Derivation of SDES for a macroevolutionary process

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September 2013, 18(7): 1777-1792. doi: 10.3934/dcdsb.2013.18.1777

Derivation of SDES for a macroevolutionary process

Ummugul Bulut ^1, and Edward J. Allen ^1,

1.	Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409-1042, United States, United States

Received September 2012 Revised February 2013 Published May 2013

Abstract
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Systems of ordinary stochastic differential equations (SDEs) are derived that describe the evolutionary dynamics of genera and species. Two different hypotheses are made in the model construction, specifically, the rate of change of the number of genera is either proportional to the number of genera in the family or is proportional to the number of species in the family. Asymptotic and exact mean numbers of species per genera are derived for both hypotheses. Computational results for the derived systems of SDEs agree well with the observed results for several families. Moreover, for each family, the SDE models yield estimates of variability in the processes which are difficult to obtain using classical methods to study the dynamics of species and genera formation.

Keywords: macroevolutionary process., Stochastic system of differential equations.

Mathematics Subject Classification: Primary: 60H10, 92D25; Secondary: 34F0.

Citation: Ummugul Bulut, Edward J. Allen. Derivation of SDES for a macroevolutionary process. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1777-1792. doi: 10.3934/dcdsb.2013.18.1777

References:

[1]	D. J. Aldous, Stochastic models and descriptive statistics for phylogenetic trees from Yule to today, Statistical Science, 16 (2001), 23-34. doi: 10.1214/ss/998929474.
[2]	E. J. Allen, "Modeling with Itô Stochastic Differential Equations," Springer, Dordrecht, 2007.
[3]	E. J. Allen, L. J. S. Allen, A. Arciniega and P. E. Greenwood, Construction of equivalent stochastic differential equation models, Stochastic Analysis and Applications, 26 (2008), 274-297. doi: 10.1080/07362990701857129.
[4]	L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology," 2nd edition, CRC Press, Florida, 2011.
[5]	T. C. Gard, "Introduction to Stochastic Differential Equations," Marcel Dekker, New York, 1988.
[6]	D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, The Journal of Physical Chemistry, 81 (1977), 2340-2361. doi: 10.1021/j100540a008.
[7]	P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," Springer-Verlag, New York, 1992.
[8]	P. E. Kloeden, E. Platen and H. Schurz, "Numerical Solution of SDE Through Computer Experiments," Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-57913-4.
[9]	T. G. Kurtz, Limit theorems for sequences of jump Markov processes approximating ordinary differential equations, The Journal of Applied Probability, 8 (1971), 344-356. doi: 10.2307/3211904.
[10]	P. Langevin, Sur la Théorie du mouvement brownien, Comptes-rendus de l'Académie des Sciences, 146 (1908), 530-533.
[11]	C. Semple and M. A. Steel, "Phylogenetics," Oxford University Press, Oxford, 2003.
[12]	G. U. Yule, A mathematical theory of evolution, based on the conclusions of Dr. J. C. Willis, Philosophical Transactions of the Royal Society of London. Series B, 213 (1925), 21-87. doi: 10.1098/rstb.1925.0002.

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References:

[1]	D. J. Aldous, Stochastic models and descriptive statistics for phylogenetic trees from Yule to today, Statistical Science, 16 (2001), 23-34. doi: 10.1214/ss/998929474.
[2]	E. J. Allen, "Modeling with Itô Stochastic Differential Equations," Springer, Dordrecht, 2007.
[3]	E. J. Allen, L. J. S. Allen, A. Arciniega and P. E. Greenwood, Construction of equivalent stochastic differential equation models, Stochastic Analysis and Applications, 26 (2008), 274-297. doi: 10.1080/07362990701857129.
[4]	L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology," 2nd edition, CRC Press, Florida, 2011.
[5]	T. C. Gard, "Introduction to Stochastic Differential Equations," Marcel Dekker, New York, 1988.
[6]	D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, The Journal of Physical Chemistry, 81 (1977), 2340-2361. doi: 10.1021/j100540a008.
[7]	P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," Springer-Verlag, New York, 1992.
[8]	P. E. Kloeden, E. Platen and H. Schurz, "Numerical Solution of SDE Through Computer Experiments," Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-57913-4.
[9]	T. G. Kurtz, Limit theorems for sequences of jump Markov processes approximating ordinary differential equations, The Journal of Applied Probability, 8 (1971), 344-356. doi: 10.2307/3211904.
[10]	P. Langevin, Sur la Théorie du mouvement brownien, Comptes-rendus de l'Académie des Sciences, 146 (1908), 530-533.
[11]	C. Semple and M. A. Steel, "Phylogenetics," Oxford University Press, Oxford, 2003.
[12]	G. U. Yule, A mathematical theory of evolution, based on the conclusions of Dr. J. C. Willis, Philosophical Transactions of the Royal Society of London. Series B, 213 (1925), 21-87. doi: 10.1098/rstb.1925.0002.

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