We study a quasi-one-dimensional classical Poisson–Nernst–Planck model for ionic flow through a membrane channel with two positively charged ion species (cations) and one negatively charged, and with zero permanent charges. We treat the model problem as a boundary value problem of a singularly perturbed differential system. Under the framework of the geometric singular perturbation theory, together with specific structures of this concrete model, the existence of solutions to the boundary value problem is established and, for a special case that the two cations have the same valences, we are able to derive approximations of the individual fluxes and the I–V (current–voltage) relation explicitly, from which, our two main focuses in this work, boundary layer effects on ionic flows and competitions between two cations, are analyzed in great details. Critical potentials are identified and their roles in characterizing these effects are studied. Nonlinear interplays among physical parameters, such as boundary concentrations and potentials, diffusion coefficients and ion valences, are characterized, which could potentially provide efficient ways to control and affect some biological functions. Numerical simulations are performed, and numerical results are consistent with our analytical ones.

我们研究了离子流经膜通道的准一维经典Poisson-Nernst-Planck模型，该膜通道具有两种带正电的离子物质（阳离子）和一种带负电的膜，且永久电荷为零。我们将模型问题视为奇摄动微分系统的边值问题。在几何奇异摄动理论的框架下，结合此具体模型的特定结构，建立了边值问题解的存在性，对于两种阳离子具有相同化合价的特殊情况，我们能够推导各个通量的近似值和I-V（电流-电压）关系明确表示，从中，我们在这项工作中的两个主要重点是边界层对离子流和详细分析了两个阳离子之间的竞争。确定了关键潜力，并研究了它们在表征这些影响方面的作用。表征物理参数之间的非线性相互作用，例如边界浓度和电势，扩散系数和离子化合价，可能为控制和影响某些生物学功能提供有效途径。进行了数值模拟，数值结果与我们的分析结果一致。