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Indefinite nonlinear diffusion problem in population genetics
Tokyo University of Marine Science and Technology, 4-5-7 Kounan, Minato-ku, Tokyo, 108-8477, Japan |
$ \left\{ {\begin{array}{*{20}{l}}\begin{array}{l}{u_t} = du'' + g(x){u^2}(1 - u)\quad {\rm{in}}\quad (0,1) \times (0,\infty ),\;\\0 \le u \le 1\quad {\rm{in}}\quad (0,1) \times (0,\infty ),\;\\u'(0,t) = u'(1,t) = 0\quad {\rm{in}}\quad (0,\infty ),\end{array}\end{array}} \right.$ |
$ g $ |
$ (0, 1) $ |
$ U_d $ |
$ d $ |
$ U_d $ |
$ \int_{0}^1\, g(x)\, dx\geq 0 $ |
References:
[1] |
Y. Lou and T. Nagylaki,
A semilinear parabolic system for migration and selection in population gentics, J. Differential Equations, 181 (2002), 388-418.
doi: 10.1006/jdeq.2001.4086. |
[2] |
Y. Lou, T. Nagylaki and W.-M. Ni,
An introduction to migration-selection PDE models, Discrete Contin. Dyn. Syst., 33 (2013), 4349-4373.
doi: 10.3934/dcds.2013.33.4349. |
[3] |
Y. Lou, W.-M. Ni and L. Su,
An indefinite nonlinear diffusion problem in population genetics. Ⅱ. Stability and multiplicity, Discrete Contin. Dyn. Syst., 27 (2010), 643-655.
doi: 10.3934/dcds.2010.27.643. |
[4] |
T. Nagylaki, Conditions for the existence of clines, Genetics, 80 (1975), 595-615. Google Scholar |
[5] |
T. Nagylaki, Polymorphism in multiallelic migration-selection models with dominance, Theoret. Population Biol., 75 (2009), 239-259.
doi: 10.1016/j.tpb.2009.01.004. |
[6] |
T. Nagylaki and Y. Lou, The dynamics of migration-selection models, in "Tutorials in Mathematical Biosciences. IV, Lecture Notes in Math., 1922, Springer, Berlin, 2008,117–170.
doi: 10.1007/978-3-540-74331-6_4. |
[7] |
K. Nakashima, W.-M. Ni and L. Su,
An indefinite nonlinear diffusion problem in population genetics. Ⅰ. Existence, Discrete Contin. Dyn. Syst., 27 (2010), 617-641.
doi: 10.3934/dcds.2010.27.617. |
[8] |
K. Nakashima,
The uniqueness of indefinite nonlinear diffusion problem in population genetics, part Ⅰ, J. Differential Equations, 261 (2016), 6233-6282.
doi: 10.1016/j.jde.2016.08.041. |
[9] |
K. Nakashima,
The uniqueness of an indefinite nonlinear diffusion problem in population genetics, part Ⅱ, J. Differential Equations, 264 (2018), 1946-1983.
doi: 10.1016/j.jde.2017.10.014. |
[10] |
K. Nakashima, Multiple existence of indefinite nonlinear diffusion problem in population genetics, J. Differential Equations, work in progress.
doi: 10.1016/j.jde.2019.11.082. |
show all references
References:
[1] |
Y. Lou and T. Nagylaki,
A semilinear parabolic system for migration and selection in population gentics, J. Differential Equations, 181 (2002), 388-418.
doi: 10.1006/jdeq.2001.4086. |
[2] |
Y. Lou, T. Nagylaki and W.-M. Ni,
An introduction to migration-selection PDE models, Discrete Contin. Dyn. Syst., 33 (2013), 4349-4373.
doi: 10.3934/dcds.2013.33.4349. |
[3] |
Y. Lou, W.-M. Ni and L. Su,
An indefinite nonlinear diffusion problem in population genetics. Ⅱ. Stability and multiplicity, Discrete Contin. Dyn. Syst., 27 (2010), 643-655.
doi: 10.3934/dcds.2010.27.643. |
[4] |
T. Nagylaki, Conditions for the existence of clines, Genetics, 80 (1975), 595-615. Google Scholar |
[5] |
T. Nagylaki, Polymorphism in multiallelic migration-selection models with dominance, Theoret. Population Biol., 75 (2009), 239-259.
doi: 10.1016/j.tpb.2009.01.004. |
[6] |
T. Nagylaki and Y. Lou, The dynamics of migration-selection models, in "Tutorials in Mathematical Biosciences. IV, Lecture Notes in Math., 1922, Springer, Berlin, 2008,117–170.
doi: 10.1007/978-3-540-74331-6_4. |
[7] |
K. Nakashima, W.-M. Ni and L. Su,
An indefinite nonlinear diffusion problem in population genetics. Ⅰ. Existence, Discrete Contin. Dyn. Syst., 27 (2010), 617-641.
doi: 10.3934/dcds.2010.27.617. |
[8] |
K. Nakashima,
The uniqueness of indefinite nonlinear diffusion problem in population genetics, part Ⅰ, J. Differential Equations, 261 (2016), 6233-6282.
doi: 10.1016/j.jde.2016.08.041. |
[9] |
K. Nakashima,
The uniqueness of an indefinite nonlinear diffusion problem in population genetics, part Ⅱ, J. Differential Equations, 264 (2018), 1946-1983.
doi: 10.1016/j.jde.2017.10.014. |
[10] |
K. Nakashima, Multiple existence of indefinite nonlinear diffusion problem in population genetics, J. Differential Equations, work in progress.
doi: 10.1016/j.jde.2019.11.082. |
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