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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 and Continuous Dynamical Systems, 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 "Partial Differential Equations and Related Topics" (Program, Tulane Univ., New Orleans, La., 1974), Lecture Notes in Math., 446, Springer, Berlin, (1975), 5-49. [2] M. Bôcher, The smallest characteristic numbers in a certain exception case, Bull. Amer. Math. Soc., 21 (1914), 6-9. doi: 10.1090/S0002-9904-1914-02560-1. [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), 112-120. doi: 10.1016/0022-247X(80)90309-1. [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), 241-249. doi: 10.1007/BF00277153. [5] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. [6] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rat. Mech. Anal., 52 (1973), 161-180. [7] P. C. Fife and L. A. Peletier, Nonlinear diffusion in population genetics, Arch. Rat. Mech. Anal., 64 (1977), 93-109. doi: 10.1007/BF00280092. [8] P. C. Fife and L. A. Peletier, Clines induced by variable selection and migration, Proc. R. Soc. Lond. B., 214 (1981), 99-123. doi: 10.1098/rspb.1981.0084. [9] R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics., 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x. [10] W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol., 2 (1975), 219-233. doi: 10.1007/BF00277151. [11] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. [12] J. B. S. Haldane, The theory of a cline, J. Genet., 48 (1948), 277-284. doi: 10.1007/BF02986626. [13] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. [14] P. Hess, "Periodic-parabolic Boundary Value Problems and Positivity," Pitman Research Notes in Mathematics Series, 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. [15] Y. Lou and T. Nagylaki, A semilinear parabolic system for migration and selection in population genetics, J. Differential Equations., 181 (2002), 388-418. doi: 10.1006/jdeq.2001.4086. [16] Y. Lou and T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection in population genetics, J. Differential Equations., 204 (2004), 292-322. doi: 10.1016/j.jde.2004.01.009. [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), 643-655. doi: 10.3934/dcds.2010.27.643. [18] H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., 15 (1979), 401-454. doi: 10.2977/prims/1195188180. [19] T. Nagylaki, Conditions for the existence of clines, Genetics, 80 (1975), 595-615. [20] T. Nagylaki and Y. Lou, The dynamics of migration-selection models, in "Tutorials in Mathematical Bioscience. IV'' (ed. A. Friedman), Lecture Notes in Math., 1922, Springer, Berlin, (2008), 117-170. doi: 10.1007/978-3-540-74331-6_4. [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), 617-641. doi: 10.3934/dcds.2010.27.617. [22] W.-M. Ni, "The Mathematics of Diffusion," CBMS-NSF Regional Conference Series in Applied Mathematics, 82, SIAM, Philedelphia, PA, 2011. doi: 10.1137/1.9781611971972. [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), 819-851. doi: 10.1002/cpa.3160440705. [24] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, Indiana Univ. Math. J., 21 (): 979. [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), 1199-1228. doi: 10.1080/03605308308820300. [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. [27] M. Slatkin, Gene flow and selection in a cline, Genetics, 75 (1973), 733-756.

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##### References:
 [1] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve propogation, in "Partial Differential Equations and Related Topics" (Program, Tulane Univ., New Orleans, La., 1974), Lecture Notes in Math., 446, Springer, Berlin, (1975), 5-49. [2] M. Bôcher, The smallest characteristic numbers in a certain exception case, Bull. Amer. Math. Soc., 21 (1914), 6-9. doi: 10.1090/S0002-9904-1914-02560-1. [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), 112-120. doi: 10.1016/0022-247X(80)90309-1. [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), 241-249. doi: 10.1007/BF00277153. [5] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. [6] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rat. Mech. Anal., 52 (1973), 161-180. [7] P. C. Fife and L. A. Peletier, Nonlinear diffusion in population genetics, Arch. Rat. Mech. Anal., 64 (1977), 93-109. doi: 10.1007/BF00280092. [8] P. C. Fife and L. A. Peletier, Clines induced by variable selection and migration, Proc. R. Soc. Lond. B., 214 (1981), 99-123. doi: 10.1098/rspb.1981.0084. [9] R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics., 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x. [10] W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol., 2 (1975), 219-233. doi: 10.1007/BF00277151. [11] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. [12] J. B. S. Haldane, The theory of a cline, J. Genet., 48 (1948), 277-284. doi: 10.1007/BF02986626. [13] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. [14] P. Hess, "Periodic-parabolic Boundary Value Problems and Positivity," Pitman Research Notes in Mathematics Series, 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. [15] Y. Lou and T. Nagylaki, A semilinear parabolic system for migration and selection in population genetics, J. Differential Equations., 181 (2002), 388-418. doi: 10.1006/jdeq.2001.4086. [16] Y. Lou and T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection in population genetics, J. Differential Equations., 204 (2004), 292-322. doi: 10.1016/j.jde.2004.01.009. [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), 643-655. doi: 10.3934/dcds.2010.27.643. [18] H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., 15 (1979), 401-454. doi: 10.2977/prims/1195188180. [19] T. Nagylaki, Conditions for the existence of clines, Genetics, 80 (1975), 595-615. [20] T. Nagylaki and Y. Lou, The dynamics of migration-selection models, in "Tutorials in Mathematical Bioscience. IV'' (ed. A. Friedman), Lecture Notes in Math., 1922, Springer, Berlin, (2008), 117-170. doi: 10.1007/978-3-540-74331-6_4. [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), 617-641. doi: 10.3934/dcds.2010.27.617. [22] W.-M. Ni, "The Mathematics of Diffusion," CBMS-NSF Regional Conference Series in Applied Mathematics, 82, SIAM, Philedelphia, PA, 2011. doi: 10.1137/1.9781611971972. [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), 819-851. doi: 10.1002/cpa.3160440705. [24] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, Indiana Univ. Math. J., 21 (): 979. [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), 1199-1228. doi: 10.1080/03605308308820300. [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. [27] M. Slatkin, Gene flow and selection in a cline, Genetics, 75 (1973), 733-756.
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