October  2013, 33(10): 4349-4373. doi: 10.3934/dcds.2013.33.4349

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

Received  September 2012 Revised  March 2013 Published  April 2013

This expository article concerns a system of semilinear parabolic partial differential equations that describes the evolution of the gene frequencies at a single locus under the joint action of migration and selection. We shall review mathematical techniques suited for the models under investigation; discuss some of the main mathematical results, including most recent developments; and also propose some open problems.
Citation: Yuan Lou, Thomas Nagylaki, Wei-Ming Ni. An introduction to migration-selection PDE models. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4349-4373. doi: 10.3934/dcds.2013.33.4349
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), 201-207.

[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), 112-120. 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, John Wiley and Sons, Chichester, UK, 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), 3861-3869. doi: 10.3934/dcds.2012.32.3861.

[5]

W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol., 2 (1975), 219-233. doi: 10.1007/BF00277151.

[6]

A. Friedman, "Partial Differential Equations," Holt, Rinehart, and Winston, New York, 1969.

[7]

K. P. Hadeler, Diffusion in Fisher's population model, Rocky Mtn. J. Math., 11 (1981), 39-45. doi: 10.1216/RMJ-1981-11-1-39.

[8]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840. Springer, Berlin, 1981.

[9]

P. Hess, "Periodic Parabolic Boundary Value Problems and Positivity," Longman Scientific & Technical, Harlow, UK, 1991.

[10]

T. Kato, Superconvexity of the spectral radius, and convexity of the spectral bound and the type, Math. Z., 180 (1982), 265-273. doi: 10.1007/BF01318910.

[11]

J. F. C. Kingman, A mathematical problem in population genetics, Proceedings of the Cambridge Philosophical Society, 57 (1961), 574-582. doi: 10.1017/S0305004100035635.

[12]

S. Liang and Y. Lou, On the dependence of the population size on the dispersal rate, Special issue on "PDE Models from Biological Processess," Disc. Cont. Dynam. Sys. Series B, 17 (2012), 2771-2788. 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), 388-418. 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), 292-322. 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), 624-665. 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), 2367-2392. 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), 643-655. doi: 10.3934/dcds.2010.27.643.

[18]

Yu. I. Lyubich, "Mathematical Structures in Population Genetics," Biomathematics, 22. Springer, Berlin. 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), 66-75.

[20]

T. Nagylaki, Conditions for the existence of clines, Genetics, 80 (1975), 595-615.

[21]

T. Nagylaki, Clines with variable migration, Genetics, 83 (1976), 867-886.

[22]

T. Nagylaki, Clines with asymmetric migration, Genetics, 88 (1978), 813-827.

[23]

T. Nagylaki, The diffusion model for migration and selection, in "Some Mathematical Questions in Biology" (Ed. A. Hastings), Lecture Notes on Mathematics in the Life Sciences, 20. American Mathematical Society, Providence, RI, (1989), 55-75

[24]

T. Nagylaki, "Introduction to Theoretical Population Genetics," Biomathematics, 21. Springer, Berlin, 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), 334-360.

[26]

T. Nagylaki, Polymorphism in multiallelic migration-selection models with dominance, Theor. Popul. Biol., 75 (2009), 239-259.

[27]

T. Nagylaki, Clines with partial panmixia, Theor. Popul. Biol., 81 (2012), 45-68.

[28]

T. Nagylaki, Clines with partial panmixia in an unbounded unidimensional habitat, Theor. Popul. Biol., 82 (2012), 22-28.

[29]

T. Nagylaki and Y. Lou, Evolution at a multiallelic locus under migration and uniform selection, J. Math. Biology, 54 (2007), 787-796. doi: 10.1007/s00285-007-0077-7.

[30]

T. Nagylaki and Y. Lou, The dynamics of migration-selection models, in "Tutor. Math. Biosci." (Ed. A. Friedman), IV Evolution and Ecology, Lecture Notes in Mathematics 1922, Springer, Berlin, (2008), 117-170 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), 617-641. doi: 10.3934/dcds.2010.27.617.

