An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity
An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity
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DOI：
10.3934/dcds.2010.27.643
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发表时间：
2010-02
影响因子：
1.1
通讯作者：
Y. Lou;W. Ni;Linlin Su
中科院分区：
文献类型：
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作者：
Y. Lou;W. Ni;Linlin Su
研究了有界光滑生境$\Omega$中具有两个等位基因$A {1}$和$A {2}$的遗传模型。等位基因$A_{1}$的频率$u$，在迁移和选择的共同影响下，服从抛物型方程$ u_{t}=d\Delta u+g（x）f（u），~0\leq u\leq 1 $ in Ω ×（0，∞），$ \frac{\partial u}{\partial\nu}=0 $ on Ω ×（0，∞），其中$\Delta$表示拉普拉斯算子，$g$可以在$\Omega$中改变符号，并且对于$s\in（0，1）$，$f（0）=f（1）=0$，$f（s）>0$。我们的主要结果包括稳定性/不稳定的平凡定态$u\equiv 0$和$u\equiv 1$，和多重的非平凡定态。这是我们工作的延续[12]。特别是Nagylaki和Lou [11，p. 152]的猜想已经基本解决。对Dirichlet和Robin边值问题也得到了类似的结果。
We study a genetic model with two alleles $A_{1}$ and $A_{2}$ in a bounded smooth habitat $\Omega$. The frequency $u$ of the allele $A_{1}$, under the combined influence of migration and selection, obeys a parabolic equation of the type $ u_{t}=d\Delta u+g(x)f(u),~0\leq u\leq 1 $ in Ω × (0, ∞), $ \frac{\partial u}{\partial\nu}=0 $ on ∂ Ω × (0, ∞), where $\Delta$ denotes the Laplace operator, $g$ may change sign in $\Omega$, and $f(0)=f(1)=0$, $f(s)>0$ for $s\in(0,1)$. Our main results include stability/instability of the trivial steady states $u\equiv 0$ and $u\equiv 1$, and the multiplicity of nontrivial steady states. This is a continuation of our work [12]. In particular, the conjecture of Nagylaki and Lou [11, p. 152] has been largely resolved. Similar results are obtained for Dirichlet and Robin boundary value problems as well.