American Institute of Mathematical Sciences

Advanced
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
References:
[1]

D. J. Aldous, Stochastic models and descriptive statistics for phylogenetic trees from Yule to today,, Statistical Science, 16 (2001), 23.  doi: 10.1214/ss/998929474.  Google Scholar

[2]

E. J. Allen, "Modeling with Itô Stochastic Differential Equations,", Springer, (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.  doi: 10.1080/07362990701857129.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

P. E. Kloeden, E. Platen and H. Schurz, "Numerical Solution of SDE Through Computer Experiments,", Springer-Verlag, (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.  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.   Google Scholar

[11]

C. Semple and M. A. Steel, "Phylogenetics,", Oxford University Press, (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.  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.  doi: 10.1214/ss/998929474.  Google Scholar

[2]

E. J. Allen, "Modeling with Itô Stochastic Differential Equations,", Springer, (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.  doi: 10.1080/07362990701857129.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

P. E. Kloeden, E. Platen and H. Schurz, "Numerical Solution of SDE Through Computer Experiments,", Springer-Verlag, (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.  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.   Google Scholar

[11]

C. Semple and M. A. Steel, "Phylogenetics,", Oxford University Press, (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.  doi: 10.1098/rstb.1925.0002.  Google Scholar

[1]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005
[2]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[3]

Tomás Caraballo, Carlos Ogouyandjou, Fulbert Kuessi Allognissode, Mamadou Abdoul Diop. Existence and exponential stability for neutral stochastic integro–differential equations with impulses driven by a Rosenblatt process. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 507-528. doi: 10.3934/dcdsb.2019251
[4]

Ellina Grigorieva, Evgenii Khailov. A nonlinear controlled system of differential equations describing the process of production and sales of a consumer good. Conference Publications, 2003, 2003 (Special) : 359-364. doi: 10.3934/proc.2003.2003.359
[5]

Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447
[6]

Ludwig Arnold, Igor Chueshov. Cooperative random and stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 1-33. doi: 10.3934/dcds.2001.7.1
[7]

Can Huang, Zhimin Zhang. The spectral collocation method for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 667-679. doi: 10.3934/dcdsb.2013.18.667
[8]

Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075
[9]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[10]

Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803
[11]

Igor Chueshov, Michael Scheutzow. Invariance and monotonicity for stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1533-1554. doi: 10.3934/dcdsb.2013.18.1533
[12]

A. Alamo, J. M. Sanz-Serna. Word combinatorics for stochastic differential equations: Splitting integrators. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2163-2195. doi: 10.3934/cpaa.2019097
[13]

Pingping Niu, Shuai Lu, Jin Cheng. On periodic parameter identification in stochastic differential equations. Inverse Problems & Imaging, 2019, 13 (3) : 513-543. doi: 10.3934/ipi.2019025
[14]

Yaozhong Hu, David Nualart, Xiaobin Sun, Yingchao Xie. Smoothness of density for stochastic differential equations with Markovian switching. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3615-3631. doi: 10.3934/dcdsb.2018307
[15]

Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103
[16]

Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115
[17]

Yayun Zheng, Xu Sun. Governing equations for Probability densities of stochastic differential equations with discrete time delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3615-3628. doi: 10.3934/dcdsb.2017182
[18]

Miroslav Bartušek. Existence of noncontinuable solutions of a system of differential equations. Conference Publications, 2009, 2009 (Special) : 54-59. doi: 10.3934/proc.2009.2009.54
[19]

Deena Schmidt, Janet Best, Mark S. Blumberg. Random graph and stochastic process contributions to network dynamics. Conference Publications, 2011, 2011 (Special) : 1279-1288. doi: 10.3934/proc.2011.2011.1279
[20]

Yuri Bakhtin. Lyapunov exponents for stochastic differential equations with infinite memory and application to stochastic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 697-709. doi: 10.3934/dcdsb.2006.6.697

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences