Integrability of stochastic birth-death processes via differential galois theory

Stochastic birth-death processes are described as continuous-time Markov processes in models of population dynamics. A system of in nite, coupled ordinary diferential equations (the so- called master equation) describes the time-dependence of the probability of each system state. Using a generating...

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Autores:: Acosta-Humánez, Primitivo B.
Capitán, José A.
Morales-Ruiz, Juan J.

Tipo de recurso:

Fecha de publicación:: 2020

Institución:: Universidad Simón Bolívar

Repositorio:: Repositorio Digital USB

Idioma:: eng

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dc.title.eng.fl_str_mv	Integrability of stochastic birth-death processes via differential galois theory
title	Integrability of stochastic birth-death processes via differential galois theory
spellingShingle	Integrability of stochastic birth-death processes via differential galois theory Diferential Galois theory Stochastic processes Population dynamics Laplace transform
title_short	Integrability of stochastic birth-death processes via differential galois theory
title_full	Integrability of stochastic birth-death processes via differential galois theory
title_fullStr	Integrability of stochastic birth-death processes via differential galois theory
title_full_unstemmed	Integrability of stochastic birth-death processes via differential galois theory
title_sort	Integrability of stochastic birth-death processes via differential galois theory
dc.creator.fl_str_mv	Acosta-Humánez, Primitivo B. Capitán, José A. Morales-Ruiz, Juan J.
dc.contributor.author.none.fl_str_mv	Acosta-Humánez, Primitivo B. Capitán, José A. Morales-Ruiz, Juan J.
dc.subject.eng.fl_str_mv	Diferential Galois theory Stochastic processes Population dynamics Laplace transform
topic	Diferential Galois theory Stochastic processes Population dynamics Laplace transform
description	Stochastic birth-death processes are described as continuous-time Markov processes in models of population dynamics. A system of in nite, coupled ordinary diferential equations (the so- called master equation) describes the time-dependence of the probability of each system state. Using a generating function, the master equation can be transformed into a partial diferential equation. In this contribution we analyze the integrability of two types of stochastic birth-death processes (with polynomial birth and death rates) using standard diferential Galois theory.We discuss the integrability of the PDE via a Laplace transform acting over the temporal variable. We show that the PDE is not integrable except for the case in which rates are linear functions of the number of individuals.
publishDate	2020
dc.date.accessioned.none.fl_str_mv	2020-12-04T17:56:50Z
dc.date.available.none.fl_str_mv	2020-12-04T17:56:50Z
dc.date.issued.none.fl_str_mv	2020
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dc.type.spa.spa.fl_str_mv	Artículo científico
dc.identifier.issn.none.fl_str_mv	17606101
dc.identifier.uri.none.fl_str_mv	https://hdl.handle.net/20.500.12442/6845
dc.identifier.doi.none.fl_str_mv	https://doi.org/10.1051/mmnp/2020005
identifier_str_mv	17606101
url	https://hdl.handle.net/20.500.12442/6845 https://doi.org/10.1051/mmnp/2020005
dc.language.iso.eng.fl_str_mv	eng
language	eng
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dc.format.mimetype.eng.fl_str_mv	pdf
dc.publisher.eng.fl_str_mv	EDP Sciences
dc.source.eng.fl_str_mv	Mathematical Modelling of Natural Phenomena. Math. Model. Nat. Phenom.
dc.source.none.fl_str_mv	Vol. 15, No. 70, (2020)
institution	Universidad Simón Bolívar
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spelling	Acosta-Humánez, Primitivo B.9439add6-3451-4b3f-b187-8b8ba4133fcdCapitán, José A.9aa6efbd-67dd-4270-a049-737ad507c8ecMorales-Ruiz, Juan J.b1dc3bcd-f9eb-4064-91b4-5ac60fd2a7512020-12-04T17:56:50Z2020-12-04T17:56:50Z202017606101https://hdl.handle.net/20.500.12442/6845https://doi.org/10.1051/mmnp/2020005Stochastic birth-death processes are described as continuous-time Markov processes in models of population dynamics. A system of in nite, coupled ordinary diferential equations (the so- called master equation) describes the time-dependence of the probability of each system state. Using a generating function, the master equation can be transformed into a partial diferential equation. In this contribution