Abstract A branching process in varying environment with generation-dependent immigration is a modification of the standard branching process in which immigration is allowed and the reproduction and immigration laws may vary over the generations. This flexibility makes the process more appropriate to model real populations due to the fact that the stability in the reproductive capacity and in the immigration laws are not usually fulfilled. In this setting, we study the extinction problem, providing a necessary and sufficient condition for the certain extinction of the population. The asymptotic behavior of the model is analyzed for those processes with critical offspring distributions, according with the classification established in [Kersting, G. A unifying approach to branching processes in varying environment. arXiv:1703.01960 2017, 1–23] , and when the immigration means stabilize to a positive value. More specifically, we establish that the asymptotic distribution of the process-under a suitable normalization- belongs to the gamma distribution family.

中文翻译：

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依赖代际移民在不同环境中的分支过程

摘要 具有世代依赖移民的不同环境中的分支过程是对标准分支过程的修改，其中允许移民并且繁殖和移民法可能会随着世代而变化。由于生殖能力和移民法的稳定性通常不会得到满足，这种灵活性使得该过程更适合模拟真实人口。在这种情况下，我们研究灭绝问题，为种群的一定灭绝提供了充分必要条件。根据 [Kersting, G. A unifying approach to branching processes in different environment. 中建立的分类，对具有临界后代分布的过程分析模型的渐近行为。arXiv:1703.01960 2017, 1–23] , 当移民手段稳定到正值时。更具体地说，我们确定该过程的渐近分布——在合适的归一化下——属于伽马分布族。