 10.11648/j.acm.20190801.12

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Antoanela Terzieva, Georgi Terziev. Model Diatom Population by Branching Stochastic Processes. Appl Comput Math. 2019;8(1):3-8. doi: 10.11648/j.acm.20190801.12

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@article{10.11648/j.acm.20190801.12,
  author = {Antoanela Terzieva and Georgi Terziev},
  title = {Model Diatom Population by Branching Stochastic Processes},
  journal = {Applied and Computational Mathematics},
  volume = {8},
  number = {1},
  pages = {3-8},
  doi = {10.11648/j.acm.20190801.12},
  url = {https://doi.org/10.11648/j.acm.20190801.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20190801.12},
  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.},
 year = {2019}
}

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TY - JOUR
T1 - Model Diatom Population by Branching Stochastic Processes
AU - Antoanela Terzieva
AU - Georgi Terziev
Y1 - 2019/02/21
PY - 2019
N1 - https://doi.org/10.11648/j.acm.20190801.12
DO - 10.11648/j.acm.20190801.12
T2 - Applied and Computational Mathematics
JF - Applied and Computational Mathematics
JO - Applied and Computational Mathematics
SP - 3
EP - 8
PB - Science Publishing Group
SN - 2328-5613
UR - https://doi.org/10.11648/j.acm.20190801.12
AB - 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.
VL - 8
IS - 1
ER -

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Author Information

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

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Antoanela Terzieva, Georgi Terziev. (2019). Model Diatom Population by Branching Stochastic Processes. Applied and Computational Mathematics, 8(1), 3-8. https://doi.org/10.11648/j.acm.20190801.12

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ACS Style

Antoanela Terzieva; Georgi Terziev. Model Diatom Population by Branching Stochastic Processes. Appl. Comput. Math. 2019, 8(1), 3-8. doi: 10.11648/j.acm.20190801.12

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AMA Style

Antoanela Terzieva, Georgi Terziev. Model Diatom Population by Branching Stochastic Processes. Appl Comput Math. 2019;8(1):3-8. doi: 10.11648/j.acm.20190801.12

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@article{10.11648/j.acm.20190801.12,
  author = {Antoanela Terzieva and Georgi Terziev},
  title = {Model Diatom Population by Branching Stochastic Processes},
  journal = {Applied and Computational Mathematics},
  volume = {8},
  number = {1},
  pages = {3-8},
  doi = {10.11648/j.acm.20190801.12},
  url = {https://doi.org/10.11648/j.acm.20190801.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20190801.12},
  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.},
 year = {2019}
}

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