we analyze the integrability of two types of stochastic birth-death processes (with polynomial birth and death rates) using standard diferential Galois theory.We discuss the integrability of the PDE via a Laplace transform acting over the temporal variable. We show that the PDE is not integrable except for the case in which rates are linear functions of the number of individuals.pdfengEDP SciencesAttribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Mathematical Modelling of Natural Phenomena. Math. Model. Nat. Phenom.Vol. 15, No. 70, (2020)Diferential Galois theoryStochastic processesPopulation dynamicsLaplace transformIntegrability of stochastic birth-death processes via differential galois theoryinfo:eu-repo/semantics/articleArtículo científicohttp://purl.org/coar/version/c_970fb48d4fbd8a85http://purl.org/coar/resource_type/c_2df8fbb1M. Abramowitz and I.A. Stegun, Handbook of mathematical functions: with formulas, graphs and mathematical tables. Dover Publications, New York (1965).P. Acosta-Humánez and D. Blázquez-Sanz, Non-integrability of some hamiltonians with rational potentials. Discr. Continu. Dyn. Syst. B 10 (2008) 265{293.P. Acosta-Humánez, J.T. Lázaro, J.J. Morales-Ruiz and C. Pantazi, Differential Galois theory and non-integrability of planar polynomial vector elds. J. Differ. Equ. 264 (2018) 7183{7212.P. Acosta-Humánez, J.J. Morales-Ruiz and J.-A. Weil, Galoisian approach to integrability of Schr odinger equation. Rep. Math. Phys. 67 (2011) 305{374.D. Alonso, R.S. Etienne and A.J. McKane, The merits of neutral theory. Trends Ecol. Evol. 21 (2006) 451{457.D. Alonso, A.J. McKane and M. Pascual, Stochastic ampli cation in epidemics. J. R. Soc. Interface 4 (2007) 575{582.J.A. Capitán, S. Cuenda and D. Alonso, How similar can co-occurring species be in the presence of competition and ecological drift? J. R. Soc. Interface 12 (2015) 20150604.J.A. Capitán, S. Cuenda and D. Alonso, Stochastic competitive exclusion leads to a cascade of species extinctions. J. Theor. Biol. 419 (2017) 137{151.T. Crespo and Z. Hajto, Algebraic Groups and Differential Galois Theory. In Vol. 122 of Graduate Studies in Mathematics. American Mathematical Society, Providence, Rhode Island (2011).W. Feller, Die grundlagen der volterrschen theorie des kampfes ums dasein in wahrscheinlichkeitstheoretischer behandlung. Acta Biometrica 5 (1939) 11{40.B. Haegeman, M. Loreau, A mathematical synthesis of niche and neutral theories in community ecology. J. Theor. Biol. 269 (2011) 150{165.S.P. Hubbell, The Uni ed Theory of Biodiversity and Biogeography. Princeton University Press, Princeton (2001).I. Kaplansky, An introduction to differential algebra. Hermann, Paris (1957).S. Karlin and H.M. Taylor, A rst course in stochastic processes. Academic Press, New York (1975).D.G. Kendall, On the generalized birth-and-death process. Ann. Math. Stat. 19 (1948) 1{15.E. Kolchin, Differential Algebra and Algebraic Groups. Academic Press, New York (1973).J.J. Kovacic, An algorithm for solving second order linear homogeneous differential equations. J. Symbolic Comput. 2 (1986) 3{43.A.G. McKendrick and M. Kesava, The rate of multiplication of micro-organisms: A mathematical study. Proc. R. Soc. Edinburgh 31 (1912) 649{653.J.J. Morales-Ruiz, Differential Galois Theory and Non-integrability of Hamiltonian Systems. In Vol. 179 of Progress in Mathematics series. Birkh ausser, Basel (1999).R.M. Nisbet and W.C.S. Gurney, Modelling uctuating populations. The Blackburn Press, Caldwell, New Jersey (1982).A.S. Novozhilov, G.P. Karev and E.V. Koonin, Biological applications of the theory of birth-and-death processes. Brief. Bioinform. 7 (2006) 70{85.A. Ronveaux, Heun's differential equations. Oxford University Press, Oxford (1995).G.U. Yule, A mathematical theory of evolution, based on the conclusions of Dr. J.C. Willis. Philos. Trans. R. Soc. Lond. B Biol. Sci. 213 (1924) 21{87.M. van der Put and M. Singer, Galois theory of linear differential equations. In Vol. 328 of Grundlehren der mathematischen Wissenschaften. Springer Verlag, New York (2003).T.L. Saati, Elements of queuing theory. McGraw-Hill, New York (1961).V. Volterra, Variazioni e uttuazioni del numero d'individui in specie animali conviventi. Mem. Acad. Naz. 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Integrability of stochastic birth-death processes via differential galois theory

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