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An introduction to migration-selection PDE models
1. | Department of Mathematics, Mathematical Bioscience Institute, Ohio State University, Columbus, Ohio 43210 |
2. | Department of Ecology and Evolution, University of Chicago, 1101 East 57th Street, Chicago, IL 60637, United States |
3. | Center for Partial Differential Equations, East China Normal University, Minhang, Shanghai, 200241 |
References:
[1] |
K. J. Brown and P. Hess, Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem,, Differential and Integral Equations, 3 (1990), 201.
|
[2] |
K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function,, J. Math. Anal. Appl., 75 (1980), 112.
doi: 10.1016/0022-247X(80)90309-1. |
[3] |
R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations,", Series in Mathematical and Computational Biology, (2003).
doi: 10.1002/0470871296. |
[4] |
E. N. Dancer, On domain perturbation for super-linear Neumann problems and a question of Y. Lou, W.-M. Ni and L. Su,, Discrete Contin. Dyn. Syst., 32 (2012), 3861.
doi: 10.3934/dcds.2012.32.3861. |
[5] |
W. H. Fleming, A selection-migration model in population genetics,, J. Math. Biol., 2 (1975), 219.
doi: 10.1007/BF00277151. |
[6] |
A. Friedman, "Partial Differential Equations,", Holt, (1969).
|
[7] |
K. P. Hadeler, Diffusion in Fisher's population model,, Rocky Mtn. J. Math., 11 (1981), 39.
doi: 10.1216/RMJ-1981-11-1-39. |
[8] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981).
|
[9] |
P. Hess, "Periodic Parabolic Boundary Value Problems and Positivity,", Longman Scientific & Technical, (1991).
|
[10] |
T. Kato, Superconvexity of the spectral radius, and convexity of the spectral bound and the type,, Math. Z., 180 (1982), 265.
doi: 10.1007/BF01318910. |
[11] |
J. F. C. Kingman, A mathematical problem in population genetics,, Proceedings of the Cambridge Philosophical Society, 57 (1961), 574.
doi: 10.1017/S0305004100035635. |
[12] |
S. Liang and Y. Lou, On the dependence of the population size on the dispersal rate,, Special issue on, 17 (2012), 2771.
doi: 10.3934/dcdsb.2012.17.2771. |
[13] |
Y. Lou and T. Nagylaki, A semilinear parabolic system for migration and selection in population genetics,, J. Diff. Eqs., 181 (2002), 388.
doi: 10.1006/jdeq.2001.4086. |
[14] |
Y. Lou and T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection in population genetics,, J. Diff. Eqs., 204 (2004), 292.
doi: 10.1016/j.jde.2004.01.009. |
[15] |
Y. Lou and T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection without dominance,, J. Diff. Eqs., 225 (2006), 624.
doi: 10.1016/j.jde.2006.01.012. |
[16] |
L. Lou, T. Nagylaki and L. Su, An Integro-PDE model from population genetics,, Journal of Differential Equations, 254 (2013), 2367.
doi: 10.1016/j.jde.2012.12.006. |
[17] |
L. Lou, W.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity,, Disc. Cont. Dynam. Sys. Series A, 27 (2010), 643.
doi: 10.3934/dcds.2010.27.643. |
[18] |
Yu. I. Lyubich, "Mathematical Structures in Population Genetics,", Biomathematics, 22 (1992).
doi: 10.1007/978-3-642-76211-6. |
[19] |
Yu. I. Lyubich, G. D. Maistrovskii and Yu. G. Ol'khovskii, Selection-induced convergence to equilibrium in a single-locus autosomal population,, Probl. Inf. Transm., 16 (1980), 66.
