In this work, we discuss a cell-cell repulsion model based on a hyperbolic Keller-Segel equation with two populations, which aims at describing the cell growth and dispersion in the co-culture experiment from the work of Pasquier et al. (Biol Direct 6(1):5, 2011). We introduce the notion of solution integrated along the characteristics, which allows us to prove the existence and uniqueness of solutions and the segregation property for the two species. From a numerical perspective, we also observe that our model admits a competitive exclusion principle which is different from the classical competitive exclusion principle for the corresponding ODE model. More importantly, our model shows the complexity of the short term (6 days) co-cultured cell distribution depending on the initial distribution of each species. Through numerical simulations, we show that the impact of the initial distribution on the proportion of each species in the final population lies in the initial number of cell clusters and that the final proportion of each species is not influenced by the precise distribution of the initial distribution. We also find that a fast dispersion rate gives a short-term advantage while the vital dynamics contributes to a long-term population advantage. When the initial condition for the two species is not segregated, the numerical simulations suggest that asymptotic segregation occurs when the dispersion coefficients are not equal for two populations.

中文翻译：

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基于双曲Keller-Segel方程的细胞排斥模型。

在这项工作中，我们讨论了基于双曲线Keller-Segel方程的细胞-细胞排斥模型，该方程具有两个种群，目的是根据Pasquier等人的工作描述共培养实验中的细胞生长和分散。（Biol Direct 6（1）：5，2011）。我们沿着特征引入了溶液集成的概念，这使我们能够证明溶液的存在性和唯一性以及两个物种的隔离性质。从数值角度来看，我们还观察到我们的模型接受竞争排除原理，该原理不同于相应ODE模型的经典竞争排除原理。更重要的是，我们的模型显示了短期（6天）共培养细胞分布的复杂性，具体取决于每种物种的初始分布。通过数值模拟 我们表明，初始分布对最终种群中每个物种的比例的影响在于细胞簇的初始数目，并且每个物种的最终比例不受初始分布的精确分布的影响。我们还发现，快速的分散速度会带来短期优势，而生命动态则有助于长期的人口优势。当两个物种的初始条件未分离时，数值模拟表明，当两个种群的扩散系数不相等时，会出现渐近分离。