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February  2014, 34(2): 821-841. doi: 10.3934/dcds.2014.34.821

A nonlinear diffusion problem arising in population genetics

1. 

Department of Mathematics, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China, China

2. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240

Received  September 2012 Revised  April 2013 Published  August 2013

In this paper we investigate a nonlinear diffusion equation with the Neumann boundary condition, which was proposed by Nagylaki in [19] to describe the evolution of two types of genes in population genetics. For such a model, we obtain the existence of nontrivial solutions and the limiting profile of such solutions as the diffusion rate $d\rightarrow0$ or $d\rightarrow\infty$. Our results show that as $d\rightarrow0$, the location of nontrivial solutions relative to trivial solutions plays a very important role for the existence and shape of limiting profile. In particular, an example is given to illustrate that the limiting profile does not exist for some nontrivial solutions. Moreover, to better understand the dynamics of this model, we analyze the stability and bifurcation of solutions. These conclusions provide a different angle to understand that obtained in [17,21].
Citation: Peng Zhou, Jiang Yu, Dongmei Xiao. A nonlinear diffusion problem arising in population genetics. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 821-841. doi: 10.3934/dcds.2014.34.821
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve propogation,, in, 446 (1975), 5.   Google Scholar

[2]

M. Bôcher, The smallest characteristic numbers in a certain exception case,, Bull. Amer. Math. Soc., 21 (1914), 6.  doi: 10.1090/S0002-9904-1914-02560-1.  Google Scholar

[3]

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

[4]

C. Conley, An application of Wazewski's method to a non-linear boundary value problem which arises in population genetics,, J. Math. Biol., 2 (1975), 241.  doi: 10.1007/BF00277153.  Google Scholar

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[6]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rat. Mech. Anal., 52 (1973), 161.   Google Scholar

[7]

P. C. Fife and L. A. Peletier, Nonlinear diffusion in population genetics,, Arch. Rat. Mech. Anal., 64 (1977), 93.  doi: 10.1007/BF00280092.  Google Scholar

[8]

P. C. Fife and L. A. Peletier, Clines induced by variable selection and migration,, Proc. R. Soc. Lond. B., 214 (1981), 99.  doi: 10.1098/rspb.1981.0084.  Google Scholar

[9]

R. A. Fisher, The wave of advance of advantageous genes,, Annals of Eugenics., 7 (1937), 355.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[10]

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

[11]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998).   Google Scholar

[12]

J. B. S. Haldane, The theory of a cline,, J. Genet., 48 (1948), 277.  doi: 10.1007/BF02986626.  Google Scholar

[13]

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

[14]

P. Hess, "Periodic-parabolic Boundary Value Problems and Positivity,", Pitman Research Notes in Mathematics Series, 247 (1991).   Google Scholar

[15]

Y. Lou and T. Nagylaki, A semilinear parabolic system for migration and selection in population genetics,, J. Differential Equations., 181 (2002), 388.  doi: 10.1006/jdeq.2001.4086.  Google Scholar

[16]

Y. Lou and T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection in population genetics,, J. Differential Equations., 204 (2004), 292.  doi: 10.1016/j.jde.2004.01.009.  Google Scholar

[17]

Y. Lou, W.-M. Ni and L. L. Su, An indefinite nonlinear diffusion problem in population genetics. II. Stability and Multiplicity,, Disc. Cont. Dyna. Syst., 27 (2010), 643.  doi: 10.3934/dcds.2010.27.643.  Google Scholar

[18]

H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations,, Publ. Res. Inst. Math. Sci., 15 (1979), 401.  doi: 10.2977/prims/1195188180.  Google Scholar

[19]

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

[20]

T. Nagylaki and Y. Lou, The dynamics of migration-selection models,, in, 1922 (2008), 117.  doi: 10.1007/978-3-540-74331-6_4.  Google Scholar

[21]

K. Nakashima, W.-M. Ni and L. L. Su, An indefinite nonlinear diffusion problem in population genetics. I. Existence and Limiting Profiles,, Disc. Cont. Dyna. Syst., 27 (2010), 617.  doi: 10.3934/dcds.2010.27.617.  Google Scholar

[22]

W.-M. Ni, "The Mathematics of Diffusion,", CBMS-NSF Regional Conference Series in Applied Mathematics, 82 (2011).  doi: 10.1137/1.9781611971972.  Google Scholar

[23]

W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem,, Comm. Pure Appl. Math., 44 (1991), 819.  doi: 10.1002/cpa.3160440705.  Google Scholar

