We study the distribution of autonomously replicating genetic elements, so-called plasmids, in a bacterial population. When a bacterium divides, the plasmids are segregated between the two daughter cells. We analyze a model for a bacterial population structured by their plasmid content. The model contains reproduction of both plasmids and bacteria, death of bacteria, and the distribution of plasmids at cell division. The model equation is a growth-fragmentation-death equation with an integral term containing a singular kernel. As we are interested in the long-term distribution of the plasmids, we consider the associated eigenproblem. Due to the singularity of the integral kernel, we do not have compactness. Thus, standard approaches to show the existence of an eigensolution like the Theorem of Krein-Rutman cannot be applied. We show the existence of an eigensolution using a fixed point theorem and the Laplace transform. The long-term dynamics of the model is analyzed using the Generalized Relative Entropy method.

中文翻译：

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分析质粒分离：非紧凑情况下本征溶液的存在和稳定性

我们研究了细菌种群中自主复制的遗传元件（所谓的质粒）的分布。当细菌分裂时，质粒分离在两个子细胞之间。我们分析了由其质粒含量构成的细菌种群的模型。该模型包含质粒和细菌的复制，细菌的死亡以及质粒在细胞分裂时的分布。该模型方程是一个增长-碎片-死亡方程，其积分项包含一个奇异核。由于我们对质粒的长期分布感兴趣，因此我们考虑了相关的本征问题。由于整数内核的奇异性，我们没有紧凑性。因此，不能采用显示本征解存在的标准方法（如Krein-Rutman定理）。我们展示了使用不动点定理和拉普拉斯变换的本征解的存在。使用广义相对熵方法分析模型的长期动力学。