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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

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 & Continuous Dynamical Systems - B, 2013, 18 (7) : 1777-1792. doi: 10.3934/dcdsb.2013.18.1777
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P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," Springer-Verlag, New York, 1992.  Google Scholar

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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.  Google Scholar

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C. Semple and M. A. Steel, "Phylogenetics," Oxford University Press, Oxford, 2003.  Google Scholar

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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.  Google Scholar

show all references

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.  Google Scholar

[2]

E. J. Allen, "Modeling with Itô Stochastic Differential Equations," Springer, Dordrecht, 2007.  Google Scholar

[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.  Google Scholar

[4]

L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology," 2nd edition, CRC Press, Florida, 2011.  Google Scholar

[5]

T. C. Gard, "Introduction to Stochastic Differential Equations," Marcel Dekker, New York, 1988.  Google Scholar

[6]

D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, The Journal of Physical Chemistry, 81 (1977), 2340-2361. doi: 10.1021/j100540a008.  Google Scholar

[7]

P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," Springer-Verlag, New York, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. Langevin, Sur la Théorie du mouvement brownien, Comptes-rendus de l'Académie des Sciences, 146 (1908), 530-533. Google Scholar

[11]

C. Semple and M. A. Steel, "Phylogenetics," Oxford University Press, Oxford, 2003.  Google Scholar

[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.  Google Scholar

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