Derivation of Several SDE Systems in One- and Two-Locus Population Genetics

References

• Allen, E.J. 2007. Modeling With Itô Stochastic Differential Equations. Springer, Dordrecht.
   Google Scholar
• Allen, E.J. 2009. Derivation of stochastic partial differential equations for size- and age-structured populations. Journal of Biological Dynamics 3:73–86.
   PubMedGoogle Scholar
• Allen, E.J., Allen, L.J.S., Arciniega, A., Greenwood, P.E. 2008. Construction of equivalent stochastic differential equation models. Stochastic Analysis and Applications 26:274–297.
   Web of Science ®Google Scholar
• Allen, L.J.S. 2011. An Introduction to Stochastic Processes with Applications to Biology. CRC Press, Boca Raton, FL.
   Google Scholar
• Blythe, R.A., McKane, A.J. 2007. Stochastic models of evolution in genetics, ecology and linguistics. Journal of Statistical Mechanics: Theory and Experiment P07018.
   Web of Science ®Google Scholar
• Choi, W., Choi, D-H. The application of stochastic differential equations to population genetic model. Bulletin of the Korean Mathematical Society 40:677–683.
   Google Scholar
• Deelstra, G., Delbaen, F. 1998. Convergence of discretized stochastic (interest rate) processes with stochastic drift term. Applied Stochastic Models Data Analysis 14:77–84.
   Google Scholar
• Dogan, E. 2011. Investigation of stochastic reaction-diffusion partial differential equations and of consistent stochastic differential equation models for one-locus and tow-loci population genetics. PhD thesis, Texas Tech University.
   Google Scholar
• Ethier, S.N., Kurtz, T.G. 1992. On the stationary distribution of the neutral diffusion model in population genetics. Annals of Applied Probability 2:24–35.
   Google Scholar
• Gillespie, D.T. 1977. Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry. 81:2340–2361.
   Web of Science ®Google Scholar
• Hedrick, P.W. 2000. Genetics of Populations. Jones and Bartlett, Boston.
   Google Scholar
• Kahl, C., and Schurz, H. 2006. Balanced milstein methods for ordinary sdesMonte Carlo Method and Applications 12:143–170.
   Google Scholar
• Kermany, A.R.R., Zhou, X., Hickey, D.A. 2008. Joint stationary moments of a two-island diffusion model of population subdivision. Theoretical Population Biology 74:226–232.
   Web of Science ®Google Scholar
• Kimura, H. 1994. Population Genetics, Molecular Evolution, and the Neutral Theory, Selected Papers.The University of Chicago Press, Chicago.
   Google Scholar
• Kloeden, P.E., and Platen, E. 1992. Numerical Solution of Stochastic Differential Equations. Springer-Verlag, New York.
   Google Scholar
• Kloeden, P.E., Platen, E., and Schurz, H. 1994. Numerical Solution of SDE Through Computer Experiments. Springer, Berlin.
   Google Scholar
• Kurtz, T.G. 1971. Limit theorems for sequences of jump markov processes approximating ordinary differential equations. Journal of Applied Probability 8:344–356.
   Web of Science ®Google Scholar
• Langevin, P. 1908. Sur la théorie du mouvement brownien. Comptes-rendus de l’Académie des Sciences 146:530–533.
   Google Scholar
• Maruyama, T. 1977. Stochastic Problems in Population Genetics. Lecture Notes in Biomathematics 17.Springer-Verlag, New York.
   Google Scholar
• Pierce, A.B., 2008. Using Genetics: A Conceptual Approach. W. H. Freeman, New York.
   Google Scholar
• Rice, S.H. 2004. Evolutionary Theory: Mathematical and Conceptual Foundations. Sinauer Associates, Sunderland, MA.
   Google Scholar
• Richter-Dyn, N., Goel, N. 1974. On the extiction of colonizing species. Theoretical Population Biology 3:406–433.
   Web of Science ®Google Scholar
• M., Ridley. 1996. Evolution. Blackwell Science, Oxan, UK.
   Google Scholar
• Russell, P.J., Wolfe, S. L., Hertz, P. E., Starr, C., and McMillan, B. 2008. Biology: The Dynamic Science. Thomson Brooks/Cole, Belmont, CA.
   Google Scholar
• Sacker, R.J., Bremen, H.F. 2003. A new approach to cycling in a 2-locus 2-allele genetic model. Journal of Difference Equations and Applications 9:441–448.
   Web of Science ®Google Scholar
• Shiga, T. 1981. Diffusion processes in population genetics. Journal of Mathematics of Kyoto University 21:133–151.
   Web of Science ®Google Scholar
• Tian, T., Burrage, K., Burrage, P.M., Carletti, M. 2007. Stochastic delay differential equations for genetic regulatory networks. Journal of Computational and Applied Mathematics 205:696–707.
   Web of Science ®Google Scholar
• Walsh, J.B. 1986. An Introduction to Stochastic Partial Differential Equations. Lecture Notes in Mathematics, Vol. 1180, pp. 265–439. . Springer-Verlag, Berlin.
   Google Scholar