|
[20] |
T. Nagylaki, Conditions for the existence of clines,, Genetics, 80 (1975), 595. Google Scholar |
[21] |
T. Nagylaki, Clines with variable migration,, Genetics, 83 (1976), 867.
|
[22] |
T. Nagylaki, Clines with asymmetric migration,, Genetics, 88 (1978), 813. Google Scholar |
[23] |
T. Nagylaki, The diffusion model for migration and selection,, in, 20 (1989), 55.
|
[24] |
T. Nagylaki, "Introduction to Theoretical Population Genetics,", Biomathematics, 21 (1992).
doi: 10.1007/978-3-642-76214-7. |
[25] |
T. Nagylaki, The diffusion model for migration and selection in a dioecious population,, J. Math. Biol., 34 (1996), 334.
|
[26] |
T. Nagylaki, Polymorphism in multiallelic migration-selection models with dominance,, Theor. Popul. Biol., 75 (2009), 239. Google Scholar |
[27] |
T. Nagylaki, Clines with partial panmixia,, Theor. Popul. Biol., 81 (2012), 45. Google Scholar |
[28] |
T. Nagylaki, Clines with partial panmixia in an unbounded unidimensional habitat,, Theor. Popul. Biol., 82 (2012), 22. Google Scholar |
[29] |
T. Nagylaki and Y. Lou, Evolution at a multiallelic locus under migration and uniform selection,, J. Math. Biology, 54 (2007), 787.
doi: 10.1007/s00285-007-0077-7. |
[30] |
T. Nagylaki and Y. Lou, The dynamics of migration-selection models,, in, IV (2008), 117.
doi: 10.1007/978-3-540-74331-6_4. |
[31] |
K. Nakashima, W.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics, I. Existence and limiting profiles,, Disc. Cont. Dynam. Sys. Series A, 27 (2010), 617.
doi: 10.3934/dcds.2010.27.617. |
[32] |
W.-M. Ni, "The Mathematics of Diffusion,", CBMS-NSF Regional Conference Series in Applied Mathematics 82, 82 (2011).
doi: 10.1137/1.9781611971972. |
[33] |
J. Piálek and N. H. Barton, The spread of an advantageous allele across a barrier: The effect of random drift and selection against heterozygotes,, Genetics, 145 (1997), 493. Google Scholar |
[34] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", 2nd ed., (1984).
doi: 10.1007/978-1-4612-1110-5_15. |
[35] |
R. Redlinger, Über die $C^2$-Kompaktheit der Bahn der Lösungen semilinearer parabolischer systeme,, Proc. Roy. Soc. Edinb. A, 93 (1983), 99.
doi: 10.1017/S0308210500031693. |
[36] |
S. Senn, On a nonlinear elliptic eigenvalue problem with Neumann boundary conditions, with an application to population genetics,, Comm. Partial Differential Equations, 8 (1983), 1199.
doi: 10.1080/03605308308820300. |
[37] |
S. Senn and P. Hess, On positive solutions of a linear elliptic boundary value problem with Neumann boundary conditions,, Math. Ann., 258 (1982), 459.
doi: 10.1007/BF01453979. |
show all references
References:
[1] |
K. J. Brown and P. Hess, Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem,, Differential and Integral Equations, 3 (1990), 201.
|
[2] |
K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function,, J. Math. Anal. Appl., 75 (1980), 112.
doi: 10.1016/0022-247X(80)90309-1. |
[3] |
R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations,", Series in Mathematical and Computational Biology, (2003).
doi: 10.1002/0470871296. |
[4] |
E. N. Dancer, On domain perturbation for super-linear Neumann problems and a question of Y. Lou, W.-M. Ni and L. Su,, Discrete Contin. Dyn. Syst., 32 (2012), 3861.
doi: 10.3934/dcds.2012.32.3861. |
[5] |
W. H. Fleming, A selection-migration model in population genetics,, J. Math. Biol., 2 (1975), 219.
doi: 10.1007/BF00277151. |
[6] |
A. Friedman, "Partial Differential Equations,", Holt, (1969).
|
[7] |
K. P. Hadeler, Diffusion in Fisher's population model,, Rocky Mtn. J. Math., 11 (1981), 39.
doi: 10.1216/RMJ-1981-11-1-39. |
[8] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981).
|
[9] |
P. Hess, "Periodic Parabolic Boundary Value Problems and Positivity,", Longman Scientific & Technical, (1991).
|
[10] |
T. Kato, Superconvexity of the spectral radius, and convexity of the spectral bound and the type,, Math. Z., 180 (1982), 265.
doi: 10.1007/BF01318910. |
[11] |
J. F. C. Kingman, A mathematical problem in population genetics,, Proceedings of the Cambridge Philosophical Society, 57 (1961), 574.
doi: 10.1017/S0305004100035635. |
[12] |
S. Liang and Y. Lou, On the dependence of the population size on the dispersal rate,, Special issue on, 17 (2012), 2771.
doi: 10.3934/dcdsb.2012.17.2771. |
[13] |
Y. Lou and T. Nagylaki, A semilinear parabolic system for migration and selection in population genetics,, J. Diff. Eqs., 181 (2002), 388.