[32]

W.-M. Ni, "The Mathematics of Diffusion," CBMS-NSF Regional Conference Series in Applied Mathematics 82, SIAM, Philedelphia, 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), 493-504.

[34]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," 2nd ed., Springer-Verlag, Berlin, 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), 99-103. 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), 1199-1228. 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), 459-470. 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), 201-207.

[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), 112-120. 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, John Wiley and Sons, Chichester, UK, 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), 3861-3869. doi: 10.3934/dcds.2012.32.3861.

[5]

W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol., 2 (1975), 219-233. doi: 10.1007/BF00277151.

[6]

A. Friedman, "Partial Differential Equations," Holt, Rinehart, and Winston, New York, 1969.

[7]

K. P. Hadeler, Diffusion in Fisher's population model, Rocky Mtn. J. Math., 11 (1981), 39-45. doi: 10.1216/RMJ-1981-11-1-39.

[8]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840. Springer, Berlin, 1981.

[9]

P. Hess, "Periodic Parabolic Boundary Value Problems and Positivity," Longman Scientific & Technical, Harlow, UK, 1991.

[10]

T. Kato, Superconvexity of the spectral radius, and convexity of the spectral bound and the type, Math. Z., 180 (1982), 265-273. doi: 10.1007/BF01318910.

[11]

J. F. C. Kingman, A mathematical problem in population genetics, Proceedings of the Cambridge Philosophical Society, 57 (1961), 574-582. doi: 10.1017/S0305004100035635.

[12]

S. Liang and Y. Lou, On the dependence of the population size on the dispersal rate, Special issue on "PDE Models from Biological Processess," Disc. Cont. Dynam. Sys. Series B, 17 (2012), 2771-2788. 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), 388-418. 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), 292-322. 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), 624-665. 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), 2367-2392. 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), 643-655. doi: 10.3934/dcds.2010.27.643.

[18]

Yu. I. Lyubich, "Mathematical Structures in Population Genetics," Biomathematics, 22. Springer, Berlin. 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), 66-75.

[20]

T. Nagylaki, Conditions for the existence of clines, Genetics, 80 (1975), 595-615.

[21]

T. Nagylaki, Clines with variable migration, Genetics, 83 (1976), 867-886.

[22]

T. Nagylaki, Clines with asymmetric migration, Genetics, 88 (1978), 813-827.

[23]

T. Nagylaki, The diffusion model for migration and selection, in "Some Mathematical Questions in Biology" (Ed. A. Hastings), Lecture Notes on Mathematics in the Life Sciences, 20. American Mathematical Society, Providence, RI, (1989), 55-75

[24]

T. Nagylaki, "Introduction to Theoretical Population Genetics," Biomathematics, 21. Springer, Berlin, 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), 334-360.

[26]

T. Nagylaki, Polymorphism in multiallelic migration-selection models with dominance, Theor. Popul. Biol., 75 (2009), 239-259.

[27]

T. Nagylaki, Clines with partial panmixia, Theor. Popul. Biol., 81 (2012), 45-68.

[28]

T. Nagylaki, Clines with partial panmixia in an unbounded unidimensional habitat, Theor. Popul. Biol., 82 (2012), 22-28.

[29]

T. Nagylaki and Y. Lou, Evolution at a multiallelic locus under migration and uniform selection, J. Math. Biology, 54 (2007), 787-796. doi: 10.1007/s00285-007-0077-7.

[30]

T. Nagylaki and Y. Lou, The dynamics of migration-selection models, in "Tutor. Math. Biosci." (Ed. A. Friedman), IV Evolution and Ecology, Lecture Notes in Mathematics 1922, Springer, Berlin, (2008), 117-170 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), 617-641. doi: 10.3934/dcds.2010.27.617.

[32]

W.-M. Ni, "The Mathematics of Diffusion," CBMS-NSF Regional Conference Series in Applied Mathematics 82, SIAM, Philedelphia, 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), 493-504.

[34]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," 2nd ed., Springer-Verlag, Berlin, 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), 99-103. 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), 1199-1228. 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), 459-470. doi: 10.1007/BF01453979.

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