[24]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, Indiana Univ. Math. J., 21 (): 979.   Google Scholar

[25]

S. Senn, On a nonlinear elliptic eigenvalue problem with Neumann boundary conditions, with an applicatiion to population genetics,, Comm. Partial Differential Equations, 8 (1983), 1199.  doi: 10.1080/03605308308820300.  Google Scholar

[26]

S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions,, Math. Ann., 258 (): 459.  doi: 10.1007/BF01453979.  Google Scholar

[27]

M. Slatkin, Gene flow and selection in a cline,, Genetics, 75 (1973), 733.   Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve propogation,, in, 446 (1975), 5.   Google Scholar

[2]

M. Bôcher, The smallest characteristic numbers in a certain exception case,, Bull. Amer. Math. Soc., 21 (1914), 6.  doi: 10.1090/S0002-9904-1914-02560-1.  Google Scholar

[3]

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

[4]

C. Conley, An application of Wazewski's method to a non-linear boundary value problem which arises in population genetics,, J. Math. Biol., 2 (1975), 241.  doi: 10.1007/BF00277153.  Google Scholar

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[6]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rat. Mech. Anal., 52 (1973), 161.   Google Scholar

[7]

P. C. Fife and L. A. Peletier, Nonlinear diffusion in population genetics,, Arch. Rat. Mech. Anal., 64 (1977), 93.  doi: 10.1007/BF00280092.  Google Scholar

[8]

P. C. Fife and L. A. Peletier, Clines induced by variable selection and migration,, Proc. R. Soc. Lond. B., 214 (1981), 99.  doi: 10.1098/rspb.1981.0084.  Google Scholar

[9]

R. A. Fisher, The wave of advance of advantageous genes,, Annals of Eugenics., 7 (1937), 355.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[10]

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

[11]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998).   Google Scholar

[12]

J. B. S. Haldane, The theory of a cline,, J. Genet., 48 (1948), 277.  doi: 10.1007/BF02986626.  Google Scholar

[13]

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

[14]

P. Hess, "Periodic-parabolic Boundary Value Problems and Positivity,", Pitman Research Notes in Mathematics Series, 247 (1991).   Google Scholar

[15]

Y. Lou and T. Nagylaki, A semilinear parabolic system for migration and selection in population genetics,, J. Differential Equations., 181 (2002), 388.  doi: 10.1006/jdeq.2001.4086.  Google Scholar

[16]

Y. Lou and T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection in population genetics,, J. Differential Equations., 204 (2004), 292.  doi: 10.1016/j.jde.2004.01.009.  Google Scholar

[17]

Y. Lou, W.-M. Ni and L. L. Su, An indefinite nonlinear diffusion problem in population genetics. II. Stability and Multiplicity,, Disc. Cont. Dyna. Syst., 27 (2010), 643.  doi: 10.3934/dcds.2010.27.643.  Google Scholar

[18]

H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations,, Publ. Res. Inst. Math. Sci., 15 (1979), 401.  doi: 10.2977/prims/1195188180.  Google Scholar

[19]

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

[20]

T. Nagylaki and Y. Lou, The dynamics of migration-selection models,, in, 1922 (2008), 117.  doi: 10.1007/978-3-540-74331-6_4.  Google Scholar

[21]

K. Nakashima, W.-M. Ni and L. L. Su, An indefinite nonlinear diffusion problem in population genetics. I. Existence and Limiting Profiles,, Disc. Cont. Dyna. Syst., 27 (2010), 617.  doi: 10.3934/dcds.2010.27.617.  Google Scholar

[22]

W.-M. Ni, "The Mathematics of Diffusion,", CBMS-NSF Regional Conference Series in Applied Mathematics, 82 (2011).  doi: 10.1137/1.9781611971972.  Google Scholar

[23]

W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem,, Comm. Pure Appl. Math., 44 (1991), 819.  doi: 10.1002/cpa.3160440705.  Google Scholar

[24]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, Indiana Univ. Math. J., 21 (): 979.   Google Scholar

[25]

S. Senn, On a nonlinear elliptic eigenvalue problem with Neumann boundary conditions, with an applicatiion to population genetics,, Comm. Partial Differential Equations, 8 (1983), 1199.  doi: 10.1080/03605308308820300.  Google Scholar

[26]

S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions,, Math. Ann., 258 (): 459.  doi: 10.1007/BF01453979.  Google Scholar

[27]

M. Slatkin, Gene flow and selection in a cline,, Genetics, 75 (1973), 733.   Google Scholar

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