doi: 10.1006/jdeq.2001.4086. |
[14] |
Y. Lou and T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection in population genetics,, J. Diff. Eqs., 204 (2004), 292.
doi: 10.1016/j.jde.2004.01.009. |
[15] |
Y. Lou and T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection without dominance,, J. Diff. Eqs., 225 (2006), 624.
doi: 10.1016/j.jde.2006.01.012. |
[16] |
L. Lou, T. Nagylaki and L. Su, An Integro-PDE model from population genetics,, Journal of Differential Equations, 254 (2013), 2367.
doi: 10.1016/j.jde.2012.12.006. |
[17] |
L. Lou, W.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity,, Disc. Cont. Dynam. Sys. Series A, 27 (2010), 643.
doi: 10.3934/dcds.2010.27.643. |
[18] |
Yu. I. Lyubich, "Mathematical Structures in Population Genetics,", Biomathematics, 22 (1992).
doi: 10.1007/978-3-642-76211-6. |
[19] |
Yu. I. Lyubich, G. D. Maistrovskii and Yu. G. Ol'khovskii, Selection-induced convergence to equilibrium in a single-locus autosomal population,, Probl. Inf. Transm., 16 (1980), 66.
|
[20] |
T. Nagylaki, Conditions for the existence of clines,, Genetics, 80 (1975), 595. Google Scholar |
[21] |
T. Nagylaki, Clines with variable migration,, Genetics, 83 (1976), 867.
|
[22] |
T. Nagylaki, Clines with asymmetric migration,, Genetics, 88 (1978), 813. Google Scholar |
[23] |
T. Nagylaki, The diffusion model for migration and selection,, in, 20 (1989), 55.
|
[24] |
T. Nagylaki, "Introduction to Theoretical Population Genetics,", Biomathematics, 21 (1992).
doi: 10.1007/978-3-642-76214-7. |
[25] |
T. Nagylaki, The diffusion model for migration and selection in a dioecious population,, J. Math. Biol., 34 (1996), 334.
|
[26] |
T. Nagylaki, Polymorphism in multiallelic migration-selection models with dominance,, Theor. Popul. Biol., 75 (2009), 239. Google Scholar |
[27] |
T. Nagylaki, Clines with partial panmixia,, Theor. Popul. Biol., 81 (2012), 45. Google Scholar |
[28] |
T. Nagylaki, Clines with partial panmixia in an unbounded unidimensional habitat,, Theor. Popul. Biol., 82 (2012), 22. Google Scholar |
[29] |
T. Nagylaki and Y. Lou, Evolution at a multiallelic locus under migration and uniform selection,, J. Math. Biology, 54 (2007), 787.
doi: 10.1007/s00285-007-0077-7. |
[30] |
T. Nagylaki and Y. Lou, The dynamics of migration-selection models,, in, IV (2008), 117.
doi: 10.1007/978-3-540-74331-6_4. |
[31] |
K. Nakashima, W.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics, I. Existence and limiting profiles,, Disc. Cont. Dynam. Sys. Series A, 27 (2010), 617.
doi: 10.3934/dcds.2010.27.617. |
[32] |
W.-M. Ni, "The Mathematics of Diffusion,", CBMS-NSF Regional Conference Series in Applied Mathematics 82, 82 (2011).
doi: 10.1137/1.9781611971972. |
[33] |
J. Piálek and N. H. Barton, The spread of an advantageous allele across a barrier: The effect of random drift and selection against heterozygotes,, Genetics, 145 (1997), 493. Google Scholar |
[34] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", 2nd ed., (1984).
doi: 10.1007/978-1-4612-1110-5_15. |
[35] |
R. Redlinger, Über die $C^2$-Kompaktheit der Bahn der Lösungen semilinearer parabolischer systeme,, Proc. Roy. Soc. Edinb. A, 93 (1983), 99.
doi: 10.1017/S0308210500031693. |
[36] |
S. Senn, On a nonlinear elliptic eigenvalue problem with Neumann boundary conditions, with an application to population genetics,, Comm. Partial Differential Equations, 8 (1983), 1199.
doi: 10.1080/03605308308820300. |
[37] |
S. Senn and P. Hess, On positive solutions of a linear elliptic boundary value problem with Neumann boundary conditions,, Math. Ann., 258 (1982), 459.
doi: 10.1007/BF01453979. |
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