Indefinite nonlinear diffusion problem in population genetics
Indefinite nonlinear diffusion problem in population genetics
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DOI：
10.3934/dcds.2020169
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
K. Nakashima
中科院分区：
文献类型：
--
作者：
K. Nakashima
We study the following Neumann problem in one dimension, \begin{document}$ \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.$\end{document} where \begin{document}$ g $\end{document} changes sign in \begin{document}$ (0, 1) $\end{document} . This equation models the "complete dominance" case in population genetics of two alleles. It is known that this equation has a nontrivial stable steady state \begin{document}$ U_d $\end{document} for \begin{document}$ d $\end{document} sufficiently small. We show that \begin{document}$ U_d $\end{document} is a unique nontrivial steady state under a condition \begin{document}$ \int_{0}^1\, g(x)\, dx\geq 0 $\end{document} and some other additional condition.
We study the following Neumann problem in one dimension, \begin{document}$ \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.$\end{document} where \begin{document}$ g $\end{document} changes sign in \begin{document}$ (0, 1) $\end{document} . This equation models the "complete dominance" case in population genetics of two alleles. It is known that this equation has a nontrivial stable steady state \begin{document}$ U_d $\end{document} for \begin{document}$ d $\end{document} sufficiently small. We show that \begin{document}$ U_d $\end{document} is a unique nontrivial steady state under a condition \begin{document}$ \int_{0}^1\, g(x)\, dx\geq 0 $\end{document} and some other additional condition.