Non-local effects in an integro-PDE model from population genetics
Non-local effects in an integro-PDE model from population genetics
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DOI：
10.1017/s0956792515000601
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发表时间：
2015-11
影响因子：
1.9
通讯作者：
F. Li;K. Nakashima;W. Ni
中科院分区：
文献类型：
--
作者：
F. Li;K. Nakashima;W. Ni
本文研究了如下非局部问题：\开始{equation*} \开始{cases} \displaystyle u_t=d{1\over\rho}\nabla\cdot（\rho V\nabla u）+B（\bar{u}-u）+ g（x）u^2（1-u）&\displaystyle \quad \textrm{in} \;\Omega\times（0，\infty），\\[3pt] \displaystyle 0\leq u\leq 1 & \quad\displaystyle \textrm{in}\ \Omega\times（0，\infty），\\[3pt] \displaystyle \nu \cdot V\nabla u=0 &\displaystyle \quad \textrm{on} \; \partial\Omega\times（0，\infty）.\ vspace*{-2pt} \end{cases} \end{equation*}这个模型是T. Nagylaki描述了两个等位基因在选择、迁移和部分泛混合（partial panmixia）的共同作用下的进化，部分泛混合是一个非局部术语，用于完全显性的情况，其中g（x）被假设至少改变一次符号以反映环境的多样性。首先，研究了一般非局部问题的性质。然后，在积分<$Ω g（x）dx的不同符号下，得到了以扩散系数d和部分泛混率B表示的非平凡定态的存在性.此外，研究了非平凡定态以及平凡定态u ∈ 0和u ∈ 1的稳定性和不稳定性。我们的研究结果说明了如何非本地术语-即部分panmixia -帮助迁移在这个模型中。
In this paper, we study the following non-local problem: \begin{equation*} \begin{cases} \displaystyle u_t=d{1\over\rho}\nabla\cdot(\rho V\nabla u)+b(\bar{u}-u)+ g(x) u^2(1-u) &\displaystyle \quad \textrm{in} \; \Omega\times (0,\infty),\\[3pt] \displaystyle 0\leq u\leq 1 & \quad\displaystyle \textrm{in}\ \Omega\times (0,\infty),\\[3pt] \displaystyle \nu \cdot V\nabla u=0 &\displaystyle \quad \textrm{on} \; \partial\Omega\times (0,\infty).\vspace*{-2pt} \end{cases} \end{equation*} This model, proposed by T. Nagylaki, describes the evolution of two alleles under the joint action of selection, migration, and partial panmixia – a non-local term, for the complete dominance case, where g(x) is assumed to change sign at least once to reflect the diversity of the environment. First, properties for general non-local problems are studied. Then, existence of non-trivial steady states, in terms of the diffusion coefficient d and the partial panmixia rate b, is obtained under different signs of the integral ∫Ω g(x)dx. Furthermore, stability and instability properties for non-trivial steady states, as well as the trivial steady states u ≡ 0 and u ≡ 1 are investigated. Our results illustrate how the non-local term – namely, the partial panmixia – helps the migration in this model.