# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol., 2 (1975), 219-233. doi: 10.1007/BF00277151.  Google Scholar [6] A. Friedman, "Partial Differential Equations," Holt, Rinehart, and Winston, New York, 1969.  Google Scholar [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.  Google Scholar [8] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840. Springer, Berlin, 1981.  Google Scholar [9] P. Hess, "Periodic Parabolic Boundary Value Problems and Positivity," Longman Scientific & Technical, Harlow, UK, 1991.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] Yu. I. Lyubich, "Mathematical Structures in Population Genetics," Biomathematics, 22. Springer, Berlin. 1992. doi: 10.1007/978-3-642-76211-6.  Google Scholar [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.  Google Scholar [20] T. Nagylaki, Conditions for the existence of clines, Genetics, 80 (1975), 595-615. Google Scholar [21] T. Nagylaki, Clines with variable migration, Genetics, 83 (1976), 867-886.  Google Scholar [22] T. Nagylaki, Clines with asymmetric migration, Genetics, 88 (1978), 813-827. Google Scholar [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  Google Scholar [24] T. Nagylaki, "Introduction to Theoretical Population Genetics," Biomathematics, 21. Springer, Berlin, 1992. doi: 10.1007/978-3-642-76214-7.  Google Scholar [25] T. Nagylaki, The diffusion model for migration and selection in a dioecious population, J. Math. Biol., 34 (1996), 334-360.  Google Scholar [26] T. Nagylaki, Polymorphism in multiallelic migration-selection models with dominance, Theor. Popul. Biol., 75 (2009), 239-259. Google Scholar [27] T. Nagylaki, Clines with partial panmixia, Theor. Popul. Biol., 81 (2012), 45-68. Google Scholar [28] T. Nagylaki, Clines with partial panmixia in an unbounded unidimensional habitat, Theor. Popul. Biol., 82 (2012), 22-28. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol., 2 (1975), 219-233. doi: 10.1007/BF00277151.  Google Scholar [6] A. Friedman, "Partial Differential Equations," Holt, Rinehart, and Winston, New York, 1969.  Google Scholar [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.  Google Scholar [8] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840. Springer, Berlin, 1981.  Google Scholar [9] P. Hess, "Periodic Parabolic Boundary Value Problems and Positivity," Longman Scientific & Technical, Harlow, UK, 1991.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] Yu. I. Lyubich, "Mathematical Structures in Population Genetics," Biomathematics, 22. Springer, Berlin. 1992. doi: 10.1007/978-3-642-76211-6.  Google Scholar [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.  Google Scholar [20] T. Nagylaki, Conditions for the existence of clines, Genetics, 80 (1975), 595-615. Google Scholar [21] T. Nagylaki, Clines with variable migration, Genetics, 83 (1976), 867-886.  Google Scholar [22] T. Nagylaki, Clines with asymmetric migration, Genetics, 88 (1978), 813-827. Google Scholar [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  Google Scholar [24] T. Nagylaki, "Introduction to Theoretical Population Genetics," Biomathematics, 21. Springer, Berlin, 1992. doi: 10.1007/978-3-642-76214-7.  Google Scholar [25] T. Nagylaki, The diffusion model for migration and selection in a dioecious population, J. Math. Biol., 34 (1996), 334-360.  Google Scholar [26] T. Nagylaki, Polymorphism in multiallelic migration-selection models with dominance, Theor. Popul. Biol., 75 (2009), 239-259. Google Scholar [27] T. Nagylaki, Clines with partial panmixia, Theor. Popul. Biol., 81 (2012), 45-68. Google Scholar [28] T. Nagylaki, Clines with partial panmixia in an unbounded unidimensional habitat, Theor. Popul. Biol., 82 (2012), 22-28. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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