American Institute of Mathematical Sciences

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

An introduction to migration-selection PDE models

Yuan Lou 1, , Thomas Nagylaki 2, and  Wei-Ming Ni 3,

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.
Keywords: migration, selection, Partial differential equations, polymorphism, geographical variation., clines.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35K57, 92D10, 92D2.
Citation: Yuan Lou, Thomas Nagylaki, Wei-Ming Ni. An introduction to migration-selection PDE models. Discrete & Continuous Dynamical Systems - A, 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.   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.  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, (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.  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.  doi: 10.1007/BF00277151.  Google Scholar

[6]

A. Friedman, "Partial Differential Equations,", Holt, (1969).   Google Scholar

[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.  Google Scholar

[8]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981).   Google Scholar

[9]

P. Hess, "Periodic Parabolic Boundary Value Problems and Positivity,", Longman Scientific & Technical, (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.  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.  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, 17 (2012), 2771.  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.  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.  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.  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.  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.  doi: 10.3934/dcds.2010.27.643.  Google Scholar

[18]

Yu. I. Lyubich, "Mathematical Structures in Population Genetics,", Biomathematics, 22 (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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[24]

T. Nagylaki, "Introduction to Theoretical Population Genetics,", Biomathematics, 21 (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.   Google Scholar

[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.  Google Scholar

[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.  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.  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, 82 (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.   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.  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.  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.  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.  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.   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.  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, (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.  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.  doi: 10.1007/BF00277151.  Google Scholar

[6]

A. Friedman, "Partial Differential Equations,", Holt, (1969).   Google Scholar

[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.  Google Scholar

[8]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981).   Google Scholar

[9]

P. Hess, "Periodic Parabolic Boundary Value Problems and Positivity,", Longman Scientific & Technical, (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.  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.  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, 17 (2012), 2771.  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.  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.  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.  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.  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.  doi: 10.3934/dcds.2010.27.643.  Google Scholar

[18]

Yu. I. Lyubich, "Mathematical Structures in Population Genetics,", Biomathematics, 22 (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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[24]

T. Nagylaki, "Introduction to Theoretical Population Genetics,", Biomathematics, 21 (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.   Google Scholar

[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.  Google Scholar

[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.  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.  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, 82 (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.   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.  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.  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.  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.  doi: 10.1007/BF01453979.  Google Scholar

[1]

Yantao Wang, Linlin Su. Monotone and nonmonotone clines with partial panmixia across a geographical barrier. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020056
[2]

Linlin Su, Thomas Nagylaki. Clines with directional selection and partial panmixia in an unbounded unidimensional habitat. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1697-1741. doi: 10.3934/dcds.2015.35.1697
[3]

Reinhard Bürger. A survey of migration-selection models in population genetics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 883-959. doi: 10.3934/dcdsb.2014.19.883
[4]

Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481
[5]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703
[6]

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053
[7]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[8]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167
[9]

Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032
[10]

Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345
[11]

Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231
[12]

Enrique Zuazua. Controllability of partial differential equations and its semi-discrete approximations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 469-513. doi: 10.3934/dcds.2002.8.469
[13]

Paul Bracken. Connections of zero curvature and applications to nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1165-1179. doi: 10.3934/dcdss.2014.7.1165
[14]

Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131
[15]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[16]

Yuhki Hosoya. First-order partial differential equations and consumer theory. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1143-1167. doi: 10.3934/dcdss.2018065
[17]

Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031
[18]

Frédéric Gibou, Doron Levy, Carlos Cárdenas, Pingyu Liu, Arthur Boyer. Partial Differential Equations-Based Segmentation for Radiotherapy Treatment Planning. Mathematical Biosciences & Engineering, 2005, 2 (2) : 209-226. doi: 10.3934/mbe.2005.2.209
[19]

Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227
[20]

Seyedeh Marzieh Ghavidel, Wolfgang M. Ruess. Flow invariance for nonautonomous nonlinear partial differential delay equations. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2351-2369. doi: 10.3934/cpaa.2012.11.2351

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences