A non-standard evolution problem arising in population genetics

Communications in Mathematical Sciences

Volumn 7, Issue 2, 2009, Pages 489-502

  Chalub, Fabio A C C ^a   Souza, Max O ^b  

^a UNIVERSIDADE NOVA DE LISBOA   (Portugal)

^b FLUMINENSE FEDERAL UNIVERSITY   (Brazil)

Author keywords

Boundary coupled weak solutions; Degenerate parabolic equations; Evolutionary dynamics; Gene fixation

Indexed keywords

EID: 70350443026     PISSN: 15396746     EISSN: None     Source Type: Journal    
DOI: 10.4310/CMS.2009.v7.n2.a11     Document Type: Article

References (28)

1. J.P. Bartier, J. Dolbeault, R. Illner and M. Kowalczyk, A qualitative study of linear drift-diffusion equations with time-dependent or degenerate coefficients, Math. Models Methods Appl. Sci., 17(3), 327-362, 2007.
2. J.W. Bruce and P.J. Giblin, Curves and Singularities, Cambridge University Press, Cambridge, second edition, 1992.
3. R. Bürger, The Mathematical Theory of Selection, Recombination, and Mutation, John Wiley & Sons Ltd., Chichester, 2000.
4. R.W. Carrol and R. Schowalter, Singular and Degenerate Cauchy Problems, Academic Press, 1976.
5. F.A.C.C. Chalub and M.O. Souza, The Continuous Limit of the Moran Process and the Diffusion of Mutant Genes in Infinite Populations, Arxiv, math/0602530, preprint, 2006.
6. F.A.C.C. Chalub and M.O. Souza, Discrete versus continuous models in evolutionary dynamics: from simple to simpler - and even simpler - models, Math. Comput. Modelling, 47(7-8), 743-754, 2008.
7. F.A.C.C. Chalub and M.O. Souza, From discrete to continuous evolution models: a unifying approach to drift-diffusion and replicator dynamics, Arxiv, q-bio/0811.0203, preprint, 2008.
8. E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw Hill, 1955.
9. E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, 1993.
10. J. Dolbeault and R. Illner, Entropy methods for kinetic models of traffic flow, Commun. Math. Sci., 1, 409-421, 2003.
11. L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 19, 1998.
12. W.J. Ewens, Mathematical Population Genetics. I, Interdisciplinary Applied Mathematics, Springer-Verlag, New York, second edition, 27, 2004.
13. W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Annals Of Mathematics, 55(3), 468-519, 1952.
14. W. Feller, Diffusion processes in one dimension, Transactions of The American Mathematical Society, 77, 1-31, 1954.
15. A. Friedman, Stochastic Differential Equations and Applications, Vol. 2, Academic Press, Harcourt Brace Jovanovich Publishers, New York, Probability and Mathematical Statistics, 1976.
16. L. Hörmander, The analysis of linear partial differential operators. I, Classics in Mathematics. Springer-Verlag, Berlin, 2003. Distribution theory and Fourier analysis, Reprint of the second edition(1990), Springer, Berlin; MR1065993 (91m:35001a).
17. M. Kimura, On the probability of fixation of mutant genes in a population, Genetics, 47, 713-719, 1962.
18. G.M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1996.
19. M.C. Lopes Filho, H.J. Nussenzveig Lopes and Z. Xin, Existence of vortex sheets with reflection symmetry in two space dimensions, Arch. Ration. Mech. Anal., 158(3), 235-257, 2001.
20. M.C. Lopes Filho, H.J. Nussenzveig Lopes and Z. Xin, Vortex sheets with reflection symmetry in exterior domains, J. Differ. Equs., 229(1), 154-171, 2006.
21. P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, Journal Of Evolution Equations, 6(2), 325-362, May 2006.
22. A.J. McKane and D. Waxman, Singular solutions of the diffusion equation of population genetics, J. Theoret. Biol., 247(4), 849-858, 2007.
23. P.A.P. Moran, The Statistical Process of Evolutionary Theory, Clarendon Press, Oxford, 1962.
24. F.W.J. Olver, Asymptotics and Special Functions, Comput. Sci. Appl. Math., Academic Press, New York and London, 1974.
25. J.D. Pryce, Numerical solution of Sturm-Liouville problems, Monographs on Numerical Analysis, The Clarendon Press Oxford University Press, Oxford Science Publications, New York, 1993.
26. L. Tartar, An introduction to Sobolev spaces and interpolation spaces, Lecture Notes of the Unione Matematica Italiana, Springer, Berlin, 3, 2007.
27. M.E. Taylor, Partial differential equations. I, Appl. Math. Sci., Springer-Verlag, New York, 115, 1996.
28. A. Traulsen, J.C. Claussen and C. Hauert, Evolutionary dynamics: from finite to infinite populations, Phys. Rev. E, 74, 011901, 2006.